**1. Introduction**

Optical pattern recognition for validation and security verifications has been one of the important issues of optical image processing for the past decade. As one technique, a method of joint transform correlation (JTC) has frequently been used for optical security systems for identification of biometric images. In that method, an image such as a fingerprint pattern is encrypted by a random key mask, and the joint Fourier transform of the image to be encrypted and the key pattern conforms an encrypted hologram. The hologram is printed on a card or a document for authenticity, such as a credit card or a passport. The hologram is read when and where necessary and decoded by using the same key that is used for the encryption. In practical optical security systems, a digital technique is used for the image encryption, since the time required to calculate an encryption pattern is not a critical factor and the encryption of an image and printing of the encrypted hologram on a card may be done offline. On the other hand, fast processing is required to decode an encrypted image and verify it. Accordingly, the optical technique is very suitable for such processing.

 In this chapter, we discuss optical security systems suitable for the use of holograms. For the congeniality of the method with real optical systems, binary holograms are frequently used in those systems. By the binarization of hologram, the reconstructed hologram is greatly degraded and, therefore, the optimization of hologram is required for the identification of a reference image. We study the method of the optimization of binary holograms based on a simulated annealing technique. The method of the simulated annealing usually takes a long time to reach a correct estimate, so that the fast optimization for binary hologram is applied. We also demonstrate an image decryption by the optimization of a binary hologram when both the hologram and the key for decryption are embedded in real electronic displays with periodic lattice structures. Finally, we discuss a technique to obtain an exact decryption image based on a phase-encoding technique, which enables easier realization for the practical applications in optical security systems. Even in this technique, the optimization of phase-encoded hologram plays an important role to obtain a good image-reconstruction.

### **2. Optical security systems**

In this section, we discuss a fundamental optical security system treated throughout this chapter. The optical security system under consideration is shown in Fig. 1 (Yamazaki &

Optimization of Hologram for Security Applications 319

established (Kobayashi & Toyoda,1999). So we do not treat them in detail in this chapter. Therefore, in the following, we will discuss how to make an encryption hologram suitable

The method of image encryption for optical security system, which is considered here, is a common one already proposed (Javidi & Horner, 1994, Refregier & Javidi, 1995, and Javidi, 1997). In the image encryption process, an image with a random phase mask is jointly Fourier transformed with another random pattern, which is a key for encryption and decryption, as shown in Fig. 2 and a hologram is formed at the Fourier plane (Yang & Kim, 1996, Javidi et al., 1996, and Unnikrishnan et al., 1998). The phase random mask put in front of the image to be embedded plays a role for scrambling the image, however it little affects the reconstruction of hologram, since the reconstruction is only the intensity of them. The

π

where *F*(*u*,*v*) and *G*(*u*,*v*) are the Fourier transformed functions of the image with a random phase, *f*(*x*,*y*), and the random pattern *g*(*x*,*y*) used as an encryption key, respectively, 2*d* is the separation between the centers of two functions, \* denotes the complex conjugate, and *H*(*u*,*v*) is the resultant hologram. We, here, only consider the AC components of the

where *f*0(*x*,*y*) is the original image function and *b*(*x*,*y*) is also a random function but different from *g*(*x*,*y*). To make an encrypted image, we assume a digital synthesis of the hologram. Fig. 3 is an example of sets of patterns used in the numerical simulations. Fig. 3(a) shows a fingerprint image with 64×64 pixels and Fig. 3(b) is a random pattern used for the encryption key. We here assume 8-bit gray scale for the fingerprint image and (1,-1) binary random pattern (i.e., equivalently (0,π) phase) as the encryption key. As a random phase mask multiplied to the image in the input plane is also assumed to be (0,π) random phase

To reconstruct the image from the encrypted pattern, the hologram is illuminated by the same random phase pattern used for the encryption as shown in Fig. 4. For the decryption

( , ) (, ) (,) (, 3)

where ⊗ denotes the convolution operation. The first term in Eq.(3) is the reconstructed image since the convolution between *g*(-*x*,-*y*) and g(*x*,*y*) reduces to a delta function due to the random nature of the function. On the other hand, the second term is the convolution between the image and the random function and it is a noise term in the reconstruction. Two terms can be spatially separated with each other. Thus, the encrypted image is successfully

*f x y gxy gxy xy d*

(,) (, ) ( , ) (,)

*pxy f xy d g x y gxy*

= − ⊗ −− ⊗

*dv*) + *F*\* (*u*,*v*)*G*(*u*,*v*)exp(*i*4

<sup>0</sup> *f x y f x y ib x y* ( , ) ( , )exp{ ( , )} <sup>=</sup> (2)

+ − − ⊗ ⊗ ⊗δ + (3)

π

*dv*) (1)

for optical decryption and how to optimize it.

holographic fringe terms are given by

different from the random key pattern.

decrypted in the image plane without noise terms.

corresponding to Eq.(1), we obtain

**3. Encryption and decryption of hologram** 

**3.1 Theory of image encryption and decryption of hologram** 

*H*(*u*,*v*) = *F*(*u*,*v*)*G*\* (*u*,*v*)exp(−*i*4

hologram. The image with a random phase is written by

Ohtsubo, 2001). The system consists of three parts. Fig. 1(a) is an encryption system of a target image. An image, for example a finger print image, is encrypted with an encoding key with holographic technique and the encoded image is binarized according to a certain rule to make it easier for reading the image and to match the post processing system. The encrypted binary image printed on a security card is optically decrypted with the decoding key in Fig. 1(b). The decoded image is compared with a test image for identification with optical joint transform correlation as shown in Fig. 1(c). The advantage of the optical method in a security system is the fast processing for decoding an encrypted image and identifying it. In practical optical security systems, a digital technique may be used for the image encryption, since the time required to calculate an encryption pattern is not essential and the encryption of an image and printing the encrypted hologram on a card may be done by offline. On the other hand, the fast processing is required for decoding an encrypted image and verifying it. Accordingly, optical technique is very suited for such processing.

Fig. 1. Schematic example of optical security system. (a) Image encryption, (b) Image decryption, and (c) identification systems.

The encrypted image is binarized to print it, for example on a credit card, and it is read when and where necessary. Binarization of hologram may be performed according to the sign of each pixel value of the encrypted hologram. By the binarization of hologram, we can avoid the load for pre-processing of an encrypted hologram for the decryption and to make a robust optical security system. Due to the binarization of hologram, the encrypted image may be degraded, so that the optimization of the encrypted binary hologram is essential. In the decryption process, the reconstruction of the encrypted image should be quickly processed. Therefore, we assume an optical image processing in the decryption system. Also, the image identification with a test image should be quickly done. We also assume an optical system for image identification. Image identification based on optical methods is an attractive issue, however the methods, especially the methods of optical correlation, are well

Ohtsubo, 2001). The system consists of three parts. Fig. 1(a) is an encryption system of a target image. An image, for example a finger print image, is encrypted with an encoding key with holographic technique and the encoded image is binarized according to a certain rule to make it easier for reading the image and to match the post processing system. The encrypted binary image printed on a security card is optically decrypted with the decoding key in Fig. 1(b). The decoded image is compared with a test image for identification with optical joint transform correlation as shown in Fig. 1(c). The advantage of the optical method in a security system is the fast processing for decoding an encrypted image and identifying it. In practical optical security systems, a digital technique may be used for the image encryption, since the time required to calculate an encryption pattern is not essential and the encryption of an image and printing the encrypted hologram on a card may be done by offline. On the other hand, the fast processing is required for decoding an encrypted image

and verifying it. Accordingly, optical technique is very suited for such processing.

