**Influence of Skin Diseases on Fingerprint Quality and Recognition**

Michal Dolezel, Martin Drahansky, Jaroslav Urbanek, Eva Brezinova and Tai-hoon Kim

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51992

## **1. Introduction**

[19] F. Miao, S. D. Bao, Y. Li (2010). *A Modified Fuzzy Vault Scheme for Biometrics-based Body*

[20] S. D. Bao and Y. T. Zhang (2005). *A new symmetric cryptosystem of body area sensor net‐ works for telemedicine,* in 6th Asian–Pacific Conference on Medical and Biological En‐

[21] Miao, F., Bao, S. D., & Li, Y. A Novel Biometric Key Distribution Solution with Ener‐ gy Distribution Information of Physiological Signals for Body Sensor Networks Se‐

[22] J.P. Berrut, L. Trefethen. *Barycentric Lagrange Interpolation.* SIAM Review 46 (3): 501–

[23] Lin Yao, Bing Liu, Guowei Wu et al. *A Biometric Key Establishment Protocol for Body*

*Area Networks, International Journal of Distributed Sensor Networks*, 2011.

*Sensor Networks Security,* IEEE Globecom.

curity. IET Information Security. Accepted.

gineering.

274 New Trends and Developments in Biometrics

517,2004.

Fingerprint recognition belongs to one of the most often used biometric technologies world‐ wide. It is believed that fingerprints could be used for the recognition of a person in nearly any case; however there exist many cases, where the fingerprint recognition could not be used. There exist some influencing factors [1] that have an impact to the process of finger‐ print recognition, e.g. the environmental influences, dirtiness on finger or the sensor, elec‐ tromagnetic radiation or diseases. This chapter deals with the circumstances which influence the quality of fingerprints – we are limited on skin diseases here, further we ex‐ plain how we can evaluate the quality of the acquired fingerprint.

The fingerprint recognition consists of five main steps (see Fig. 1) [2, 3, 4]:


**•** *Thinning* or *Skeletization* – the papillary lines from the previous step have varying thick‐ ness. To make the algorithm for minutiae extraction as simple as possible, we prefer the thickness of all papillary lines in all parts having only one pixel.

ease has attacked and destroyed the structure of papillary lines in the epidermis and underlying dermis (top two layers of the skin), the papillary lines will not grow in the same form as before (if at all) and therefore such user could be restricted in their future life by being excluded from the use of fingerprint recognition systems, though their fingers do not

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277

Skin makes up to 12-15% of an adult's body weight. Each square centimeter has 6 million cells, 5,000 sensory points, 100 sweat glands and 15 sebaceous glands. It consists of three lay‐

Skin is constantly being regenerated. A keratinocyte ("skin cell") starts its life at the lower layer of epidermis (the basal layer), which is nourished by blood vessels and is supplied with nerve endings from dermis. The cell migrates upward from basal layer to stratum cor‐ neum (the outermost skin layer). During four weeks the cell undergoes a series of changes, gradually flattening out and moving toward the surface. Then it dies and is shed. The epi‐ dermis is not supplied with blood vessels, but has nerve endings. The shape of dermoepi‐

**•** *Sensation* – the nerve endings in the skin identify touch, heat, cold, pain and light pres‐

**•** *Heat regulation* – the skin helps to regulate the body temperature by sweating to cool the body down when it overheats and by shivering creating "goose bumps" when it is cold. Shivering closes the pores. The tiny hair that stands on end traps warm air and thus helps

**•** *Absorption* – absorption of ultraviolet rays from the sun helps to form vitamin D in the body, which is vital for bone formation. Some creams, essential oils and medicines (e.g.

**•** *Protection* – the skin protects the body from ultraviolet light – too much of it is harmful to the body – by producing a pigment called melanin. It also protects us from the invasion of bacteria and germs by forming an acid mantle (formed by the skin sebum and sweat).

**•** *Excretion* – waste products and toxins are eliminated from the body through the sweat

**•** *Secretion* – sebum and sweat are secreted onto the skin surface. The sebum keeps the skin lubricated and soft, and the sweat combines with the sebum to form an acid mantle which

There are a lot of skin diseases, which can affect palms and fingers. We find plenty of skin diseases including description of their influence on the structure and color of the skin in spe‐ cialized medical literature, e.g. [16]. In the following subchapters we describe some of these diseases together with photographs. These clearly show that these diseases may cause many

glands. It is a very important function which helps to keep the body "clean".

creates the right pH-balance for the skin to fight off infection.

anti-smoking patches) can also be absorbed through the skin into the blood stream.

have any symptoms of the skin disease anymore.

ers [18]: *epidermis* (the outer layer), *dermis* and *subcutaneous* (fat) layer.

dermal junction basically forms the structure of papillary lines.

There are several skin functions [19]:

This barrier also prevents moisture loss.

problems in automatic biometric systems.

keep the body warm.

sure.

**•** *Minutiae extraction* – this algorithm detects and extracts all minutiae found in the finger‐ print. We distinguish between minutiae in verification systems (here are generally used 2 minutiae – ridge ending and bifurcation [2]) and identification (dactyloscopic) systems [5], where many special minutiae are used.

**Figure 1.** An overview of the fingerprint recognition.

The fingerprint recognition technology is well accepted in our society [6]. Fingerprints could be used not only for the known user verification / identification tasks, but also e.g. for cryp‐ tographic key generation [7, 8, 9, 10], computerized patient record [11] or for use with credit cards [12] etc. Anyway, the influence of skin diseases to fingerprint recognition in biometric systems has not been discussed sufficiently till today, therefore we hope that this chapter brings you closer information to this topic.

In the chapter 2, the categorization into three groups of skin diseases is done and the most important diseases in each group are briefly described. The chapter 3 describes how these skin diseases could influence the process of automatic fingerprint recognition. Chapters 4 and 5 deal with fingerprint image enhancement and estimation of their quality.

## **2. Skin diseases**

Skin diseases represent very important, but often neglected factor of the fingerprint acquire‐ ment. It is impossible to say in general how many people suffer from skin diseases, because there are so many various skin diseases [4]. In a general medical practice about 20-25% of patients with skin complaints are referred. When discussing whether the fingerprint recog‐ nition technology is a perfect solution capable to resolve all our security problems, we should always keep in mind those potential users who suffer from some skin disease.

In the following text, several skin diseases, which attack hand palms and fingertips, are in‐ troduced from the medical point of view.

The situation after successful recovery of a potential user from such skin diseases is, howev‐ er, very important for the possible further use of fingerprint recognition devices. If the dis‐ ease has attacked and destroyed the structure of papillary lines in the epidermis and underlying dermis (top two layers of the skin), the papillary lines will not grow in the same form as before (if at all) and therefore such user could be restricted in their future life by being excluded from the use of fingerprint recognition systems, though their fingers do not have any symptoms of the skin disease anymore.

Skin makes up to 12-15% of an adult's body weight. Each square centimeter has 6 million cells, 5,000 sensory points, 100 sweat glands and 15 sebaceous glands. It consists of three lay‐ ers [18]: *epidermis* (the outer layer), *dermis* and *subcutaneous* (fat) layer.

Skin is constantly being regenerated. A keratinocyte ("skin cell") starts its life at the lower layer of epidermis (the basal layer), which is nourished by blood vessels and is supplied with nerve endings from dermis. The cell migrates upward from basal layer to stratum cor‐ neum (the outermost skin layer). During four weeks the cell undergoes a series of changes, gradually flattening out and moving toward the surface. Then it dies and is shed. The epi‐ dermis is not supplied with blood vessels, but has nerve endings. The shape of dermoepi‐ dermal junction basically forms the structure of papillary lines.

There are several skin functions [19]:

**•** *Thinning* or *Skeletization* – the papillary lines from the previous step have varying thick‐ ness. To make the algorithm for minutiae extraction as simple as possible, we prefer the

**•** *Minutiae extraction* – this algorithm detects and extracts all minutiae found in the finger‐ print. We distinguish between minutiae in verification systems (here are generally used 2 minutiae – ridge ending and bifurcation [2]) and identification (dactyloscopic) systems

**Fingerprint Acquirement Image Enhancement Thresholding Thinning Minutiae Exraction**

The fingerprint recognition technology is well accepted in our society [6]. Fingerprints could be used not only for the known user verification / identification tasks, but also e.g. for cryp‐ tographic key generation [7, 8, 9, 10], computerized patient record [11] or for use with credit cards [12] etc. Anyway, the influence of skin diseases to fingerprint recognition in biometric systems has not been discussed sufficiently till today, therefore we hope that this chapter

In the chapter 2, the categorization into three groups of skin diseases is done and the most important diseases in each group are briefly described. The chapter 3 describes how these skin diseases could influence the process of automatic fingerprint recognition. Chapters 4

Skin diseases represent very important, but often neglected factor of the fingerprint acquire‐ ment. It is impossible to say in general how many people suffer from skin diseases, because there are so many various skin diseases [4]. In a general medical practice about 20-25% of patients with skin complaints are referred. When discussing whether the fingerprint recog‐ nition technology is a perfect solution capable to resolve all our security problems, we

In the following text, several skin diseases, which attack hand palms and fingertips, are in‐

The situation after successful recovery of a potential user from such skin diseases is, howev‐ er, very important for the possible further use of fingerprint recognition devices. If the dis‐

should always keep in mind those potential users who suffer from some skin disease.

and 5 deal with fingerprint image enhancement and estimation of their quality.

thickness of all papillary lines in all parts having only one pixel.

[5], where many special minutiae are used.

276 New Trends and Developments in Biometrics

**Figure 1.** An overview of the fingerprint recognition.

brings you closer information to this topic.

troduced from the medical point of view.

**2. Skin diseases**


There are a lot of skin diseases, which can affect palms and fingers. We find plenty of skin diseases including description of their influence on the structure and color of the skin in spe‐ cialized medical literature, e.g. [16]. In the following subchapters we describe some of these diseases together with photographs. These clearly show that these diseases may cause many problems in automatic biometric systems.

The fingerprint recognition systems are usually used only for adults. There is almost no in‐ formation from appropriate tests with children. Although we know that papillary lines emerge on infant's fingers already in the mother's uterus [24], i.e. we might be able to recog‐ nize the fingerprints of infants, the common fingerprint recognition systems are suitable for adults only (due to the area and resolution of fingerprint sensors, etc.). It should not be for‐ gotten that a skin disease in early childhood could have an influence on the skin in adult years (example is *incontinentia pigmenti* [25] on a small child hand), i.e. there could be some problems with fingerprint acquirement caused by such skin disease in a young age.

phony. Hand eczema was more common among people reporting occupational exposure. The most harmful exposure was to chemicals, water and detergents, dust, and dry dirt.

Influence of Skin Diseases on Fingerprint Quality and Recognition

http://dx.doi.org/10.5772/51992

279

*Fingertip eczema* [17] is very dry, chronic form of eczema of the palmar surface of the finger‐ tips, it may be result of an allergic reaction or may occur in children and adults as an isolat‐ ed phenomenon of unknown cause. One finger or several fingers may be involved. Initially the skin may be moist and then become dry, cracked, and scaly. The skin peels from the fin‐ gertips distally, exposing a very dry, red, cracked, fissured, tender, or painful surface with‐

*Pomfolyx (dishydrosis)* [16] is a distinctive reaction pattern of unknown etiology presenting as symmetric vesicular hand and foot dermatitis. Itching precedes the appearance of vesicles on the palms and sides of the fingers. The skin may be red and wet. The vesicles slowly re‐ solve and are replaced by rings of scale. Chronic eczematous changes with erythema, scal‐

*Tinea of the palm* [17] is dry, diffuse, keratotic form of tinea. The dry keratotic form may be asymptomatic and the patient may be unaware of the infection, attributing the dry, thick, scaly surface to hard physical labor. It is frequently seen in association with tineapedis

*Pyoderma* [22] is a sign of bacterial infection of the skin. It is caused by *Staphylococcus aureus* and *Streptococcus pyogenes*. Some people are more susceptible to these diseases (such as dia‐

out skin lines – see Figure 2.

**Figure 2.** Fingertip eczema [17].

ing, and lichenification may follow.

which prevalence is 10 to 30%.

betics, alcoholics, etc.) – see Figure 3.

**Figure 3.** Abscess on finger of patient with diabetes [16] and pyoderma [23].

The subcategory of skin diseases affecting only the skin color are the least dangerous for the quality of the fingerprint image. In fact, only one fingerprint technology can be considered as sensitive to such diseases – the optical technology [26], but if FTIR-based optical sensors are used, the change of skin color may have no influence on the quality of the resulting im‐ ages. The case of the other two subcategories (influence of skin structure and combination of influence of skin color and structure) is different. If the structure of papillary lines has changed, it is often impossible to recognize the original curvatures of papillary lines and therefore it is impossible to decide whether the claimed identity is the user's identity. Un‐ fortunately, there are many such skin diseases which attack papillary line structure. Nearly all sensor technologies, namely optical, capacitive, e-field, electro-optical, pressure sensitive and thermal are exposed to such risk [26]. Only one sensor technology is missing here – the ultrasound technology. This technology has an advantage: the ultrasound waves can pene‐ trate under the upper skin layer to the curvatures in dermoepidermal junction forming the papillary lines structures and therefore it might be possible to reconstruct the real finger‐ print image, but only if the disease has not attacked this underlying structure. If yes, there is no chance to get an original papillary lines structure.

The situation after successful recovery of a potential user from such skin diseases is, howev‐ er, very important for the possible further use of fingerprint recognition devices. If the dis‐ ease has attacked and destroyed the structure of papillary lines in dermoepidermal junction, the papillary lines will not grow in the same form as before (if at all) and therefore such user could be restricted in his/her future life by being excluded from the use of fingerprint recog‐ nition systems, though his fingers don't have any symptoms of a skin disease any more.

#### **2.1. Diseases causing histopathological changes of epidermis and dermis**

These diseases may cause problems for the most types of sensors, because color of the skin and structure of epidermis and dermis are influenced.

*Hand eczema* [17] is an inflammatory non-infectious long-lasting disease with relapsing course. It is one of the most common problems encountered by the dermatologist. Hand der‐ matitis causes discomfort and embarrassment and, because of its locations, interferes signifi‐ cantly with normal daily activities. Hand dermatitis is common in industrial occupations. The prevalence of hand eczema was approximately 5.4% and was twice as common in fe‐ males as in males. The most common type of hand eczema was irritant contact dermatitis (35%), followed by atopic eczema (22%), and allergic contact dermatitis (19%). The most common contact allergies were to nickel, cobalt, fragnance mix, balsam of Peru, and colo‐ phony. Hand eczema was more common among people reporting occupational exposure. The most harmful exposure was to chemicals, water and detergents, dust, and dry dirt.

*Fingertip eczema* [17] is very dry, chronic form of eczema of the palmar surface of the finger‐ tips, it may be result of an allergic reaction or may occur in children and adults as an isolat‐ ed phenomenon of unknown cause. One finger or several fingers may be involved. Initially the skin may be moist and then become dry, cracked, and scaly. The skin peels from the fin‐ gertips distally, exposing a very dry, red, cracked, fissured, tender, or painful surface with‐ out skin lines – see Figure 2.

**Figure 2.** Fingertip eczema [17].

The fingerprint recognition systems are usually used only for adults. There is almost no in‐ formation from appropriate tests with children. Although we know that papillary lines emerge on infant's fingers already in the mother's uterus [24], i.e. we might be able to recog‐ nize the fingerprints of infants, the common fingerprint recognition systems are suitable for adults only (due to the area and resolution of fingerprint sensors, etc.). It should not be for‐ gotten that a skin disease in early childhood could have an influence on the skin in adult years (example is *incontinentia pigmenti* [25] on a small child hand), i.e. there could be some

The subcategory of skin diseases affecting only the skin color are the least dangerous for the quality of the fingerprint image. In fact, only one fingerprint technology can be considered as sensitive to such diseases – the optical technology [26], but if FTIR-based optical sensors are used, the change of skin color may have no influence on the quality of the resulting im‐ ages. The case of the other two subcategories (influence of skin structure and combination of influence of skin color and structure) is different. If the structure of papillary lines has changed, it is often impossible to recognize the original curvatures of papillary lines and therefore it is impossible to decide whether the claimed identity is the user's identity. Un‐ fortunately, there are many such skin diseases which attack papillary line structure. Nearly all sensor technologies, namely optical, capacitive, e-field, electro-optical, pressure sensitive and thermal are exposed to such risk [26]. Only one sensor technology is missing here – the ultrasound technology. This technology has an advantage: the ultrasound waves can pene‐ trate under the upper skin layer to the curvatures in dermoepidermal junction forming the papillary lines structures and therefore it might be possible to reconstruct the real finger‐ print image, but only if the disease has not attacked this underlying structure. If yes, there is

The situation after successful recovery of a potential user from such skin diseases is, howev‐ er, very important for the possible further use of fingerprint recognition devices. If the dis‐ ease has attacked and destroyed the structure of papillary lines in dermoepidermal junction, the papillary lines will not grow in the same form as before (if at all) and therefore such user could be restricted in his/her future life by being excluded from the use of fingerprint recog‐ nition systems, though his fingers don't have any symptoms of a skin disease any more.

These diseases may cause problems for the most types of sensors, because color of the skin

*Hand eczema* [17] is an inflammatory non-infectious long-lasting disease with relapsing course. It is one of the most common problems encountered by the dermatologist. Hand der‐ matitis causes discomfort and embarrassment and, because of its locations, interferes signifi‐ cantly with normal daily activities. Hand dermatitis is common in industrial occupations. The prevalence of hand eczema was approximately 5.4% and was twice as common in fe‐ males as in males. The most common type of hand eczema was irritant contact dermatitis (35%), followed by atopic eczema (22%), and allergic contact dermatitis (19%). The most common contact allergies were to nickel, cobalt, fragnance mix, balsam of Peru, and colo‐

**2.1. Diseases causing histopathological changes of epidermis and dermis**

problems with fingerprint acquirement caused by such skin disease in a young age.

no chance to get an original papillary lines structure.

278 New Trends and Developments in Biometrics

and structure of epidermis and dermis are influenced.

*Pomfolyx (dishydrosis)* [16] is a distinctive reaction pattern of unknown etiology presenting as symmetric vesicular hand and foot dermatitis. Itching precedes the appearance of vesicles on the palms and sides of the fingers. The skin may be red and wet. The vesicles slowly re‐ solve and are replaced by rings of scale. Chronic eczematous changes with erythema, scal‐ ing, and lichenification may follow.

*Tinea of the palm* [17] is dry, diffuse, keratotic form of tinea. The dry keratotic form may be asymptomatic and the patient may be unaware of the infection, attributing the dry, thick, scaly surface to hard physical labor. It is frequently seen in association with tineapedis which prevalence is 10 to 30%.

*Pyoderma* [22] is a sign of bacterial infection of the skin. It is caused by *Staphylococcus aureus* and *Streptococcus pyogenes*. Some people are more susceptible to these diseases (such as dia‐ betics, alcoholics, etc.) – see Figure 3.

**Figure 3.** Abscess on finger of patient with diabetes [16] and pyoderma [23].

*Pitted keratolysis* [17] is a disease mimicking tinea, especially for people who swelter and wear rubber gloves in the hot, humid environment. Hyperhydrosis is the most frequently observed symptom. The disease is bacterial in origin, characterized by many circular or lon‐ gitudinal, punched out depressions in the skin surface. The eruption is limited to the stra‐ tum corneum.

Drug induced skin reactions [17] are among the most common adverse drug reactions. They occur in many forms and can mimic virtually any dermatosis. Occur in 2-3% of hospitalized patients. Sulfonamides, NSAIDs and anticonvulsants are most often applied in the etiology.

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*Herpes simplex virus* [16] in the host with systemic immune-compromise may cause chronic

Herpetic infection may uncommonly occur on the fingers or periungually. Lesions begin with tenderness and erythema and deepseated blisters develop 24 to 48 hours after symp‐

*Scabies* [21] is highly contagious disease caused by the mite *Sarcoptes scabiei*. It is character‐ ized by red papules, vesicles and crusts located usually on the areas with tender skin, palms

ulcerations as you can see by patient with advanced HIV disease in Figure 5 (left).

**Figure 5.** Herpes simplex virus:patient with HIV (left) [20]; deepseated blisters (right) [16].

toms begin (see Figure 5, right).

and soles especially in infants.

**Figure 6.** Erythmea multiforme.

*Keratolysis exfoliativa* [17] is a common, chronic, asymptomatic, non-inflamatory, bilateral peeling of the palms of the hands. Its cause is unknown. The eruption is most common dur‐ ing the summer months and is often associated with sweaty palms and soles. It is character‐ ized by scaling and peeling, the central area becomes slightly red and tender.

*Lichen planus* [22] is quite common, unique inflammatory cutaneous and mucous membrane reaction pattern of unknown etiology. LP of the palm and soles generally occurs as an isolat‐ ed phenomenon. The lesions are papules aggregated into semitranslucent plaques with globular waxy surface, ulceration may occur.

*Acanthosis nigricans* [16] is non-specific reaction pattern that may accompany obesity, diabe‐ tes, tumors. AN is classified into benign and malignant forms. In all cases the disease presents with symmetric, brown thickening of the skin. During the process there is papillary hypertrophy, hyperkeratosis, and increased number of melanocytes in the epidermis.

*Pyogenic granuloma* [17] is a benign acquired vascular lesion of the skin that is common in children and young adults. It often appears as a response to an injury or hormonal factors. Lesions are small rapidly growing, yellow-to-bright red, dome-shaped.

*Systemic sclerosis* [20] is a chronic autoimmune disease characterized by sclerosis of the skin or other organs. Emergence of acrosclerosis is decisive for fingerprinting. Initially the skin is infused with edema mainly affecting hands. With the progressive edema stiff skin appears and necrosis of fingers may form. The disease leads to sclerodactyly with contractures of the fingers. For more than 90% of patients is typical Raynaud's phenomenon (see below). The typical patient is a woman over 50 years of age.

*Raynaud's phenomenon*[17] represents an episodic vasoconstriction of the digital arteries and arterioles that is precipitated by cold and stress. It is much more common in women. There are three stages during a single episode: pallor (white), cyanosis (blue), and hyperemia (red).

**Figure 4.** Different types of eczema [17] (3× left) and acanthosis nigricans [16] (right).

Drug induced skin reactions [17] are among the most common adverse drug reactions. They occur in many forms and can mimic virtually any dermatosis. Occur in 2-3% of hospitalized patients. Sulfonamides, NSAIDs and anticonvulsants are most often applied in the etiology.

**Figure 5.** Herpes simplex virus:patient with HIV (left) [20]; deepseated blisters (right) [16].

*Herpes simplex virus* [16] in the host with systemic immune-compromise may cause chronic ulcerations as you can see by patient with advanced HIV disease in Figure 5 (left).

Herpetic infection may uncommonly occur on the fingers or periungually. Lesions begin with tenderness and erythema and deepseated blisters develop 24 to 48 hours after symp‐ toms begin (see Figure 5, right).

*Scabies* [21] is highly contagious disease caused by the mite *Sarcoptes scabiei*. It is character‐ ized by red papules, vesicles and crusts located usually on the areas with tender skin, palms and soles especially in infants.

**Figure 6.** Erythmea multiforme.

*Pitted keratolysis* [17] is a disease mimicking tinea, especially for people who swelter and wear rubber gloves in the hot, humid environment. Hyperhydrosis is the most frequently observed symptom. The disease is bacterial in origin, characterized by many circular or lon‐ gitudinal, punched out depressions in the skin surface. The eruption is limited to the stra‐

*Keratolysis exfoliativa* [17] is a common, chronic, asymptomatic, non-inflamatory, bilateral peeling of the palms of the hands. Its cause is unknown. The eruption is most common dur‐ ing the summer months and is often associated with sweaty palms and soles. It is character‐

*Lichen planus* [22] is quite common, unique inflammatory cutaneous and mucous membrane reaction pattern of unknown etiology. LP of the palm and soles generally occurs as an isolat‐ ed phenomenon. The lesions are papules aggregated into semitranslucent plaques with

*Acanthosis nigricans* [16] is non-specific reaction pattern that may accompany obesity, diabe‐ tes, tumors. AN is classified into benign and malignant forms. In all cases the disease presents with symmetric, brown thickening of the skin. During the process there is papillary

*Pyogenic granuloma* [17] is a benign acquired vascular lesion of the skin that is common in children and young adults. It often appears as a response to an injury or hormonal factors.

*Systemic sclerosis* [20] is a chronic autoimmune disease characterized by sclerosis of the skin or other organs. Emergence of acrosclerosis is decisive for fingerprinting. Initially the skin is infused with edema mainly affecting hands. With the progressive edema stiff skin appears and necrosis of fingers may form. The disease leads to sclerodactyly with contractures of the fingers. For more than 90% of patients is typical Raynaud's phenomenon (see below). The

*Raynaud's phenomenon*[17] represents an episodic vasoconstriction of the digital arteries and arterioles that is precipitated by cold and stress. It is much more common in women. There are three stages during a single episode: pallor (white), cyanosis (blue), and hyperemia

hypertrophy, hyperkeratosis, and increased number of melanocytes in the epidermis.

Lesions are small rapidly growing, yellow-to-bright red, dome-shaped.

**Figure 4.** Different types of eczema [17] (3× left) and acanthosis nigricans [16] (right).

ized by scaling and peeling, the central area becomes slightly red and tender.

globular waxy surface, ulceration may occur.

typical patient is a woman over 50 years of age.

tum corneum.

280 New Trends and Developments in Biometrics

(red).

*Erythema multiforme* [22] is quite common skin disorder with multifactorial cause (see Figure 6). The most common triggering agents are infections (in the first place herpes virus) and drugs. Minor and major variant of this disease is described. Both forms are characterized by erythematous target-shaped lesions with a center with hemorrhage, blistering, necrosis or crust. When the trigger is herpetic infection, frequent recurrences come.

*Dermatitis artifacta* [25] are changes of skin due to the manipulation by patient. Patients often have psychosomatic, psychiatric or drug abuse problems.

## **2.2. Diseases causing skin discoloration**

*Hand, foot, and mouth disease* (HFMD) [16] is contagious enteroviral infection occurring pri‐ marily in children and characterized by a vesicular palmoplantar eruption. The skin lesions begin as red macules that rapidly become pale, white, oval vesicles with red areola.

**Figure 8.** Hereditary hemorrhagic teleangiectasia [16].

*Hand eczema* – particularly chronic forms (see above).

ble from chronic eczema.

**Figure 9.** Psoriasis (left) [21]; scarlet fever (right) [17].

gual regions. HPVs induce hyperplasia and hyperkeratosis.

*Hereditary hemorrhagic teleangiectasia* [20] is an autosomal dominant condition affecting blood vessels, especially in the mucous membranes of the mouth and the gastrointestinal tract. The diagnostic lesions are small, pulsating, macular and papular, usually punctuate. Teleangiec‐ tases are present on different parts of the body, palms and soles including (see Figure 8).

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**2.3. Diseases causing histopathological changes in junction of epidermis and dermis**

These diseases are focused mainly on ultrasonic sensors, which detect the base of papillary lines on the border of epidermis and dermis. The diagnoses also belong to the first group.

*Warts (verruca vulgaris)* [22] are benign epidermal neoplasms that are caused by human pap‐ illoma viruses (HPVs). Warts commonly appear at sites of trauma, on the hand, in periun‐

*Psoriasis* [20] is characterized by scaly papules and plaques. It occurs in 1% to 3% of the pop‐ ulation. The disease is transmitted genetically; environmental factors are needed to precipi‐ tate the disease. The disease is lifelong and characterized by chronic, recurrent exacerbations and remissions that are emotionally and physically debilitating. Psoriasis of the palms and fingertips is characterized by red plaques with thick brown scale and may be indistinguisha‐

*Xantomas* [17] are lipid deposits in the skin and tendons that occur secondary to a lipid ab‐ normality. These localized deposits are yellow and are frequently very firm.

*Scarlet fever (scarlatina)* [17] is contagious disease produced by streptococcal, erythrogenic toxin. It is most common in children (ages 1 to 10 years). In the ending stages of the disease large sheats of epidermis may be shed from the palms in glovelike cast, exposing new ten‐ der and red epidermis beneath.

*Kawasaki's disease* [20] is an acute febrile illness of infants and children, characterized by cu‐ taneous and mucosal erythema and edema with subsequent desquamation, cervical lym‐ phadenitis, and complicated by coronary artery aneurysms (20%). Most cases of Kawasaki's disease in adults represent toxic shock syndrome. Erytematous macules appear 1 to 3 days after onset of fever, enlarge and become more numerous, then desquamation beginning on tips of fingers is highly characteristic.

*Secondary syphilis* [20]is characterized by mucocutaneous lesions, which may assume a varie‐ ty of shapes, including round, elliptic, or annular. The color is characteristic, resembling a "clean-cut ham" or having a copery tint.

*Carotenosis* [16] is yellowish discoloration of the skin, especially of the palms and soles that is sometimes seen in diabetic patients.

**Figure 8.** Hereditary hemorrhagic teleangiectasia [16].

*Erythema multiforme* [22] is quite common skin disorder with multifactorial cause (see Figure 6). The most common triggering agents are infections (in the first place herpes virus) and drugs. Minor and major variant of this disease is described. Both forms are characterized by erythematous target-shaped lesions with a center with hemorrhage, blistering, necrosis or

*Dermatitis artifacta* [25] are changes of skin due to the manipulation by patient. Patients often

*Hand, foot, and mouth disease* (HFMD) [16] is contagious enteroviral infection occurring pri‐ marily in children and characterized by a vesicular palmoplantar eruption. The skin lesions

*Xantomas* [17] are lipid deposits in the skin and tendons that occur secondary to a lipid ab‐

*Scarlet fever (scarlatina)* [17] is contagious disease produced by streptococcal, erythrogenic toxin. It is most common in children (ages 1 to 10 years). In the ending stages of the disease large sheats of epidermis may be shed from the palms in glovelike cast, exposing new ten‐

*Kawasaki's disease* [20] is an acute febrile illness of infants and children, characterized by cu‐ taneous and mucosal erythema and edema with subsequent desquamation, cervical lym‐ phadenitis, and complicated by coronary artery aneurysms (20%). Most cases of Kawasaki's disease in adults represent toxic shock syndrome. Erytematous macules appear 1 to 3 days after onset of fever, enlarge and become more numerous, then desquamation beginning on

*Secondary syphilis* [20]is characterized by mucocutaneous lesions, which may assume a varie‐ ty of shapes, including round, elliptic, or annular. The color is characteristic, resembling a

*Carotenosis* [16] is yellowish discoloration of the skin, especially of the palms and soles that is

begin as red macules that rapidly become pale, white, oval vesicles with red areola.

normality. These localized deposits are yellow and are frequently very firm.

**Figure 7.** Hand, foot and mouth syndrome[16]; xantomas [20]; epidermolysis bullosa [21].

crust. When the trigger is herpetic infection, frequent recurrences come.

have psychosomatic, psychiatric or drug abuse problems.

**2.2. Diseases causing skin discoloration**

282 New Trends and Developments in Biometrics

der and red epidermis beneath.

tips of fingers is highly characteristic.

"clean-cut ham" or having a copery tint.

sometimes seen in diabetic patients.

*Hereditary hemorrhagic teleangiectasia* [20] is an autosomal dominant condition affecting blood vessels, especially in the mucous membranes of the mouth and the gastrointestinal tract. The diagnostic lesions are small, pulsating, macular and papular, usually punctuate. Teleangiec‐ tases are present on different parts of the body, palms and soles including (see Figure 8).

#### **2.3. Diseases causing histopathological changes in junction of epidermis and dermis**

These diseases are focused mainly on ultrasonic sensors, which detect the base of papillary lines on the border of epidermis and dermis. The diagnoses also belong to the first group.

#### *Hand eczema* – particularly chronic forms (see above).

*Warts (verruca vulgaris)* [22] are benign epidermal neoplasms that are caused by human pap‐ illoma viruses (HPVs). Warts commonly appear at sites of trauma, on the hand, in periun‐ gual regions. HPVs induce hyperplasia and hyperkeratosis.

*Psoriasis* [20] is characterized by scaly papules and plaques. It occurs in 1% to 3% of the pop‐ ulation. The disease is transmitted genetically; environmental factors are needed to precipi‐ tate the disease. The disease is lifelong and characterized by chronic, recurrent exacerbations and remissions that are emotionally and physically debilitating. Psoriasis of the palms and fingertips is characterized by red plaques with thick brown scale and may be indistinguisha‐ ble from chronic eczema.

**Figure 9.** Psoriasis (left) [21]; scarlet fever (right) [17].

*Systemic lupus erytematosus* (SLE) [17] is a multisystem disease of unknown origin character‐ ized by production of numerous diverse of antibodies that cause several combinations of clinical signs, symptoms and laboratory abnormalities. The prevalence of LE in North Amer‐ ica and northern Europe is about 40 per 100,000 population. In the case of acute cutaneous LE indurated erythematous lesions may be presented on palms.

lary lines is then captured by an integrated CCD or CMOS camera. The oldest and most commonly used type of optical fingerprint scanner is the type which uses the Frustrated To‐

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There also exist models which use another image acquisition techniques like FTIR technique with sheet prism made of a number of prismlets adjacent to each other instead of a single

Fingerprint scanners based on a capacitive sensing technology are also very common type of fingerprint scanners. The sensor itself is a two-dimensional array of conductive plates. By placing the finger on sensor surface, each small plate and the corresponding part of skin over it start behave like a micro-capacitor. By measuring the small electrical charges be‐ tween plates and finger, it is possible to reconstruct the profile of papillary lines ridges and

Thermal fingerprint scanners contain special, so called pyro-electric cell which detects the thermal changes and converts them into an electrical charge. The main idea is that finger‐ print papillary line ridges produce a higher temperature differential to the valleys. The tem‐ perature difference produces an image when a contact occurs, but this image soon disappears because the thermal equilibrium is quickly reached and the pixel temperature is stabilized [28]. Therefore the thermal sensors are usually made in sweep variant in which

Pressure fingerprint scanners are made from two parallel electro-conductive layers with non/conductive gel between them. Ridges of papillary lines unlike the valleys by pressing the first flexible conductive layer create the contact of these two conductive layers. The con‐ ductive layers are in contact only in sensor parts where papillary line ridges are. By measur‐ ing the electrical charge between connected layers, it is possible to reconstruct the original

Electro-optical scanners contain two layers. First layer is made from a special polymer, which emits light when connected to the proper voltage [28]. Proper voltage can be obtained by contact with finger skin, which is conductive enough. Only the ridges are touching the polymer so on the other side of the polymer we could see light pattern of the fingerprint. The light pattern is than captured by the second layer, which is composed of an array of

tal Internal Reflection (FTIR) for the fingerprint image acquisition.

large prism or a model which uses optical fibers [28].

valleys and thus to reconstruct the fingerprint image.

*3.1.2. Capacitive fingerprint scanners*

*3.1.3. Thermal fingerprint scanners*

this disappearing problem does not occur.

*3.1.4. Pressure fingerprint scanners*

*3.1.5. Electro-optical fingerprint scanners*

fingerprint image.

photodiodes.

**Figure 10.** Psoriasis vulgaris [23].

*Epidermolysis bullosa* [20] is a term given to groups of genetic diseases in which minor trauma causes non-inflammatory blistering (mechanobullosus diseases). Repetitive trauma may lead to a mitten-like deformity with digits encased in an epidermal "cocoon". These diseases are classified as scarring and non-scarring and histologically by the level of blister forma‐ tion. Approximately 50 epidermolysis cases occur per million live births in the United States.

## **3. Influence of skin diseases to fingerprint pattern**

The process of analysis and further elimination of influence of dermatologic diseases to fin‐ gerprint recognition process begins with analysis of influence to the fingerprint pattern. Im‐ age of fingerprint pattern can be obtained either by classic manual way using dactyloscopic card and special ink or using the electronic sensors. Both ways have their advantages and disadvantages and both of them could have been influenced by skin diseases in different ways. It will be necessary to analyze the influence on both of these capturing methods.

#### **3.1. Ability of fingerprint scanners to scan a finger distorted by skin disease**

For acquiring the digital image of a fingerprint pattern in the most cases the so called finger‐ print scanners are used. These scanners are called "live-scan" fingerprint capture devices [28]. This term reflexes the fact that these sensors cannot be used for latent fingerprint scan‐ ning and for the scanning the live finger is needed. These scanners can be divided into sev‐ eral groups upon their sensing technology [4, 27, 28] – see the following subchapters

#### *3.1.1. Optical fingerprint scanners*

Basic principle of optical fingerprint scanners works in the following way: finger is placed on the sensor platen and it's illuminated by a light source. The pattern of fingerprint papil‐ lary lines is then captured by an integrated CCD or CMOS camera. The oldest and most commonly used type of optical fingerprint scanner is the type which uses the Frustrated To‐ tal Internal Reflection (FTIR) for the fingerprint image acquisition.

There also exist models which use another image acquisition techniques like FTIR technique with sheet prism made of a number of prismlets adjacent to each other instead of a single large prism or a model which uses optical fibers [28].

## *3.1.2. Capacitive fingerprint scanners*

*Systemic lupus erytematosus* (SLE) [17] is a multisystem disease of unknown origin character‐ ized by production of numerous diverse of antibodies that cause several combinations of clinical signs, symptoms and laboratory abnormalities. The prevalence of LE in North Amer‐ ica and northern Europe is about 40 per 100,000 population. In the case of acute cutaneous

*Epidermolysis bullosa* [20] is a term given to groups of genetic diseases in which minor trauma causes non-inflammatory blistering (mechanobullosus diseases). Repetitive trauma may lead to a mitten-like deformity with digits encased in an epidermal "cocoon". These diseases are classified as scarring and non-scarring and histologically by the level of blister forma‐ tion. Approximately 50 epidermolysis cases occur per million live births in the United

The process of analysis and further elimination of influence of dermatologic diseases to fin‐ gerprint recognition process begins with analysis of influence to the fingerprint pattern. Im‐ age of fingerprint pattern can be obtained either by classic manual way using dactyloscopic card and special ink or using the electronic sensors. Both ways have their advantages and disadvantages and both of them could have been influenced by skin diseases in different ways. It will be necessary to analyze the influence on both of these capturing methods.

For acquiring the digital image of a fingerprint pattern in the most cases the so called finger‐ print scanners are used. These scanners are called "live-scan" fingerprint capture devices [28]. This term reflexes the fact that these sensors cannot be used for latent fingerprint scan‐ ning and for the scanning the live finger is needed. These scanners can be divided into sev‐

Basic principle of optical fingerprint scanners works in the following way: finger is placed on the sensor platen and it's illuminated by a light source. The pattern of fingerprint papil‐

**3.1. Ability of fingerprint scanners to scan a finger distorted by skin disease**

eral groups upon their sensing technology [4, 27, 28] – see the following subchapters

LE indurated erythematous lesions may be presented on palms.

**3. Influence of skin diseases to fingerprint pattern**

**Figure 10.** Psoriasis vulgaris [23].

284 New Trends and Developments in Biometrics

*3.1.1. Optical fingerprint scanners*

States.

Fingerprint scanners based on a capacitive sensing technology are also very common type of fingerprint scanners. The sensor itself is a two-dimensional array of conductive plates. By placing the finger on sensor surface, each small plate and the corresponding part of skin over it start behave like a micro-capacitor. By measuring the small electrical charges be‐ tween plates and finger, it is possible to reconstruct the profile of papillary lines ridges and valleys and thus to reconstruct the fingerprint image.

#### *3.1.3. Thermal fingerprint scanners*

Thermal fingerprint scanners contain special, so called pyro-electric cell which detects the thermal changes and converts them into an electrical charge. The main idea is that finger‐ print papillary line ridges produce a higher temperature differential to the valleys. The tem‐ perature difference produces an image when a contact occurs, but this image soon disappears because the thermal equilibrium is quickly reached and the pixel temperature is stabilized [28]. Therefore the thermal sensors are usually made in sweep variant in which this disappearing problem does not occur.

## *3.1.4. Pressure fingerprint scanners*

Pressure fingerprint scanners are made from two parallel electro-conductive layers with non/conductive gel between them. Ridges of papillary lines unlike the valleys by pressing the first flexible conductive layer create the contact of these two conductive layers. The con‐ ductive layers are in contact only in sensor parts where papillary line ridges are. By measur‐ ing the electrical charge between connected layers, it is possible to reconstruct the original fingerprint image.

## *3.1.5. Electro-optical fingerprint scanners*

Electro-optical scanners contain two layers. First layer is made from a special polymer, which emits light when connected to the proper voltage [28]. Proper voltage can be obtained by contact with finger skin, which is conductive enough. Only the ridges are touching the polymer so on the other side of the polymer we could see light pattern of the fingerprint. The light pattern is than captured by the second layer, which is composed of an array of photodiodes.

## *3.1.6. E-field fingerprint scanners*

The sensor consists of a drive ring that generates an RF (radio frequency) sinusoidal signal and a matrix of active antennas that receives a very small amplitude signal transmitted by the drive ring and modulated by the derma structure (subsurface of the finger skin) [28]. By analyzing the signal response received by antennas array, the reconstruction of fingerprint image is performed. The fingerprint pattern is acquired by simply measuring the electric field in subsurface of finger skin.

**•** application has to be able to communicate with all connected sensors and with connected

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**•** application has to contain united and practical interface for easy and fast capturing of multiple samples of each patients of the affected finger by each dactyloscopic sensor,

The newest version of capturing application also contains full multi-language support in‐ cluding runtime dynamic language switching. At the moment, the application supports

The first created station is installed in Faculty Hospital in Olomouc in the Czech Republic. The capturing is performed by a medical specialist from the Dermatologic and Venerologic Clinic at the Palacky University and Faculty Hospital in Olomouc. In the nearest future the process of the second station creation will be finished and the second station (see Fig. 11)

Very significant and inseparable part of skin diseases influence analysis plays the process of suitable testing data acquirement. By these data it is meant a database of fingerprints affect‐ ed and influenced by at least one of various dermatologic diseases. Having done the data‐ base acquirement it will be possible to analyze the influence of specific skin diseases and/or

Obtaining a high quality biometric database usually is a long time consuming task which demands a big amount of patience. Biometric algorithms cannot be tested only on few sam‐ ples from a small group of individuals of similar age and employment like research collea‐ gues. High quality biometric database has to contain samples from wide spectrum of individuals categorized by all factors which may have influence on reason for which the da‐

**•** application has to contain controls for entering the patients diagnosis by a doctor.

digital microscope,

Czech, English and German languages.

**Figure 11.** Second version of the capturing station.

**3.3. Creation of suitable database of diseased fingerprints**

test the designed and implemented correction algorithms.

will be installed at a dermatologist in Darmstadt, Germany.

#### *3.1.7. Ultrasonic fingerprint scanners*

Ultrasonic scanners are based on sending acoustic signals toward the fingertip and captur‐ ing the response. The received response is analyzed and the fingerprint is reconstructed.

## **3.2. Creation of station for diseased fingerprint capturing**

For analyzing the influence of skin diseases on the process of fingerprint recognition it will be necessary for the capturing station to contain as much dactyloscopic sensors as possible, ideally each of them based on different scanning technology. It is also presumable that some very invasive skin disease deforms the fingerprint pattern in a way that no connected sensor will be able to scan this fingerprint. For these situations the capturing station has to be equipped with tools for manual dactyloscopic fingerprinting. Another significant and insep‐ arable part of capturing station creation process is creation of capturing application. This capturing application has to be able to communicate with all connected sensors and to fast fingerprint capturing of all patient's fingers. The capturing station should also contain some device for affected finger photo-documentation like camera, video-camera or digital micro‐ scope. This device should also be controllable by the capturing application.

The final version of capturing station consists of the following components:


Due to available financial, technological and implementation resources the following dacty‐ loscopic scanners were chosen: **Sagem MSO 300** (optical touch), **UPEK EikonTouch 500** (ca‐ pacitive touch), **UPEK Eikon II Fingerprint Reader** (capacitive sweep), **TBS 3D Enroll Series 2011** (touchless optical multispectral) and the **digital microscope DinoLite Pro.**

After obtaining all necessary hardware the next step was to design and implement the cap‐ turing application. During the design and implementation process the following require‐ ments had to be considered:


The newest version of capturing application also contains full multi-language support in‐ cluding runtime dynamic language switching. At the moment, the application supports Czech, English and German languages.

The first created station is installed in Faculty Hospital in Olomouc in the Czech Republic. The capturing is performed by a medical specialist from the Dermatologic and Venerologic Clinic at the Palacky University and Faculty Hospital in Olomouc. In the nearest future the process of the second station creation will be finished and the second station (see Fig. 11) will be installed at a dermatologist in Darmstadt, Germany.

**Figure 11.** Second version of the capturing station.

*3.1.6. E-field fingerprint scanners*

286 New Trends and Developments in Biometrics

field in subsurface of finger skin.

*3.1.7. Ultrasonic fingerprint scanners*

**•** laptop and its accessories

**•** digital microscope

ments had to be considered:

**•** capturing application installed on laptop

**•** laboratory stand with boss and clamp for microscope

**•** set of electronic dactyloscopic sensors

**•** dactyloscopic card and special ink

**3.2. Creation of station for diseased fingerprint capturing**

The sensor consists of a drive ring that generates an RF (radio frequency) sinusoidal signal and a matrix of active antennas that receives a very small amplitude signal transmitted by the drive ring and modulated by the derma structure (subsurface of the finger skin) [28]. By analyzing the signal response received by antennas array, the reconstruction of fingerprint image is performed. The fingerprint pattern is acquired by simply measuring the electric

Ultrasonic scanners are based on sending acoustic signals toward the fingertip and captur‐ ing the response. The received response is analyzed and the fingerprint is reconstructed.

For analyzing the influence of skin diseases on the process of fingerprint recognition it will be necessary for the capturing station to contain as much dactyloscopic sensors as possible, ideally each of them based on different scanning technology. It is also presumable that some very invasive skin disease deforms the fingerprint pattern in a way that no connected sensor will be able to scan this fingerprint. For these situations the capturing station has to be equipped with tools for manual dactyloscopic fingerprinting. Another significant and insep‐ arable part of capturing station creation process is creation of capturing application. This capturing application has to be able to communicate with all connected sensors and to fast fingerprint capturing of all patient's fingers. The capturing station should also contain some device for affected finger photo-documentation like camera, video-camera or digital micro‐

Due to available financial, technological and implementation resources the following dacty‐ loscopic scanners were chosen: **Sagem MSO 300** (optical touch), **UPEK EikonTouch 500** (ca‐ pacitive touch), **UPEK Eikon II Fingerprint Reader** (capacitive sweep), **TBS 3D Enroll Series 2011** (touchless optical multispectral) and the **digital microscope DinoLite Pro.**

After obtaining all necessary hardware the next step was to design and implement the cap‐ turing application. During the design and implementation process the following require‐

scope. This device should also be controllable by the capturing application. The final version of capturing station consists of the following components:

#### **3.3. Creation of suitable database of diseased fingerprints**

Very significant and inseparable part of skin diseases influence analysis plays the process of suitable testing data acquirement. By these data it is meant a database of fingerprints affect‐ ed and influenced by at least one of various dermatologic diseases. Having done the data‐ base acquirement it will be possible to analyze the influence of specific skin diseases and/or test the designed and implemented correction algorithms.

Obtaining a high quality biometric database usually is a long time consuming task which demands a big amount of patience. Biometric algorithms cannot be tested only on few sam‐ ples from a small group of individuals of similar age and employment like research collea‐ gues. High quality biometric database has to contain samples from wide spectrum of individuals categorized by all factors which may have influence on reason for which the da‐ tabase is acquired. Also there has to be enough samples in each of such category. For exam‐ ple the ideal database of fingerprints has to contain enough fingerprints of men and women of all age groups and such database should also contain so called critical samples, i.e. sam‐ ples from individuals whose job or hobby affects their fingerprint pattern like mountain climbers or people working with chemicals.

dermatologic diseases. The most common and typical that are present in the database are: light atopic eczema, advanced atopic eczema, verruca vulgaris, psoriasis and cut wound. The last sample does not belong to the dermatologic diseases it is related to them because it

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In Figures 13 to 17 we show several samples of acquired fingerprints. Each set of finger‐ prints begins with photography of the affected finger from the digital microscope and fin‐ gerprints made manually by using the dactyloscopic card and ink. After them there are

can negatively affect the process of fingerprint recognition.

**Figure 13.** Light atopic eczema – influence on fingerprints.

**Figure 14.** Advanced atopic eczema – influence on fingerprints.

**Figure 15.** Verruca vulgaris – influence on fingerprints.

fingerprints from electronic sensors if the sensor capturing was successful.

For our developing and testing purposes it is necessary to create a database of fingerprints affected by a dermatologic disease. In the presence there exists no such special database so it will be necessary to create a new and first one. The most promising and reliable sources of such data are dermatological departments in hospital. It is also necessary to agree coopera‐ tion with dermatological specialists from such departments which will be willing to scan theirs patient's fingerprints and to provide reliable diagnosis of the scanned disease. For the purpose of database categorization the following factors are considered and recorded: age, gender, job and kind of dermatologic disease.

**Figure 12.** ER diagram of current diseased fingerprint database.

Current version of the database contains 594 fingerprints of 19 different patients. These amounts are from the first acquired set from the hospital in Olomouc. There are two more sets of fingerprints but they were not processed yet. The real number of fingerprints in our database is two or three times higher. In Figure 12 you can see the entity relationship dia‐ gram of actual version of database. The database contains fingerprints of eleven different dermatologic diseases. The most common and typical that are present in the database are: light atopic eczema, advanced atopic eczema, verruca vulgaris, psoriasis and cut wound. The last sample does not belong to the dermatologic diseases it is related to them because it can negatively affect the process of fingerprint recognition.

In Figures 13 to 17 we show several samples of acquired fingerprints. Each set of finger‐ prints begins with photography of the affected finger from the digital microscope and fin‐ gerprints made manually by using the dactyloscopic card and ink. After them there are fingerprints from electronic sensors if the sensor capturing was successful.

**Figure 13.** Light atopic eczema – influence on fingerprints.

tabase is acquired. Also there has to be enough samples in each of such category. For exam‐ ple the ideal database of fingerprints has to contain enough fingerprints of men and women of all age groups and such database should also contain so called critical samples, i.e. sam‐ ples from individuals whose job or hobby affects their fingerprint pattern like mountain

For our developing and testing purposes it is necessary to create a database of fingerprints affected by a dermatologic disease. In the presence there exists no such special database so it will be necessary to create a new and first one. The most promising and reliable sources of such data are dermatological departments in hospital. It is also necessary to agree coopera‐ tion with dermatological specialists from such departments which will be willing to scan theirs patient's fingerprints and to provide reliable diagnosis of the scanned disease. For the purpose of database categorization the following factors are considered and recorded: age,

Current version of the database contains 594 fingerprints of 19 different patients. These amounts are from the first acquired set from the hospital in Olomouc. There are two more sets of fingerprints but they were not processed yet. The real number of fingerprints in our database is two or three times higher. In Figure 12 you can see the entity relationship dia‐ gram of actual version of database. The database contains fingerprints of eleven different

climbers or people working with chemicals.

288 New Trends and Developments in Biometrics

gender, job and kind of dermatologic disease.

**Figure 12.** ER diagram of current diseased fingerprint database.

**Figure 14.** Advanced atopic eczema – influence on fingerprints.

**Figure 15.** Verruca vulgaris – influence on fingerprints.

the quality of the ridge structures of input fingerprint images, is thus necessary. Generally, for a given fingerprint image, fingerprint regions can be assigned to one of the following

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**•** *Well-defined regions*, in which ridges and furrows are clearly visible for a minutia extrac‐

**•** *Recoverable corrupted regions*, in which ridges and furrows are corrupted by a small amount of creases, smudges, etc. But they can still be correctly recovered by an enhance‐

**•** *Unrecoverable corrupted regions*, in which ridges and furrows are corrupted by such a se‐

**Figure 18.** Examples of fingerprint regions [29]: a) Well-defined region (left); b) Recoverable region (middle); c) Unre‐

The interoperability among sensors from different vendors, or using different sensing tech‐ nologies, plays a relevant role. The resulting images from different technologies vary very much in the representation of the grayscale levels, sharpness of valleys and ridges and reso‐ lution. Fortunately, it is often possible to compensate these factors to achieve a good intero‐

Based on filtering domains, most fingerprint enhancement schemes can be roughly classi‐ fied using two major approaches [31]: *spatial-domain and frequency-domain*. The filtering in a spatial-domain applies a convolution directly to the fingerprint image. On the other hand, the filtering in a frequency-domain needs the Fourier analysis and synthesis. Thus a finger‐ print image is transformed than multiplied by filter coefficients and in the end inverse-trans‐ formed by Fourier coefficients back to an enhanced fingerprint image. In fact, if the employed filters are the same, enhancement results from both domains should be exactly the same according to the signal processing theorem. However, in a practical implementa‐ tion, these two approaches are different in terms of enhancement quality and computational

In the following subchapters, some important and often used fingerprint enhancement methods will be introduced. Nevertheless, the list of such methods cannot be complete, as

the amount of such methods exceeds the scope and possibilities of this chapter.

vere amount of noise and distortion that it is impossible to recover them.

three categories (Fig. 18) [29]:

ment algorithm.

coverable region (right).

complexity of algorithms.

perability among such sensors, e.g. see [30].

tion algorithm to operate reliably.

**Figure 16.** Psoriasis – influence on fingerprints.

**Figure 17.** Cut wound – influence on fingerprints.

## **4. Fingerprint image enhancement algorithm**

Currently the most widely used and the most accurate automatic fingerprint verification/ identification techniques use minutiae-based automatic fingerprint matching algorithms. Reliably extracting minutiae from the input fingerprint images is critical to fingerprint matching. The performance of current minutiae extraction algorithms depends heavily on the quality of input fingerprint images [29]. In an ideal fingerprint image, ridges and valleys alternate and flow in a locally constant direction and minutiae are anomalies of ridges. In practice, due to variations in impression conditions, ridge configurations, skin conditions (dryness, moist finger, aberrant formations in epidermal ridges of fingerprints, postnatal marks, occupational marks, skin diseases), acquisition devices, and non-cooperative atti‐ tudes of subjects, etc., a significant percentage of acquired fingerprint images (approximate‐ ly 10% according to [29]) is of a poor quality. The ridge structures in poor-quality fingerprint images are not always well defined and hence they cannot be always correctly detected. This could result in failures of minutiae extraction algorithms; a significant number of spuri‐ ous minutiae may be created, a large percentage of genuine minutiae may be undetected, and a significant amount of error in position and orientation may be introduced.

To ensure that the performance of the minutiae extraction algorithms is robust with respect to the quality of input fingerprint images, an *enhancement algorithm* [42], which can improve the quality of the ridge structures of input fingerprint images, is thus necessary. Generally, for a given fingerprint image, fingerprint regions can be assigned to one of the following three categories (Fig. 18) [29]:


**Figure 16.** Psoriasis – influence on fingerprints.

290 New Trends and Developments in Biometrics

**Figure 17.** Cut wound – influence on fingerprints.

**4. Fingerprint image enhancement algorithm**

Currently the most widely used and the most accurate automatic fingerprint verification/ identification techniques use minutiae-based automatic fingerprint matching algorithms. Reliably extracting minutiae from the input fingerprint images is critical to fingerprint matching. The performance of current minutiae extraction algorithms depends heavily on the quality of input fingerprint images [29]. In an ideal fingerprint image, ridges and valleys alternate and flow in a locally constant direction and minutiae are anomalies of ridges. In practice, due to variations in impression conditions, ridge configurations, skin conditions (dryness, moist finger, aberrant formations in epidermal ridges of fingerprints, postnatal marks, occupational marks, skin diseases), acquisition devices, and non-cooperative atti‐ tudes of subjects, etc., a significant percentage of acquired fingerprint images (approximate‐ ly 10% according to [29]) is of a poor quality. The ridge structures in poor-quality fingerprint images are not always well defined and hence they cannot be always correctly detected. This could result in failures of minutiae extraction algorithms; a significant number of spuri‐ ous minutiae may be created, a large percentage of genuine minutiae may be undetected,

and a significant amount of error in position and orientation may be introduced.

To ensure that the performance of the minutiae extraction algorithms is robust with respect to the quality of input fingerprint images, an *enhancement algorithm* [42], which can improve

**Figure 18.** Examples of fingerprint regions [29]: a) Well-defined region (left); b) Recoverable region (middle); c) Unre‐ coverable region (right).

The interoperability among sensors from different vendors, or using different sensing tech‐ nologies, plays a relevant role. The resulting images from different technologies vary very much in the representation of the grayscale levels, sharpness of valleys and ridges and reso‐ lution. Fortunately, it is often possible to compensate these factors to achieve a good intero‐ perability among such sensors, e.g. see [30].

Based on filtering domains, most fingerprint enhancement schemes can be roughly classi‐ fied using two major approaches [31]: *spatial-domain and frequency-domain*. The filtering in a spatial-domain applies a convolution directly to the fingerprint image. On the other hand, the filtering in a frequency-domain needs the Fourier analysis and synthesis. Thus a finger‐ print image is transformed than multiplied by filter coefficients and in the end inverse-trans‐ formed by Fourier coefficients back to an enhanced fingerprint image. In fact, if the employed filters are the same, enhancement results from both domains should be exactly the same according to the signal processing theorem. However, in a practical implementa‐ tion, these two approaches are different in terms of enhancement quality and computational complexity of algorithms.

In the following subchapters, some important and often used fingerprint enhancement methods will be introduced. Nevertheless, the list of such methods cannot be complete, as the amount of such methods exceeds the scope and possibilities of this chapter.

#### **4.1.1. Spatial domain filtering algorithm**

The *spatial domain filtering algorithm* [29] adaptively enhances the clarity of ridge and valley structures using a bank of Gabor filters (see below) that are tuned to the local ridge orienta‐ tion and ridge frequency. The local ridge orientation and ridge frequency are estimated di‐ rectly from input images in the spatial domain.

A 2D Gabor filter [32] can be thought of as a complex plane wave modulated by a 2D Gaus‐ sian envelope [33]. These filters optimally capture both the local orientation and frequency information and their development has been initiated by observing the linear response of the receptive field in simple striate cortex cells. By tuning a Gabor filter to a specific frequen‐ cy and direction, the local frequency and orientation information can be obtained. Thus, they are well suited for extracting the texture information from images.

An even symmetric Gabor filter has the following general form in the spatial domain [33]:

$$\mathbf{G}\_{\theta,f}(\mathbf{x},\mathbf{y}) = e^{-\frac{1}{2}\left[\frac{\mathbf{x}'^2}{\delta\_x^2} + \frac{\mathbf{y}'^2}{\delta\_y^2}\right]} \cos(2\pi f \mathbf{x}') \tag{1}$$

frequency is defined as the average frequency in the neighborhood. The local ridge fre‐

**•** *Estimation of region mask*. The region mask is used to indicate the category of pixels. A pix‐ el could be either a non-ridge-and-valley (unrecoverable) pixel or a ridge-and-valley (re‐ coverable) pixel. A pixel (or a block of pixels) in an input fingerprint image could be either in a recoverable region or in an unrecoverable region. The classification of pixels into recoverable and unrecoverable categories can be performed based on the assessment

**•** Filtering. A bank of Gabor filters tuned to the local ridge orientation and ridge frequency is applied to the ridge-and-valley pixels in the normalized input fingerprint image to ob‐

**Filtering**

The fingerprint enhancement approach in a frequency domain introduced in [31] consists of four concatenated processes: discrete cosine transform of sub-blocks of partitioning finger‐ print, ridge orientation and frequency parameters estimation, filtering in DCT (*Discrete Co‐ sine Transform*) domain and inverse discrete cosine transform of sub-blocks. The advantages

**•** Fingerprint ridges form a natural sinusoid image – its spectrums are packed or localized in a frequency domain. Hence these spectrums can be easily shaped or filtered in this do‐ main. Moreover, a filter can be specially designed in order to handle high curvature ridge area such as singular points. This is a great advantage over the spatial-domain filtering

**•** When comparing with the discrete Fourier transform, the discrete cosine transform per‐ forms better in terms of energy compaction. Moreover, DCT coefficients are real numbers in comparison with complex numbers of discrete Fourier transform (DFT) coefficients. Therefore, we can handle DCT coefficients easier than DFT coefficients. Besides, the fast

**Region mask generation**

**Frequency images estimation**

**Orientation image estimation**

**Normalization**

**Enhanced image**

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quency represents another intrinsic property of a fingerprint image.

of the shape of the wave formed by local ridges and valleys.