Encoded Image

Image Decryption

Fig. 1. Schematic example of optical security system. (a) Image encryption, (b) Image

The encrypted image is binarized to print it, for example on a credit card, and it is read when and where necessary. Binarization of hologram may be performed according to the sign of each pixel value of the encrypted hologram. By the binarization of hologram, we can avoid the load for pre-processing of an encrypted hologram for the decryption and to make a robust optical security system. Due to the binarization of hologram, the encrypted image may be degraded, so that the optimization of the encrypted binary hologram is essential. In the decryption process, the reconstruction of the encrypted image should be quickly processed. Therefore, we assume an optical image processing in the decryption system. Also, the image identification with a test image should be quickly done. We also assume an optical system for image identification. Image identification based on optical methods is an attractive issue, however the methods, especially the methods of optical correlation, are well

Decoding Key

JTC Result

*CREDIT CARD*

Decoded Image

M V

(a) Encryption of Image

(b) Decryption of Image

(c) Ide ntification

*CREDIT CARD*

Fingerprint Image

> M V

decryption, and (c) identification systems.

Encryption

Encoding Key

Encoded

Decoded Image

 Test Image established (Kobayashi & Toyoda,1999). So we do not treat them in detail in this chapter. Therefore, in the following, we will discuss how to make an encryption hologram suitable for optical decryption and how to optimize it.

#### **3. Encryption and decryption of hologram**

#### **3.1 Theory of image encryption and decryption of hologram**

The method of image encryption for optical security system, which is considered here, is a common one already proposed (Javidi & Horner, 1994, Refregier & Javidi, 1995, and Javidi, 1997). In the image encryption process, an image with a random phase mask is jointly Fourier transformed with another random pattern, which is a key for encryption and decryption, as shown in Fig. 2 and a hologram is formed at the Fourier plane (Yang & Kim, 1996, Javidi et al., 1996, and Unnikrishnan et al., 1998). The phase random mask put in front of the image to be embedded plays a role for scrambling the image, however it little affects the reconstruction of hologram, since the reconstruction is only the intensity of them. The holographic fringe terms are given by

$$H(\mu, \upsilon) = F(\mu, \upsilon) \mathcal{G}^{\ \ast}(\mu, \upsilon) \exp(-i4.4\pi t\upsilon) + F^{\ \ast}(\mu, \upsilon) \mathcal{G}(\mu, \upsilon) \exp(i4.4\pi t\upsilon) \tag{1}$$

where *F*(*u*,*v*) and *G*(*u*,*v*) are the Fourier transformed functions of the image with a random phase, *f*(*x*,*y*), and the random pattern *g*(*x*,*y*) used as an encryption key, respectively, 2*d* is the separation between the centers of two functions, \* denotes the complex conjugate, and *H*(*u*,*v*) is the resultant hologram. We, here, only consider the AC components of the hologram. The image with a random phase is written by

$$f(\mathbf{x}, y) = f\_o(\mathbf{x}, y) \exp\{ib(\mathbf{x}, y)\} \tag{2}$$

where *f*0(*x*,*y*) is the original image function and *b*(*x*,*y*) is also a random function but different from *g*(*x*,*y*). To make an encrypted image, we assume a digital synthesis of the hologram. Fig. 3 is an example of sets of patterns used in the numerical simulations. Fig. 3(a) shows a fingerprint image with 64×64 pixels and Fig. 3(b) is a random pattern used for the encryption key. We here assume 8-bit gray scale for the fingerprint image and (1,-1) binary random pattern (i.e., equivalently (0,π) phase) as the encryption key. As a random phase mask multiplied to the image in the input plane is also assumed to be (0,π) random phase different from the random key pattern.

To reconstruct the image from the encrypted pattern, the hologram is illuminated by the same random phase pattern used for the encryption as shown in Fig. 4. For the decryption corresponding to Eq.(1), we obtain

$$\begin{aligned} p(\mathbf{x}, y) &= f(\mathbf{x}, y - d) \otimes g(-\mathbf{x}, -y) \otimes g(\mathbf{x}, y) \\ &+ f(-\mathbf{x}, -y) \otimes g(\mathbf{x}, y) \otimes g(\mathbf{x}, y) \otimes \delta(\mathbf{x}, y + 3d) \end{aligned} \tag{3}$$

where ⊗ denotes the convolution operation. The first term in Eq.(3) is the reconstructed image since the convolution between *g*(-*x*,-*y*) and g(*x*,*y*) reduces to a delta function due to the random nature of the function. On the other hand, the second term is the convolution between the image and the random function and it is a noise term in the reconstruction. Two terms can be spatially separated with each other. Thus, the encrypted image is successfully decrypted in the image plane without noise terms.

Optimization of Hologram for Security Applications 321

In actual applications, such as in credit card identification, a binary hologram is suited for acquiring electronic image and digital-electronic pre-processing. Thus, the use of the binarization of hologram is one of excellent methods to make a robust security system, so that we employed a binary hologram as an encrypted pattern in the following. The calculated hologram is binarized according to the sign of each element of the composite pattern. Fig. 5 is the result of the encrypted binary hologram. In the figure, the hologram that has a (0,1) binary distribution is printed on a card as a black and white pattern. However, for the reconstruction of the binary hologram, the value of each pixel is assigned to +1 ( exp( 0) *i* ) when the pixel hologram has a value of 1, while it is -1 ( exp( ) *i*π ) for 0. A hologram that has (0,π) phase distribution can be easily realized by using a phase modulation spatial light modulator such as a parallel aligned liquid crystal display. The hologram that has 0 and π phase distribution has the advantage in the reconstruction, since the zero-th order diffraction is eliminated in the reconstruction pattern. Fig. 5(a) shows the calculated binary hologram corresponding to the original image with the random key pattern in Fig. 3. Fig. 5(b) is the decrypted pattern. The hologram and the reconstructed image have the size of 256×256 pixels. The lower noisy part in the figure is the second term in Eq.(3). The binarization of hologram is suited for printing it on a credit card in practical use. However the image is not completely reconstructed because of the binarization for the original hologram as shown in Fig. 5(b). As a result, the ability for the identification between the reconstructed and reference images is deteriorated. Therefore, the optimization of the binary hologram is expected to obtain a good reconstructed image. The method is discussed

(a) (b)

For an optical security system considered here such as in a credit card identification system, the image encryption may be performed in off-line. In that case, the hologram to be printed such as on a credit card may not necessarily be made by the optical method. The encryption of an image and the optimization of the hologram to reconstruct a good image can be performed on digital computer. We here discuss the optimization of the encrypted hologram to obtain a

Fig. 5. (a) Binary phase hologram and (b) reconstruction of hologram with random key

**3.2 Reconstruction of binary hologram** 

in the following section.