**Figure 19.** The flowchart of the spatial domain fingerprint enhancement algorithm [34].

tain an enhanced fingerprint image.

**Input image**

**4.1.2. Frequency domain filtering algorithm**

of the proposed approach are as follows [31]:

approach.

$$\mathbf{x} \text{ and } \mathbf{x}' = \mathbf{x}\sin\theta + y\cos\theta, \ y' = \mathbf{x}\cos\theta - y\sin\theta \tag{2}$$

where f is the frequency of the sinusoidal plane wave at the angle θ with the x-axis, and δ<sup>x</sup> and δy are the standard deviations of the Gaussian envelope along the x and y axes, respec‐ tively.

The main steps of the enhancement algorithm are shown in Fig. 19 and are listed below [29]:


frequency is defined as the average frequency in the neighborhood. The local ridge fre‐ quency represents another intrinsic property of a fingerprint image.


**Figure 19.** The flowchart of the spatial domain fingerprint enhancement algorithm [34].

#### **4.1.2. Frequency domain filtering algorithm**

**4.1.1. Spatial domain filtering algorithm**

292 New Trends and Developments in Biometrics

rectly from input images in the spatial domain.

tively.

The *spatial domain filtering algorithm* [29] adaptively enhances the clarity of ridge and valley structures using a bank of Gabor filters (see below) that are tuned to the local ridge orienta‐ tion and ridge frequency. The local ridge orientation and ridge frequency are estimated di‐

A 2D Gabor filter [32] can be thought of as a complex plane wave modulated by a 2D Gaus‐ sian envelope [33]. These filters optimally capture both the local orientation and frequency information and their development has been initiated by observing the linear response of the receptive field in simple striate cortex cells. By tuning a Gabor filter to a specific frequen‐ cy and direction, the local frequency and orientation information can be obtained. Thus,

An even symmetric Gabor filter has the following general form in the spatial domain [33]:

2 2 2 2 1 2 , (,) cos(2 ) *x y x y G xy e <sup>f</sup> fx* d d

é ù - + ê ú ê ú

and*xx y yx y* =+ =- sin cos , cos sin

 q

where f is the frequency of the sinusoidal plane wave at the angle θ with the x-axis, and δ<sup>x</sup> and δy are the standard deviations of the Gaussian envelope along the x and y axes, respec‐

The main steps of the enhancement algorithm are shown in Fig. 19 and are listed below [29]:

**•** *Normalization*. An input image needs to be normalized so that it has a pre-specified mean and variance. The normalization is a pixel-wise operation, in which an output pixel value depends only on the corresponding input pixel. It does not change the clarity of the ridge and valley structures. The main purpose of normalization is to reduce the variations in

**•** *Local ridge orientation estimation*. The local orientation indicates the major ridge orientation tendency in a local neighborhood. It represents an intrinsic property of a fingerprint im‐ age and defines an invariant coordinate for ridges and valleys in a local neighborhood. In neighboring ridges, the local ridge orientation changes slowly. Therefore, it is usually a specified block-wise property. In addition, there is no difference between a local ridge ori‐ entation of 90° and 270°, since the ridges oriented at 90° and the ridges oriented at 270° in

**•** *Local ridge frequency estimation*. Local ridge frequency is the frequency of the ridge and val‐ ley structures in a local neighborhood along a direction normal to the local ridge orienta‐ tion. The ridge and valley structures in a local neighborhood, where minutiae or singular points appear, do not form a well-defined sinusoidal-shaped wave. In such situations, the

gray-level values along ridges and valleys what facilitates the subsequent steps.

a local neighborhood cannot be differentiated from each other.

q

p

 qq

ë û <sup>=</sup> (1)

(2)

they are well suited for extracting the texture information from images.

q

> The fingerprint enhancement approach in a frequency domain introduced in [31] consists of four concatenated processes: discrete cosine transform of sub-blocks of partitioning finger‐ print, ridge orientation and frequency parameters estimation, filtering in DCT (*Discrete Co‐ sine Transform*) domain and inverse discrete cosine transform of sub-blocks. The advantages of the proposed approach are as follows [31]:


DCT requires less computational complexity and less memory usage when comparing with the fast Fourier transform (FFT).

The Hd(ϕ) filter, which performs the ridge orientation filtering, is given by [31]:

<sup>0</sup> ( ,, )

jj s j

form. The quadrant correction filter, Hq(u,v), is given by [31]:

ì

*H uv*

band of the DCT domain as shown in Fig. 20.

plied only in the principal region.

( ) cos <sup>2</sup>

<sup>ï</sup> é ù - <sup>ï</sup> <sup>×</sup> ê ú <sup>ï</sup> î ë û

*<sup>q</sup> u v*

rameter, and ϕBW is the angular bandwidth.

by [31]:

j

2 0 2 ( )

where ϕ0 is the peak orientation for the bandpass filter, σϕ is the directional bandwidth pa‐

*Filtering by Convolution* [31]: Since θ and π-θ ridge orientation coefficients are projected into the same DCT domain region, both directional coefficients still remain from the previous fil‐ tering. In order to truncate inappropriate directional coefficients, two diagonal Gabor filters are exploited by the convolution operation. The finally enhanced DCT coefficients are given

where Fenh(u,v) are enhanced DCT coefficients in rectangular-form, Ffd(u,v) is the previous result of enhanced DCT coefficients in rectangular-form converted from Ffd(ρ,ϕ) in polar-

> 2 2

( ) 2

*u v*

<sup>+</sup> -

*q*

s

( ) 2


*q*

s

where σq is the quadratic parameter and cos(nπ/2) can attain only one of the three following values: -1, 0 or 1. Indeed, this convolution operation requires less computing because most of bandpass filtered coefficients are truncated to zero from the previous operation. In case of highly curved ridges, the transformed coefficients are projected into widely curved sub-

From Fig. 20, we can approximate the orientation range from θ1 to θ2 by a non-coherence factor from Eq. (2). The curved sub-band can be classified as one of two regions, either the principal region (R1) or the reflection region (R2). The principal region R1 contains only one diagonal component (45° or 135°) as mentioned before. The 45° or 135° diagonal compo‐ nents correspond to the phase pattern of the oriented ridges in the range of 0° to 90° or 90° to 180°, respectively. The reflection region R2 is composed of both 45° and 135° diagonal components from the reflection property of DCT coefficients. Then the convolution is ap‐

( ) cos <sup>2</sup> (,) <sup>2</sup>

<sup>ï</sup> é ù <sup>+</sup> <sup>×</sup> <sup>ï</sup> ê ú <sup>ï</sup> ë û <sup>³</sup> <sup>=</sup> <sup>í</sup>

*u v <sup>e</sup>*

p

p

2 2

*otherwise u v <sup>e</sup>*

q p

1 *BW Hd BW <sup>e</sup> otherwise* j

s

j j


ï î 2 0

<sup>ï</sup> - ³ <sup>=</sup> <sup>í</sup>

jj

 j

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(,) (,) (,) *enh fd <sup>q</sup> F uv F uv H uv* = \* (6)

(5)

295

(7)

**•** By partitioning a fingerprint into sub-blocks, the proposed approach utilizes the spatially contextual information including the instantaneous frequency and orientation. Intrinsic features such as ridge frequency, ridge orientation, and angular bandwidth can be simply analyzed directly from DCT coefficients.

Conventional fingerprint enhancement schemes, when applied with non-overlapping blocks of partitioning fingerprint, often encounter blocking artifacts such as ridge discontinuities and spurious minutiae [31]. To preserve the ridge continuity and eliminate blocking arti‐ facts, an overlapping block is applied to both DCT decomposition and reconstruction proce‐ dures. However, there is no need to apply any smooth spectral window for DCT because the overlapping area is large enough to prevent any blocking effects, corresponding with its en‐ ergy compaction property.

#### **4.1.3. Enhancement filtering in DCT domain**

In the DCT domain, the filtering process is not simply the same as in the DFT domain [31] which required only the multiplication of coefficients. The Gabor filter is modified in order to cooperate with the DCT domain based on the Cartesian-form representation. The en‐ hancement filtering in the DCT domain can be divided into two arithmetic manipulations, i.e. multiplication and convolution.

*Filtering by Multiplication* [31]: The enhancement filter can be expressed in terms of the prod‐ uct of separable Gaussian functions what is similar to the frequency-domain filtering techni‐ que [31]:

$$F\_{j\underline{l}}(\rho,\varphi) = F(\rho,\varphi)H\_f(\rho)H\_d(\varphi) \tag{3}$$

where *F*(ρ,ϕ) are DCT coefficients in polar-form representation, directly related to DCT coef‐ ficients *F*(*u*,*v*) in rectangular-form representation. *F*fd(ρ,ϕ) are DCT coefficients of the filter‐ ing output. The *H*<sup>f</sup> (ρ) filter, which performs the ridge frequency filtering in Gaussian shape, is given by [31]:

$$H\_f(\rho \mid \rho\_0, \sigma\_{\rho'}, Z) = e^{-\frac{(\rho \cdot \rho\_0)^2}{2\sigma\_\rho^2}}, \rho\_0 = \sqrt{\mu\_0^2 + \upsilon\_0^2} \tag{4}$$
 
$$\rho\_{\text{min}} \le \rho\_0 \le \rho\_{\text{max}}$$

where ρ0 and σρ are the center of the high-peak frequency group and the filtering bandwidth parameter, respectively. The ρmin and ρmax parameters are minimum and maximum cut-off frequency constraints which suppress the effects of lower and higher frequencies such as ink, sweat gland holes or scratches in the fingerprint. Z is a filtering normalization factor de‐ pending on the filtering energy result.

The Hd(ϕ) filter, which performs the ridge orientation filtering, is given by [31]:

DCT requires less computational complexity and less memory usage when comparing

**•** By partitioning a fingerprint into sub-blocks, the proposed approach utilizes the spatially contextual information including the instantaneous frequency and orientation. Intrinsic features such as ridge frequency, ridge orientation, and angular bandwidth can be simply

Conventional fingerprint enhancement schemes, when applied with non-overlapping blocks of partitioning fingerprint, often encounter blocking artifacts such as ridge discontinuities and spurious minutiae [31]. To preserve the ridge continuity and eliminate blocking arti‐ facts, an overlapping block is applied to both DCT decomposition and reconstruction proce‐ dures. However, there is no need to apply any smooth spectral window for DCT because the overlapping area is large enough to prevent any blocking effects, corresponding with its en‐

In the DCT domain, the filtering process is not simply the same as in the DFT domain [31] which required only the multiplication of coefficients. The Gabor filter is modified in order to cooperate with the DCT domain based on the Cartesian-form representation. The en‐ hancement filtering in the DCT domain can be divided into two arithmetic manipulations,

*Filtering by Multiplication* [31]: The enhancement filter can be expressed in terms of the prod‐ uct of separable Gaussian functions what is similar to the frequency-domain filtering techni‐

> r j

(ρ) filter, which performs the ridge frequency filtering in Gaussian shape,

<sup>2</sup> <sup>+</sup> *<sup>v</sup>*<sup>0</sup> 2 ,

,*ρ*<sup>0</sup> = *u*<sup>0</sup>

where *F*(ρ,ϕ) are DCT coefficients in polar-form representation, directly related to DCT coef‐ ficients *F*(*u*,*v*) in rectangular-form representation. *F*fd(ρ,ϕ) are DCT coefficients of the filter‐

= (3)

(4)

(,) (,) () () *fd f d F F HH*

− (*ρ*−*ρ*0)<sup>2</sup> 2*σρ* 2

min 0 max

where ρ0 and σρ are the center of the high-peak frequency group and the filtering bandwidth parameter, respectively. The ρmin and ρmax parameters are minimum and maximum cut-off frequency constraints which suppress the effects of lower and higher frequencies such as ink, sweat gland holes or scratches in the fingerprint. Z is a filtering normalization factor de‐

 rr£ £

 rj

rj

*Hf* (*ρ* |*ρ*0, *σρ*, *Z*)=*e*

r

with the fast Fourier transform (FFT).

294 New Trends and Developments in Biometrics

analyzed directly from DCT coefficients.

**4.1.3. Enhancement filtering in DCT domain**

i.e. multiplication and convolution.

pending on the filtering energy result.

que [31]:

ing output. The *H*<sup>f</sup>

is given by [31]:

ergy compaction property.

$$H\_d(\boldsymbol{\varphi} \left| \boldsymbol{\varphi}\_0, \boldsymbol{\sigma}\_{\boldsymbol{\varphi}}, \boldsymbol{\varphi}\_{\text{BW}} \right> = \begin{cases} \frac{(\boldsymbol{\varphi} - \boldsymbol{\varphi}\_0)^2}{2\sigma\_{\boldsymbol{\varphi}}^2} & \left| \boldsymbol{\varphi} - \boldsymbol{\varphi}\_0 \right| \ge \boldsymbol{\varphi}\_{\text{BW}} \\\ 1 & \text{otherwise} \end{cases} \tag{5}$$

where ϕ0 is the peak orientation for the bandpass filter, σϕ is the directional bandwidth pa‐ rameter, and ϕBW is the angular bandwidth.

*Filtering by Convolution* [31]: Since θ and π-θ ridge orientation coefficients are projected into the same DCT domain region, both directional coefficients still remain from the previous fil‐ tering. In order to truncate inappropriate directional coefficients, two diagonal Gabor filters are exploited by the convolution operation. The finally enhanced DCT coefficients are given by [31]:

$$F\_{emh}(\mu, \upsilon) = F\_{fd}(\mu, \upsilon) \* H\_q(\mu, \upsilon) \tag{6}$$

where Fenh(u,v) are enhanced DCT coefficients in rectangular-form, Ffd(u,v) is the previous result of enhanced DCT coefficients in rectangular-form converted from Ffd(ρ,ϕ) in polarform. The quadrant correction filter, Hq(u,v), is given by [31]:

$$H\_q(\mu, \upsilon) = \begin{cases} \cos\left[\frac{(\mu + \upsilon)\pi}{2}\right] \cdot e^{-\frac{(\mu + \upsilon)^2}{2\sigma\_q^2}} & \theta \ge \pi \Big/ \\\cos\left[\frac{(\mu - \upsilon)\pi}{2}\right] \cdot e^{-\frac{(\mu - \upsilon)^2}{2\sigma\_q^2}} & \text{otherwise} \end{cases} \tag{7}$$

where σq is the quadratic parameter and cos(nπ/2) can attain only one of the three following values: -1, 0 or 1. Indeed, this convolution operation requires less computing because most of bandpass filtered coefficients are truncated to zero from the previous operation. In case of highly curved ridges, the transformed coefficients are projected into widely curved subband of the DCT domain as shown in Fig. 20.

From Fig. 20, we can approximate the orientation range from θ1 to θ2 by a non-coherence factor from Eq. (2). The curved sub-band can be classified as one of two regions, either the principal region (R1) or the reflection region (R2). The principal region R1 contains only one diagonal component (45° or 135°) as mentioned before. The 45° or 135° diagonal compo‐ nents correspond to the phase pattern of the oriented ridges in the range of 0° to 90° or 90° to 180°, respectively. The reflection region R2 is composed of both 45° and 135° diagonal components from the reflection property of DCT coefficients. Then the convolution is ap‐ plied only in the principal region.

**•** to attach weights for each minutiae according the quality of fingerprint area in which they are located so during the minutiae based comparison process the weights of each minuti‐

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According to the source [28] the methods for fingerprint quality estimation can be divided into two categories: approaches based on local features extraction and approaches based on

Fingerprint area segmentation, sometimes also known as fingerprint foreground / back‐

The human finger and its fingerprint have a typical rounded shape. Fingerprint images from electronic dactyloscopic scanners are usually rectangles containing the rounded-shape fin‐ gerprint and of course a background. For the fingerprint processing purposes, mainly for the quality estimation process based on local feature extraction it is important to exclude the fingerprint background from the process. This needs to be done not only for speeding up the calculation time but also for making the estimation process more precise. Therefore, the main fingerprint foreground feature extraction is needed before the fingerprint area seg‐

Generally the segmentation is a process of dividing the input image into several nonoverlapping parts where usually the individual objects and the background are meant by the parts. If there is only one object in the image the segmentation can be called foreground/background detection. The fingerprint area segmentation is a process of detection which part of an image belongs to the fingerprint and which part of the

For our research we decided to test the following fingerprint area segmentation techniques: block grayscale variance method [37], *directional method* [37] and the *Gabor filter method* [38]. The block grayscale variance method computes variance of pixel intensity in each block and by using a set up threshold the decision logic marks each block as a background block or as a foreground (fingerprint) block. The block grayscale variance method is based on the idea that image blocks which contain fingerprint pattern have a high pixel intensity variance and the blocks with background have a low pixel intensity variance. The directional method de‐ scribed in [37] makes a foreground/background decision based on the block nominal direc‐ tion computed from pixel nominal direction. This approach is based on the idea that the block with fingerprint has a high value of the block nominal direction and the others have a low value of the block nominal direction. The third method (Gabor filter method) [38] uses

Generally the process of fingerprint quality estimation consists of three phases:

**2.** Fingerprint foreground feature extraction (local or global)

ae are considered as well.

**1.** Fingerprint area segmentation

**5.1. Fingerprint area segmentation**

image belongs to the background.

ground detection, is complex and difficult task.

**3.** Quality value estimation

mentation.

global features extraction.

**Figure 20.** Highly curved ridges in spatial and frequency (DCT) domain. The signal is localized in a widely curved subband which can be classified as either the principal region (R1) or the reflection region (R2) [31].

It is then possible to compute a quality index of the fingerprint in the frequency domain [35] which gives us the information about the fingerprint image quality.

## **5. Fingerprint quality estimation**

Quality of fingerprint image has a strong influence on biometric system performance. There exist several factors on which the final fingerprint quality depends: skin conditions, sensor quality and conditions, user cooperation and proper use of sensing device [36]. For the skin conditions the most influencing factors are dryness, presence of dirt and smudge and main‐ ly the presence of skin disease. By the quality of the fingerprint image usually the "clarity" of ridge and valley pattern is meant. Because of existence of different fingerprint quality measures and indexes the standardization of fingerprint image quality as a precisely de‐ fined unit was needed. Fingerprint image quality according the international standard ISO/IEC 19794-4:2005 is defined as an integer from interval <0, 100>, where 0 corresponds to the lowest possible fingerprint image quality and 100 corresponds to the best possible fin‐ gerprint image quality. Transformation of values from other quality indexes can be per‐ formed by, for example, normalization.

The process of fingerprint image quality estimation is very important part of fingerprint rec‐ ognition system, because it enables for example:


**•** to attach weights for each minutiae according the quality of fingerprint area in which they are located so during the minutiae based comparison process the weights of each minuti‐ ae are considered as well.

According to the source [28] the methods for fingerprint quality estimation can be divided into two categories: approaches based on local features extraction and approaches based on global features extraction.

Generally the process of fingerprint quality estimation consists of three phases:


**Figure 20.** Highly curved ridges in spatial and frequency (DCT) domain. The signal is localized in a widely curved sub-

It is then possible to compute a quality index of the fingerprint in the frequency domain [35]

Quality of fingerprint image has a strong influence on biometric system performance. There exist several factors on which the final fingerprint quality depends: skin conditions, sensor quality and conditions, user cooperation and proper use of sensing device [36]. For the skin conditions the most influencing factors are dryness, presence of dirt and smudge and main‐ ly the presence of skin disease. By the quality of the fingerprint image usually the "clarity" of ridge and valley pattern is meant. Because of existence of different fingerprint quality measures and indexes the standardization of fingerprint image quality as a precisely de‐ fined unit was needed. Fingerprint image quality according the international standard ISO/IEC 19794-4:2005 is defined as an integer from interval <0, 100>, where 0 corresponds to the lowest possible fingerprint image quality and 100 corresponds to the best possible fin‐ gerprint image quality. Transformation of values from other quality indexes can be per‐

The process of fingerprint image quality estimation is very important part of fingerprint rec‐

**•** to reject fingerprints with very low quality during the enrolment process and force the

**•** to reject fingerprints with very low quality during the comparison process – for the non-

**•** appropriate choosing the comparing algorithm in systems having different algorithms for

band which can be classified as either the principal region (R1) or the reflection region (R2) [31].

which gives us the information about the fingerprint image quality.

**5. Fingerprint quality estimation**

296 New Trends and Developments in Biometrics

formed by, for example, normalization.

differently quality fingerprints,

ognition system, because it enables for example:

user to perform a new attempt to enroll a quality fingerprint,

forensic applications it is better way than false accept decision,

Fingerprint area segmentation, sometimes also known as fingerprint foreground / back‐ ground detection, is complex and difficult task.

#### **5.1. Fingerprint area segmentation**

The human finger and its fingerprint have a typical rounded shape. Fingerprint images from electronic dactyloscopic scanners are usually rectangles containing the rounded-shape fin‐ gerprint and of course a background. For the fingerprint processing purposes, mainly for the quality estimation process based on local feature extraction it is important to exclude the fingerprint background from the process. This needs to be done not only for speeding up the calculation time but also for making the estimation process more precise. Therefore, the main fingerprint foreground feature extraction is needed before the fingerprint area seg‐ mentation.

Generally the segmentation is a process of dividing the input image into several nonoverlapping parts where usually the individual objects and the background are meant by the parts. If there is only one object in the image the segmentation can be called foreground/background detection. The fingerprint area segmentation is a process of detection which part of an image belongs to the fingerprint and which part of the image belongs to the background.

For our research we decided to test the following fingerprint area segmentation techniques: block grayscale variance method [37], *directional method* [37] and the *Gabor filter method* [38]. The block grayscale variance method computes variance of pixel intensity in each block and by using a set up threshold the decision logic marks each block as a background block or as a foreground (fingerprint) block. The block grayscale variance method is based on the idea that image blocks which contain fingerprint pattern have a high pixel intensity variance and the blocks with background have a low pixel intensity variance. The directional method de‐ scribed in [37] makes a foreground/background decision based on the block nominal direc‐ tion computed from pixel nominal direction. This approach is based on the idea that the block with fingerprint has a high value of the block nominal direction and the others have a low value of the block nominal direction. The third method (Gabor filter method) [38] uses eight different oriented Gabor filters for computing the vector of eight Gabor features. The standard deviation of these features can be used as a threshold for the foreground/back‐ ground decision.

gerprint image quality estimation. The main requirements were capability of batch fingerprint image batch processing using different processing algorithms and reusable ex‐ tendable code design. During the design process the whole task was split into two impor‐ tant sub-tasks: the fingerprint quality estimation library and the batch processing tool

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The fingerprint quality estimation library was implemented in C++ programming language with the use of OpenCV 2.1 library for some image processing operation. The library pro‐ vides three fingerprint segmentation algorithms (variance, directional and Gabor filter) and four fingerprint quality estimation algorithms (check ratio, directional contrast, directional

The graphical user interface was implemented in C++ programming language with the use of Qt 4.6 framework. The application observes the Model-View-Controller design model. The application uses the cooperating library algorithms for fingerprint segmentation and fingerprint quality estimation and due to fact that the check ratio quality estimation algo‐ rithm can be used as add-on for other quality estimation methods the application can batch process the all fingerprint images in specified directory by 21 different fingerprint quality estimation pipelines. Results of the fingerprint quality estimation are then normalized into 0 – 100 interval according the standard and can be exported into .csv or .xml file format.

The created database of fingerprints with a skin disease has been tested for the quality in the closest past. For the testing the implemented tool with all 21 local feature extraction based methods and the global NFIQ method were used. The results were normalized into interval from 0 to 100 where 100 means the best possible fingerprint quality. The example of ob‐

*Quality estimation method Cut wound Verruca vulgaris Atopic*

NFIQ 61 45 5 7

*eczema Psoriasis*

48 53 41 37

10 11 10 13

48 52 51 48

tained results of the most promising methods can be seen in Table 1.

graphical user interface.

and Gabor filter).

**5.5. Diseased fingerprints quality testing**

Variance method segmentation + Gabor filter quality

Variance method segmentation + Directional quality

Directional method segmentation + Directional contrast

**Table 1.** Example of results of diseased fingerprint quality testing.

estimation + Check ratio

quality + Check ratio

estimation

#### **5.2. Local feature extraction based methods**

Local feature extraction methods divide the input image into rectangular blocks of a specific size. Next step is to estimate fingerprint quality value for each block and by using these val‐ ues to compute the fingerprint quality value for the whole image. The final fingerprint im‐ age quality value is usually computed as a rate of count of blocks with high fingerprint quality value divided by the count of all blocks. These blocks may also contain information about their weight. The weight of each block then corresponds to its quality.

In the presence we have the following implementations of the local feature based quality es‐ timation algorithms: *directional contrast method, directional method, Gabor filter method and the check ratio method*. These methods work with blocks and compute a feature which character‐ izes the block and works as a quality index. The directional contrast method [39] computes the directional contrast of a local ridge flow orientation. The directional method uses the block nominal direction value and the Gabor filter method makes the quality value estima‐ tion by using the standard deviation of several different orientated Gabor filter responses. The check ratio method [40] is very simple and not precise method which presumes that the high quality fingerprints have a higher rate of foreground block count to the background block count. The success of the basic check ratio method mainly depends on the quality of the previous fingerprint image segmentation. However the check ratio method has a much better utilization because it can be used for weighting the result of previous three fingerprint quality estimation algorithms in order to make the quality estimation result more precise.

#### **5.3. Methods on global feature extraction**

The methods based on global feature extraction estimate the fingerprint quality value from the features extracted from the whole fingerprint image, not only some image block. The most important representative of this group is the method developed by the US National In‐ stitute of Standards and Technology (NIST) and its name is NFIQ *(NIST Fingerprint Image Quality)* rate. The NFIQ rate divides the fingerprint images into five categories according to their estimated quality. NFIQ defines the quality of an input image as a prediction of a com‐ parison performance [28]. The fingerprints with a high quality will probably achieve a high comparison score. The NFIQ implementation uses a special vector of features for fingerprint quality estimation, created by the fingerprint quality map and statistics of its internal algo‐ rithm for minutiae extraction. These feature vectors are then used as an input for the multilayer perceptron neural network [41] which decides about the resulting fingerprint quality.

#### **5.4. Creation of batch fingerprint quality estimation tool**

One of the crucial tasks for the mapping the influence of dermatologic diseases on the fin‐ gerprint image quality was to design and implement an application for automatic batch fin‐ gerprint image quality estimation. The main requirements were capability of batch fingerprint image batch processing using different processing algorithms and reusable ex‐ tendable code design. During the design process the whole task was split into two impor‐ tant sub-tasks: the fingerprint quality estimation library and the batch processing tool graphical user interface.

The fingerprint quality estimation library was implemented in C++ programming language with the use of OpenCV 2.1 library for some image processing operation. The library pro‐ vides three fingerprint segmentation algorithms (variance, directional and Gabor filter) and four fingerprint quality estimation algorithms (check ratio, directional contrast, directional and Gabor filter).

The graphical user interface was implemented in C++ programming language with the use of Qt 4.6 framework. The application observes the Model-View-Controller design model. The application uses the cooperating library algorithms for fingerprint segmentation and fingerprint quality estimation and due to fact that the check ratio quality estimation algo‐ rithm can be used as add-on for other quality estimation methods the application can batch process the all fingerprint images in specified directory by 21 different fingerprint quality estimation pipelines. Results of the fingerprint quality estimation are then normalized into 0 – 100 interval according the standard and can be exported into .csv or .xml file format.

#### **5.5. Diseased fingerprints quality testing**

eight different oriented Gabor filters for computing the vector of eight Gabor features. The standard deviation of these features can be used as a threshold for the foreground/back‐

Local feature extraction methods divide the input image into rectangular blocks of a specific size. Next step is to estimate fingerprint quality value for each block and by using these val‐ ues to compute the fingerprint quality value for the whole image. The final fingerprint im‐ age quality value is usually computed as a rate of count of blocks with high fingerprint quality value divided by the count of all blocks. These blocks may also contain information

In the presence we have the following implementations of the local feature based quality es‐ timation algorithms: *directional contrast method, directional method, Gabor filter method and the check ratio method*. These methods work with blocks and compute a feature which character‐ izes the block and works as a quality index. The directional contrast method [39] computes the directional contrast of a local ridge flow orientation. The directional method uses the block nominal direction value and the Gabor filter method makes the quality value estima‐ tion by using the standard deviation of several different orientated Gabor filter responses. The check ratio method [40] is very simple and not precise method which presumes that the high quality fingerprints have a higher rate of foreground block count to the background block count. The success of the basic check ratio method mainly depends on the quality of the previous fingerprint image segmentation. However the check ratio method has a much better utilization because it can be used for weighting the result of previous three fingerprint quality estimation algorithms in order to make the quality estimation result more precise.

The methods based on global feature extraction estimate the fingerprint quality value from the features extracted from the whole fingerprint image, not only some image block. The most important representative of this group is the method developed by the US National In‐ stitute of Standards and Technology (NIST) and its name is NFIQ *(NIST Fingerprint Image Quality)* rate. The NFIQ rate divides the fingerprint images into five categories according to their estimated quality. NFIQ defines the quality of an input image as a prediction of a com‐ parison performance [28]. The fingerprints with a high quality will probably achieve a high comparison score. The NFIQ implementation uses a special vector of features for fingerprint quality estimation, created by the fingerprint quality map and statistics of its internal algo‐ rithm for minutiae extraction. These feature vectors are then used as an input for the multilayer perceptron neural network [41] which decides about the resulting fingerprint quality.

One of the crucial tasks for the mapping the influence of dermatologic diseases on the fin‐ gerprint image quality was to design and implement an application for automatic batch fin‐

about their weight. The weight of each block then corresponds to its quality.

ground decision.

298 New Trends and Developments in Biometrics

**5.2. Local feature extraction based methods**

**5.3. Methods on global feature extraction**

**5.4. Creation of batch fingerprint quality estimation tool**

The created database of fingerprints with a skin disease has been tested for the quality in the closest past. For the testing the implemented tool with all 21 local feature extraction based methods and the global NFIQ method were used. The results were normalized into interval from 0 to 100 where 100 means the best possible fingerprint quality. The example of ob‐ tained results of the most promising methods can be seen in Table 1.


**Table 1.** Example of results of diseased fingerprint quality testing.

## **6. Conclusion and future challenges**

The dermatologic diseases have a strong negative influence on the process of fingerprint rec‐ ognition and are causing problems to people who are suffering from them. These people are discriminated because they cannot use the fingerprint recognition systems which are very common these days. Currently there exists no usable database of fingerprints affected by a skin disease. As a first step in this wide and complex research we designed and developed a special diseased fingerprints capturing station. With this station the medical experts cap‐ tured the first version of this special fingerprint database. A special algorithm for fingerprint image enhancement has been designed and fingerprint batch processing tool has been im‐ plemented. The first diseased fingerprint quality testing has been realized. Our greatest challenge in the nearest future is to develop an algorithm for distinguishing diseased finger‐ prints from the other fingerprints with a low quality.

**References**

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In the nearest future the several challenges will be needed to face. First,the global analysis of the results of testing of the quality estimation algorithms has to be done. We will try to find out why some quality estimation algorithm failed and why some did not fail. The biggest challenge for us will now be to design and implement an algorithm for dermatologic disease presence detection. This algorithm should be able to detect whether the fingerprint with a low quality is diseased or if it is not and the low quality is caused for example by dirt, dry skin, mud etc.

## **Acknowledgement**

This research has been realized under the support of the following grants: "*Security-Oriented Research in Information Technology*" – *MSM0021630528 (CZ), "Information Technology in Bio‐ medical Engineering*" – GD102/09/H083 (CZ), "*Advanced secured, reliable and adaptive IT*" – FIT-S-11-1 (CZ) and "*The IT4Innovations Centre of Excellence*" – IT4I-CZ 1.05/1.1.00/02.0070 (CZ).

Authors of this research thank to the student Bc. Tomas Korec who created the implementa‐ tion of batch fingerprint image quality estimation application for us.

## **Author details**

Michal Dolezel1 , Martin Drahansky1 , Jaroslav Urbanek2 , Eva Brezinova3 and Tai-hoon Kim4

1 Faculty of Information Technology, Brno University of Technology, Brno, Czech Republic

2 Faculty of Medicine and Dentistry, Palacky University and Faculty Hospital, Olomouc, Czech Republic

3 Faculty of Medicine, Masaryk University, Brno, Czech Republic

4 Department of Multimedia Engineering, Hannam University, Deadeok-gu, South Korea

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**6. Conclusion and future challenges**

300 New Trends and Developments in Biometrics

prints from the other fingerprints with a low quality.

**Acknowledgement**

**Author details**

Michal Dolezel1

Czech Republic

it is not and the low quality is caused for example by dirt, dry skin, mud etc.

tion of batch fingerprint image quality estimation application for us.

3 Faculty of Medicine, Masaryk University, Brno, Czech Republic

, Martin Drahansky1

The dermatologic diseases have a strong negative influence on the process of fingerprint rec‐ ognition and are causing problems to people who are suffering from them. These people are discriminated because they cannot use the fingerprint recognition systems which are very common these days. Currently there exists no usable database of fingerprints affected by a skin disease. As a first step in this wide and complex research we designed and developed a special diseased fingerprints capturing station. With this station the medical experts cap‐ tured the first version of this special fingerprint database. A special algorithm for fingerprint image enhancement has been designed and fingerprint batch processing tool has been im‐ plemented. The first diseased fingerprint quality testing has been realized. Our greatest challenge in the nearest future is to develop an algorithm for distinguishing diseased finger‐

In the nearest future the several challenges will be needed to face. First,the global analysis of the results of testing of the quality estimation algorithms has to be done. We will try to find out why some quality estimation algorithm failed and why some did not fail. The biggest challenge for us will now be to design and implement an algorithm for dermatologic disease presence detection. This algorithm should be able to detect whether the fingerprint with a low quality is diseased or if

This research has been realized under the support of the following grants: "*Security-Oriented Research in Information Technology*" – *MSM0021630528 (CZ), "Information Technology in Bio‐ medical Engineering*" – GD102/09/H083 (CZ), "*Advanced secured, reliable and adaptive IT*" – FIT-S-11-1 (CZ) and "*The IT4Innovations Centre of Excellence*" – IT4I-CZ 1.05/1.1.00/02.0070 (CZ). Authors of this research thank to the student Bc. Tomas Korec who created the implementa‐

, Jaroslav Urbanek2

1 Faculty of Information Technology, Brno University of Technology, Brno, Czech Republic

2 Faculty of Medicine and Dentistry, Palacky University and Faculty Hospital, Olomouc,

4 Department of Multimedia Engineering, Hannam University, Deadeok-gu, South Korea

, Eva Brezinova3

and Tai-hoon Kim4


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[36] Boulgouris N. V., Plataniotis K. N., Micheli-Tzanakou E. *Biometrics: Theory, Methods and Applications*. Hoboken, N.J.: Wiley, 2010, p. 745, ISBN 978-0470-24782-2.

[37] Mehtre B., Chatterjee B. *Segmentation of Fingerprint images – A composite method*. In:

[38] Shen L., Kot A., Koo W. *Quality Measures of Fingerprint Images*. In: Audio- and Video-Based Biometric Person Authentication. Springer Berlin / Heidelberg, 2001, p.266,

[39] Wu C., Tulyakov S., Govindaraju V. *Image Quality Measures for Fingerprint Image En‐ hancement*. In: Multimedia Content Representation, Classification and Security,

[40] Joun S., Kim H., Chung Y. et al. *An Experimental Study on Measuring Image Quality of Infant Fingerprints*. In: Knowledge-Based Intelligent Information and Engineering

[41] Noriega L. *Multilayer Perceptron Tutorial*, tutorial, School of Computing, Staffordshire

[42] Yang J.C., Xiong N., Vasilakos A.V. *Two-stage Enhancement Scheme for Low-quality Fin‐ gerprint Images by Learning from the Image*. IEEE Transactions on Systems, Man, and

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[18] Iyad J. and Hao Y. *New algorithms for contrast enhancement in grayscale images based on the variational definition of histogram equalization*. Integrated Computer-Aided Engi‐

[19] *The Science of the Skin* [online]. [cit. 2012-05-30]. Available at: <http://www.naturalrus‐

[20] Wolff K., Johnson R. A., Suurmond D. *Fitzpatrick's Color Atlas and Synopsis of Clinical Dermatology*. 5th Edition, McGraw-Hill, 2005, p. 1085, ISBN 0-07-144019-4.

[21] Weston W. L., Lane A. T., Morelli J. G. *Color Textbook of Pediatric Dermatology*. Mosby

[22] Štork J., et al. *Dermatovenerologie*. Galén, Prague, 2008, p. 502, ISBN 978-80-7262-371-6.

[23] Niedner R., Adler Y.: *Kožní choroby – kapesní obrazový atlas*. Triton, Prague, 2005, p.

[25] Benáková N. (Ed.) et al. *Dermatovenerologie, dětstká dermatologie a korektivní dermatolo‐ gie* (*Dermatovenerology, Pediatric Dermatology and Corrective Dermatology*). Triton, Pra‐

[26] Drahanský M. *Fingerprint Recognition Technology – Liveness Detection*. Image Quality

[27] Ratha N. K., Govindaraju V. *Advances in Biometrics: Sensors, Algorithms and Systems*.

[28] Maltoni D., Maio D., Jain A.K., Prabhakar S. *Handbook of Fingerprint Recognition*.

[29] Ratha N., Bolle R. *Automatic Fingerprint Recognition Systems*. Springer-Verlag, 2004, p.

[30] Jang J., Elliott S.J., Kim H. *On Improving Interoperability of Fingerprint Recognition Us‐ ing Resolution Compensation Based on Sensor Evaluation*. In: S.-W. Lee and S.Z. Li (Eds.): ICB 2007, LNCS 4642, 2007, pp. 455-463, Springer-Verlag Berlin Heidelberg, 2007,

[31] Jirachaweng S., Areekul V. *Fingerprint Enhancement Based on Discrete Cosine Trans‐ form*. In: Proceedings of ICB 2007, LNCS 4642, Springer-Verlag Berlin Heidelberg,

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**Chapter 13**

**Provisional chapter**

**Algorithms for Processing Biometric Data Oriented to**

**to Privacy Protection and Preservation of Significant**

We address a general theory of transformations of biometric data, independently on particular biometric features taken into the processing. An application assumes representation of data in the specified format and the use of the proposed technique.

The content of a biometric database can be used for different purposes, and their complete list is usually unknown in advance. These purposes include different variants of processing noisy data, such as authentication and identification of a person on the basis of his biometric observations, detection of certain features, etc. General requirements to data processing schemes in any of these applications contain the design of a verification scheme consisting of the enrollment and the verification stages. At the enrollment stage, the input data should be converted to the data stored in the database under the identifier of the person. At the verification stage, the presented data have to be analyzed to make the decision whether they belong to the chosen person or not. Privacy protection of the data includes protection against

• a blind attacker, who wants to guess the content of the database using his knowledge

• an attacker, who has access to the database and wants to find the input biometric vector

• an attacker, who has access to the database and wants to generate artificial data to pass

©2012 Balakirsky and Han Vinck, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 B. Balakirsky and Han Vinck; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Balakirsky and Han Vinck; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

about the probability distribution over input biometric data;

through the verification stage with the acceptance decision.

We will present the class of transformations having the following features.

**Algorithms for Processing Biometric Data Oriented**

**Privacy Protection and Preservation of Significant**

**Parameters**

**1. Introduction**

attackers of three types:

processed at the enrollment stage;

**Parameters**

http://dx.doi.org/10.5772/51800

Vladimir B. Balakirsky and A. J. Han Vinck

Additional information is available at the end of the chapter

Vladimir B. Balakirsky and A. J. Han Vinck

Additional information is available at the end of the chapter

**Provisional chapter**

## **Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters**

Vladimir B. Balakirsky and A. J. Han Vinck Vladimir B. Balakirsky and A. J. Han Vinck

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51800

## **1. Introduction**

We address a general theory of transformations of biometric data, independently on particular biometric features taken into the processing. An application assumes representation of data in the specified format and the use of the proposed technique.

The content of a biometric database can be used for different purposes, and their complete list is usually unknown in advance. These purposes include different variants of processing noisy data, such as authentication and identification of a person on the basis of his biometric observations, detection of certain features, etc. General requirements to data processing schemes in any of these applications contain the design of a verification scheme consisting of the enrollment and the verification stages. At the enrollment stage, the input data should be converted to the data stored in the database under the identifier of the person. At the verification stage, the presented data have to be analyzed to make the decision whether they belong to the chosen person or not. Privacy protection of the data includes protection against attackers of three types:


We will present the class of transformations having the following features.

©2012 Balakirsky and Han Vinck, licensee InTech. This is an open access chapter distributed under the terms of

• If input data are represented by the vector whose components are generated by a stationary memoryless source having the known probability distribution, then they can be converted to the vector x whose components are uniformly distributed over the (−1, +1) interval. Therefore, the scheme has a perfect protection against blind attackers. A generalized version of the procedure brings the same property for non-stationary sources. Moreover, if the source has memory or the input probability distribution is unknown, an approximate uniform distribution can be created.

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 3

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

processing. Furthermore, the addressed issues are relevant to constructing fault–tolerant passwords from biometric data [17]–[19]. The cited list of publications is certainly far from being complete, but we only indicated directions that are relevant to the material of the chapter. We also understand that specific applications of the presented results need research on probability distributions of the measured biometric data. In particular, the approaches can be fruitful for the schemes where different lengths are processed (for example, distances

The chapter is organized as follows. We begin with the introduction and the notation sections and then introduce the *F*-transformation for the input data and their noisy observations in Sections 3,4. A possible implementation of the verification scheme, constructed using the derived properties, is presented in Section 5. Some general conclusions are included in

Suppose that outcomes of biometric measurements of a person, received at the enrollment stage, are represented by a float–valued vector <sup>r</sup> = (*r*1,...,*rn*) <sup>∈</sup> <sup>R</sup>*n*. The *<sup>t</sup>*-th component of

> 

1, if *rt* ≥ 0 0, if *rt* < 0

+*rt*, if *rt* ≥ 0 −*rt*, if *rt* < 0

Thus, the specification of the component *rt* is equivalent to the specification of the pair (sgn(*rt*), |*rt*|). Such a representation is introduced, because we assume that sgn(*rt*) and |*rt*| are independent parameters: if one knows the sign, then he has no information about the magnitude; if one knows the magnitude, then he has no information about the sign. Furthermore, we will assume that the magnitude |*rt*| characterizes "significance" of the *t*-th component: for example, if *rt* is defined as the difference between the value of the measured parameter and the average or the expected value, then |*rt*| determines the deviation of the

We will analyze the scheme where one of results of the processing of the vector r at the enrollment stage is expressed by a float–valued vector ˜<sup>x</sup> = (*x*˜1,..., *<sup>x</sup>*˜*n*) <sup>∈</sup> <sup>F</sup>*<sup>n</sup>* whose

*x* ∈ R : |*x*| < 1

The vector ˜x is stored in the database under the identifier of the person, associated with the vector r. We call it the wrapped version of the input data. The transformation r → x˜ is not a one-to-one mapping, and some data are lost. This transformation is divided into two steps. We first introduce a one-to-one mapping <sup>r</sup> <sup>↔</sup> <sup>x</sup>, where <sup>x</sup> = (*x*1,..., *xn*) <sup>∈</sup> <sup>F</sup>*n*, and

�

(1)

http://dx.doi.org/10.5772/51800

307

(2)

(3)

sgn(*rt*) =


F = �  

between certain points on the face or hand).

the vector r has the sign, which is understood as

*t*-th parameter for the particular person.

components have magnitudes less than 1, i.e.,

Section 6.

and the magnitude


Verification algorithms are considered in the most of publications, related to biometrics (see, in particular [1]–[4]), and they are usually introduced as algorithms of combinatorial matching of the outcomes of observations received at the enrollment and at the verification stages. However, secrecy requirements to a biometric system to be designed do not allow storage of the outcomes received at the enrollment stage, and lead to the schemes where only relatively few outcomes characterizing the person are taken into account. In the present chapter, these outcomes are referred to as significant components. The algorithms with similar understanding are presented in [6]–[8] where significance is determined by the values of the probability distribution function computed at the observed values. This approach follows the lines of information theory when data processing is represented as secret sharing [9]–[12]. Some part of the secret is published, while another part is hidden by the so–called one-way hash function [13] and it has to be decoded after a noisy version of the observation data is available. The transformation of the outcomes, received at the enrollment stage, to a published part of the secret is also viewed as wrapping in a number of applications where the unwrapping is possible only when outcomes, received at the verification stage, are close to the wrapped data [14]. The use of possibly non-stationary probability distributions also allows us to include multi–biometric measurements (see, for example [15], [16]) into the processing. Furthermore, the addressed issues are relevant to constructing fault–tolerant passwords from biometric data [17]–[19]. The cited list of publications is certainly far from being complete, but we only indicated directions that are relevant to the material of the chapter. We also understand that specific applications of the presented results need research on probability distributions of the measured biometric data. In particular, the approaches can be fruitful for the schemes where different lengths are processed (for example, distances between certain points on the face or hand).

The chapter is organized as follows. We begin with the introduction and the notation sections and then introduce the *F*-transformation for the input data and their noisy observations in Sections 3,4. A possible implementation of the verification scheme, constructed using the derived properties, is presented in Section 5. Some general conclusions are included in Section 6.

Suppose that outcomes of biometric measurements of a person, received at the enrollment stage, are represented by a float–valued vector <sup>r</sup> = (*r*1,...,*rn*) <sup>∈</sup> <sup>R</sup>*n*. The *<sup>t</sup>*-th component of the vector r has the sign, which is understood as

$$\text{sgn}(r\_t) = \begin{cases} 1, \text{ if } r\_t \ge 0 \\ 0, \text{ if } r\_t < 0 \end{cases} \tag{1}$$

and the magnitude

2 New Trends and Developments in Biometrics

306 New Trends and Developments in Biometrics

approximate uniform distribution can be created.

corresponding vector stored in the database.

properties.

• If input data are represented by the vector whose components are generated by a stationary memoryless source having the known probability distribution, then they can be converted to the vector x whose components are uniformly distributed over the (−1, +1) interval. Therefore, the scheme has a perfect protection against blind attackers. A generalized version of the procedure brings the same property for non-stationary sources. Moreover, if the source has memory or the input probability distribution is unknown, an

• The enrollment can be organized in such a way that the constructed vector x is encrypted and mapped to the vector ˆx, which is stored in the database. The encryption is understood as replacement of certain components of the vector x by randomly chosen components. As a result, the input biometric data cannot be reconstructed from the vector ˆx. We show that the encryption can be organized in such a way that the properties of the constructed uniform distribution are conserved, but the union of (−1, −*a*) and (+*a*, +1) intervals replaces the (−1, +1) interval. The value of parameter *a* ∈ (0, 1) controls the trade-off between privacy protection and the verification performance, and it has be assigned in advance. As a result, the scheme becomes protected against attackers, who have access to the database and want to guess the input vector by reading the

• The case, when biometrics of the same person is measured at the enrollment and the verification stages, is simulated as transmission of outcomes of the enrollment stage measurements over an observation channel. We create another channel between results of transformations of these outcomes. It turns out that this channel has the property that the level of noise essentially depends on the magnitude of the transmitted component, and it is very low when the magnitude is large. Since a large magnitude at the output of the transformer is obtained when the input magnitude is large, we talk about the preservation of significant parameters under the noise, provided that these parameters are represented in a vector of outcomes of biometric measurements by components having large magnitudes. We present a verification algorithm designed on the basis of these

Verification algorithms are considered in the most of publications, related to biometrics (see, in particular [1]–[4]), and they are usually introduced as algorithms of combinatorial matching of the outcomes of observations received at the enrollment and at the verification stages. However, secrecy requirements to a biometric system to be designed do not allow storage of the outcomes received at the enrollment stage, and lead to the schemes where only relatively few outcomes characterizing the person are taken into account. In the present chapter, these outcomes are referred to as significant components. The algorithms with similar understanding are presented in [6]–[8] where significance is determined by the values of the probability distribution function computed at the observed values. This approach follows the lines of information theory when data processing is represented as secret sharing [9]–[12]. Some part of the secret is published, while another part is hidden by the so–called one-way hash function [13] and it has to be decoded after a noisy version of the observation data is available. The transformation of the outcomes, received at the enrollment stage, to a published part of the secret is also viewed as wrapping in a number of applications where the unwrapping is possible only when outcomes, received at the verification stage, are close to the wrapped data [14]. The use of possibly non-stationary probability distributions also allows us to include multi–biometric measurements (see, for example [15], [16]) into the

$$|r\_t| = \begin{cases} +r\_{t\prime} \text{ if } r\_t \ge 0 \\ -r\_{t\prime} \text{ if } r\_t < 0 \end{cases} \tag{2}$$

Thus, the specification of the component *rt* is equivalent to the specification of the pair (sgn(*rt*), |*rt*|). Such a representation is introduced, because we assume that sgn(*rt*) and |*rt*| are independent parameters: if one knows the sign, then he has no information about the magnitude; if one knows the magnitude, then he has no information about the sign. Furthermore, we will assume that the magnitude |*rt*| characterizes "significance" of the *t*-th component: for example, if *rt* is defined as the difference between the value of the measured parameter and the average or the expected value, then |*rt*| determines the deviation of the *t*-th parameter for the particular person.

We will analyze the scheme where one of results of the processing of the vector r at the enrollment stage is expressed by a float–valued vector ˜<sup>x</sup> = (*x*˜1,..., *<sup>x</sup>*˜*n*) <sup>∈</sup> <sup>F</sup>*<sup>n</sup>* whose components have magnitudes less than 1, i.e.,

$$\mathsf{F} = \left\{ \mathbf{x} \in \mathsf{R} : |\mathbf{x}| < 1 \right\} \tag{3}$$

The vector ˜x is stored in the database under the identifier of the person, associated with the vector r. We call it the wrapped version of the input data. The transformation r → x˜ is not a one-to-one mapping, and some data are lost. This transformation is divided into two steps. We first introduce a one-to-one mapping <sup>r</sup> <sup>↔</sup> <sup>x</sup>, where <sup>x</sup> = (*x*1,..., *xn*) <sup>∈</sup> <sup>F</sup>*n*, and

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 5

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

y ∈ F*<sup>n</sup>*

309

y ∈ F*<sup>n</sup>*

,x, y.

(a)

*V<sup>n</sup>*

❄

(b)

**Figure 2.** Schemes for transmission of the vector x to the verifier: (a) Legitimate case. (b) Attack.

, *X*,*Y* denote random variables and *r*,*r*′

extended to the vectors of random variables *Rn*,(*R*′

Encryptor

✲ ✲

constructed as an encrypted version of the vector x, we get the situation, when the vector x is sent to the verifier over two parallel channels. If the level of noise in the physical channel is not high, then the verifier is supposed to make the acceptance decision. As an alternative, there is a possibility that the vector r is generated independently of the vector x (see Figure 2b where we delete the *F*<sup>−</sup>1-transformation). In this case, the verifier is supposed to make

In the basic model, we assume that the vector r is generated by a stationary memoryless source having the probability distribution (PD) *F*<sup>∗</sup> and the probability density function (PDF)

Encryptor

❄

Transformer Channel Transformer

✲ ✲ ✲ ✲

✲ ✲

*V<sup>n</sup>*

x˜ ∈ F*<sup>n</sup>*

x˜ ∈ F*<sup>n</sup>*

✲ ✲ ✲

Channel Transformer

❄

❄

, *x*, *y* denote their values. This notation is also

)*n*, *Xn*,*Y<sup>n</sup>* and their values r, r′

*F*

*F*

http://dx.doi.org/10.5772/51800

<sup>x</sup> ∈ <sup>F</sup>*<sup>n</sup>* <sup>r</sup> ∈ <sup>R</sup>*<sup>n</sup>* <sup>r</sup>′ ∈ <sup>R</sup>*<sup>n</sup>*

<sup>x</sup> ∈ <sup>F</sup>*<sup>n</sup>* <sup>r</sup> ∈ <sup>R</sup>*<sup>n</sup>* <sup>r</sup>′ ∈ <sup>R</sup>*<sup>n</sup>*

❄

the rejection decision.

**2. Notation**

Let *R*, *R*′

*P*∗,

*F*−1

**Figure 1.** (a) Transformations of the vector r to the vector x˜ stored in the database under the identifier of the person. (b) Transformation of the vector r′ to the vector y at the verification stage for its comparison with the vector x˜ and investigation of the closeness of the vectors r and r′ .

then encrypt the vector x in such a way that the change of the *t*-th component *xt* is possible only if |*rt*| is small.

At the verification stage, the verifier observes a vector r′ = (*r*′ 1,...,*r*′ *<sup>n</sup>*) ∈ R*n*. This vector is mapped to the vector y in the same way as r was mapped to x, and there should be an algorithm that analyzes the pair (y, ˜x) to find the closeness of the vector r′ and some vector r that could be used as an origin of the vector ˜x. If the verifier decides that these vectors are close enough, then components of the vector r′ are considered as possible outcomes of biometric measurements of the person whose identifier corresponds to the vector ˜x.

The procedures above describe a general structure of the verification scheme under constraints that the transformation r → x˜ is divided into two steps and that components of the constructed vectors belong to the set F. These procedures are illustrated in Figure 1 where we use some additional notation. Namely, we parameterize the mappings r ↔ x and r′ ↔ y by a monotone increasing function *F*(*r*),*r* ∈ R, approaching 0 when *r* → −∞ and *r* → +∞, respectively. Formally,

$$\frac{d}{dr}F(r) > 0, \ r \in \mathbb{R} \tag{4}$$

and

$$(F(-\\\infty), F(+\\\infty)) = (0,1)\tag{5}$$

By (4), (5), the function *F* has a uniquely determined inverse function *F*<sup>−</sup>1, and we can simulate the data received at the verification stage, as the result of transmission of the vector x over a channel consisting of the deterministic *F*<sup>−</sup>1-transformation x → r, the stochastic mapping r → r′ introduced by a physical *V<sup>n</sup>* channel, and the deterministic *F*-transformation r′ → y (see Figure 2a). As the verifier also has access to the vector ˜x

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 5 Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters http://dx.doi.org/10.5772/51800 309

**Figure 2.** Schemes for transmission of the vector x to the verifier: (a) Legitimate case. (b) Attack.

constructed as an encrypted version of the vector x, we get the situation, when the vector x is sent to the verifier over two parallel channels. If the level of noise in the physical channel is not high, then the verifier is supposed to make the acceptance decision. As an alternative, there is a possibility that the vector r is generated independently of the vector x (see Figure 2b where we delete the *F*<sup>−</sup>1-transformation). In this case, the verifier is supposed to make the rejection decision.

## **2. Notation**

4 New Trends and Developments in Biometrics

308 New Trends and Developments in Biometrics

of the closeness of the vectors r and r′

*r* → +∞, respectively. Formally,

and

only if |*rt*| is small.

(a)

(b)

1,...,*r*′

*dr <sup>F</sup>*(*r*) <sup>&</sup>gt; 0, *<sup>r</sup>* <sup>∈</sup> <sup>R</sup> (4)

(*F*(−∞), *F*(+∞)) = (0, 1) (5)

**Figure 1.** (a) Transformations of the vector r to the vector x˜ stored in the database under the identifier of the person. (b) Transformation of the vector r′ to the vector y at the verification stage for its comparison with the vector x˜ and investigation

then encrypt the vector x in such a way that the change of the *t*-th component *xt* is possible

is mapped to the vector y in the same way as r was mapped to x, and there should be an algorithm that analyzes the pair (y, ˜x) to find the closeness of the vector r′ and some vector r that could be used as an origin of the vector ˜x. If the verifier decides that these vectors are close enough, then components of the vector r′ are considered as possible outcomes of

The procedures above describe a general structure of the verification scheme under constraints that the transformation r → x˜ is divided into two steps and that components of the constructed vectors belong to the set F. These procedures are illustrated in Figure 1 where we use some additional notation. Namely, we parameterize the mappings r ↔ x and r′ ↔ y by a monotone increasing function *F*(*r*),*r* ∈ R, approaching 0 when *r* → −∞ and

By (4), (5), the function *F* has a uniquely determined inverse function *F*<sup>−</sup>1, and we can simulate the data received at the verification stage, as the result of transmission of the vector x over a channel consisting of the deterministic *F*<sup>−</sup>1-transformation x → r, the stochastic mapping r → r′ introduced by a physical *V<sup>n</sup>* channel, and the deterministic *F*-transformation r′ → y (see Figure 2a). As the verifier also has access to the vector ˜x

biometric measurements of the person whose identifier corresponds to the vector ˜x.

*d*

✲ Transformer ✲ Encryptor ✲

x˜ ∈ F*<sup>n</sup>*

*<sup>n</sup>*) ∈ R*n*. This vector

<sup>r</sup> <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>x</sup> <sup>∈</sup> <sup>F</sup>*<sup>n</sup>*

<sup>r</sup>′ <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>y</sup> <sup>∈</sup> <sup>F</sup>*<sup>n</sup>* ❄

.

At the verification stage, the verifier observes a vector r′ = (*r*′

✲ Transformer ✲

*F*

❄

*F*

Let *R*, *R*′ , *X*,*Y* denote random variables and *r*,*r*′ , *x*, *y* denote their values. This notation is also extended to the vectors of random variables *Rn*,(*R*′ )*n*, *Xn*,*Y<sup>n</sup>* and their values r, r′ ,x, y.

In the basic model, we assume that the vector r is generated by a stationary memoryless source having the probability distribution (PD) *F*<sup>∗</sup> and the probability density function (PDF) *P*∗,

$$F^\* = \left( F^\*(r) = \Pr\_{\text{data}} \{ R < r \} , r \in \mathbb{R} \right) \tag{6}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 7

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

*F*(*r*) = Pr{ *R* < *r* }, *r* ∈ R

*dr <sup>F</sup>*(*r*), *<sup>r</sup>* <sup>∈</sup> <sup>R</sup>

data{ <sup>2</sup>*F*(*R*) <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>x</sup>* }, *<sup>x</sup>* <sup>∈</sup> <sup>F</sup>

*dx <sup>f</sup>* <sup>∗</sup>(*x*), *<sup>x</sup>* <sup>∈</sup> <sup>F</sup>

*F*−1(*z*), *z* ∈ (0, 1)

 *x* + 1 2 

Let us fix the function *F*, satisfying (4), (5), and map an *r* ∈ R to the *x* ∈ F using the rule

*x* = 2*F*(*r*) − 1 (20)

<sup>2</sup> <sup>⇒</sup> *<sup>r</sup>*(*x*) = *<sup>ρ</sup>*

√

2 · erf−1(*x*) (23)

(16)

311

http://dx.doi.org/10.5772/51800

(17)

(18)

(19)

(21)

(22)

The function *F*, satisfying (4), (5), is the PD of some random variable *R*, and we write

*<sup>P</sup>*(*r*) = *<sup>d</sup>*

**3. The** *F***-transformation of the input data**

The corresponding PDF is defined as

*F* = 

> *P* =

*f* <sup>∗</sup>(*x*) = Pr

*<sup>p</sup>*∗(*x*) = *<sup>d</sup>*

*F*−<sup>1</sup> = 

Some values of *r*(*x*) for Gaussian data are given in Table 1. Notice that

erf *<sup>r</sup>*(*x*) *ρ* √2 

*r*(*x*) = *F*−<sup>1</sup>

The *F*-transformation of the vector r ∈ R*<sup>n</sup>* will be understood as the result of *n*

<sup>=</sup> *<sup>x</sup>* <sup>+</sup> <sup>1</sup>

The data in Table 1 illustrate the point that (20) is a non-linear mapping. In particular, if *ρ* = 2, then the values belonging to the interval (0, 0.64) are mapped to the values between 0 and 0.25, . . . , the values greater than 3.92 are mapped to the values between 0.95 and 1. The

Notice that (*F*∗, *P*∗) �= (*F*, *P*) in general case and denote

*f* <sup>∗</sup> = 

*p*<sup>∗</sup> = 

The function *F* has a uniquely determined inverse function

and the equality (20) implies *r* = *r*(*x*), where

component-wise applications of the mapping (20).