**4. Optimization of binary hologram** 

**4.1 Procedure for optimization of binary hologram** 

pattern.

Fig. 2. Optical encryption system using joint Fourier transform

Fig. 3. (a) Original input image and (b) random key pattern for encryption.

Fig. 4. Optical decryption system

#### **3.2 Reconstruction of binary hologram**

320 Holograms – Recording Materials and Applications

**Lens**

**(a)** (b)

*v*

**Hologram** *H*(*u*, *v*)

**Input plane Fourier plane Output plane**

*u*

*x*

*y*

**Dummy area**

**Decrypted image**

**Output image** *f* (*x*, *y*)

Fig. 3. (a) Original input image and (b) random key pattern for encryption.

**Input plane Fourier plane**

*f*

*x*

**Focal length**

Fig. 2. Optical encryption system using joint Fourier transform

*f*

*y*

**Image to be encrypted**

**Encryption key**

*x*

*y*

**Decryption key** *g*(*x*, *y*)

Fig. 4. Optical decryption system

*g*(*x*, *y*)

2*d*

*f*(*x, y*)

*u*

*v*

**Hologram** *H*(*u*, *v*)

In actual applications, such as in credit card identification, a binary hologram is suited for acquiring electronic image and digital-electronic pre-processing. Thus, the use of the binarization of hologram is one of excellent methods to make a robust security system, so that we employed a binary hologram as an encrypted pattern in the following. The calculated hologram is binarized according to the sign of each element of the composite pattern. Fig. 5 is the result of the encrypted binary hologram. In the figure, the hologram that has a (0,1) binary distribution is printed on a card as a black and white pattern. However, for the reconstruction of the binary hologram, the value of each pixel is assigned to +1 ( exp( 0) *i* ) when the pixel hologram has a value of 1, while it is -1 ( exp( ) *i*π ) for 0. A hologram that has (0,π) phase distribution can be easily realized by using a phase modulation spatial light modulator such as a parallel aligned liquid crystal display. The hologram that has 0 and π phase distribution has the advantage in the reconstruction, since the zero-th order diffraction is eliminated in the reconstruction pattern. Fig. 5(a) shows the calculated binary hologram corresponding to the original image with the random key pattern in Fig. 3. Fig. 5(b) is the decrypted pattern. The hologram and the reconstructed image have the size of 256×256 pixels. The lower noisy part in the figure is the second term in Eq.(3). The binarization of hologram is suited for printing it on a credit card in practical use. However the image is not completely reconstructed because of the binarization for the original hologram as shown in Fig. 5(b). As a result, the ability for the identification between the reconstructed and reference images is deteriorated. Therefore, the optimization of the binary hologram is expected to obtain a good reconstructed image. The method is discussed in the following section.

Fig. 5. (a) Binary phase hologram and (b) reconstruction of hologram with random key pattern.

#### **4. Optimization of binary hologram**

#### **4.1 Procedure for optimization of binary hologram**

For an optical security system considered here such as in a credit card identification system, the image encryption may be performed in off-line. In that case, the hologram to be printed such as on a credit card may not necessarily be made by the optical method. The encryption of an image and the optimization of the hologram to reconstruct a good image can be performed on digital computer. We here discuss the optimization of the encrypted hologram to obtain a

Optimization of Hologram for Security Applications 323

Initial Hologram

 Perturbation for Each Pixel

Reconstruction of Hologram

Evaluation of Cost Function

Δ*E*<0 ?

Yes

Yes

All Pixels ? Reject

he process is almost the same as a usual simulated annealing method except for the random flipping of the (0,π) phase pattern at the second step. When the cost function becomes small and temperature is sufficiently cool down, the obtained pattern should be a good estimate for the hologram that well reproduces the original image to be reconstructed and thus the

 In the simulations, the optimization of the binary phase hologram for the fingerprint image as shown in Fig. 3(a) is performed. The reference for the reconstruction of the hologram is the random (1,-1) pattern in Fig. 3(b). Three cooling schedules of the temperature are used in the simulations, i.e., *T*=0 (this corresponds to no annealing), *T*=1/exp(*n*), and *T*=0.5/(1+*n*) (*n* being the iteration number). Fig. 7(a) shows the variations of the cost functions for the iteration number during the simulated annealing. When the annealing process exists ( 0 *T* ≠ ), the value of the cost function once increases with increase of the iteration number and reaches its maximum point. Then it decreases for further increase of the iteration number. In usual simulated annealing, the cost function monotonically decreases with increase of the iteration. However, in our case, the annealing process once trapped to a local minimum, since the perturbation to be added to the image is different from that of ordinary

No

exp(-Δ*E*/*T*)<*r* ?

Perturbation

No

Accept Perterbation

Lower the Temperature

Fig. 6. Algorithm for optimizing hologram by simulated annealing method.

No

optimization of the hologram is realized.

**4.2 Results of optimization** 

Yes

good reconstruction of it based on the numerical method. The method employed for the optimization is a simulated annealing like technique (Kirkpatrick et al., 1983, Ohtsubo & Nakajima, 1991, and Bättig et al., 1992). In ordinary sense, a small perturbation is applied to each pixel of a analogue-valued hologram in a simulated annealing method and the reconstructed image is gradually improved through the iterations (Metropolis et al., 1953, and Aarts and Korst, 1990). However, the method proposed here is somewhat different from the strict simulated annealing technique, since a large perturbation for each pixel value of 1 (zerophase) or -1 (pi-phase) flipping is applied to each pixel.

In the simulated annealing in the present method, (0,π) phase of each pixel in the hologram elements is flipped 0 to π or vice versa as a perturbation. Then the cost function is calculated and the perturbation is accepted or not according to the simulated annealing. The cost function defined here is the mean-square error between the original image intensity to be reconstructed and the estimated one and is given by

$$E = \iint \left| \|f\_o(\mathbf{x}, y)\|^2 - \alpha \|f\_s(\mathbf{x}, y)\|^2 \right|^2 dx dy \tag{4}$$

where *f*(*x*,*y*) is the amplitude of the original image to be reconstructed (defined by Eq.(2)), *fn*(*x*,*y*) is the *n*-th estimate, and the scaling factor α is defined by

$$\alpha = \frac{\iint |f\_o(\mathbf{x}, y)|^2 \, dx dy}{\iint |f\_u(\mathbf{x}, y)|^2 \, dx dy} \tag{5}$$

The cost function is evidently zero when the estimate converges to the original image.

The basic flow of the simulated annealing employed here is shown in Fig. 6. According to the diagram, each step in the simulated annealing is described as follows;


$$P = \exp(-\frac{\Delta E}{T})\tag{6}$$

where *T* is the temperature of the annealing. If *P*<*r* (*r* is a random number between 0 and 1), the perturbation is accepted. On the other hand, it is rejected when *P r* ≥


good reconstruction of it based on the numerical method. The method employed for the optimization is a simulated annealing like technique (Kirkpatrick et al., 1983, Ohtsubo & Nakajima, 1991, and Bättig et al., 1992). In ordinary sense, a small perturbation is applied to each pixel of a analogue-valued hologram in a simulated annealing method and the reconstructed image is gradually improved through the iterations (Metropolis et al., 1953, and Aarts and Korst, 1990). However, the method proposed here is somewhat different from the strict simulated annealing technique, since a large perturbation for each pixel value of 1 (zero-

In the simulated annealing in the present method, (0,π) phase of each pixel in the hologram elements is flipped 0 to π or vice versa as a perturbation. Then the cost function is calculated and the perturbation is accepted or not according to the simulated annealing. The cost function defined here is the mean-square error between the original image intensity to be

where *f*(*x*,*y*) is the amplitude of the original image to be reconstructed (defined by Eq.(2)),


**Step 1.** As an initial input for the iteration, a binary phase hologram calculated from Eq.(1) is

hologram is reconstructed and the new cost function *Enew* is calculated. **Step 3.** The difference between the cost functions before and after the perturbation

exp( ) *<sup>E</sup> <sup>P</sup>*

**Step 5.** If the cost function of each iteration has still a large value, the temperature for the

0

The cost function is evidently zero when the estimate converges to the original image. The basic flow of the simulated annealing employed here is shown in Fig. 6. According to

α =

the diagram, each step in the simulated annealing is described as follows;

2

*f x y dxdy f x y dxdy*

2

used. The hologram is reconstructed and the initial cost function *E* (*Eold*) is calculated. The reference for the reconstruction is the random key pattern used for the encryption. The temperature for the annealing is set with a relatively high value. **Step 2.** The perturbation is applied to one of the pixels of the hologram and the other pixels

are remained unchanged. The phase is flipped 0 to π or π to 0. Then the estimated

Δ*E*=*Enew*-*Eold* is calculated. If Δ*E*<0, the new phase is accepted and the cost function is retained as an old cost function for the next perturbation. Otherwise ( 0 Δ ≥ *E* ), the acceptance or rejection is stochastically determined according to the Boltzmann

*T*

where *T* is the temperature of the annealing. If *P*<*r* (*r* is a random number between 0 and 1), the perturbation is accepted. On the other hand, it is rejected when *P r* ≥

annealing is lowered and the next iteration is performed again. If the cost function

<sup>2</sup> 2 2 *E f x y f x y dxdy* <sup>=</sup> | ( , )| | ( , )| <sup>0</sup> −α *<sup>n</sup>* (4)

(5)

<sup>Δ</sup> = − (6)

phase) or -1 (pi-phase) flipping is applied to each pixel.

reconstructed and the estimated one and is given by

distribution

**Step 4.** Step 4: Steps 2 and 3 are repeated for every pixel.

is lowered enough, the iteration is stopped.

*fn*(*x*,*y*) is the *n*-th estimate, and the scaling factor α is defined by

Fig. 6. Algorithm for optimizing hologram by simulated annealing method.

he process is almost the same as a usual simulated annealing method except for the random flipping of the (0,π) phase pattern at the second step. When the cost function becomes small and temperature is sufficiently cool down, the obtained pattern should be a good estimate for the hologram that well reproduces the original image to be reconstructed and thus the optimization of the hologram is realized.

#### **4.2 Results of optimization**

 In the simulations, the optimization of the binary phase hologram for the fingerprint image as shown in Fig. 3(a) is performed. The reference for the reconstruction of the hologram is the random (1,-1) pattern in Fig. 3(b). Three cooling schedules of the temperature are used in the simulations, i.e., *T*=0 (this corresponds to no annealing), *T*=1/exp(*n*), and *T*=0.5/(1+*n*) (*n* being the iteration number). Fig. 7(a) shows the variations of the cost functions for the iteration number during the simulated annealing. When the annealing process exists ( 0 *T* ≠ ), the value of the cost function once increases with increase of the iteration number and reaches its maximum point. Then it decreases for further increase of the iteration number. In usual simulated annealing, the cost function monotonically decreases with increase of the iteration. However, in our case, the annealing process once trapped to a local minimum, since the perturbation to be added to the image is different from that of ordinary

Optimization of Hologram for Security Applications 325

Correlation

Correlation

Correlation

image with optimization.

**5. Fast optimization method 5.1 Algorithm of fast optimization** 

where the coefficient β is defined by

can be given by

introduce the error function *exy* (,) for the decrypted image;

Fig. 8. Reconstructed images (left) and correlations with original image (right). (a) Original fingerprint image, (b) reconstructed image without optimization, and (c) reconstructed

 Though the method based on the simulated annealing algorithm discussed in the previous section is very effective for the optimization of encrypted holograms, it is time consuming. Here, we propose an error correction method by which we can perform very fast optimization of encrypted binary holograms (Nakayama & Ohtsubo, 2007). The simulated annealing technique introduced in the previous section is a like Boltzmann machine system, while the proposed error correction method here is similar to a back propagation technique in neural net works. Before discussing the method, we first define two terms; 'decrypted image' is the part of the decrypted area where the pattern corresponding to the encrypted image is reconstructed as shown in Fig. 4. The 'dummy area' is the rest of the pattern in the output plane. The complex amplitude of the output image reconstructed from the binary encrypted hologram after *n* iterations is given by <sup>0</sup> ( , ) ( , )exp{ ( , )} *nn n f x y f x y ib x y* = . Then, we

Pixel -64 64 0 (a)

(b)

(c)

2 2 | ( , ) ( , )| | ( , )| *<sup>n</sup>* <sup>0</sup> β+ = *f xy exy f xy* (7)

(8)

2

*f xy x y*



2

*f xy x y*

0

The function *exy* (,) defines the difference between the decrypted and original images and

decrypted area

β =

simulated annealing. From the comparison between the two cooling schedules (*T*=1/exp(*n*) and *T*=0.5/(1+*n*)), a rapid cooling rate is rather effective for the optimization of the hologram. The cost function monotonically and rapidly decreases without trapping any local minimum when *T*=0, that is, no annealing process. The point of the simulated annealing method is the moderate perturbation with random fluctuations to escape local minima in the energy function. The reference to construct the hologram in the method is a binary random pattern and the hologram itself is also a random like pattern, so that random fluctuations are automatically given without introducing the stochastic process (Step 3) in the iteration. This may come from a random nature of the reconstruction process in the present method. For either case, the optimized hologram well reproduces a fingerprint image very close to the original one. Fig. 7(b) is the result of the optimized hologram at *T*=0.