1 2 + 1 2

*F* = *F*G0,*<sup>ρ</sup>* ⇒

mapping *r* ↔ *x* is also illustrated in Figure 3.

$$P^\* = \left(P^\*(r) = \frac{d}{dr}F^\*(r), \ r \in \mathbb{R}\right) \tag{7}$$

The value of the PDF, associated with the vector r, is expressed as

$$\text{PDF}(r) = \prod\_{t=1}^{n} P^\*(r\_t) \tag{8}$$

We will simulate the stochastic mapping r → r′ as transmission of the vector r over an observation channel, which is introduced as a stationary memoryless channel specified by the collection of the conditional PDs Φ*<sup>r</sup>* and the collection of the conditional PDFs *Vr*,

$$\Phi\_{\mathcal{T}} = \left( \Phi(r'|r) = \Pr\_{\text{noise}} \left\{ R' < r' \, | \, R = r \right\}, r' \in \mathbb{R} \right) \tag{9}$$

$$V\_r = \left( V(r'|r) = \frac{d}{dr'} \Phi(r'|r), r' \in \mathbb{R} \right) \tag{10}$$

for all *r* ∈ R. The value of the conditional PDF, associated with the output vector r′ , given the input vector r, is expressed as

$$\text{PDF}(r'|r) = \prod\_{t=1}^{n} V(r'\_t|r\_t) \tag{11}$$

Let

$$\text{FG}\_{\mathfrak{m},\gamma} = \left( \text{FG}\_{\mathfrak{m},\gamma}(r) = \frac{1}{2} + \frac{1}{2} \text{erf}\left(\frac{r-m}{\gamma\sqrt{2}}\right), r \in \mathbb{R} \right) \tag{12}$$

$$\mathcal{G}\_{\mathfrak{m},\gamma} = \left( \mathcal{G}\_{\mathfrak{m},\gamma}(r) = \frac{1}{\gamma \sqrt{2\pi}} \exp\left\{ -\frac{(r-m)^2}{2\gamma^2} \right\}, r \in \mathbb{R} \right) \tag{13}$$

where

$$\text{erf}(r) = \frac{2}{\sqrt{\pi}} \int\_0^r \exp\{-\tilde{r}^2\} \,d\tilde{r}, \,\, r \in \mathbb{R} \tag{14}$$

is the erf-function, denote the Gaussian PD and PDF, when *m* is the mean and *γ*<sup>2</sup> is the variance. We will also use the function

$$\psi\_c(\mathbf{x}) = \frac{1}{2} + \frac{1}{2} \text{erf}\left(c \cdot \text{erf}^{-1}(\mathbf{x})\right) \tag{15}$$

## **3. The** *F***-transformation of the input data**

The function *F*, satisfying (4), (5), is the PD of some random variable *R*, and we write

$$F = \left( F(r) = \Pr\{R < r\}, r \in \mathbb{R} \right) \tag{16}$$

The corresponding PDF is defined as

6 New Trends and Developments in Biometrics

310 New Trends and Developments in Biometrics

*F*<sup>∗</sup> = 

*P*<sup>∗</sup> = 

Φ*<sup>r</sup>* = Φ(*r*′

*Vr* = *V*(*r*′

*F*G*m*,*<sup>γ</sup>* =

G*m*,*<sup>γ</sup>* =

variance. We will also use the function

the input vector r, is expressed as

Let

where

The value of the PDF, associated with the vector r, is expressed as

*F*∗(*r*) = Pr

*<sup>P</sup>*∗(*r*) = *<sup>d</sup>*

PDF(r) =


<sup>|</sup>*r*) = *<sup>d</sup>*

PDF(r′

*<sup>F</sup>*G*m*,*γ*(*r*) = <sup>1</sup>

<sup>G</sup>*m*,*γ*(*r*) = <sup>1</sup>

√*π*

erf(*r*) = <sup>2</sup>

*<sup>ψ</sup>c*(*x*) = <sup>1</sup>

*dr*′ <sup>Φ</sup>(*r*′

for all *r* ∈ R. The value of the conditional PDF, associated with the output vector r′


2 + 1 2 erf

*γ* <sup>√</sup>2*<sup>π</sup>*

 *r* 0

2 + 1 2 erf 

*n* ∏ *t*=1

> exp

exp{−*r*˜

is the erf-function, denote the Gaussian PD and PDF, when *m* is the mean and *γ*<sup>2</sup> is the

data{ *<sup>R</sup>* <sup>&</sup>lt; *<sup>r</sup>* }, *<sup>r</sup>* <sup>∈</sup> <sup>R</sup>

noise{ *<sup>R</sup>*′ <sup>&</sup>lt; *<sup>r</sup>*′ <sup>|</sup> *<sup>R</sup>* <sup>=</sup> *<sup>r</sup>*}, *<sup>r</sup>*′ <sup>∈</sup> <sup>R</sup>


*V*(*r*′

*<sup>r</sup>* <sup>−</sup> *<sup>m</sup> γ* √2 , *r* ∈ R 

*c* · erf−1(*x*)

<sup>−</sup>(*<sup>r</sup>* <sup>−</sup> *<sup>m</sup>*)<sup>2</sup> 2*γ*<sup>2</sup>

 , *r* ∈ R 

*dr <sup>F</sup>*∗(*r*), *<sup>r</sup>* <sup>∈</sup> <sup>R</sup>

*n* ∏ *t*=1

We will simulate the stochastic mapping r → r′ as transmission of the vector r over an observation channel, which is introduced as a stationary memoryless channel specified by the collection of the conditional PDs Φ*<sup>r</sup>* and the collection of the conditional PDFs *Vr*,

*P*∗(*rt*) (8)

*<sup>t</sup>*|*rt*) (11)

<sup>2</sup>} *dr*˜, *r* ∈ R (14)

(6)

(7)

(9)

(10)

(12)

(13)

(15)

, given

$$P = \left( P(r) = \frac{d}{dr} F(r), \; r \in \mathbb{R} \right) \tag{17}$$

Notice that (*F*∗, *P*∗) �= (*F*, *P*) in general case and denote

$$f^\* = \left( f^\*(\mathbf{x}) = \Pr\_{\text{data}} \left\{ 2F(\mathbf{R}) - 1 < \mathbf{x} \right\}, \mathbf{x} \in \mathsf{F} \right) \tag{18}$$

$$p^\* = \left(p^\*(\mathbf{x}) = \frac{d}{d\mathbf{x}} f^\*(\mathbf{x}) , \mathbf{x} \in \mathsf{F}\right) \tag{19}$$

Let us fix the function *F*, satisfying (4), (5), and map an *r* ∈ R to the *x* ∈ F using the rule

$$\mathbf{x} = \mathbf{2}F(r) - \mathbf{1} \tag{20}$$

The function *F* has a uniquely determined inverse function

$$F^{-1} = \left( F^{-1}(z), z \in (0, 1) \right) \tag{21}$$

and the equality (20) implies *r* = *r*(*x*), where

$$r(\mathbf{x}) = F^{-1}\left(\frac{\mathbf{x} + 1}{2}\right) \tag{22}$$

The *F*-transformation of the vector r ∈ R*<sup>n</sup>* will be understood as the result of *n* component-wise applications of the mapping (20).

Some values of *r*(*x*) for Gaussian data are given in Table 1. Notice that

$$F = F \mathbf{G}\_{0, \rho} \implies \frac{1}{2} + \frac{1}{2} \text{erf} \left( \frac{r(\mathbf{x})}{\rho \sqrt{2}} \right) = \frac{\mathbf{x} + 1}{2} \implies r(\mathbf{x}) = \rho \sqrt{2} \cdot \text{erf}^{-1}(\mathbf{x}) \tag{23}$$

The data in Table 1 illustrate the point that (20) is a non-linear mapping. In particular, if *ρ* = 2, then the values belonging to the interval (0, 0.64) are mapped to the values between 0 and 0.25, . . . , the values greater than 3.92 are mapped to the values between 0.95 and 1. The mapping *r* ↔ *x* is also illustrated in Figure 3.


**Table 1.** Some values of *r*(*x*) for the function *F* = *F*G0,*ρ*.

**Figure 3.** Illustration of the one-to-one mapping *x* ↔ *r* = *r*(*x*).

In the following considerations we will restrict ourselves to the class of functions *F*, satisfying (4), (5) and having additional symmetric properties:

$$r(-\mathbf{x}) = -r(+\mathbf{x})\tag{24}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 9

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

As 2*F*(*R*) − 1 is a deterministic function of the random variable *R*, the value of *x* is the observation of the random variable *X* = 2*F*(*R*) − 1 having the PD *f* <sup>∗</sup> and the PDF *p*∗. We

*F*(*R*) <

*x* + 1 2

*dx <sup>r</sup>*(*x*) = *<sup>P</sup>*∗(*r*(*x*)) · *<sup>d</sup>*

<sup>−</sup><sup>1</sup> · *d dx*  = Pr data 

= *<sup>F</sup>*∗(*r*(*x*)) (26)

*x* + 1 <sup>2</sup> <sup>=</sup> <sup>1</sup> 2 ·

*<sup>P</sup>*∗(*r*(*x*))

2*P*(*r*(*x*)), *<sup>x</sup>* <sup>∈</sup> <sup>F</sup>

*dx <sup>F</sup>*−<sup>1</sup>

*R* < *F*−<sup>1</sup>

 *x* + 1 2 

*P*∗(*r*(*x*))

<sup>2</sup>*P*(*r*(*xt*)) (29)

<sup>√</sup><sup>2</sup> · erf−1(*x*)

*r*2(*x*) 2*ρ*<sup>2</sup>

> <sup>2</sup>(*x*) , *x* ∈ F

*ρ*∗ √2  *x* + 1 2

http://dx.doi.org/10.5772/51800

*<sup>P</sup>*(*r*(*x*)) (27)

(28)

(30)

(31)

(32)

313

data 

*dx <sup>F</sup>*∗(*r*(*x*))

 , *p*<sup>∗</sup> =

By (8) and the fact that the *F*-transformation is a component-wise mapping, the value of the

*n* ∏ *t*=1

<sup>2</sup> · erf−1(*x*)) = <sup>1</sup>

2*ρ*<sup>∗</sup> <sup>√</sup>2*<sup>π</sup>*

<sup>√</sup>2*<sup>π</sup>*

*ρ*

where *ψρ*/*ρ*<sup>∗</sup> (*x*) is defined by (15) with *<sup>c</sup>* <sup>=</sup> *<sup>ρ</sup>*/*ρ*∗. Examples of the functions *<sup>f</sup>* <sup>∗</sup>(*x*) and *<sup>p</sup>*∗(*x*)

<sup>2</sup>*ρ*<sup>∗</sup> exp

*P*∗(*r*(*xt*))

2 + 1 2 erf *<sup>ρ</sup>*

> <sup>−</sup> *<sup>r</sup>*2(*x*) <sup>2</sup>(*ρ*∗)<sup>2</sup> <sup>+</sup>

<sup>−</sup>*ρ*<sup>2</sup> <sup>−</sup> (*ρ*∗)<sup>2</sup> <sup>2</sup>*ρ*2(*ρ*∗)<sup>2</sup> *<sup>r</sup>*

exp

write

and

Therefore

for all x ∈ F*n*.

Hence,

*f* <sup>∗</sup> = 

are given in Figures 4, 5.

*f* <sup>∗</sup>(*x*) = Pr

= *F*<sup>∗</sup> *F*−<sup>1</sup>

data{ <sup>2</sup>*F*(*R*) <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>x</sup>* } <sup>=</sup> Pr

*dx <sup>f</sup>* <sup>∗</sup>(*x*) = *<sup>d</sup>*

 *d dr*˜ *F*(*r*˜) *r*˜=*r*(*x*)

*F*∗(*r*(*x*)), *x* ∈ F

PDF(x) =

= *P*∗(*r*(*x*)) ·

 *x* + 1 2

*<sup>p</sup>*∗(*x*) = *<sup>d</sup>*

= *d dr*˜ *F*∗(*r*˜) *r*˜=*r*(*x*) · *d*

*f* <sup>∗</sup> = 

PDF, associated with the vector x, is expressed as

If (*F*∗, *<sup>F</sup>*)=(*F*G0,*ρ*<sup>∗</sup> , *<sup>F</sup>*G0,*ρ*), then we use (23) and write

√

2G0,*ρ*(*r*(*x*)) <sup>=</sup> *<sup>ρ</sup>*

 , *p*<sup>∗</sup> =

*<sup>f</sup>* <sup>∗</sup>(*x*) = *<sup>F</sup>*G0,*ρ*<sup>∗</sup> (*<sup>ρ</sup>*

*<sup>p</sup>*∗(*x*) = G0,*ρ*<sup>∗</sup> (*r*(*x*))

*ψρ*/*ρ*<sup>∗</sup> (*x*), *<sup>x</sup>* <sup>∈</sup> <sup>F</sup>

and

$$P(r(-\mathbf{x})) = P(r(+\mathbf{x})), \ \mathbf{x} \in \mathsf{F} \tag{25}$$

The *F*-transformation of input data can be viewed as an application of the inverse to the well-known construction of pseudorandom generators having a fixed probability distribution *F*. In this case, an algorithm generates the value *z* of a random variable "uniformly distributed over the (0, 1) interval" and outputs *F*−1(*z*) ∈ (0, 1). Our scheme receives an *r* ∈ R and outputs 2*F*(*r*) − 1 where the multiplication by 2 and the subtraction of 1 are introduced because of further processing of the signs and the magnitudes. Notice that a similar approach can be used to generate Gaussian variables having the mean 0 and the fixed variance *ρ*2; the transformer has to output *F*G−<sup>1</sup> 0,*<sup>ρ</sup>* (*F*(*r*)) in this case.

As 2*F*(*R*) − 1 is a deterministic function of the random variable *R*, the value of *x* is the observation of the random variable *X* = 2*F*(*R*) − 1 having the PD *f* <sup>∗</sup> and the PDF *p*∗. We write

$$f^\*(\mathbf{x}) = \Pr\_{\text{data}}\left\{ 2F(\mathbf{R}) - 1 < \mathbf{x} \right\} = \Pr\_{\text{data}}\left\{ F(\mathbf{R}) < \frac{\mathbf{x} + 1}{2} \right\} = \Pr\_{\text{data}}\left\{ \mathbf{R} < F^{-1}\left(\frac{\mathbf{x} + 1}{2}\right) \right\}$$

$$= F^\*\left(F^{-1}\left(\frac{\mathbf{x} + 1}{2}\right)\right) = F^\*(r(\mathbf{x})) \tag{26}$$

and

8 New Trends and Developments in Biometrics

312 New Trends and Developments in Biometrics

**Table 1.** Some values of *r*(*x*) for the function *F* = *F*G0,*ρ*.

−1

and

**Figure 3.** Illustration of the one-to-one mapping *x* ↔ *r* = *r*(*x*).

(4), (5) and having additional symmetric properties:

fixed variance *ρ*2; the transformer has to output *F*G−<sup>1</sup>

+1

**x** = **0 0.25 0.50 0.75 0.80 0.85 0.90 0.95 1** *ρ* = 2 0 0.64 1.35 2.30 2.56 2.88 3.29 3.92 +∞ *ρ* = 1 0 0.32 0.67 1.15 1.28 1.44 1.64 1.96 +∞ *ρ* = 1/2 0 0.16 0.34 0.58 0.64 0.72 0.82 0.98 +∞

<sup>2</sup>*F*(*r*˜) <sup>−</sup> <sup>1</sup> ✻

*<sup>F</sup>*−1( *<sup>x</sup>*+<sup>1</sup> <sup>2</sup> )

*r*(−*x*) = −*r*(+*x*) (24)

*P*(*r*(−*x*)) = *P*(*r*(+*x*)), *x* ∈ F (25)

0,*<sup>ρ</sup>* (*F*(*r*)) in this case.

*x*

.............................................................................................. ............. ............................................. ... ... ... ... ... ... ................................................... ............. ......... ...............................................................................

In the following considerations we will restrict ourselves to the class of functions *F*, satisfying

The *F*-transformation of input data can be viewed as an application of the inverse to the well-known construction of pseudorandom generators having a fixed probability distribution *F*. In this case, an algorithm generates the value *z* of a random variable "uniformly distributed over the (0, 1) interval" and outputs *F*−1(*z*) ∈ (0, 1). Our scheme receives an *r* ∈ R and outputs 2*F*(*r*) − 1 where the multiplication by 2 and the subtraction of 1 are introduced because of further processing of the signs and the magnitudes. Notice that a similar approach can be used to generate Gaussian variables having the mean 0 and the

✲ *r*˜

$$\begin{split} p^\*(\mathbf{x}) &= \frac{d}{dx} f^\*(\mathbf{x}) = \frac{d}{dx} F^\*(r(\mathbf{x})) \\ &= \left(\frac{d}{d\tilde{r}} F^\*(\tilde{r})\Big|\_{\tilde{r}=r(\mathbf{x})}\right) \cdot \frac{d}{d\mathbf{x}} r(\mathbf{x}) = P^\*(r(\mathbf{x})) \cdot \frac{d}{d\mathbf{x}} F^{-1}\left(\frac{\mathbf{x}+1}{2}\right) \\ &= P^\*(r(\mathbf{x})) \cdot \left(\frac{d}{d\tilde{r}} F(\tilde{r})\Big|\_{\tilde{r}=r(\mathbf{x})}\right)^{-1} \cdot \frac{d}{d\mathbf{x}} \frac{\mathbf{x}+1}{2} = \frac{1}{2} \cdot \frac{P^\*(r(\mathbf{x}))}{P(r(\mathbf{x}))} \end{split} \tag{27}$$

Therefore

$$f^\* = \left( F^\*(r(\mathbf{x})), \; \mathbf{x} \in \mathsf{F} \right), \; p^\* = \left( \frac{P^\*(r(\mathbf{x}))}{2P(r(\mathbf{x}))}, \; \mathbf{x} \in \mathsf{F} \right) \tag{28}$$

By (8) and the fact that the *F*-transformation is a component-wise mapping, the value of the PDF, associated with the vector x, is expressed as

$$\text{PDF}(x) = \prod\_{t=1}^{n} \frac{P^\*(r(\mathbf{x}\_t))}{2P(r(\mathbf{x}\_t))} \tag{29}$$

for all x ∈ F*n*.

If (*F*∗, *<sup>F</sup>*)=(*F*G0,*ρ*<sup>∗</sup> , *<sup>F</sup>*G0,*ρ*), then we use (23) and write

$$f^\*(\mathbf{x}) = \mathrm{FG}\_{0, \rho^\*} (\rho \sqrt{2} \cdot \mathrm{erf}^{-1}(\mathbf{x})) = \frac{1}{2} + \frac{1}{2} \mathrm{erf} \left( \frac{\rho \sqrt{2} \cdot \mathrm{erf}^{-1}(\mathbf{x})}{\rho^\* \sqrt{2}} \right) \tag{30}$$

$$p^\*(\mathbf{x}) = \frac{\mathcal{G}\_{0, \rho^\*}(r(\mathbf{x}))}{\mathcal{Z}\mathcal{G}\_{0, \rho}(r(\mathbf{x}))} = \frac{\rho\sqrt{2\pi}}{2\rho^\*\sqrt{2\pi}} \exp\left\{-\frac{r^2(\mathbf{x})}{2(\rho^\*)^2} + \frac{r^2(\mathbf{x})}{2\rho^2}\right\} \tag{31}$$

Hence,

$$f^\* = \left(\psi\_{\rho/\rho^\*}(\mathbf{x}), \; \mathbf{x} \in \mathsf{F}\right), \; p^\* = \left(\frac{\rho}{2\rho^\*} \exp\left\{-\frac{\rho^2 - (\rho^\*)^2}{2\rho^2 (\rho^\*)^2} r^2(\mathbf{x})\right\}, \; \mathbf{x} \in \mathsf{F}\right) \tag{32}$$

where *ψρ*/*ρ*<sup>∗</sup> (*x*) is defined by (15) with *<sup>c</sup>* <sup>=</sup> *<sup>ρ</sup>*/*ρ*∗. Examples of the functions *<sup>f</sup>* <sup>∗</sup>(*x*) and *<sup>p</sup>*∗(*x*) are given in Figures 4, 5.

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 11

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

*<sup>p</sup>*∗(*x*) ✻

*ρ*<sup>∗</sup> = 1/2 *ρ*<sup>∗</sup> = 2

*ρ*<sup>∗</sup> = 1

.. .. .. . .. .. . .. .. ... .. ... .. .. .. .. .. .. .. .. .. . .. ... .. .. ... .. .. .. ... .. .. .. .. .. .. .. .. .. .. ... .. .. . .. .. .. .. .. .. . .. .. .. .. .. .. . .. .. .. . .. ... ... .. .. .. . .. .. .. .. .. .. . .. .. ... .. ... .............................

1

. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. . .. .. .. .. .. .. .. .. .. .. . .. .. .. .. ..

........................... ... ... .. .. .. .. .. .. ... ....... .. .. .. .. .. ... .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. .. ... .. ... .. .. .. ..

**Figure 5.** Examples of the PDFs *p*∗, when (*F*, *F*∗)=(*F*G0,1, *F*G0,*ρ*<sup>∗</sup> ).

the practice. If the source is non–stationary, and *P*<sup>∗</sup>

*Ft* = 

where r*t*−<sup>1</sup> = (*r*1,...,*rt*−1) and *t* = 1, . . . , *n*. Then

1/2

1/4

function *F* with the functions

*<sup>F</sup>*∗(r*t*−1) =

*<sup>P</sup>*∗(r*t*−1) =

−1 0 +1

Our assumption that the vector r is generated by a known stationary memoryless source, and the value of the PDF, associated with this vector, is expressed by (8), can contradict to

the right-hand side of (8), then considerations above are directly extended by replacing the

that approximate the PDs of components of the input vector. A more general assignment

*dr <sup>F</sup>*(*r*|r*t*−1), *<sup>r</sup>* <sup>∈</sup> <sup>R</sup>

*n* ∏ *t*=1

*Ft*(*r*) = Pr{ *Rt* < *r* }, *r* ∈ R

corresponds to the sources with memory, when the PDs and the PDFs are given as

PDF(r) =

*F*(*r*|r*t*−1) = Pr

*<sup>P</sup>*(*r*|r*t*−1) = *<sup>d</sup>*

<sup>1</sup> (*r*1),..., *<sup>P</sup>*<sup>∗</sup>

data{ *Rt* <sup>&</sup>lt; *<sup>r</sup>* <sup>|</sup>(*R*1,..., *Rt*−1) = <sup>r</sup>*t*−<sup>1</sup> }, *<sup>r</sup>* <sup>∈</sup> <sup>R</sup>

... ................................................................................................................................................................................................................................................................. ... ...

..... .............................................................. ... ... ..

✲ *x*

http://dx.doi.org/10.5772/51800

315

*<sup>n</sup>* (*rn*) replace the multipliers at

, *t* = 1, . . . , *n* (35)

*<sup>P</sup>*∗(*rt*|r*t*−1) (38)

(36)

(37)

.. .. .. .. .. .. .. ..... .. .. .. .. ... .. ..... .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. ..

**Figure 4.** Examples of the PDs *f* <sup>∗</sup>, when (*F*, *F*∗)=(*F*G0,1, *F*G0,*ρ*<sup>∗</sup> ).

Suppose that *F*<sup>∗</sup> = *F*. Then

$$f^\*(\mathbf{x}) = F\left(F^{-1}\left(\frac{\mathbf{x} + 1}{2}\right)\right) = \frac{\mathbf{x} + 1}{2}, \ p^\*(\mathbf{x}) = \frac{d}{d\mathbf{x}}\frac{\mathbf{x} + 1}{2} = \frac{1}{2} \tag{33}$$

and

$$f^\* = \left(\frac{\mathbf{x} + \mathbf{1}}{2}, \mathbf{x} \in \mathsf{F}\right), \ p^\* = \left(\frac{1}{2}, \mathbf{x} \in \mathsf{F}\right) \tag{34}$$

i.e., *X* is a random variable, uniformly distributed over the set F. If the vector x is stored, then the database is perfectly protected against attackers, who want to guess its content. If *F*<sup>∗</sup> �= *F*, then we get an approximate uniform distribution. However, r → x is a one-to-one mapping, and the attackers, who have access to the database, can reconstruct the vector r.

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 11 Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters http://dx.doi.org/10.5772/51800 315

**Figure 5.** Examples of the PDFs *p*∗, when (*F*, *F*∗)=(*F*G0,1, *F*G0,*ρ*<sup>∗</sup> ).

10 New Trends and Developments in Biometrics

314 New Trends and Developments in Biometrics

1

1/2

1

1/2

Suppose that *F*<sup>∗</sup> = *F*. Then

and

. .. .. .. .. .. .. .. .. .. .... ... *<sup>f</sup>* <sup>∗</sup>(*x*) ✻

*ρ*<sup>∗</sup> = 1

*ρ*<sup>∗</sup> = 2

*ρ*<sup>∗</sup> = 1

... ... ... .............. ... ... ... ... ........... ... ........................................................................................................

... ............. .. .. .. .. ..

−1 0 +1

−1 0 +1

<sup>=</sup> *<sup>x</sup>* <sup>+</sup> <sup>1</sup>

 , *p*<sup>∗</sup> =

i.e., *X* is a random variable, uniformly distributed over the set F. If the vector x is stored, then the database is perfectly protected against attackers, who want to guess its content. If *F*<sup>∗</sup> �= *F*, then we get an approximate uniform distribution. However, r → x is a one-to-one mapping, and the attackers, who have access to the database, can reconstruct the vector r.

<sup>2</sup> , *<sup>p</sup>*∗(*x*) = *<sup>d</sup>*

 1 2 *dx*

, *x* ∈ F 

*x* + 1 <sup>2</sup> <sup>=</sup> <sup>1</sup>

... .. .. . ... ...

.......................................................................................................... ... ......... ... ... ... ... ............ ... ... ... ...

**Figure 4.** Examples of the PDs *f* <sup>∗</sup>, when (*F*, *F*∗)=(*F*G0,1, *F*G0,*ρ*<sup>∗</sup> ).

 *F*−<sup>1</sup>

*f* <sup>∗</sup> =

 *x* + 1 2

*x* + 1

<sup>2</sup> , *<sup>x</sup>* <sup>∈</sup> <sup>F</sup>

*f* <sup>∗</sup>(*x*) = *F*

*<sup>f</sup>* <sup>∗</sup>(*x*) ✻

*ρ*<sup>∗</sup> = 1/2

....................................................................................................................................................................................................................................................................................................................................................

✲ *x*

✲ *x*

<sup>2</sup> (33)

(34)

Our assumption that the vector r is generated by a known stationary memoryless source, and the value of the PDF, associated with this vector, is expressed by (8), can contradict to the practice. If the source is non–stationary, and *P*<sup>∗</sup> <sup>1</sup> (*r*1),..., *<sup>P</sup>*<sup>∗</sup> *<sup>n</sup>* (*rn*) replace the multipliers at the right-hand side of (8), then considerations above are directly extended by replacing the function *F* with the functions

$$F\_t = \left( F\_t(r) = \Pr\{\ R\_t < r \}, r \in \mathbb{R} \right), \ t = 1, \ldots, n \tag{35}$$

that approximate the PDs of components of the input vector. A more general assignment corresponds to the sources with memory, when the PDs and the PDFs are given as

$$F^\*(\mathbf{r}\_{t-1}) = \left( F(\mathbf{r}|\mathbf{r}\_{t-1}) = \Pr\_{\text{data}} \left\{ R\_t < r \, \middle| \, (\mathbf{R}\_1, \dots, \mathbf{R}\_{t-1}) = \mathbf{r}\_{t-1} \right\}, \, r \in \mathbb{R} \right) \tag{36}$$

$$P^\*(\boldsymbol{r}\_{t-1}) = \left(P(\boldsymbol{r}|\boldsymbol{r}\_{t-1}) = \frac{d}{dr}F(\boldsymbol{r}|\boldsymbol{r}\_{t-1}), \; \boldsymbol{r} \in \mathbb{R}\right) \tag{37}$$

where r*t*−<sup>1</sup> = (*r*1,...,*rt*−1) and *t* = 1, . . . , *n*. Then

$$\text{PDF}(r) = \prod\_{t=1}^{n} P^\*(r\_t | r\_{t-1}) \tag{38}$$

Let

$$\mathbf{x}\_{t} = 2\mathbf{F}\_{t}(r\_{t}) - \mathbf{1}, \ \ r\_{t}(\mathbf{x}\_{t}) = \mathbf{F}\_{t}^{-1}\left(\frac{\mathbf{x}\_{t} + 1}{2}\right) \tag{39}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 13

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

<sup>2</sup>*F*(*r*˜) <sup>−</sup> <sup>1</sup> ✻

*r*(*y*) *r*(*x*)

*dy* <sup>Φ</sup>(*r*(*y*)|*r*(*x*))

*y* + 1

*V*(*r*(*y*)|*r*(*x*))

<sup>2</sup>*P*(*r*(*y*)) , *<sup>y</sup>* <sup>∈</sup> <sup>F</sup>

*dy <sup>r</sup>*(*y*) = *<sup>V</sup>*(*r*(*y*)|*r*(*x*)) · *<sup>d</sup>*

*V*(*r*(*yt*)|*r*(*xt*))

<sup>−</sup><sup>1</sup> · *d dy*

.. .. .. . .. .. .. ..... ... .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. ... .. .. .. ..... .. .. .. .. .. .. ... .. .. .. .. .. .. .. ..... .. ... .. .. .. .. .. .. .. .. ... .. .. ... .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. . .. .. .. ... .. .. .. .

*x y*

..................................................................................................... ..... ...... .... .... .... .................................................... ... ... ... ... ... ... ........................................................ .... .... ...... ..... ..... ...... ...... ...................................................................................

**Figure 6.** Illustration of the computation of the probability Φ(*r*(*y*)|*r*(*x*)) for a given pair (*x*, *y*) ∈ F × F, when *F* = *F*G0,1.

2Φ(*r*(*y*)|*r*(*x*)) − 1

noise{ *<sup>Y</sup>* <sup>&</sup>lt; *<sup>y</sup>* <sup>|</sup> *<sup>X</sup>* <sup>=</sup> *<sup>x</sup>* } <sup>=</sup> *<sup>d</sup>*

 *d dr*˜ *F*(*r*˜) *r*˜=*r*(*y*)

Therefore the conditional PDs and the conditional PDFs are specified as

PDF(y|x) =

functions constructed using this procedure are given in Figures 7, 8.