Fig. 7. (a) Variations of cost functions for each cooling schedule and (b) optimized binary hologram at *T*=0.

The degree of the reconstruction for the optimized hologram is tested by a joint transform correlation method. Fig. 8 shows the results. Fig. 8(a) is the original fingerprint image (left) and the joint transform correlation between the same patterns (right). The result of the correlation is a one-dimensional scan along the correlation peaks. For the calculation of the correlation function, the power spectrum is filtered by a band-pass filter in the Fourier plane to eliminate the unwanted noise floor. The zero-th order correlation peak is normalized to 255 level and the value of the correlation peak is 63. Fig. 8(b) is the decrypted image from the original binary hologram (not optimized one) and its correlation with the original fingerprint image. The fingerprint image is vague compared with the original one due to the binarization of the hologram and its correlation peak value is only 39. Starting from the hologram corresponding to Fig. 8(b), the simulated annealing is performed along the procedure discussed in the previous section. Fig. 8(c) shows the reconstructed fingerprint image by the optimized hologram for *T*=0 and its correlation with the original image. The image is perfectly recovered (compare the pattern in Fig. 8(a)) and its correlation peak is 63, which is completely the same value with the correlation between the original images. For the iteration cycle of n ≥ 5 the hologram is almost optimized. The optimization is also successful for the cooling schedule of *T*=1/exp(*n*) and the fingerprint image having the correlation value of 63 is obtained for n ≥ 10. The reconstructed image is much improved for the cooling schedule of *T*=0.5/(1+*n*), however, the optimization speed is slower and the value of the correlation is 58 at *n*=20.

simulated annealing. From the comparison between the two cooling schedules (*T*=1/exp(*n*) and *T*=0.5/(1+*n*)), a rapid cooling rate is rather effective for the optimization of the hologram. The cost function monotonically and rapidly decreases without trapping any local minimum when *T*=0, that is, no annealing process. The point of the simulated annealing method is the moderate perturbation with random fluctuations to escape local minima in the energy function. The reference to construct the hologram in the method is a binary random pattern and the hologram itself is also a random like pattern, so that random fluctuations are automatically given without introducing the stochastic process (Step 3) in the iteration. This may come from a random nature of the reconstruction process in the present method. For either case, the optimized hologram well reproduces a fingerprint image very close to the original one. Fig. 7(b) is the result of the optimized hologram at *T*=0.

*T*=0 (a) (b)

5 10 15 20 Ieratation Number *n*

▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

▲ ■

*T*=1/exp(*n*) *T*=0.5(1+*n*)

Fig. 7. (a) Variations of cost functions for each cooling schedule and (b) optimized binary

The degree of the reconstruction for the optimized hologram is tested by a joint transform correlation method. Fig. 8 shows the results. Fig. 8(a) is the original fingerprint image (left) and the joint transform correlation between the same patterns (right). The result of the correlation is a one-dimensional scan along the correlation peaks. For the calculation of the correlation function, the power spectrum is filtered by a band-pass filter in the Fourier plane to eliminate the unwanted noise floor. The zero-th order correlation peak is normalized to 255 level and the value of the correlation peak is 63. Fig. 8(b) is the decrypted image from the original binary hologram (not optimized one) and its correlation with the original fingerprint image. The fingerprint image is vague compared with the original one due to the binarization of the hologram and its correlation peak value is only 39. Starting from the hologram corresponding to Fig. 8(b), the simulated annealing is performed along the procedure discussed in the previous section. Fig. 8(c) shows the reconstructed fingerprint image by the optimized hologram for *T*=0 and its correlation with the original image. The image is perfectly recovered (compare the pattern in Fig. 8(a)) and its correlation peak is 63, which is completely the same value with the correlation between the original images. For the iteration cycle of n ≥ 5 the hologram is almost optimized. The optimization is also successful for the cooling schedule of *T*=1/exp(*n*) and the fingerprint image having the correlation value of 63 is obtained for n ≥ 10. The reconstructed image is much improved for the cooling schedule of *T*=0.5/(1+*n*), however, the optimization speed is slower and the

0

hologram at *T*=0.

0

100

Cost Function

*E*

■

▲

■ ■ ■

value of the correlation is 58 at *n*=20.

▲

▲

▲ ▲

■

200

Fig. 8. Reconstructed images (left) and correlations with original image (right). (a) Original fingerprint image, (b) reconstructed image without optimization, and (c) reconstructed image with optimization.

#### **5. Fast optimization method**

#### **5.1 Algorithm of fast optimization**

 Though the method based on the simulated annealing algorithm discussed in the previous section is very effective for the optimization of encrypted holograms, it is time consuming. Here, we propose an error correction method by which we can perform very fast optimization of encrypted binary holograms (Nakayama & Ohtsubo, 2007). The simulated annealing technique introduced in the previous section is a like Boltzmann machine system, while the proposed error correction method here is similar to a back propagation technique in neural net works. Before discussing the method, we first define two terms; 'decrypted image' is the part of the decrypted area where the pattern corresponding to the encrypted image is reconstructed as shown in Fig. 4. The 'dummy area' is the rest of the pattern in the output plane. The complex amplitude of the output image reconstructed from the binary encrypted hologram after *n* iterations is given by <sup>0</sup> ( , ) ( , )exp{ ( , )} *nn n f x y f x y ib x y* = . Then, we introduce the error function *exy* (,) for the decrypted image;

$$|\beta f\_n(\mathbf{x}, \boldsymbol{y}) + e(\mathbf{x}, \boldsymbol{y})|^2 = |f\_n(\mathbf{x}, \boldsymbol{y})|^2 \tag{7}$$

where the coefficient β is defined by

$$\beta = \sqrt{\frac{\iint |f\_{\circ}(\mathbf{x}, y)|^{2} \, \mathrm{d}x \mathrm{d}y}{\iint\_{\mathrm{d} \mathrm{d} \mathrm{expytad}\,\mathrm{a}u} |\, f\_{\circ}(\mathbf{x}, y)|^{2} \, \mathrm{d}x \mathrm{d}y}} \tag{8}$$

The function *exy* (,) defines the difference between the decrypted and original images and can be given by

Optimization of Hologram for Security Applications 327

theory. This flipping trial is continued until it reaches a certain number. **Step 3.** When the image decrypted from , (,) *H uv n new* is closer to the original image than that

The steps 1~3 are the essentials of the proposed algorithm. The method has many advantages over the existing optimization methods. Firstly, the most effective pixel to be corrected is selected for the optimization. Secondly, not only one pixel but also multiple pixels can be flipped at the same time on the basis of the priority order for the error correction. Therefore, we can expect faster calculation for the optimization of encrypted

In ordinary single flipping for pixel elements in the simulated annealing like method, it usually takes a long time to obtain a good reconstruction. On the other hand, the error correction method has a merit of simultaneous flipping of multi-pixels for the correction of the binary hologram. We could perform a faster optimization for encrypted hologram by the method of the error correction. In the multiple-flip algorithm, many pixels of the encrypted hologram are selected according to the priority order calculated from the error function and the respective pixels are simultaneously flipped. Then, the hologram is reconstructed and the iteration is either accepted or rejected for the evaluation of the cost function. The iteration is stopped either when the cost function is sufficiently lowered or when the process

Fig. 9 shows the results of the cost function for a fixed multiple-flip algorithm. Two multiple-flip schemes are plotted; one is a 7-flip and the other is a 655-flip. The optimization stopped at 1818 iterations for the 7-flip, while it stopped only at 19 iterations for the 655-flip. The number of iterations to reach an optimized hologram is drastically reduced for a large number of simultaneous flipping, however the iteration stopped at a higher value of the cost function and the quality of image is rather poor compared with that for a lower number of the multiple-flip algorithm. The final iteration number for the 655-flip algorithm is 1/96 of that for the 7-flip algorithm. The actual calculation time on the computer is 1/86. Therefore, as a rough estimate, the calculation time is inversely proportional to the number of pixels to be flipped at the same time. The final costs for the 655-flip and the 7-flip are 117.67 and 67.78, respectively, while it is only 9.47 for the single-flip algorithm. Accordingly, the quality of the decrypted image for a higher number of the multiple-flip is worse than that of the lower case. There exists a trade-off between the final quality of the decrypted image and the calculation speed for the optimization. To show the trade-off, the dependence of the final iteration number and the cost function were investigated for various numbers of the multiple-flip algorithm. Fig. 10 is the results for two different fingerprint images. Similar

hologram and define the cost function in the decryption plane as

The optimization is performed so as to lower the above cost function.

trends have been observed for respective fingerprint images.

holograms.

**5.2 Multiple-flip algorithm** 

is trapped to a local minimum.

where , (,) *H uv n new* is a temporal hologram to be tested for the new reconstruction, which is generally called *neighborhood* in the field of the combinatorial optimization

from (,) *H uv <sup>n</sup>* , , (,) *H uv n new* is adopted as a new (,) *H uv <sup>n</sup>* . Then, the above process is repeated. In this stage, we introduce the measure for the optimization of binary

<sup>2</sup> *E ex* <sup>=</sup> | ( , )|d d *<sup>y</sup> <sup>x</sup> <sup>y</sup>* . (13)

$$\begin{aligned} e(\mathbf{x}, y) &= (|f\_o(\mathbf{x}, y)| - \\$) |f\_s(\mathbf{x}, y)| ) \exp\{ib\_s(\mathbf{x}, y)\} & \text{within accepted image} \\ e(\mathbf{x}, y) &= 0 & \text{in dummy area} \end{aligned} \tag{9}$$

In this method, the error information is used not only for evaluating the intensity of the decrypted image but also for selecting the flipping pixels in the hologram plane. For the concern of intensity, the phase of the decrypted image may be ignored as far as the optimization is performed only in the decryption plane. However, the phase (,) *<sup>n</sup> b xy* is included in the error function. The error projected back into the hologram plane, namely generated from *exy* (,) by the inverse Fourier transform operation, is affected by the phase as shown in the following and it plays a crucial role in this algorithm.

 Also, the error function is accounted in the hologram plane. The expression of the error function Δ*H*(*u*,*v*) in the hologram plane is deduced from the following relation;

$$\begin{aligned} \mathsf{\bf{\bf{\mathcal{B}}f}}(\mathbf{x},\boldsymbol{\nu},\boldsymbol{\nu}) + e(\mathbf{x},\boldsymbol{\nu}) &= \mathsf{\bf{\mathcal{H}FT}}[\boldsymbol{H}\_{\boldsymbol{u}}(\boldsymbol{\mu},\boldsymbol{\nu})\mathbf{G}(\boldsymbol{u},\boldsymbol{\nu})] + \mathsf{\bf{\mathcal{F}FT}}[\mathsf{FT}[e(\mathbf{x},\boldsymbol{\nu})]] \\ &= \mathsf{\bf{\mathcal{F}FT}}[\mathsf{\bf{\mathcal{H}}H}\_{\boldsymbol{u}}(\boldsymbol{\mu},\boldsymbol{\nu})\mathbf{G}(\boldsymbol{u},\boldsymbol{\nu}) + \mathsf{FT}[e(\mathbf{x},\boldsymbol{\nu})]] \\ &= \mathsf{\bf{\mathcal{H}FT}}\left[\left\{H\_{\boldsymbol{u}}(\boldsymbol{u},\boldsymbol{\nu}) + \frac{\mathsf{FT}[e(\mathbf{x},\boldsymbol{\nu})]}{\mathsf{\bf{\mathcal{G}}G}(\boldsymbol{u},\boldsymbol{\nu})}\right\}\mathbf{G}(\boldsymbol{u},\boldsymbol{\nu})\right] \\ &= \mathsf{\bf{\mathcal{H}FT}}[\left\{H\_{\boldsymbol{u}}(\boldsymbol{u},\boldsymbol{\nu}) + \Delta H(\boldsymbol{u},\boldsymbol{\nu})\right\}\mathbf{G}(\boldsymbol{u},\boldsymbol{\nu})] \end{aligned} \tag{10}$$

where *Hn*(*u*,*v*) is the hologram after *n*-th iterations and FT and IFT are the forward and inverse Fourier transform operations. Then, the error function in the hologram plane reads

$$
\Delta H(\mu, v) = \frac{\text{FT}[e(x, y)]}{\beta G(\mu, v)} \tag{11}
$$

Equation (11) indicates that Δ*Huv* (,) is like as a gradient vector to the locally or globally optimal solution with respect to (,) *H uv <sup>n</sup>* . However, we cannot directly use the error information Δ*H* for the correction for (,) *H uv <sup>n</sup>* , since the hologram is a binary nature and Δ*Huv* (,) is a continuous complex valued function. Therefore, we require some modifications for the application of the error correction method for the binary encrypted hologram.