Φ(*r*(*y*)|*r*(*x*)), *y* ∈ F

· *d*

 , *vx* =

for all *x* ∈ F. By (11) and the fact that the *F*-transformation is a component-wise mapping, the value of the PDF, associated with the vector y, given the vector x, is expressed as

> *n* ∏ *t*=1

The computation of the PDs *ϕx* is illustrated in Figure 6 for Gaussian data, and examples of

 *r*˜=*r*(*y*)

. ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .................................................... .... ....

−1

*<sup>v</sup>*(*y*|*x*) = *<sup>d</sup>*

<sup>=</sup> *<sup>d</sup> dr*˜

*ϕ<sup>x</sup>* = 

for all x, y ∈ F*n*.

*dy* Pr

Φ(*r*˜|*r*(*x*))

= *V*(*r*(*y*)|*r*(*x*)) ·

and

+1

✲ *r*˜

) = 2*F*(*r*˜) − 1

http://dx.doi.org/10.5772/51800

317

<sup>√</sup><sup>2</sup> ) = <sup>2</sup>Φ(*r*˜|*r*(*x*)) <sup>−</sup> <sup>1</sup>

erf( *<sup>r</sup>*˜ *ρ* √2

*dy <sup>F</sup>*−<sup>1</sup>

<sup>2</sup> <sup>=</sup> *<sup>V</sup>*(*r*(*y*)|*r*(*x*))

<sup>2</sup>*P*(*r*(*yt*)) (49)

 *y* + 1 2 

<sup>2</sup>*P*(*r*(*y*)) (47)

(48)

.... ... ..... .............................................. ..... ..... ..... ..... ..... ..... .....

erf(*r*˜−*r*(*x*) *σ*

and x*t*−<sup>1</sup> = (*x*1,..., *xt*−1), *r*(x*t*−1)=(*r*1(*x*1),...,*rt*−1(*xt*−1)). We also denote

$$f^\*(\mathbf{z}\_{t-1}) = \left( f(\mathbf{x}|\mathbf{z}\_{t-1}) = \Pr\_{\text{data}} \left\{ X\_t < \mathbf{x} \, \middle| \, (X\_1, \dots, X\_{t-1}) = \mathbf{z}\_{t-1} \right\}, \mathbf{x} \in \mathsf{F} \right) \tag{40}$$

$$p^\*(x\_{t-1}) = \left( p(\mathbf{x}|x\_{t-1}) = \frac{d}{d\mathbf{x}} f(\mathbf{x}|x\_{t-1}), \; \mathbf{x} \in \mathsf{F} \right) \tag{41}$$

Then

$$f^\*(x\_{t-1}) = \left(F\_t^\*(r\_t(\mathbf{x})|r(x\_{t-1})), \ x \in \mathsf{F}\right) \tag{42}$$

$$p^\*(x\_{t-1}) = \left(\frac{P^\*(r\_t(\mathbf{x})|r(x\_{t-1}))}{2P\_t(r\_t(\mathbf{x}))}, \mathbf{x} \in \mathsf{F}\right) \tag{43}$$

These formulas allow us to extend statistical properties of the *F*-transformation of the input data to the general case.

#### **4. The** *F***-transformation of noisy observations of the input data**

Let us denote

$$\varphi\_{\mathbf{x}} = \left( \varphi(y|\mathbf{x}) = \Pr\_{\text{noise}} \left\{ 2F(\mathbf{R}') - 1 < y \, | \, 2F(\mathbf{R}) - 1 = \mathbf{x} \right\}, y \in \mathsf{F} \right) \tag{44}$$

$$v\_{\mathbf{x}} = \left( v(y|\mathbf{x}) = \frac{d}{dy} \varphi(y|\mathbf{x}), y \in \mathsf{F} \right) \tag{45}$$

for all *<sup>x</sup>* ∈ <sup>F</sup>. If (Φ*r*, *Vr*)=(*F*G*r*,*σ*, G*r*,*σ*) for all *<sup>r</sup>* ∈ <sup>R</sup>, then the observation channel is an additive white Gaussian noise channel having the variance of the noise equal to *σ*2.

Let us map an *r*′ ∈ R to the *y* ∈ F in the same way as an *r* ∈ R was mapped to the *x* ∈ F, i.e., *y* = 2*F*(*r*′ ) − 1 and *r*′ = *r*(*y*), where *r*(*y*) = *F*−<sup>1</sup> *y*+1 2 . Notice that the value of *y* is the observation of the random variable *Y* = 2*F*(*R*′ ) − 1 having the conditional PD *ϕ<sup>x</sup>* and the conditional PDF *vx*, given *X* = *x*, since 2*F*(*R*′ ) − 1 is a deterministic function of the random variable *R*′ . The result of the *F*-transformation of the vector r′ ∈ R*n*, will be understood is the vector (2*F*(*r*′ <sup>1</sup>) <sup>−</sup> 1, . . . , 2*F*(*r*′ *<sup>n</sup>*) − 1).

One can see that the stochastic dependence between random variables *R* and *R*′ is translated to the stochastic dependence between random variables *X* and *Y* in such a way that

$$\begin{split} \varphi(y|\mathbf{x}) &= \Pr\_{\text{noise}} \left\{ Y < y \, | \, \mathbf{X} = \mathbf{x} \right\} = \Pr\_{\text{noise}} \left\{ 2F(\mathbf{R}') - 1 < y \, | \, 2F(\mathbf{R}) - 1 = \mathbf{x} \right\} \\ &= \Pr\_{\text{noise}} \left\{ \mathbf{R}' < F^{-1} \left( \frac{y+1}{2} \right) \, \middle| \, \mathbf{R} = F^{-1} \left( \frac{\mathbf{x}+1}{2} \right) \, \right\} \\ &= \Pr\_{\text{noise}} \left\{ \mathbf{R}' < r(y) \, | \, \mathbf{R} = r(\mathbf{x}) \, \right\} = \Phi(r(y)|r(\mathbf{x})) \end{split} \tag{46}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 13 Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters http://dx.doi.org/10.5772/51800 317

**Figure 6.** Illustration of the computation of the probability Φ(*r*(*y*)|*r*(*x*)) for a given pair (*x*, *y*) ∈ F × F, when *F* = *F*G0,1.

and

12 New Trends and Developments in Biometrics

316 New Trends and Developments in Biometrics

*<sup>f</sup>* <sup>∗</sup>(x*t*−1) =

*<sup>p</sup>*∗(x*t*−1) =

data to the general case.

*ϕ<sup>x</sup>* = 

*vx* = 

*ϕ*(*y*|*x*) = Pr

= Pr noise 

= Pr

Let us denote

i.e., *y* = 2*F*(*r*′

variable *R*′

the vector (2*F*(*r*′

*xt* = 2*Ft*(*rt*) − 1, *rt*(*xt*) = *F*−<sup>1</sup>

*dx <sup>f</sup>*(*x*|x*t*−1), *<sup>x</sup>* <sup>∈</sup> <sup>F</sup>

*<sup>P</sup>*∗(*rt*(*x*)|*r*(x*t*−1))

for all *<sup>x</sup>* ∈ <sup>F</sup>. If (Φ*r*, *Vr*)=(*F*G*r*,*σ*, G*r*,*σ*) for all *<sup>r</sup>* ∈ <sup>R</sup>, then the observation channel is an

Let us map an *r*′ ∈ R to the *y* ∈ F in the same way as an *r* ∈ R was mapped to the *x* ∈ F,

One can see that the stochastic dependence between random variables *R* and *R*′ is translated

noise 2*F*(*R*′

*R* = *F*−<sup>1</sup>

 *x* + 1 2

noise{ *<sup>R</sup>*′ <sup>&</sup>lt; *<sup>r</sup>*(*y*)<sup>|</sup> *<sup>R</sup>* <sup>=</sup> *<sup>r</sup>*(*x*) } <sup>=</sup> <sup>Φ</sup>(*r*(*y*)|*r*(*x*)) (46)

 

to the stochastic dependence between random variables *X* and *Y* in such a way that

 *y* + 1 2

 *y*+1 2 

. The result of the *F*-transformation of the vector r′ ∈ R*n*, will be understood is

These formulas allow us to extend statistical properties of the *F*-transformation of the input

and x*t*−<sup>1</sup> = (*x*1,..., *xt*−1), *r*(x*t*−1)=(*r*1(*x*1),...,*rt*−1(*xt*−1)). We also denote

 *F*∗

**4. The** *F***-transformation of noisy observations of the input data**

noise{ <sup>2</sup>*F*(*R*′

*dy <sup>ϕ</sup>*(*y*|*x*), *<sup>y</sup>* <sup>∈</sup> <sup>F</sup>

additive white Gaussian noise channel having the variance of the noise equal to *σ*2.

*f*(*x*|x*t*−1) = Pr

*<sup>p</sup>*(*x*|x*t*−1) = *<sup>d</sup>*

*<sup>f</sup>* <sup>∗</sup>(x*t*−1) =

*<sup>p</sup>*∗(x*t*−1) =

*ϕ*(*y*|*x*) = Pr

*<sup>v</sup>*(*y*|*x*) = *<sup>d</sup>*

observation of the random variable *Y* = 2*F*(*R*′

<sup>1</sup>) <sup>−</sup> 1, . . . , 2*F*(*r*′

conditional PDF *vx*, given *X* = *x*, since 2*F*(*R*′

) − 1 and *r*′ = *r*(*y*), where *r*(*y*) = *F*−<sup>1</sup>

*<sup>n</sup>*) − 1).

noise{ *<sup>Y</sup>* <sup>&</sup>lt; *<sup>y</sup>* <sup>|</sup> *<sup>X</sup>* <sup>=</sup> *<sup>x</sup>* } <sup>=</sup> Pr

*R*′ < *F*−<sup>1</sup>

*t*

 *xt* + 1 2

data{ *Xt* <sup>&</sup>lt; *<sup>x</sup>* <sup>|</sup>(*X*1,..., *Xt*−1) = <sup>x</sup>*t*−<sup>1</sup> }, *<sup>x</sup>* <sup>∈</sup> <sup>F</sup>

*<sup>t</sup>* (*rt*(*x*)|*r*(x*t*−1)), *<sup>x</sup>* ∈ <sup>F</sup>

<sup>2</sup>*Pt*(*rt*(*x*)) , *<sup>x</sup>* <sup>∈</sup> <sup>F</sup>

) − 1 < *y* | 2*F*(*R*) − 1 = *x*}, *y* ∈ F

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(44)

(45)

. Notice that the value of *y* is the

) − 1 having the conditional PD *ϕ<sup>x</sup>* and the

) − 1 is a deterministic function of the random

) − 1 < *y* | 2*F*(*R*) − 1 = *x*

Let

Then

$$\begin{split} v(y|\mathbf{x}) &= \frac{d}{dy} \operatorname\*{Pr}\_{\text{noise}}\left\{ Y < y \,|\, \mathbf{X} = \mathbf{x} \right\} = \frac{d}{dy} \Phi(r(y)|r(\mathbf{x})) \\ &= \frac{d}{d\tilde{r}} \Phi(\tilde{r}|r(\mathbf{x})) \Big|\_{\mathbf{r}=r(y)} \cdot \frac{d}{dy} r(y) = V(r(y)|r(\mathbf{x})) \cdot \frac{d}{dy} \mathcal{F}^{-1}\left(\frac{y+1}{2}\right) \\ &= V(r(y)|r(\mathbf{x})) \cdot \left(\frac{d}{d\tilde{r}} F(\tilde{r}) \Big|\_{\mathbf{r}=r(y)}\right)^{-1} \cdot \frac{d}{dy} \frac{y+1}{2} = \frac{V(r(y)|r(\mathbf{x}))}{2P(r(y))} \end{split} \tag{47}$$

Therefore the conditional PDs and the conditional PDFs are specified as

$$\varphi\_{\mathbf{x}} = \left( \Phi(r(y)|r(\mathbf{x})), y \in \mathsf{F} \right), \ v\_{\mathbf{x}} = \left( \frac{V(r(y)|r(\mathbf{x}))}{2P(r(y))}, y \in \mathsf{F} \right) \tag{48}$$

for all *x* ∈ F. By (11) and the fact that the *F*-transformation is a component-wise mapping, the value of the PDF, associated with the vector y, given the vector x, is expressed as

$$\text{PDF}(y|\mathbf{x}) = \prod\_{t=1}^{n} \frac{V(r(y\_t)|r(\mathbf{x}\_t))}{2P(r(y\_t))} \tag{49}$$

for all x, y ∈ F*n*.

The computation of the PDs *ϕx* is illustrated in Figure 6 for Gaussian data, and examples of functions constructed using this procedure are given in Figures 7, 8.

**Figure 7.** Examples of the conditional PDs *ϕx*, when (*F*, Φ*r*)=(*F*G0,1, *F*G*r*,1/4).

If the observation channel is an additive channel, then *V*(*r*′ |*r*) depends only on the absolute value of the difference *r*′ − *r*. It is also usually assumed that

$$V(r'|r) \text{ is a monotone decreasing function of } |r'-r|\tag{50}$$

and

$$\max\_{r' \in \mathbb{R}} V(r'|r) = V(r|r) = \text{Const} \tag{51}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 15

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

*<sup>v</sup>*(*y*|*x*) ✻

*x* = 0

*x* = +3/4

.. .. .. .. .. .. ..... .. .. .. .. .. .. ... .. .. .. .. ..... .. .. .. .. ... .. .. ..... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. ... .. .. .. ... .. .. ... ... .......... . .. .. .. .. ... .. .. .. .. .. . .. .. .. . .. .. .. .. .. .. .. .. .. .. ... .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... ... ....... .. .... .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .........

−1 0 +1

and the expression at the right-hand side tends to infinity, as *ε* → 0, i.e., the created channel becomes noiseless. In other words, we preserve significant components of the vector r in

erf−1(*y*) − erf−1(*x*)

erf−1(*y*) − erf−1(*x*)

, *<sup>v</sup>*(*y*|*x*) = <sup>1</sup>

2 + 

2*σ*

<sup>2</sup>

Const <sup>2</sup>*P*(*r*(*x*)) <sup>=</sup> *<sup>ρ</sup>*

= exp erf−1(*x*)

..... ....... .... ... ... ... ... ... ... ... ... ... ... ...................................... .......................................................... .......... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .............................. ...

... ........... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .......... ....................................... ........................................................ .... .... .... .... .... .......................... ..

*x* = −3/4

**Figure 8.** Examples of the conditional PDFs *vx*, when (*F*, Φ*r*)=(*F*G0,1, *F*G*r*,1/4).

noisy observations for a wide class of observation channels.

and if *<sup>F</sup>* = *<sup>F</sup>*G0,*ρ*, then these equalities can be continued as

2 + 1 2 erf *<sup>ρ</sup> σ* 

<sup>2</sup>*<sup>σ</sup>* exp

.. .. .. .. .. .. . .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. ... .. . .. ........ .. ..... ......................... .. .. .... . .... . .... . .. .. .. .. .. .. ... .. .. . .. .. .. ... ... .. .. .. .. .. . .. .. .. ..

erf*<sup>r</sup>*(*y*) <sup>−</sup> *<sup>r</sup>*(*x*) *σ* √2

> − *ρ*2 *σ*2

(*F*, <sup>Φ</sup>*r*)=(*F*G0,*ρ*, *<sup>F</sup>*G*r*,*σ*) ⇒

*<sup>E</sup>*(*x*) = exp *<sup>r</sup>*2(*x*)

2*ρ*<sup>2</sup>

........ .. ... .. .. .. .. .. .. .. .. ... .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. ........ ..... .. .. .... .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. ...... .. .. .. .. .. ... .. .. .. .. .. .. ... .. .. ... .. .. .. .. ... .. .. ... .......... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. ... .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. ..

*<sup>ϕ</sup>*(*y*|*x*) = <sup>1</sup>

*<sup>ϕ</sup>*(*y*|*x*) = <sup>1</sup>

*<sup>v</sup>*(*y*|*x*) = *<sup>ρ</sup>*

as it follows from (23). We also write

2 + 1 2

If <sup>Φ</sup>*<sup>r</sup>* = *<sup>F</sup>*G*r*,*σ*, then

where

✲ *y*

http://dx.doi.org/10.5772/51800

319

2*P*(*r*(*y*))G*r*(*x*),*σ*(*r*(*y*)) (54)

(55)

<sup>2</sup>

*E*(*x*) (57)

(56)

(58)

erf−1(*y*)

In particular, Const = 1/(*σ* <sup>√</sup>2*π*) for additive white Gaussian noise channels having the variance of the noise equal to *σ*2.

We include transformations at the input/output of the channel and create another channel x → y whose conditional PDFs essentially depend on the magnitude of the input symbols. This point is illustrated in Figures 7, 8: the slope of the function *ϕ*(*x*|*x*) increases with |*x*|, and *v*(*x*|*x*) tends to the *δ*-function, as *x* → ±1. Notice that the behavior of functions under considerations is completely determined by the description of physical channel and the chosen function *F*, and it is not affected by the PD *F*∗, which can be unknown.

By (51),

$$\frac{V(r(y)|r(\mathbf{x}))}{2P(r(y))}\Big|\_{y=\mathbf{x}} = \frac{V(r(\mathbf{x})|r(\mathbf{x}))}{2P(r(\mathbf{x}))} = \frac{\text{Const}}{2P(r(\mathbf{x}))}\tag{52}$$

Suppose that, for any *<sup>ε</sup>* > 0, there is an *<sup>r</sup><sup>ε</sup>* ∈ <sup>R</sup> such that |*r*| > *<sup>r</sup><sup>ε</sup>* implies *<sup>P</sup>*(*r*) ≤ *<sup>ε</sup>*. Then

$$|r(\mathbf{x})| > r\_{\varepsilon} \Rightarrow \frac{V(r(\mathbf{y})|r(\mathbf{x}))}{2P(r(\mathbf{y}))} \Big|\_{\mathbf{y}=\mathbf{x}} \ge \frac{\text{Const}}{2\varepsilon} \tag{53}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 15 Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters http://dx.doi.org/10.5772/51800 319

**Figure 8.** Examples of the conditional PDFs *vx*, when (*F*, Φ*r*)=(*F*G0,1, *F*G*r*,1/4).

and the expression at the right-hand side tends to infinity, as *ε* → 0, i.e., the created channel becomes noiseless. In other words, we preserve significant components of the vector r in noisy observations for a wide class of observation channels.

If <sup>Φ</sup>*<sup>r</sup>* = *<sup>F</sup>*G*r*,*σ*, then

✲ *y*


<sup>2</sup>*P*(*r*(*x*)) (52)

<sup>2</sup>*<sup>ε</sup>* (53)

14 New Trends and Developments in Biometrics

318 New Trends and Developments in Biometrics

1

1/2

and

By (51),

................. .. .. .. .. .. .. .. ... .. ..... .. ... .. .. .. .. .. .. .. ... .. ... .. .. ... .. .. .. .. .. .. ... .. .. .. .. ...... .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. ..

*V*(*r*′

In particular, Const = 1/(*σ*

variance of the noise equal to *σ*2.

*<sup>ϕ</sup>*(*y*|*x*) ✻

*x* = 0

*x* = +3/4

........ ...................................................................... .... .... .... ....

.. .. .. .. .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. ... .. ... .. .. .. .. .. .. .. .. .. ... ..... .. ... ... .. .. .. .. .. .. .. .. .. ..... ... .. ... .. .. .. .. ....................


<sup>√</sup>2*π*) for additive white Gaussian noise channels having the


−1 +1

..... .................................................................. .......... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ........ ... ... ... ... ... ... ... ... ... ... ... ... ... ... ........................................................................ ... ...

.... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .......... .................................................................... ..

.. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. ... .. .. ...... .. .. ..... .. ... .. .. .... ..... .. .. .. ...... .. .. ... .. .. ..... .. ... .. .. ... .. .. ..... .. .. .. .. .. .. .. .. .. .. . .. ..

−3/4 0 +3/4

We include transformations at the input/output of the channel and create another channel x → y whose conditional PDFs essentially depend on the magnitude of the input symbols. This point is illustrated in Figures 7, 8: the slope of the function *ϕ*(*x*|*x*) increases with |*x*|, and *v*(*x*|*x*) tends to the *δ*-function, as *x* → ±1. Notice that the behavior of functions under considerations is completely determined by the description of physical channel and

<sup>=</sup> *<sup>V</sup>*(*r*(*x*)|*r*(*x*))

*V*(*r*(*y*)|*r*(*x*)) 2*P*(*r*(*y*))

<sup>2</sup>*P*(*r*(*x*)) <sup>=</sup> Const

 *y*=*x* <sup>≥</sup> Const

the chosen function *F*, and it is not affected by the PD *F*∗, which can be unknown.

Suppose that, for any *<sup>ε</sup>* > 0, there is an *<sup>r</sup><sup>ε</sup>* ∈ <sup>R</sup> such that |*r*| > *<sup>r</sup><sup>ε</sup>* implies *<sup>P</sup>*(*r*) ≤ *<sup>ε</sup>*. Then

 *y*=*x*

*x* = −3/4

**Figure 7.** Examples of the conditional PDs *ϕx*, when (*F*, Φ*r*)=(*F*G0,1, *F*G*r*,1/4).

If the observation channel is an additive channel, then *V*(*r*′

value of the difference *r*′ − *r*. It is also usually assumed that

max *r*′∈R

*V*(*r*(*y*)|*r*(*x*)) 2*P*(*r*(*y*))


*V*(*r*′

$$\varphi(y|\mathbf{x}) = \frac{1}{2} + \frac{1}{2} \text{erf}\left(\frac{r(y) - r(\mathbf{x})}{\sigma\sqrt{2}}\right), \ v(y|\mathbf{x}) = \frac{1}{2P(r(y))} \text{G}\_{r(\mathbf{x})\varphi}(r(y)) \tag{54}$$

and if *<sup>F</sup>* = *<sup>F</sup>*G0,*ρ*, then these equalities can be continued as

$$\varphi(y|\mathbf{x}) = \frac{1}{2} + \frac{1}{2} \text{erf}\left(\frac{\rho}{\sigma} \left(\text{erf}^{-1}(y) - \text{erf}^{-1}(\mathbf{x})\right)\right) \tag{55}$$

$$v(y|\mathbf{x}) = \frac{\rho}{2\sigma} \exp\left\{-\frac{\rho^2}{\sigma^2} \left(\text{erf}^{-1}(y) - \text{erf}^{-1}(\mathbf{x})\right)^2 + \left(\text{erf}^{-1}(y)\right)^2\right\} \tag{56}$$

as it follows from (23). We also write

$$E(F\_r\Phi\_r) = (FG\_{0,\rho\nu}FG\_{r\mathcal{F}}) \implies \frac{\text{Const}}{2P(r(\mathbf{x}))} = \frac{\rho}{2\sigma}E(\mathbf{x})\tag{57}$$

where

$$E(\mathbf{x}) = \exp\left\{\frac{r^2(\mathbf{x})}{2\rho^2}\right\} = \exp\left\{\left(\text{erf}^{-1}(\mathbf{x})\right)^2\right\} \tag{58}$$

**Figure 9.** Processing the vector r at the enrollment stage with zone encryption.

does not depend on *ρ*. By (52), the expression at the right-hand side of (57) specifies the value of *v*(*x*|*x*). The function *E*(*x*) is minimized at *x* = 0 and tends to infinity, as *x* → ±1 (see Table 2). Thus, the PDF *vx* tends to the *δ*-function, as *x* → ±1. The speed of convergence is proportional to the ratio *ρ*/(2*σ*).

The derived formulas will be used in the evaluation of verification performance, but some preliminary estimates can be already presented. Namely, let us fix a *λ* ∈ (0, 1) and, for all *x* ∈ (0, 1), denote

$$
\Delta\_{\mathbf{x}}(\lambda) = \mathbf{x} - y\_{\lambda} \tag{59}
$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 17

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321

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

**x E**(**x**) **Λ<sup>x</sup> ∆x**(**10**−**2**) **∆x**(**10**−**4**) **∆x**(**10**−**8**) **∆x**(**10**−**16**)

0 1.00 0.5 0.44 0.65 0.84 0.96 0.25 1.05 0.1 0.46 0.71 0.97 1.17 0.50 1.25 3.5 · 10−<sup>3</sup> 0.43 0.70 1.03 1.33 0.75 1.94 2.1 · 10−<sup>6</sup> 0.32 0.58 0.95 1.38 0.80 2.22 1.5 · 10−<sup>7</sup> 0.28 0.52 0.90 1.36 0.85 2.82 4.3 · 10−<sup>9</sup> 0.24 0.46 0.82 1.31 0.90 3.87 2.4 · 10−<sup>11</sup> 0.19 0.37 0.71 1.22 0.91 4.21 5.9 · 10−<sup>12</sup> 0.18 0.35 0.68 1.19 0.92 4.63 1.3 · 10−<sup>12</sup> 0.16 0.33 0.65 1.16 0.93 5.16 2.1 · 10−<sup>13</sup> 0.15 0.31 0.61 1.12 0.94 5.86 2.7 · 10−<sup>14</sup> 0.13 0.28 0.57 1.08 0.95 6.82 2.3 · 10−<sup>15</sup> 0.12 0.25 0.53 1.03 0.96 8.24 < 10−<sup>15</sup> 0.10 0.22 0.48 0.96 0.97 10.53 < 10−<sup>15</sup> 0.08 0.18 0.41 0.88 0.98 14.97 < 10−<sup>15</sup> 0.06 0.14 0.34 0.77 0.99 27.59 < 10−<sup>15</sup> 0.04 0.09 0.23 0.59

**Table 2.** Some values of *E*(*x*), Λ*x*, and ∆*<sup>x</sup>* (*λ*) for (*ρ*, *σ*)=(1, 1/4).

**Data**

ID(r)

r ∈ R*<sup>n</sup>*

**Figure 10.** Mapping of the input data to the data stored in an open access database.

**Input** ✲

**Open Access Database**

ID(r)

*a*, *a*<sup>0</sup> ∈ (0, 1)

x˜ ∈ F*<sup>n</sup>*

Hash(s)

where *<sup>y</sup><sup>λ</sup>* is the solution to the equation *<sup>ϕ</sup>*(*yλ*|*x*) = *<sup>λ</sup>* and the function *<sup>ϕ</sup>*(*y*|*x*) is expressed in (55). Furthermore, let

$$\Lambda\_{\mathbf{x}} = \Pr\_{\text{noise}}\left\{ Y < 0 \, | \, \mathbf{X} = \mathbf{x} \right\} = \boldsymbol{\varrho}(\mathbf{0}|\mathbf{x}) = \boldsymbol{\Psi}\_{-\boldsymbol{\rho}/\sigma}(\mathbf{x}) \tag{60}$$

where the function *ψ*(*x*) is defined in (15) for *c* = −*ρ*/*σ*. By the symmetry properties, Λ*<sup>x</sup>* is the probability that the input symbol, having the magnitude *x*, changes the sign after the symbol is transmitted over the *X* → *Y* channel. The probability that *Y* < *x* − ∆*x*(*λ*) is equal to *λ*, when +*x* is transmitted. The probability that *Y* > −*x* + ∆*x*(*λ*) is also equal to *λ*, when −*x* is transmitted. The numerical illustration is included in Table 2 for (*ρ*, *σ*)=(1, 1/4). For example, if *x* = 0.90, then *Y* < 0 with the probability 2.4 · 10−<sup>11</sup> and components of the vector (10<sup>−</sup>2, 10<sup>−</sup>4, 10<sup>−</sup>8, 10<sup>−</sup>16) are the probabilities of the events *Y* < 0.90 − (0.19, 0.37, 0.71, 1.22) = (+0.71, +0.53, +0.19, −0.32). Comparison of parameters above for different *x* allows us to conclude that the increase of the magnitude of the transmitted symbol leads to essential improvement over the channel having the input alphabet (-1,+1) and the output alphabet {-,+}.

## **5. Constructing the wrapped versions of input data and verification over noisy observations**

We believe that there is a large variety of verification schemes that can be constructed on the basis of statistical properties of the probabilistic ensemble (*X*,*Y*). We modify the scheme in

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 17 Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters http://dx.doi.org/10.5772/51800 321


**Table 2.** Some values of *E*(*x*), Λ*x*, and ∆*<sup>x</sup>* (*λ*) for (*ρ*, *σ*)=(1, 1/4).

320 New Trends and Developments in Biometrics

is proportional to the ratio *ρ*/(2*σ*).

*x* ∈ (0, 1), denote

(55). Furthermore, let

{-,+}.

**noisy observations**

<sup>r</sup> <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>x</sup> <sup>∈</sup> <sup>F</sup>*<sup>n</sup>*

**Figure 9.** Processing the vector r at the enrollment stage with zone encryption.

Λ*<sup>x</sup>* = Pr

❄

✲ ✲ ✲

Encryptor

<sup>∆</sup>*x*(*λ*) = *<sup>x</sup>* <sup>−</sup> *<sup>y</sup><sup>λ</sup>* (59)

noise{*<sup>Y</sup>* <sup>&</sup>lt; <sup>0</sup> <sup>|</sup> *<sup>X</sup>* <sup>=</sup> *<sup>x</sup>*} <sup>=</sup> *<sup>ϕ</sup>*(0|*x*) = *<sup>ψ</sup>*−*ρ*/*σ*(*x*) (60)

Transformer Zone

does not depend on *ρ*. By (52), the expression at the right-hand side of (57) specifies the value of *v*(*x*|*x*). The function *E*(*x*) is minimized at *x* = 0 and tends to infinity, as *x* → ±1 (see Table 2). Thus, the PDF *vx* tends to the *δ*-function, as *x* → ±1. The speed of convergence

The derived formulas will be used in the evaluation of verification performance, but some preliminary estimates can be already presented. Namely, let us fix a *λ* ∈ (0, 1) and, for all

where *<sup>y</sup><sup>λ</sup>* is the solution to the equation *<sup>ϕ</sup>*(*yλ*|*x*) = *<sup>λ</sup>* and the function *<sup>ϕ</sup>*(*y*|*x*) is expressed in

where the function *ψ*(*x*) is defined in (15) for *c* = −*ρ*/*σ*. By the symmetry properties, Λ*<sup>x</sup>* is the probability that the input symbol, having the magnitude *x*, changes the sign after the symbol is transmitted over the *X* → *Y* channel. The probability that *Y* < *x* − ∆*x*(*λ*) is equal to *λ*, when +*x* is transmitted. The probability that *Y* > −*x* + ∆*x*(*λ*) is also equal to *λ*, when −*x* is transmitted. The numerical illustration is included in Table 2 for (*ρ*, *σ*)=(1, 1/4). For example, if *x* = 0.90, then *Y* < 0 with the probability 2.4 · 10−<sup>11</sup> and components of the vector (10<sup>−</sup>2, 10<sup>−</sup>4, 10<sup>−</sup>8, 10<sup>−</sup>16) are the probabilities of the events *Y* < 0.90 − (0.19, 0.37, 0.71, 1.22) = (+0.71, +0.53, +0.19, −0.32). Comparison of parameters above for different *x* allows us to conclude that the increase of the magnitude of the transmitted symbol leads to essential improvement over the channel having the input alphabet (-1,+1) and the output alphabet

**5. Constructing the wrapped versions of input data and verification over**

We believe that there is a large variety of verification schemes that can be constructed on the basis of statistical properties of the probabilistic ensemble (*X*,*Y*). We modify the scheme in

*F*

*a*, *a*<sup>0</sup> ∈ (0, 1)

x˜ ∈ F*<sup>n</sup>*

✲

✲

s ∈ {0, 1}*<sup>n</sup>*

**Figure 10.** Mapping of the input data to the data stored in an open access database.

Figure 1a and introduce the so–called zone encryption (see Figure 9),

$$\left(\mathbf{x}\,\,\, \rightarrow \left( (a\_{\prime}a\_{0})\_{\prime}\tilde{\mathbf{x}}, \mathbf{s} \right) \,\, \right) \tag{61}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 19

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

*xt*

*x*˜*t*

**Figure 11.** Partitioning of the (−1, +1) interval into zones and illustration of the mapping *xt* → *x*˜*t*, when *xt* belongs to the

The zone encryption can be introduced as representation of the vector x by the sum of three

(Zr)

*<sup>t</sup>* = *xt*, if *xt* ∈ Sg<sup>+</sup> ∪ Sg<sup>−</sup>

*<sup>t</sup>* = *xt*, if *xt* ∈ Bf<sup>+</sup> ∪ Bf<sup>−</sup>

*<sup>t</sup>* = *xt*, if *xt* ∈ Zr

❄

*xt*

x = x(Sg) + x(Bf) + x(Zr) (66)

x˜ = x˜(Sg) + x˜(Bf) + x˜(Zr) (68)

*<sup>t</sup>* = 0 for all *t* = 1, . . . , *n* and use the rules

(67)

*x*˜*t*

✻

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323

+1

Significant<sup>+</sup> zone

Buffer<sup>+</sup> zone

Zero zone

Buffer<sup>−</sup> zone

Significant<sup>−</sup> zone

(Sg) *<sup>t</sup>* = *x*

> 

*x* (Sg)

*x* (Bf)

*x* (Zr)

The vector ˜x can be also represented by the sum of three vectors,

(Bf) *<sup>t</sup>* = *x*

+*a*

+*a*<sup>0</sup>

−*a*<sup>0</sup>

−*a*

−1

constructed as follows: set *x*

Buffer zones.

vectors,

where *<sup>a</sup>*, *<sup>a</sup>*<sup>0</sup> <sup>∈</sup> (0, 1), *<sup>a</sup>* <sup>&</sup>gt; *<sup>a</sup>*0; ˜<sup>x</sup> = (*x*˜1,..., *<sup>x</sup>*˜*n*) <sup>∈</sup> <sup>F</sup>*n*; <sup>s</sup> = (*s*1,...,*sn*) ∈ {0, 1}*n*. If ID(r) is the identifier of the person whose data are given by the vector r, then the enrollment is represented as the mapping

$$\left(\left(\mathrm{ID}(r), r\right) \rightarrow \left(\mathrm{ID}(r), (a\_\prime a\_0), \tilde{x}, \mathrm{Hash}(s)\right)\right) \tag{62}$$

where the function Hash is a cryptographic "one–way" function having the property that one can easily find the value of the function for the given argument, but the inversion (finding the argument for the known value of the function) is practically impossible. The parameters at the right-hand side are stored in an open access database (see Figure 10).

The use of "the zone encryption", is caused by the point that we partition the (−1, +1) interval into 5 zones: Significant+/<sup>−</sup> zones (Sg+, Sg−), Buffer <sup>+</sup>/<sup>−</sup> zones (Bf+, Bf−), Zero zone (Zr), where

$$\begin{cases} \text{Sg}^+ = (+a\_\prime + 1) \text{,} \text{Sg}^- = (-1 \text{,} -a) \\\\ \text{Bf}^+ = (+a\_{0\prime} + a) \text{,} \text{Bf}^- = (-a\_\prime - a\_0) \\\\ \text{Zr} = (-a\_{0\prime} + a\_0) \end{cases} \tag{63}$$

Let the notation *x*˜*<sup>t</sup>* ∼ U(+*a*, +1) be understood in such a way that the value of *x*˜*<sup>t</sup>* is chosen at random using a uniform PD over the (+*a*, +1) interval. Similarly, if *x*˜*<sup>t</sup>* ∼ U(−1, −*a*), then the value of *x*˜*<sup>t</sup>* is chosen at random using a uniform PD over the (−1, −*a*) interval. If x is the wrapped version of the vector r, constructed by the *F*-transformation, then we set

$$\begin{cases} \pounds\_{t} = \mathbf{x}\_{t\prime} & \text{if } \mathbf{x}\_{t} \in \mathbf{S} \mathbf{g}^{+} \cup \mathbf{S} \mathbf{g}^{-} \\\\ \pounds\_{t} \sim \mathbf{U}(-1, -a)\_{\prime} \text{ if } \mathbf{x}\_{t} \in \mathbf{B} \mathbf{f}^{+} \\\\ \pounds\_{t} \sim \mathbf{U}(+a, +1)\_{\prime} \text{ if } \mathbf{x}\_{t} \in \mathbf{B} \mathbf{f}^{-} \\\\ \pounds\_{t} = 0, & \text{if } \mathbf{x}\_{t} \in \mathbf{Z} \mathbf{r} \end{cases} \tag{64}$$

Therefore components of the vector x, belonging to the Significant zones, are unchanged and components, belonging to the Zero zone, are set to zero. Components, belonging to the Buffer zones, are changed in such a way that the results belong to the Significant zones with different signs. The presented procedure is illustrated in Figure 11, and the binary vector s having components

$$s\_{\mathbf{f}} = \begin{cases} 1, \text{ if } \mathbf{x}\_{\mathbf{f}} \in \mathbf{S} \mathbf{g}^+ \cup \mathbf{S} \mathbf{g}^- \\ 0, \text{ if } \mathbf{x}\_{\mathbf{f}} \notin \mathbf{S} \mathbf{g}^+ \cup \mathbf{S} \mathbf{g}^- \end{cases} \tag{65}$$

specifies the Significant zones.

322 New Trends and Developments in Biometrics

represented as the mapping

zone (Zr), where

having components

specifies the Significant zones.

�

ID(r), r

 

 

*st* =

 

� → �

at the right-hand side are stored in an open access database (see Figure 10).

Figure 1a and introduce the so–called zone encryption (see Figure 9),

x → �

(*a*, *a*0), ˜x, s

where *<sup>a</sup>*, *<sup>a</sup>*<sup>0</sup> <sup>∈</sup> (0, 1), *<sup>a</sup>* <sup>&</sup>gt; *<sup>a</sup>*0; ˜<sup>x</sup> = (*x*˜1,..., *<sup>x</sup>*˜*n*) <sup>∈</sup> <sup>F</sup>*n*; <sup>s</sup> = (*s*1,...,*sn*) ∈ {0, 1}*n*. If ID(r) is the identifier of the person whose data are given by the vector r, then the enrollment is

where the function Hash is a cryptographic "one–way" function having the property that one can easily find the value of the function for the given argument, but the inversion (finding the argument for the known value of the function) is practically impossible. The parameters

The use of "the zone encryption", is caused by the point that we partition the (−1, +1) interval into 5 zones: Significant+/<sup>−</sup> zones (Sg+, Sg−), Buffer <sup>+</sup>/<sup>−</sup> zones (Bf+, Bf−), Zero

> Sg<sup>+</sup> = (+*a*, +1), Sg<sup>−</sup> = (−1, −*a*) Bf<sup>+</sup> = (+*a*0, +*a*), Bf<sup>−</sup> = (−*a*, −*a*0) Zr = (−*a*0, +*a*0)

Let the notation *x*˜*<sup>t</sup>* ∼ U(+*a*, +1) be understood in such a way that the value of *x*˜*<sup>t</sup>* is chosen at random using a uniform PD over the (+*a*, +1) interval. Similarly, if *x*˜*<sup>t</sup>* ∼ U(−1, −*a*), then the value of *x*˜*<sup>t</sup>* is chosen at random using a uniform PD over the (−1, −*a*) interval. If x is the wrapped version of the vector r, constructed by the *F*-transformation, then we set

*x*˜*<sup>t</sup>* = *xt*, if *xt* ∈ Sg<sup>+</sup> ∪ Sg<sup>−</sup>

Therefore components of the vector x, belonging to the Significant zones, are unchanged and components, belonging to the Zero zone, are set to zero. Components, belonging to the Buffer zones, are changed in such a way that the results belong to the Significant zones with different signs. The presented procedure is illustrated in Figure 11, and the binary vector s

1, if *xt* ∈ Sg<sup>+</sup> ∪ Sg<sup>−</sup>

0, if *xt* �∈ Sg<sup>+</sup> <sup>∪</sup> Sg<sup>−</sup> (65)

*x*˜*<sup>t</sup>* ∼ U(−1, −*a*), if *xt* ∈ Bf<sup>+</sup> *x*˜*<sup>t</sup>* ∼ U(+*a*, +1), if *xt* ∈ Bf<sup>−</sup> *x*˜*<sup>t</sup>* = 0, if *xt* ∈ Zr

�

ID(r),(*a*, *a*0), ˜x, Hash(s)

�

(61)

(62)

(63)

(64)

**Figure 11.** Partitioning of the (−1, +1) interval into zones and illustration of the mapping *xt* → *x*˜*t*, when *xt* belongs to the Buffer zones.

The zone encryption can be introduced as representation of the vector x by the sum of three vectors,

$$\mathbf{x} = \mathbf{z}^{(\text{Sg})} + \mathbf{z}^{(\text{Bf})} + \mathbf{z}^{(\text{Zr})} \tag{66}$$

constructed as follows: set *x* (Sg) *<sup>t</sup>* = *x* (Bf) *<sup>t</sup>* = *x* (Zr) *<sup>t</sup>* = 0 for all *t* = 1, . . . , *n* and use the rules

$$\begin{cases} \mathbf{x}\_t^{(\mathbf{S}\mathbf{g})} = \mathbf{x}\_{t\prime} \text{ if } \mathbf{x}\_t \in \mathbf{S}\mathbf{g}^+ \cup \mathbf{S}\mathbf{g}^-\\ \mathbf{x}\_t^{(\mathbf{B}\mathbf{f})} = \mathbf{x}\_{t\prime} \text{ if } \mathbf{x}\_t \in \mathbf{B}\mathbf{f}^+ \cup \mathbf{B}\mathbf{f}^-\\ \mathbf{x}\_t^{(\mathbf{Z}\mathbf{r})} = \mathbf{x}\_{t\prime} \text{ if } \mathbf{x}\_t \in \mathbf{Z}\mathbf{r} \end{cases} \tag{67}$$

The vector ˜x can be also represented by the sum of three vectors,

$$
\tilde{\mathbf{x}} = \tilde{\mathbf{x}}^{(\text{Sg})} + \tilde{\mathbf{x}}^{(\text{Bf})} + \tilde{\mathbf{x}}^{(\text{Zr})} \tag{68}
$$


**Table 3.** Example of the zone encryption where gaps contain zeroes.

where we first set *x*˜ (Sg) *<sup>t</sup>* = *x*˜ (Bf) *<sup>t</sup>* = *x*˜ (Zr) *<sup>t</sup>* = 0 for all *t* = 1, . . . , *n* and then set

$$\begin{cases} \begin{aligned} \tilde{\mathbf{x}}\_{t}^{(\text{Sg})} &= \mathbf{x}\_{t}^{(\text{Sg})}, & \text{if } \mathbf{x}\_{t} \in \text{Sg}^{+} \cup \text{Sg}^{-} \\ \tilde{\mathbf{x}}\_{t}^{(\text{Bf})} &\sim \mathcal{U}(-1, -a), \text{ if } \mathbf{x}\_{t} \in \text{Bf}^{+} \\ \tilde{\mathbf{x}}\_{t}^{(\text{Bf})} &\sim \mathcal{U}(+a, +1), \text{ if } \mathbf{x}\_{t} \in \text{Bf}^{-} \\ \tilde{\mathbf{x}}\_{t}^{(\text{Zr})} &= 0, & \text{if } \mathbf{x}\_{t} \in \text{Zr} \end{aligned} \tag{69}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 21

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

x˜ = (0, +.90, +.91, 0, −.98, 0, 0, +.95) (74)

(75)

325

http://dx.doi.org/10.5772/51800

(x˜). All

(x)+*n*(Bf)(x)) (77)

→ <sup>s</sup><sup>ˆ</sup> ∈ {0, 1}*<sup>n</sup>* (78)

zones. The encryptor replaces these components by numbers chosen at random from the

and the hidden version of the vector s = (0, 0, 1, 0, 1, 0, 0, 0) are stored in the database.

rSg<sup>+</sup> = (*r*(+*a*),*r*(+1)), rSg<sup>−</sup> = (*r*(−1),*r*(−*a*)) rBf<sup>+</sup> = (*r*(+*a*0),*r*(+*a*)), rBf<sup>−</sup> = (*r*(−*a*),*r*(−*a*0)) rZr = (*r*(−*a*0),*r*(+*a*0))

where the function *r*(*x*), *x* ∈ F, is defined in (22). It is important to notice that the same

If *x*˜*<sup>t</sup>* ∈ Sg+, then the conclusion, whether *rt* belongs to the sets rSg<sup>+</sup> or rBf−, is specified by the *t*-th component of the vector s, which is hidden from the attacker. Similarly, if *x*˜*<sup>t</sup>* ∈ Sg−, then the conclusion, whether *rt* belongs to the sets rSg<sup>−</sup> or rBf+, is also specified by

of them are presented in Table 4 for the vector ˜x, defined in (74). For example, if *F* = *F*G0,1, then (*r*(.88),*r*(.76)) = (1.55, 1.17) and *r*2,*r*3,*r*<sup>8</sup> ∈ (+1.55, +∞) ∪ (−1.55, −1.17), *r*<sup>5</sup> ∈

If we construct a uniform probability distribution at the enrolment stage by using the *F*-transformation, then all variants are equivalent, and the probability of the correct guess

(x˜) <sup>=</sup> <sup>2</sup>−(*n*(Sg)

and the value of the sum *<sup>n</sup>*(Sg)(x) + *<sup>n</sup>*(Bf)(x) is a function of the vector <sup>x</sup> and the pair (*a*, *<sup>a</sup>*0). Notice that the randomization at the enrollment stage can be replaced by deterministic mappings Bf<sup>+</sup> → Sg<sup>−</sup> and Bf<sup>−</sup> → Sg+. For example, if the Significant and the Buffer zones have equal sizes, i.e., if 1 − *<sup>a</sup>* = *<sup>a</sup>* − *<sup>a</sup>*0, then one can follow the rules: if *xt* ∈ Bf+, then

�

the *<sup>t</sup>*-th component of the vector <sup>s</sup>. The total number of variants is equal to 2*n*(Sg)

(−∞, −1.55) ∪ (+1.17, +1.55), *r*1,*r*4,*r*6,*r*<sup>7</sup> ∈ (−1.17, +1.17).

*<sup>x</sup>*˜*<sup>t</sup>* = −*<sup>a</sup>* − (*xt* − *<sup>a</sup>*0); if *xt* ∈ Bf−, then *<sup>x</sup>*˜*<sup>t</sup>* = +*<sup>a</sup>* + (*a*<sup>0</sup> − *xt*).

�

y,(*a*, *a*0), ˜x

Let us introduce the decoding as the mapping

of the vector s by an attacker, who knows the vector ˜x, is equal to

*<sup>P</sup>*att(x˜) = <sup>2</sup>−*n*(Sg)

*x*˜*<sup>t</sup>* ∈ Sg<sup>+</sup> ⇒ *rt* ∈ rSg<sup>+</sup> ∪ rBf−, *x*˜*<sup>t</sup>* ∈ Sg<sup>−</sup> ⇒ *rt* ∈ rSg<sup>−</sup> ∪ rBf+, *x*˜*<sup>t</sup>* = 0 ⇒ *rt* ∈ rZr (76)

(+.88, +1) interval. For example, −.77 → +.90 and −.81 → +.95. The vector

Similarly to (63), let us introduce the sets

 

vector ˜x encrypts many vectors r and

Thus, the *t*-th component of the vector ˜x is either equal to 0, or belongs to the set Sg<sup>+</sup> ∪ Sg−. Furthermore, *x*˜*<sup>t</sup>* > 0 implies *x*˜*<sup>t</sup>* ∈ Sg+, and *x*˜*<sup>t</sup>* < 0 implies *x*˜*<sup>t</sup>* ∈ Sg−. Thus, *n* = *n*(Sg)(x˜) + *n*(Zr)(x˜), where

$$n^{\langle \mathbf{Sg} \rangle}(\tilde{\mathbf{x}}) = \left\{ t \in \{1, \ldots, n\} \, : \, \mathfrak{x}\_t \in \mathbf{Sg}^+ \cup \mathbf{Sg}^- \right\} \tag{70}$$

$$m^{(\mathbf{Zr})}(\tilde{\mathbf{z}}) = \left\{ t \in \{1, \ldots, n\} \, : \, \mathfrak{x}\_t = \mathbf{0} \right\} \tag{71}$$

Moreover, *n*(Sg)(x˜) = *n*(Sg)(x) + *n*(Bf)(x), where *n*(Sg)(x) = | T (Sg)(x)|, *n*(Bf)(x) = | T (Bf)(x)|, and

$$\mathcal{T}^{(\text{Sg})}(\mathbf{z}) = \left\{ t \in \{1, \ldots, n\} \, : \, \mathbf{x}\_t \in \text{Sg}^+ \cup \text{Sg}^- \right\} \tag{72}$$

$$\mathcal{T}^{(\mathbf{Bf})}(\mathbf{z}) = \left\{ t \in \{1, \ldots, n\} \, : \, \mathbf{x}\_t \in \mathbf{Bf}^+ \cup \mathbf{Bf}^- \right\} \tag{73}$$

The numerical example of the zone encryption is given in Table 3. Since | − .77|, | − .81| ∈ (.76, .88)=(*a*0, *a*), the 2-nd and the 8-th components of the vector x belong to the Buffer zones. The encryptor replaces these components by numbers chosen at random from the (+.88, +1) interval. For example, −.77 → +.90 and −.81 → +.95. The vector

20 New Trends and Developments in Biometrics

324 New Trends and Developments in Biometrics

(**a**, **a0**)=(.**88**, .**76**)

x +.68 −.77 +.91 +.24 −.98 +.08 −.52 −.81

x(Bf) −.77 −.81

x˜(Bf) +.90 +.95

x˜ 0 +.90 +.91 0 −.98 0 0 +.95 s 00101000

*<sup>t</sup>* = 0 for all *t* = 1, . . . , *n* and then set

(69)

(70)

(71)

(72)

(73)

*<sup>t</sup>* , if *xt* ∈ Sg<sup>+</sup> ∪ Sg<sup>−</sup>

*<sup>t</sup>* ∈ {1, . . . , *<sup>n</sup>*} : *<sup>x</sup>*˜*<sup>t</sup>* ∈ Sg<sup>+</sup> ∪ Sg<sup>−</sup> �

*<sup>t</sup>* ∈ {1, . . . , *<sup>n</sup>*} : *xt* ∈ Sg<sup>+</sup> ∪ Sg<sup>−</sup> �

*<sup>t</sup>* ∈ {1, . . . , *<sup>n</sup>*} : *xt* ∈ Bf<sup>+</sup> ∪ Bf<sup>−</sup> �

�

x(Zr) +.68 +.24 +.08 −.52

x˜(Zr) 0 0 00

x(Sg) +.91 −.98

x˜(Sg) +.91 −.98

**Table 3.** Example of the zone encryption where gaps contain zeroes.

(Bf) *<sup>t</sup>* = *x*˜

> *x*˜ (Sg) *<sup>t</sup>* = *x*

*x*˜ (Bf)

*x*˜ (Bf)

*x*˜ (Zr)

(x˜) = �

(x˜) = �

(x) = �

(x) = �

 

*n*(Sg)

*n*(Zr)

T (Sg)

T (Bf)

(Zr)

(Sg)

*<sup>t</sup>* ∼ U(−1, −*a*), if *xt* ∈ Bf<sup>+</sup>

*<sup>t</sup>* ∼ U(+*a*, +1), if *xt* ∈ Bf<sup>−</sup>

*<sup>t</sup>* = 0, if *xt* ∈ Zr

Thus, the *t*-th component of the vector ˜x is either equal to 0, or belongs to the set Sg<sup>+</sup> ∪ Sg−. Furthermore, *x*˜*<sup>t</sup>* > 0 implies *x*˜*<sup>t</sup>* ∈ Sg+, and *x*˜*<sup>t</sup>* < 0 implies *x*˜*<sup>t</sup>* ∈ Sg−. Thus, *n* = *n*(Sg)(x˜) +

*t* ∈ {1, . . . , *n*} : *x*˜*<sup>t</sup>* = 0

Moreover, *n*(Sg)(x˜) = *n*(Sg)(x) + *n*(Bf)(x), where *n*(Sg)(x) = | T (Sg)(x)|, *n*(Bf)(x) =

The numerical example of the zone encryption is given in Table 3. Since | − .77|, | − .81| ∈ (.76, .88)=(*a*0, *a*), the 2-nd and the 8-th components of the vector x belong to the Buffer

(Sg) *<sup>t</sup>* = *x*˜

where we first set *x*˜

*n*(Zr)(x˜), where


$$
\tilde{x} = (0, +.90, +.91, 0, -.98, 0, 0, +.95) \tag{74}
$$

and the hidden version of the vector s = (0, 0, 1, 0, 1, 0, 0, 0) are stored in the database. Similarly to (63), let us introduce the sets

$$\begin{cases} \mathbf{r} \mathbf{S} \mathbf{g}^+ = (r(+a), r(+1)), \mathbf{r} \mathbf{S} \mathbf{g}^- = (r(-1), r(-a)) \\\\ \mathbf{r} \mathbf{B} \mathbf{f}^+ = (r(+a\_0), r(+a)), \mathbf{r} \mathbf{B} \mathbf{f}^- = (r(-a), r(-a\_0)) \\\\ \mathbf{r} \mathbf{Z} \mathbf{r} = (r(-a\_0), r(+a\_0)) \end{cases} \tag{75}$$

where the function *r*(*x*), *x* ∈ F, is defined in (22). It is important to notice that the same vector ˜x encrypts many vectors r and

$$\text{r.f.} \in \text{Sg}^+ \Rightarrow r\_l \in \text{rSg}^+ \cup \text{rBf}^-, \ \text{\color{red}{x}\_l \in \text{Sg}^-} \Rightarrow r\_l \in \text{rSg}^- \cup \text{rBf}^+, \ \text{\color{red}{x}\_l = 0} \Rightarrow r\_l \in \text{rZr} \tag{76}$$

If *x*˜*<sup>t</sup>* ∈ Sg+, then the conclusion, whether *rt* belongs to the sets rSg<sup>+</sup> or rBf−, is specified by the *t*-th component of the vector s, which is hidden from the attacker. Similarly, if *x*˜*<sup>t</sup>* ∈ Sg−, then the conclusion, whether *rt* belongs to the sets rSg<sup>−</sup> or rBf+, is also specified by the *<sup>t</sup>*-th component of the vector <sup>s</sup>. The total number of variants is equal to 2*n*(Sg) (x˜). All of them are presented in Table 4 for the vector ˜x, defined in (74). For example, if *F* = *F*G0,1, then (*r*(.88),*r*(.76)) = (1.55, 1.17) and *r*2,*r*3,*r*<sup>8</sup> ∈ (+1.55, +∞) ∪ (−1.55, −1.17), *r*<sup>5</sup> ∈ (−∞, −1.55) ∪ (+1.17, +1.55), *r*1,*r*4,*r*6,*r*<sup>7</sup> ∈ (−1.17, +1.17).

If we construct a uniform probability distribution at the enrolment stage by using the *F*-transformation, then all variants are equivalent, and the probability of the correct guess of the vector s by an attacker, who knows the vector ˜x, is equal to

$$P\_{\mathbf{att}}(\tilde{\mathbf{x}}) = \mathbf{2}^{-n^{(\mathbf{S}\mathbf{g})}(\mathbf{a})} = \mathbf{2}^{-(n^{(\mathbf{S}\mathbf{g})}(\mathbf{a}) + n^{(\mathbf{Bf})}(\mathbf{a}))} \tag{77}$$

and the value of the sum *<sup>n</sup>*(Sg)(x) + *<sup>n</sup>*(Bf)(x) is a function of the vector <sup>x</sup> and the pair (*a*, *<sup>a</sup>*0).

Notice that the randomization at the enrollment stage can be replaced by deterministic mappings Bf<sup>+</sup> → Sg<sup>−</sup> and Bf<sup>−</sup> → Sg+. For example, if the Significant and the Buffer zones have equal sizes, i.e., if 1 − *<sup>a</sup>* = *<sup>a</sup>* − *<sup>a</sup>*0, then one can follow the rules: if *xt* ∈ Bf+, then *<sup>x</sup>*˜*<sup>t</sup>* = −*<sup>a</sup>* − (*xt* − *<sup>a</sup>*0); if *xt* ∈ Bf−, then *<sup>x</sup>*˜*<sup>t</sup>* = +*<sup>a</sup>* + (*a*<sup>0</sup> − *xt*).

Let us introduce the decoding as the mapping

$$\left(y\_{\prime}(a, a\_0), \check{\mathbf{x}}\right) \to \hat{\mathbf{s}} \in \{0, 1\}^n \tag{78}$$


**Table 4.** The list of vectors r that are encrypted by the vector x˜, defined in (74).

Let the verifier set *s*ˆ1 = ··· = *s*ˆ*<sup>n</sup>* = 0 and, for all *t* = 1, . . . , *n*, use the rule:

$$\begin{aligned} \mathfrak{x}\_t &\neq 0 \\ |y\_t| &> T \\ \text{sgn}(\mathfrak{x}\_t) &= \text{sgn}(y\_t) \end{aligned} \Rightarrow \quad \mathfrak{x}\_t = \mathbf{1} \tag{79}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 23

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

The decoding error in the case, when the vector r′ is a noisy version of the vector r, occurs

E10: (*st*,*s*ˆ*t*)=(1, 0). Then *xt* ∈ (+*a*, +1), as it follows from (80) and *st* = 1. Furthermore, *yt* < +*T*, since *s*ˆ*<sup>t</sup>* = 0 implies that the conditions at the left-hand side of (79) are not

E01: (*st*,*s*ˆ*t*)=(0, 1). Then *xt* ∈ (+*a*0, +*a*), as it follows from (80) and *st* = 0, *s*ˆ*<sup>t</sup>* �= 0. Furthermore, *yt* < −*T*, since *s*ˆ*<sup>t</sup>* = 1 implies that the conditions at the left-hand side

If the channel *Xt* → *Yt* would be an additive channel, then its probabilistic description does not depend on the input to the channel, and the differences at the left-hand sides of (81), (82) specify the magnitudes of the noise. The differences at the right-hand sides give lower bounds on these magnitudes. If *T* = (*a* − *a*0)/2, then these differences are equal. However, as the created channel is not an additive channel, we will use another assignment of *T*. The decoding error probability, denoted by Λ(*T*|x), can be bounded from above as

in one of two situations (see Figure 12).

of (79) are satisfied. Hence,

Λ10(*T*|x) = Pr

= 1 − Pr

= <sup>1</sup> − ∏

≤ <sup>1</sup> − ∏

= 1 − 

*<sup>t</sup>*∈T (Sg)

*<sup>t</sup>*∈T (Sg)

(x)

(x)

1 − *ϕ*(+*T* | + *a*)

Pr

Pr

satisfied. Hence,

where

*<sup>x</sup>*1,..., *xn* ∈ Sg<sup>+</sup> ∪ Bf<sup>+</sup> ∪ Zr (80)

http://dx.doi.org/10.5772/51800

327

*xt* − *yt* ≥ (+*a*) − (+*T*) (81)

*xt* − *yt* ≥ (+*a*0) − (−*T*) (82)

Λ(*T*|x) ≤ Λ10(*T*|x) + Λ01(*T*|x) (83)

noise{*Yt* <sup>&</sup>lt; <sup>+</sup>*T*, for some *<sup>t</sup>* ∈ T (Sg)(x)<sup>|</sup> *<sup>X</sup><sup>n</sup>* <sup>=</sup> <sup>x</sup>}

noise{*Yt* <sup>≥</sup> <sup>+</sup>*T*, for all *<sup>t</sup>* ∈ T (Sg)(x)<sup>|</sup> *<sup>X</sup><sup>n</sup>* <sup>=</sup> <sup>x</sup>}

noise{*Yt* <sup>≥</sup> <sup>+</sup>*<sup>T</sup>* <sup>|</sup> *Xt* <sup>=</sup> *xt*}

noise{*Yt* <sup>≥</sup> <sup>+</sup>*<sup>T</sup>* <sup>|</sup> *Xt* = +*a*}

(84)

*<sup>n</sup>*(Sg) (x)

where the value of the threshold *T* is a function of (*a*, *a*0), which will be specified later. The verifier can then check whether Hash(sˆ) is equal to the value of Hash(s), stored in the database, or not. If the answer is positive, then the acceptance decision is made. If the answer is negative, then the rejection decision is made.

Without loss of generality, let us suppose that

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 23 Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters http://dx.doi.org/10.5772/51800 327

$$\mathbf{x}\_{1}, \ldots, \mathbf{x}\_{n} \in \mathbf{S} \mathbf{g}^{+} \cup \mathbf{B} \mathbf{f}^{+} \cup \mathbf{Z} \mathbf{r} \tag{80}$$

The decoding error in the case, when the vector r′ is a noisy version of the vector r, occurs in one of two situations (see Figure 12).