In accordance with the above discussion, we adopt the following processes for the optimization of a binary hologram;


$$\begin{aligned} \text{if } H\_\*(\mu, v) &= 1 \text{ and } \operatorname{Re}[\Delta H(\mu, v)] > 0, \text{ then } H\_{\*, \mu w}(\mu, v) = 1\\ \text{if } H\_\*(\mu, v) &= 1 \text{ and } \operatorname{Re}[\Delta H(\mu, v)] < 0, \text{ then } H\_{\*, \mu w}(\mu, v) = -1\\ \text{else } \text{no flip} \end{aligned} \tag{12}$$

<sup>0</sup> ( , ) (| ( , )| | ( , )|)exp{ ( , )} within decrypted image

In this method, the error information is used not only for evaluating the intensity of the decrypted image but also for selecting the flipping pixels in the hologram plane. For the concern of intensity, the phase of the decrypted image may be ignored as far as the optimization is performed only in the decryption plane. However, the phase (,) *<sup>n</sup> b xy* is included in the error function. The error projected back into the hologram plane, namely generated from *exy* (,) by the inverse Fourier transform operation, is affected by the phase

Also, the error function is accounted in the hologram plane. The expression of the error

( , ) ( , ) IFT[ ( , ) ( , )]+IFT[FT[ ( , )]]

*n*

*n*

where *Hn*(*u*,*v*) is the hologram after *n*-th iterations and FT and IFT are the forward and inverse Fourier transform operations. Then, the error function in the hologram plane reads

> FT[ ( , )] (,) (,) *e x <sup>y</sup> Huv*

Equation (11) indicates that Δ*Huv* (,) is like as a gradient vector to the locally or globally optimal solution with respect to (,) *H uv <sup>n</sup>* . However, we cannot directly use the error information Δ*H* for the correction for (,) *H uv <sup>n</sup>* , since the hologram is a binary nature and Δ*Huv* (,) is a continuous complex valued function. Therefore, we require some modifications for the application of the error correction method for the binary encrypted

In accordance with the above discussion, we adopt the following processes for the

**Step 1.** At first, Δ*Huv* (,) is calculated by Eq. (11). Then, each pixel of the hologram is

**Step 2.** On the descending order of the priority for the error correction, each pixel value is

if ( , )=-1 and Re[ ( , )] 0, then ( , ) 1 if ( , )= 1 and Re[ ( , )] 0, then ( , ) 1

*n n new n n new H uv Huv H uv H uv Huv H uv*

ranked by the magnitude of the value |Re[ ( , )]| Δ*Huv* for the optimization, namely, the pixel with the largest value of |Re[ ( , )]| Δ*Huv* has the highest priority for the

> , ,

Δ < = − (12)

Δ> =

*Guv*

= β Δ

*n*

*f x y ex y H uvGuv exy*

IFT[ ( , ) ( , )+FT[ ( , )]]

*H uvGuv exy*

IFT[{ ( , )+ ( , )} ( , )]

FT[ ( , )] IFT ( , ) (,) (,)

= β <sup>+</sup> <sup>β</sup>

*H uv Huv Guv*

*exy H uv Guv Guv*

<sup>β</sup> (11)

(10)

<sup>=</sup> (9)

( , ) 0 in dummy area

*n n e x y f x y f x y ib x y*

as shown in the following and it plays a crucial role in this algorithm.

*n n*

β + =β

function Δ*H*(*u*,*v*) in the hologram plane is deduced from the following relation;

= β

Δ =

= −β

*exy*

hologram.

flip.

optimization of a binary hologram;

flipped according to the following rules;

else o flip

*n*

where , (,) *H uv n new* is a temporal hologram to be tested for the new reconstruction, which is generally called *neighborhood* in the field of the combinatorial optimization theory. This flipping trial is continued until it reaches a certain number.

**Step 3.** When the image decrypted from , (,) *H uv n new* is closer to the original image than that from (,) *H uv <sup>n</sup>* , , (,) *H uv n new* is adopted as a new (,) *H uv <sup>n</sup>* . Then, the above process is repeated. In this stage, we introduce the measure for the optimization of binary hologram and define the cost function in the decryption plane as

$$E = \iint |e(\mathbf{x}, y)|^2 \mathrm{d}x \mathrm{d}y \,\,. \tag{13}$$

The optimization is performed so as to lower the above cost function.

The steps 1~3 are the essentials of the proposed algorithm. The method has many advantages over the existing optimization methods. Firstly, the most effective pixel to be corrected is selected for the optimization. Secondly, not only one pixel but also multiple pixels can be flipped at the same time on the basis of the priority order for the error correction. Therefore, we can expect faster calculation for the optimization of encrypted holograms.

#### **5.2 Multiple-flip algorithm**

In ordinary single flipping for pixel elements in the simulated annealing like method, it usually takes a long time to obtain a good reconstruction. On the other hand, the error correction method has a merit of simultaneous flipping of multi-pixels for the correction of the binary hologram. We could perform a faster optimization for encrypted hologram by the method of the error correction. In the multiple-flip algorithm, many pixels of the encrypted hologram are selected according to the priority order calculated from the error function and the respective pixels are simultaneously flipped. Then, the hologram is reconstructed and the iteration is either accepted or rejected for the evaluation of the cost function. The iteration is stopped either when the cost function is sufficiently lowered or when the process is trapped to a local minimum.

Fig. 9 shows the results of the cost function for a fixed multiple-flip algorithm. Two multiple-flip schemes are plotted; one is a 7-flip and the other is a 655-flip. The optimization stopped at 1818 iterations for the 7-flip, while it stopped only at 19 iterations for the 655-flip. The number of iterations to reach an optimized hologram is drastically reduced for a large number of simultaneous flipping, however the iteration stopped at a higher value of the cost function and the quality of image is rather poor compared with that for a lower number of the multiple-flip algorithm. The final iteration number for the 655-flip algorithm is 1/96 of that for the 7-flip algorithm. The actual calculation time on the computer is 1/86. Therefore, as a rough estimate, the calculation time is inversely proportional to the number of pixels to be flipped at the same time. The final costs for the 655-flip and the 7-flip are 117.67 and 67.78, respectively, while it is only 9.47 for the single-flip algorithm. Accordingly, the quality of the decrypted image for a higher number of the multiple-flip is worse than that of the lower case. There exists a trade-off between the final quality of the decrypted image and the calculation speed for the optimization. To show the trade-off, the dependence of the final iteration number and the cost function were investigated for various numbers of the multiple-flip algorithm. Fig. 10 is the results for two different fingerprint images. Similar trends have been observed for respective fingerprint images.

Optimization of Hologram for Security Applications 329

holograms by using the error correction method with the variable multiple-flip algorithm without loosing the quality of the reconstruction. Finally, the variable multi-flip algorithm was applied to other types of images. Fig. 12 shows the results. It is clear that the proposed algorithm is applicable not only to fingerprint image but also to other images (binary logo of

(a) (b)

655 66 7 4 2

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> Iteration number

Without Optimization With Optimization

(a)

(b)

(c)

(c)

Fig. 11. Results for variable multiple-flip algorithm. Decrypted images from optimized hologram (a) with the encryption key and (b) with a wrong key. (c) Cost function for the

Fig. 12. Decrypted images from optimized holograms for various image structures in

Shizuoka University and stamp of Chinese characters).

University Logo

Stamp

variable multiple-flip algorithm.

Fingerprint

Cost function

iteration number.

Fig. 9. Cost function for flip number in a fixed multiple-flip algorithm

Fig. 10. Dependence of final iteration number and its cost on the number of simultaneous multiple flipping pixels. (a) corresponds to the fingerprint image in Fig. 3(a), and (b) is also for the fingerprint image in Fig. 12(a).