E10: (*st*,*s*ˆ*t*)=(1, 0). Then *xt* ∈ (+*a*, +1), as it follows from (80) and *st* = 1. Furthermore, *yt* < +*T*, since *s*ˆ*<sup>t</sup>* = 0 implies that the conditions at the left-hand side of (79) are not satisfied. Hence,

$$
\omega\_t - y\_t \ge (+a) - (+T) \tag{81}
$$

E01: (*st*,*s*ˆ*t*)=(0, 1). Then *xt* ∈ (+*a*0, +*a*), as it follows from (80) and *st* = 0, *s*ˆ*<sup>t</sup>* �= 0. Furthermore, *yt* < −*T*, since *s*ˆ*<sup>t</sup>* = 1 implies that the conditions at the left-hand side of (79) are satisfied. Hence,

$$
\varepsilon x\_t - y\_t \ge (+a\_0) - (-T) \tag{82}
$$

If the channel *Xt* → *Yt* would be an additive channel, then its probabilistic description does not depend on the input to the channel, and the differences at the left-hand sides of (81), (82) specify the magnitudes of the noise. The differences at the right-hand sides give lower bounds on these magnitudes. If *T* = (*a* − *a*0)/2, then these differences are equal. However, as the created channel is not an additive channel, we will use another assignment of *T*.

The decoding error probability, denoted by Λ(*T*|x), can be bounded from above as

$$
\Lambda(T|\mathbf{x}) \le \Lambda\_{10}(T|\mathbf{x}) + \Lambda\_{01}(T|\mathbf{x}) \tag{83}
$$

where

22 New Trends and Developments in Biometrics

326 New Trends and Developments in Biometrics

(**a**, **a0**)=(.**88**, .**76**)

*x*˜1 = *x*˜4 = *x*˜6 = *x*˜7 = 0 *r*1,*r*4,*r*6,*r*<sup>7</sup> ∈ rZr

⇒ *s*ˆ*<sup>t</sup>* = 1 (79)

**Table 4.** The list of vectors r that are encrypted by the vector x˜, defined in (74).

is negative, then the rejection decision is made. Without loss of generality, let us suppose that

Let the verifier set *s*ˆ1 = ··· = *s*ˆ*<sup>n</sup>* = 0 and, for all *t* = 1, . . . , *n*, use the rule:

sgn(*x*˜*t*) = sgn(*yt*)

*x*˜*<sup>t</sup>* �= 0 |*yt*| > *T*

where the value of the threshold *T* is a function of (*a*, *a*0), which will be specified later. The verifier can then check whether Hash(sˆ) is equal to the value of Hash(s), stored in the database, or not. If the answer is positive, then the acceptance decision is made. If the answer

 

*s*<sup>2</sup> *s*<sup>3</sup> *s*<sup>5</sup> *s*<sup>8</sup> *x*˜2 = +.90 *x*˜3 = +.91 *x*˜5 = −.98 *x*˜8 = +.95 *<sup>r</sup>*<sup>2</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>3</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>5</sup> ∈ rBf<sup>+</sup> *<sup>r</sup>*<sup>8</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>2</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>3</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>5</sup> ∈ rBf<sup>+</sup> *<sup>r</sup>*<sup>8</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>2</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>3</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>5</sup> ∈ rSg<sup>−</sup> *<sup>r</sup>*<sup>8</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>2</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>3</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>5</sup> ∈ rSg<sup>−</sup> *<sup>r</sup>*<sup>8</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>2</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>3</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>5</sup> ∈ rBf<sup>+</sup> *<sup>r</sup>*<sup>8</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>2</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>3</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>5</sup> ∈ rBf<sup>+</sup> *<sup>r</sup>*<sup>8</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>2</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>3</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>5</sup> ∈ rSg<sup>−</sup> *<sup>r</sup>*<sup>8</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>2</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>3</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>5</sup> ∈ rSg<sup>−</sup> *<sup>r</sup>*<sup>8</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>2</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>3</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>5</sup> ∈ rBf<sup>+</sup> *<sup>r</sup>*<sup>8</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>2</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>3</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>5</sup> ∈ rBf<sup>+</sup> *<sup>r</sup>*<sup>8</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>2</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>3</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>5</sup> ∈ rSg<sup>−</sup> *<sup>r</sup>*<sup>8</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>2</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>3</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>5</sup> ∈ rSg<sup>−</sup> *<sup>r</sup>*<sup>8</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>2</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>3</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>5</sup> ∈ rBf<sup>+</sup> *<sup>r</sup>*<sup>8</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>2</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>3</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>5</sup> ∈ rBf<sup>+</sup> *<sup>r</sup>*<sup>8</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>2</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>3</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>5</sup> ∈ rSg<sup>−</sup> *<sup>r</sup>*<sup>8</sup> ∈ rBf<sup>−</sup> *<sup>r</sup>*<sup>2</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>3</sup> ∈ rSg<sup>+</sup> *<sup>r</sup>*<sup>5</sup> ∈ rSg<sup>−</sup> *<sup>r</sup>*<sup>8</sup> ∈ rSg<sup>+</sup>

$$\Lambda\_{10}(T|\mathbf{z}) = \Pr\_{\text{noise}}\left\{Y\_{t} < +T, \text{ for some } t \in \mathcal{T}^{(\text{S\S})}(x) \mid X^{n} = x\right\}$$

$$= 1 - \Pr\_{\text{noise}}\left\{Y\_{t} \ge +T, \text{ for all } t \in \mathcal{T}^{(\text{S\S})}(x) \mid X^{n} = x\right\}$$

$$= 1 - \prod\_{t \in \mathcal{T}^{(\text{S\S})}(x)} \Pr\_{\text{noise}}\left\{Y\_{t} \ge +T \mid X\_{t} = x\_{t}\right\}$$

$$\le 1 - \prod\_{t \in \mathcal{T}^{(\text{S\S})}(x)} \Pr\_{\text{noise}}\left\{Y\_{t} \ge +T \mid X\_{t} = +a\right\}$$

$$= 1 - \left(1 - q(+T \mid +a)\right)^{n^{(\text{S\S})}(x)}\tag{84}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 25

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

erf−1(*T*) − erf−1(*a*)

erf−1(*T*) + erf−1(*a*0)

Λ10(*T*|x) = Λ01(*T*|x) (88)

(x˜), since the verifier "punctures" *n* − *n*(Sg)(x˜)

(*<sup>a</sup>* − *<sup>a</sup>*0)*i*<sup>0</sup> (<sup>1</sup> − *<sup>a</sup>* − *<sup>a</sup>*0)*n*−*i*−*i*<sup>0</sup> (89)

= ( *w*, *w*<sup>0</sup> − *w*) (91)

(1 − *a*)*n*, (*a* − *a*0)*n*

(86)

http://dx.doi.org/10.5772/51800

(87)

,

329

(90)

The products on *t* at the right-hand sides are written because the observation channel is memoryless and the inequalities follow from the assumption (50). Notice that, by (55),

Suppose that there is a blind attacker, who does not have access to the database and wants to pass through the verification stage with the acceptance decision. We also assume that the attacker can fix a small *δ* > 0 in such a way that 1 − *δ* > *a*. Let the attacker submit a vector r′

which is mapped to the vector y ∈ {−1 + *δ*, +1 − *δ*}*n*, i.e., the *t*-th component of the vector y is either equal to −1 + *δ* or +1 − *δ*. Let the decision, whether the *t*-th component of the vector r′ is equal to *r*(−1 + *δ*) or *r*(+1 − *δ*), be made with probabilities 1/2. The probability

components *x*˜*<sup>t</sup>* equal to 0 and sets the *t*-th component of the binary vector equal to 1 if and only if sgn(*x*˜*t*) = sgn(*yt*). By (77), the obtained probability is exactly the same as the probability of success of an attacker, who knows the vector ˜x. Thus, we attain the property of a perfect algorithmic secrecy of the designed verification scheme: although the database is open and an attacker may know the vector ˜x, he cannot include this information into the

Suppose that the vector x ∈ F*<sup>n</sup>* is generated by a memoryless source according to a uniform probability distribution over the set F. One can see that the probability that there are *i*, *i*0, *n* − *i* − *i*<sup>0</sup> components of the vector x whose magnitudes belong to the (*a*, 1),(*a*0, *a*),(0, *a*)

(1 − *a*)*<sup>i</sup>*

*Qa*,*a*<sup>0</sup> (*i*, *<sup>i</sup>*0) =

Hence, the typical distribution of magnitudes under considerations (see Figure 13) is

(1 − *a*)*n*, (*a* − *a*0)*n*

*<sup>ϕ</sup>*(+*T*<sup>|</sup> <sup>+</sup> *<sup>a</sup>*) = <sup>1</sup>

*<sup>ϕ</sup>*(−*T*<sup>|</sup> <sup>+</sup> *<sup>a</sup>*0) = <sup>1</sup>

and assign *T* in such a way that

guessing strategy.

and

specified as

intervals, respectively, is equal to

(*a*, *<sup>a</sup>*0) = *<sup>n</sup>* <sup>−</sup> *<sup>w</sup>*

*Qa*,*a*<sup>0</sup> (*i*, *<sup>i</sup>*0) = *<sup>n</sup>*!

arg max (*i*,*i*0)∈{0,...,*n*}, *<sup>i</sup>*+*i*0≤*<sup>n</sup>*

*<sup>n</sup>* ,

*n* − *w*<sup>0</sup> *n*

 ⇒ 

where *w*, *w*<sup>0</sup> ∈ {0, . . . , *n*} are integers, fixed in such a way that *w* ≤ *w*0.

*i*!*i*0!(*n* − *i* − *i*0)!

of the acceptance decision is equal to 2−*n*(Sg)

2 + 1 2 erf *<sup>ρ</sup> σ* 

<sup>2</sup> <sup>−</sup> <sup>1</sup> 2 erf *<sup>ρ</sup> σ* 

**Figure 12.** Illustration of the events, when the decoding error occurs and *xt* ∈ Sg<sup>+</sup> ∪ Bf+.

and

$$\Lambda\_{\mathrm{01}}(T|\mathbf{x}) = \Pr\_{\mathrm{noise}}\left\{Y\_{\mathbf{f}} < -T, \text{ for some } t \in \mathcal{T}^{(\mathbf{Bf})}(\mathbf{x}) \mid X^{n} = \mathbf{x}\right\}$$

$$= 1 - \Pr\_{\mathrm{noise}}\left\{Y\_{\mathbf{f}} \ge -T, \text{ for all } t \in \mathcal{T}^{(\mathbf{Bf})}(\mathbf{x}) \mid X^{n} = \mathbf{x}\right\}$$

$$= 1 - \prod\_{t \in \mathcal{T}^{(\mathbf{Bf})}(\mathbf{x})} \Pr\_{\mathrm{noise}}\left\{Y\_{\mathbf{f}} \ge -T \mid X\_{\mathbf{f}} = \mathbf{x}\_{\mathbf{f}}\right\}$$

$$\le 1 - \prod\_{t \in \mathcal{T}^{(\mathbf{Bf})}(\mathbf{x})} \Pr\_{\mathrm{noise}}\left\{Y\_{\mathbf{f}} \ge -T \mid X\_{\mathbf{f}} = +a\_{0}\right\}$$

$$= 1 - \left(1 - q(-T \mid +a\_{0})\right)^{\mathrm{nF}(\mathbf{Bf})}\tag{85}$$

The products on *t* at the right-hand sides are written because the observation channel is memoryless and the inequalities follow from the assumption (50). Notice that, by (55),

$$\log(+T|+a) = \frac{1}{2} + \frac{1}{2}\text{erf}\left(\frac{\rho}{\sigma}\left(\text{erf}^{-1}(T) - \text{erf}^{-1}(a)\right)\right) \tag{86}$$

$$\varphi(-T|+a\_0) = \frac{1}{2} - \frac{1}{2} \text{erf}\left(\frac{\rho}{\sigma}\left(\text{erf}^{-1}(T) + \text{erf}^{-1}(a\_0)\right)\right) \tag{87}$$

and assign *T* in such a way that

24 New Trends and Developments in Biometrics

328 New Trends and Developments in Biometrics

+1

Significant<sup>+</sup> zone

+*T*

−*T*

*st* = 1 *s*ˆ*<sup>t</sup>* = 0

*xt* = *x*˜*<sup>t</sup>*

*yt*

❄

∗

noise{*Yt* <sup>&</sup>lt; <sup>−</sup>*T*, for some *<sup>t</sup>* ∈ T (Bf)(x)<sup>|</sup> *<sup>X</sup><sup>n</sup>* <sup>=</sup> <sup>x</sup>}

noise{*Yt* ≥ −*T*, for all *<sup>t</sup>* ∈ T (Bf)(x)<sup>|</sup> *<sup>X</sup><sup>n</sup>* <sup>=</sup> <sup>x</sup>}

noise{*Yt* ≥ −*<sup>T</sup>* <sup>|</sup> *Xt* <sup>=</sup> *xt*}

noise{*Yt* ≥ −*<sup>T</sup>* <sup>|</sup> *Xt* = +*a*0}

*<sup>n</sup>*(Bf)(x)

(85)

*st* = 0 *s*ˆ*<sup>t</sup>* = 1

*xt*

*x*˜*t*

*yt*

❄

∗

Buffer<sup>+</sup> zone

Zero zone

Buffer<sup>−</sup> zone

Significant<sup>−</sup> zone

**Figure 12.** Illustration of the events, when the decoding error occurs and *xt* ∈ Sg<sup>+</sup> ∪ Bf+.

+*a*

+*a*<sup>0</sup>

−*a*<sup>0</sup>

−*a*

−1

Λ01(*T*|x) = Pr

= 1 − Pr

= <sup>1</sup> − ∏

≤ <sup>1</sup> − ∏

= 1 − 

*<sup>t</sup>*∈T (Bf)(x)

*<sup>t</sup>*∈T (Bf)(x)

Pr

Pr

1 − *ϕ*(−*T* | + *a*0)

and

$$
\Lambda\_{10}(T|\mathbf{z}) = \Lambda\_{01}(T|\mathbf{z})\tag{88}
$$

Suppose that there is a blind attacker, who does not have access to the database and wants to pass through the verification stage with the acceptance decision. We also assume that the attacker can fix a small *δ* > 0 in such a way that 1 − *δ* > *a*. Let the attacker submit a vector r′ , which is mapped to the vector y ∈ {−1 + *δ*, +1 − *δ*}*n*, i.e., the *t*-th component of the vector y is either equal to −1 + *δ* or +1 − *δ*. Let the decision, whether the *t*-th component of the vector r′ is equal to *r*(−1 + *δ*) or *r*(+1 − *δ*), be made with probabilities 1/2. The probability of the acceptance decision is equal to 2−*n*(Sg) (x˜), since the verifier "punctures" *n* − *n*(Sg)(x˜) components *x*˜*<sup>t</sup>* equal to 0 and sets the *t*-th component of the binary vector equal to 1 if and only if sgn(*x*˜*t*) = sgn(*yt*). By (77), the obtained probability is exactly the same as the probability of success of an attacker, who knows the vector ˜x. Thus, we attain the property of a perfect algorithmic secrecy of the designed verification scheme: although the database is open and an attacker may know the vector ˜x, he cannot include this information into the guessing strategy.

Suppose that the vector x ∈ F*<sup>n</sup>* is generated by a memoryless source according to a uniform probability distribution over the set F. One can see that the probability that there are *i*, *i*0, *n* − *i* − *i*<sup>0</sup> components of the vector x whose magnitudes belong to the (*a*, 1),(*a*0, *a*),(0, *a*) intervals, respectively, is equal to

$$Q\_{a,a\_0}(i, i\_0) = \frac{n!}{i!i\_0!(n-i-i\_0)!}(1-a)^i(a-a\_0)^{i\_0}(1-a-a\_0)^{n-i-i\_0} \tag{89}$$

and

$$\arg\max\_{(i,i\_0)\in\{0,\ldots,n\},\ i+i\_0\le n} Q\_{a,a\_0}(i,i\_0) = \left( (1-a)n, (a-a\_0)n \right) \tag{90}$$

Hence, the typical distribution of magnitudes under considerations (see Figure 13) is specified as

$$(a, a\_0) = \left(\frac{n - w}{n}, \frac{n - w\_0}{n}\right) \implies \left((1 - a)n, (a - a\_0)n\right) = (w, w\_0 - w) \tag{91}$$

where *w*, *w*<sup>0</sup> ∈ {0, . . . , *n*} are integers, fixed in such a way that *w* ≤ *w*0.

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 27

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters

*n aa*<sup>0</sup> *T* 1/2 1/3 1/4 1/5 1/6 128 .750 .500 .188 > 1 1.9 · 10−<sup>1</sup> 8.4 · 10−<sup>3</sup> 1.6 · 10−<sup>4</sup> 1.4 · 10−<sup>6</sup> 256 .875 .750 .152 2.2 · 10−<sup>1</sup> 1.8 · 10−<sup>3</sup> 2.5 · 10−<sup>6</sup> 6.2 · 10−<sup>10</sup> 2.8 · 10−<sup>14</sup> 512 .938 .875 .131 2.2 · 10−<sup>2</sup> 1.1 · 10−<sup>5</sup> 3.5 · 10−<sup>10</sup> < 10−<sup>15</sup> < 10−<sup>15</sup> 1024 .969 .938 .116 1.9 · 10−<sup>3</sup> 5.4 · 10−<sup>8</sup> 2.8 · 10−<sup>14</sup> < 10−<sup>15</sup> < 10−<sup>15</sup>

**Table 5.** Values of the upper bound on the decoding error probability, when (*w*, *w*0)=(32, 64), (*a*, *a*0) are expressed by the

 <sup>−</sup>*a*<sup>0</sup> −*a*

We believe that there is a request for general theory of processing biometric data, caused by a large variety of parameters that can be taken into account and their different descriptions. It is usually difficult to find a probabilistic models for biometric observations received both at the enrollment and the verification stages that agree with practical situations. One of the most important features is privacy protection, which makes the reconstruction of outcomes of biometric measurements on the basis of the corresponding record record, stored in the database difficult. Another part of privacy protection should make the generation of artificial outcomes of the measurements that allow an attacker to pass through the verification with the acceptance decision difficult. The algorithms presented above can be considered as candidates for the inclusion into such a theory. We introduce the *F*-transformation of input data, where *F* specifies some probability distribution. As a result, the data are mapped into the (−1, +1) interval and, if the actual probability distribution is memoryless and it coincides with the multiplicative extension of *F*, then we attain a uniform probability distribution over the (−1, +1) interval. Otherwise, a uniform distribution can be approached using a generalized version of the *F*-transformation. The use of the proposed technique for noisy observations at the verification stage leads to an interesting effect that rare outcomes over the artificial ensemble, defined by the probability distribution *F* and the corresponding probability density function *P*, are reliably transmitted over any additive observation channel, since *P*(*r*(*y*)) appears in the denominator of the constructed probability density function (see 48). This property allows us to transmit large magnitudes over the constructed *X* → *Y* channel with high reliability. Notice that this claim is not affected by the match of the actual probability distribution of input data and the introduced probability

The points above can be translated to different verification strategies. Our verification algorithm can be viewed as a secret sharing scheme where the input vector r is converted to a pair of vectors (x˜, s). The vector ˜x is published, while the vector s is supposed to be

*<sup>v</sup>*(*yt*|*x*) *dx*, <sup>1</sup>

*a* − *a*<sup>0</sup>

 <sup>+</sup>*<sup>a</sup>* +*a*<sup>0</sup>

*<sup>v</sup>*(*yt*|*x*) *dx*

http://dx.doi.org/10.5772/51800

(94)

331

*a* − *a*<sup>0</sup>

left-hand side of (91), and (*F*, Φ*r*)=(*F*G0,1, *F*G*r*,*σ*).

( *va*,*a*<sup>0</sup> (*yt*|1), *va*,*a*<sup>0</sup> (*yt*|0))= <sup>1</sup>

where

**6. Conclusion**

distribution *F*.

*σ*

**Figure 13.** The typical partitioning of the values of magnitudes in three zones.

Let (*w*, *w*0)=(32, 64). Compute (*a*, *a*0) using the expressions at the left-hand side of (91), and consider the typical case when the distribution of magnitudes of components, belonging to the Significant and the Buffer zones, is given by the right-hand side of (91), i.e., there are 32 magnitudes belonging to the Significant zones, 64 − 32 = 32 magnitudes belonging to the Buffer zones, and *n* − 64 magnitudes belonging to the Zero zone. Thus, the vectors x under consideration are such that *n*(Sg)(x) = *n*(Bf)(x) = 32. Then (88) is equivalent to the equality *ϕ*(+*T*| + *a*) = *ϕ*(−*T*| + *a*0). One can see that the equality above implies the assignment

$$\begin{aligned} \left( \mathbf{F}, \Phi\_r \right) &= \left( \mathbf{F} \mathbf{G}\_{0, \rho \prime} \mathbf{F} \mathbf{G}\_{r, r} \right) \\\ n^{\langle \mathbf{S} \mathbf{g} \rangle} \left( \mathbf{z} \right) &= n^{\langle \mathbf{B} \mathbf{f} \rangle} \left( \mathbf{z} \right) \end{aligned} \Rightarrow \begin{aligned} \mathbf{T} &= \operatorname{erf} \left( \frac{\mathbf{erf}^{-1} (a) - \mathbf{erf}^{-1} (a\_0)}{2} \right) \end{aligned} \tag{92}$$

independently of *ρ* and *σ*. The corresponding vectors ˜x contain 32 + 32 = 64 non–zero components, and the secrecy of the verification scheme, evaluated by the probability of correct guess of the vector s on the basis of the vector ˜x (see (77)), is equal to 2<sup>−</sup>64. Notice that if the attacker would know that *n*(Sg)(x) = *n*(Bf)(x) = 32, then this probability is equal to ( 64 32) −1 . Some numerical results are included in Table 5, and one can see that the decoding error probability is very small even for noisy channels and relatively small lengths.

The presented version of the verification algorithm uses only the signs of components of the vector *x*˜, and this vector can be transformed further to a ternary vector whose contain either 0 or the sign. An improvement of the performance is attained by using the maximum likelihood decoding rule, when we construct the vector ˆs in such a way that

$$\mathfrak{s}\_{t} = \begin{cases} 1, \text{ if } \mathfrak{x}\_{t} \neq 0, \ v(y\_{t}|\mathfrak{x}\_{t}) \ge v\_{a, a\_{0}}(y\_{t}|\text{sgn}(\mathfrak{x}\_{t})) \\ 0, \text{ if } \mathfrak{x}\_{t} = 0 \text{ or } \mathfrak{x}\_{t} \ne 0, \ v(y\_{t}|\mathfrak{x}\_{t}) < v\_{a, a\_{0}}(y\_{t}|\text{sgn}(\mathfrak{x}\_{t})) \end{cases} \tag{93}$$

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 27 Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters http://dx.doi.org/10.5772/51800 331


**Table 5.** Values of the upper bound on the decoding error probability, when (*w*, *w*0)=(32, 64), (*a*, *a*0) are expressed by the left-hand side of (91), and (*F*, Φ*r*)=(*F*G0,1, *F*G*r*,*σ*).

where

26 New Trends and Developments in Biometrics

330 New Trends and Developments in Biometrics

1

*w* components (Significant zone)

*w*<sup>0</sup> − *w* components (Buffer zone)

*n* − *w*<sup>0</sup> components (Zero zone)

�erf−1(*a*) <sup>−</sup> erf−1(*a*0) 2

�

(92)

(93)

0

Let (*w*, *w*0)=(32, 64). Compute (*a*, *a*0) using the expressions at the left-hand side of (91), and consider the typical case when the distribution of magnitudes of components, belonging to the Significant and the Buffer zones, is given by the right-hand side of (91), i.e., there are 32 magnitudes belonging to the Significant zones, 64 − 32 = 32 magnitudes belonging to the Buffer zones, and *n* − 64 magnitudes belonging to the Zero zone. Thus, the vectors x under consideration are such that *n*(Sg)(x) = *n*(Bf)(x) = 32. Then (88) is equivalent to the equality *ϕ*(+*T*| + *a*) = *ϕ*(−*T*| + *a*0). One can see that the equality above implies the assignment

⇒ *T* = erf

. Some numerical results are included in Table 5, and one can see that the decoding

independently of *ρ* and *σ*. The corresponding vectors ˜x contain 32 + 32 = 64 non–zero components, and the secrecy of the verification scheme, evaluated by the probability of correct guess of the vector s on the basis of the vector ˜x (see (77)), is equal to 2<sup>−</sup>64. Notice that if the attacker would know that *n*(Sg)(x) = *n*(Bf)(x) = 32, then this probability is equal

The presented version of the verification algorithm uses only the signs of components of the vector *x*˜, and this vector can be transformed further to a ternary vector whose contain either 0 or the sign. An improvement of the performance is attained by using the maximum

1, if *x*˜*<sup>t</sup>* �= 0, *v*(*yt*|*x*˜*t*) ≥ *va*,*a*<sup>0</sup> (*yt*|sgn(*x*˜*t*))

0, if *x*˜*<sup>t</sup>* = 0 or *x*˜*<sup>t</sup>* �= 0, *v*(*yt*|*x*˜*t*) < *va*,*a*<sup>0</sup> (*yt*|sgn(*x*˜*t*))

 

error probability is very small even for noisy channels and relatively small lengths.

likelihood decoding rule, when we construct the vector ˆs in such a way that

**Figure 13.** The typical partitioning of the values of magnitudes in three zones.

(*F*, <sup>Φ</sup>*r*)=(*F*G0,*ρ*, *<sup>F</sup>*G*r*,*σ*) *n*(Sg)(x) = *n*(Bf)(x)

*s*ˆ*<sup>t</sup>* =

 

to ( 64 32) −1 *a* = (*n* − *w*)/*n*

*a*<sup>0</sup> = (*n* − *w*0)/*n*

$$\left( (v\_{a,a\_0}(y\_t|\mathbf{1}), v\_{a,a\_0}(y\_t|0)) \right) = \left( \frac{1}{a-a\_0} \int\_{-a}^{-a\_0} v(y\_t|\mathbf{x}) \, d\mathbf{x}, \frac{1}{a-a\_0} \int\_{+a\_0}^{+a} v(y\_t|\mathbf{x}) \, d\mathbf{x} \right) \tag{94}$$

## **6. Conclusion**

We believe that there is a request for general theory of processing biometric data, caused by a large variety of parameters that can be taken into account and their different descriptions. It is usually difficult to find a probabilistic models for biometric observations received both at the enrollment and the verification stages that agree with practical situations. One of the most important features is privacy protection, which makes the reconstruction of outcomes of biometric measurements on the basis of the corresponding record record, stored in the database difficult. Another part of privacy protection should make the generation of artificial outcomes of the measurements that allow an attacker to pass through the verification with the acceptance decision difficult. The algorithms presented above can be considered as candidates for the inclusion into such a theory. We introduce the *F*-transformation of input data, where *F* specifies some probability distribution. As a result, the data are mapped into the (−1, +1) interval and, if the actual probability distribution is memoryless and it coincides with the multiplicative extension of *F*, then we attain a uniform probability distribution over the (−1, +1) interval. Otherwise, a uniform distribution can be approached using a generalized version of the *F*-transformation. The use of the proposed technique for noisy observations at the verification stage leads to an interesting effect that rare outcomes over the artificial ensemble, defined by the probability distribution *F* and the corresponding probability density function *P*, are reliably transmitted over any additive observation channel, since *P*(*r*(*y*)) appears in the denominator of the constructed probability density function (see 48). This property allows us to transmit large magnitudes over the constructed *X* → *Y* channel with high reliability. Notice that this claim is not affected by the match of the actual probability distribution of input data and the introduced probability distribution *F*.

The points above can be translated to different verification strategies. Our verification algorithm can be viewed as a secret sharing scheme where the input vector r is converted to a pair of vectors (x˜, s). The vector ˜x is published, while the vector s is supposed to be

decoded on the basis of the vector ˜x and a noisy version of the vector r. An important ingredient of the presented algorithms is the dependence of the threshold *T* on the pair (*a*, *a*0). This dependence assumes that the verifier assigns this pair depending on the vector x received at the enrollment stage. Some other verification schemes are described in [20].

Algorithms for Processing Biometric Data Oriented to Privacy Protection and Preservation of Significant Parameters 29

http://dx.doi.org/10.5772/51800

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## **Author details**

Vladimir B. Balakirsky<sup>1</sup> and A. J. Han Vinck<sup>2</sup>

1 Data Security Association "Confident", St-Petersburg, Russia American University of Armenia, Yerevan, Armenia 2 Institute for Experimental Mathematics, University of Duisburg-Essen, Essen, Germany

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Vladimir B. Balakirsky<sup>1</sup> and A. J. Han Vinck2

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2 Institute for Experimental Mathematics, University of Duisburg-Essen, Essen, Germany

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*West, Electronic Imaging, Media Forensics and Security XIII*. San Fransisco, U.S.A.


## *Edited by Jucheng Yang, Shan Juan Xie*

In recent years, biometrics has developed rapidly with its worldwide applications for daily life. New trends and novel developments have been proposed to acquire and process many different biometric traits. The ignored challenges in the past and potential problems need to be thought together and deeply integrated. The key objective of the book is to keep up with the new technologies on some recent theoretical development as well as new trends of applications in biometrics. The topics covered in this book reflect well both aspects of development. They include the new development in forensic speaker recognition, 3D and thermo face recognition, finger vein recognition, contact-less biometric system, hand geometry recognition, biometric performance evaluation, multi-biometric template protection, and novel subfields in the new challenge fields. The book consists of 13 chapters. It is divided into four sections, namely, theory and method, performance evaluation, security and template protection, and other applications. The book was reviewed by editors Dr. Jucheng Yang and Dr. Shanjuan Xie. We deeply appreciate the efforts of our guest editors: Dr. Norman Poh, Dr. Loris Nanni, Dr. Dongsun Park, Dr. Sook Yoon and Ms. Congcong Xiong, as well as a number of anonymous reviewers.

New Trends and Developments in Biometrics

New Trends and

Developments in Biometrics

*Edited by Jucheng Yang, Shan Juan Xie*

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