#### **5.3 Variable multiple-flip algorithm**

To avoid the trap of a local minimum and obtain a good decryption, we employ a scheme of a variable multiple-flip algorithm. In this algorithm, the flip number is dynamically changed. We first set the flip number at a certain value and started the iteration. When the iteration stops due to the trapping of a local minimum, the flip number is lowered and the next iteration is repeated. This process is repeated until the cost function is much lowered.

Fig. 11 is the results of the algorithm for the variable multiple-flip number. Fig. 11(a) is the decrypted image for the final optimized hologram with the encryption key and Fig. 11(b) is that with a wrong key. Fig. 11(c) is the change of the cost function for the iteration number. The flip number was at first chosen to be 655 and it was lowered as 66 after the trapping a local minimum. Then, the iteration was continued as lowering the flip number as 7, 4, and finally 2. The boundaries of the changing points are indicated as broken lines. We calculated the joint transform correlation between the obtained pattern and the original fingerprint image. The value of the correlation peaks calculated in Fig. 11(a) is 0.243 (normalized by the initial cost to unity), while that of the simulated annealing method of single flipping is 0.247. Almost the same quality of the reconstruction as the simulated annealing method is achieved by the current technique. We also compared the calculation time both for the proposed and the previous methods. For the optimization of the simulated annealing method, the calculation time at which the value of cost function became 0.243 was evaluated. The calculation times for the proposed and previous methods are 1149 and 39364 s, respectively, i.e., the calculation time of the proposed method is only 2.9 % of that of the simulated annealing method. Thus, we can attain a very fast optimization of encrypted

0 5 10 15 20 Iteration Number for 655-flip algorithm

> 655-flip algorithm 7-flip algorithm

0 500 1000 1500 Iteration Number for 7-flip algorithm

Cost Function

Fig. 10. Dependence of final iteration number and its cost on the number of simultaneous multiple flipping pixels. (a) corresponds to the fingerprint image in Fig. 3(a), and (b) is also

To avoid the trap of a local minimum and obtain a good decryption, we employ a scheme of a variable multiple-flip algorithm. In this algorithm, the flip number is dynamically changed. We first set the flip number at a certain value and started the iteration. When the iteration stops due to the trapping of a local minimum, the flip number is lowered and the next iteration is repeated. This process is repeated until the cost function is much lowered. Fig. 11 is the results of the algorithm for the variable multiple-flip number. Fig. 11(a) is the decrypted image for the final optimized hologram with the encryption key and Fig. 11(b) is that with a wrong key. Fig. 11(c) is the change of the cost function for the iteration number. The flip number was at first chosen to be 655 and it was lowered as 66 after the trapping a local minimum. Then, the iteration was continued as lowering the flip number as 7, 4, and finally 2. The boundaries of the changing points are indicated as broken lines. We calculated the joint transform correlation between the obtained pattern and the original fingerprint image. The value of the correlation peaks calculated in Fig. 11(a) is 0.243 (normalized by the initial cost to unity), while that of the simulated annealing method of single flipping is 0.247. Almost the same quality of the reconstruction as the simulated annealing method is achieved by the current technique. We also compared the calculation time both for the proposed and the previous methods. For the optimization of the simulated annealing method, the calculation time at which the value of cost function became 0.243 was evaluated. The calculation times for the proposed and previous methods are 1149 and 39364 s, respectively, i.e., the calculation time of the proposed method is only 2.9 % of that of the simulated annealing method. Thus, we can attain a very fast optimization of encrypted

1 10 100 1000 10000 Flip Number

Cost Function

Iteration Number

100

1 10 100 1000 10000 Flip Number

for the fingerprint image in Fig. 12(a).

**5.3 Variable multiple-flip algorithm** 

(a)

Iteration Number

Fig. 9. Cost function for flip number in a fixed multiple-flip algorithm

(b) <sup>2000</sup>

holograms by using the error correction method with the variable multiple-flip algorithm without loosing the quality of the reconstruction. Finally, the variable multi-flip algorithm was applied to other types of images. Fig. 12 shows the results. It is clear that the proposed algorithm is applicable not only to fingerprint image but also to other images (binary logo of Shizuoka University and stamp of Chinese characters).

Fig. 11. Results for variable multiple-flip algorithm. Decrypted images from optimized hologram (a) with the encryption key and (b) with a wrong key. (c) Cost function for the iteration number.

Fig. 12. Decrypted images from optimized holograms for various image structures in variable multiple-flip algorithm.

Optimization of Hologram for Security Applications 331

Using the hologram and the random key pattern in Figs. 14(a) and (b), the decryption was performed. The result is shown in Fig. 14(c). We cannot see any information of the original pattern of the fingerprint image. In the optical security system discussed here, the decryption is not a simple reconstruction of hologram, such as illumination by a plane wave. The hologram is illuminated by the Fourier transform of the random key pattern. Therefore the illumination of the Fourier transform of the periodic lattice structure greatly affects the performance of the reconstruction of hologram. It has less redundancy compared with a simple holographic reconstruction done by a plain wave illumination. Without considering the lattice structure, at worst case, we cannot extract any information from the reconstructed

(a) (b) (c) Fig. 14. Image decryption in the presence of opaque lattice structures on the hologram and the decryption key pattern. (a) Hologram, (b) decryption key pattern, and (c) decryption of

The procedure for optimization of hologram in the presence of a lattice structure is almost the same as discussed in section 3. Starting from a hologram as shown in Fig. 14(a) together with a decryption key in Fig. 14(b) all including the lattice structures, the optimization of hologram to obtain good reconstruction is performed following the step 1~5 as discussed in section 3. Throughout the following optimization, the decryption key pattern, which also has a periodic lattice structure, is not changed and is assumed to be the same pattern as shown in Fig. 14(b). On the other hand, starting from the encryption hologram shown in Fig. 14(a), the value of each pixel of the hologram is modified by flipping from +1 to –1 or vice versa. In each flipping, we test the newly decrypted image as to whether it gives rise to a good reconstruction or not. The flipping is successively repeated for every pixel. If the cost function for the optimization still has a large value, the next iteration is performed. When the value of the cost function is sufficiently lowered, the iteration stops. Then, the image is

 In the numerical simulation for the optimization, the area of the image to be compared with the decrypted pattern is expanded to 64x64 pixels due to the presence of the periodic opaque lattice structure. Therefore, we used the fingerprint image with 64x64 pixels as the ideal target image as shown in Fig. 15(a). Fig. 15(b) is the optimized hologram calculated by the proposed method when it contains the lattice structure. Using the optimized hologram together with the random key pattern in Fig. 14(b), we obtain the decrypted pattern as shown in Fig. 15(c). We can successfully decrypt an image close to the original one, though the periodic multiple images are reconstructed. The multiple images are originated due to

**6.3 Optimization of hologram in the presence of lattice structure** 

pattern as shown in Fig. 14(c).

optimized to reach a good estimation.

image.
