**1. Introduction**

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In recent years, the wavelet transform emerged in the field of image/signal processing as an alternative to the well-known Fourier Transform (FT) and its related transforms, namely, the Discrete Cosine Transform (DCT) and the Discrete Sine Transform (DST). In the Fourier theory, a signal (an image is considered as a finite 2-D signal) is expressed as a sum, theoretically infinite, of sines and cosines, making the FT suitable for infinite and periodic signal analysis. For several years, the FT dominated the field of signal processing, however, if it succeeded well in providing the frequency information contained in the analysed signal; it failed to give any information about the occurrence time. This shortcoming, but not the only one, motivated the scientists to scrutinise the transform horizon for a "messiah" transform. The first step in this long research journey was to cut the signal of interest in several parts and then to analyse each part separately. The idea at a first glance seemed to be very promising since it allowed the extraction of time information and the localisation of different frequency components. This approach is known as the Short-Time Fourier Transform (STFT). The fundamental question, which arises here, is how to cut the signal? The best solution to this dilemma was of course to find a fully scalable modulated window in which no signal cutting is needed anymore. This goal was achieved successfully by the use of the wavelet transform.

Formally, the wavelet transform is defined by many authors as a mathematical technique in which a particular signal is analysed (or synthesised) in the time domain by using different versions of a dilated (or contracted) and translated (or shifted) basis function called the wavelet prototype or the mother wavelet. However, in reality, the wavelet transform found its essence and emerged from different disciplines and was not, as stated by Mallat, totally new to mathematicians working in harmonic analysis, or to computer vision researchers studying multiscale image processing (Mallat, 1989).

At the beginning of the 20th century, Haar, a German mathematician introduced the first wavelet transform named after him (almost a century after the introduction of the FT, by the French J. Fourier). The Haar wavelet basis function has compact support and integer coefficients. Later, the Haar basis was used in physics to study Brownian motion (Graps,

The Wavelet Transform for Image Processing Applications 397

As it is apparent from equation (3), even if the integral limits are infinite, the analysis is always limited to a portion of the signal, bounded by the limits [-, ] of the sliding window. The time-frequency plane of a fixed window STFT transform is illustrated in

Although, this approach (using STFT transform) succeeds well in giving both time and frequency information about a portion of the signal, however, as its predecessor, it has a major drawback. The fact is that the choice of the window size is crucial. As stated by Starck and al (Starck et al., 1998): " The smaller the window size, the better the time-resolution. However, the smaller the window size also, the more the number of discrete frequencies which can be represented in the frequency domain will be reduced, and therefore the more weakened will be the discrimination potential among frequencies". This problem is closely linked to the Heisenberg's uncertainty principle, which states that a signal (e.g. a very short

portion of the signal) cannot be represented as a point in the time-frequency domain.

This shortcoming brings us to rise the fundamental question: how to size then the sliding window? Not surprisingly, the answer to this question leads us by means of certain transformations to the wavelet transform. In fact, by considering the convolution of the sliding window with the time-dependant exponential *e-jwt* within the integral of equation (3):

And replacing the frequency by a scaling factor a, and the window bound by a shifting factor b, leads us to the first step leading to the Continuous Wavelet Transform (CWT), as

1 tb K (t) <sup>ψ</sup> ( ) a R ,b R

\*

a a

a,b

jwt ,ωθ <sup>θ</sup>)ew(t(t)K (4)

(5)

(3)

 w),STFT( f(t)w(t θ)e dt jwt

Fig. 1. Fourier time-frequency plane (Graps, 1995)

represented in equation (5):

Figure 1.

1995). Since then, different works have been carried out either in the development of the theory related to the wavelet, or towards its application in different fields. In the field of signal processing, the great achievements reached in different studies by Mallat, Meyer and Daubechies have allowed the emergence of a wide range of wavelet-based applications. In fact, inspired by the work developed by Mallat on the relationships between the Quadrature Mirror Filters (QMF), pyramid algorithms and orthonormal wavelet bases (Mallat, 1989), Meyer constructed the first non-trivial wavelets (Meyer, 1989). However, the most important work was carried out by Ingrid Daubechies. Based on Mallat's work, Daubechies succeeded to construct a set of wavelet orthonormal basis functions, which have become the cornerstone of many applications (Daubechies, 1988). Few years later, the same author, in collaboration with others (Cody, 1994), presented a set of wavelet biorthogonal basis function, which later found their use in different applications, especially in image coding. Recently, JPEG2000, a biorthogonal wavelet-based compression has been adopted as the new compression standard (Ebrahimi et al., 2002).

### **2. Continuous Wavelet Transform**

Different ways to introduce the wavelet transform can be envisaged (Starck et al., 1998). However, the traditional method to achieve this goal remains the use of the Fourier theory (more precisely, STFT). The Fourier theory uses sine and cosine as basis functions to analyse a particular signal. Due to the infinite expansion of the basis functions, the FT is more appropriate for signals of the same nature, which generally are assumed to be periodic. Hence, the Fourier theory is purely a frequency domain approach, which means that a particular signal f(t) can be represented by the frequency spectrum F(w), as follows:

$$\mathbf{F}(\alpha) = \int\_{-\infty}^{+\infty} \mathbf{f}(\mathbf{t}) \mathbf{e}^{-j\alpha \cdot \mathbf{t}} \mathbf{d}\mathbf{t} \tag{1}$$

The original signal can be recovered, under certain conditions, by the inverse Fourier Transform as follows:

$$f(t) = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} F(o)e^{j\alpha t} dco\tag{2}$$

Obviously, discrete-time versions of both direct and inverse forms of the Fourier transform are possible.

Due to the non-locality and the time-independence of the basis functions in the Fourier analysis, as represented by the exponential factor of equation (1), the FT can only suit signals with "time-independent" statistical properties. In other words, the FT can only provide global information of a signal and fails in dealing with local patterns like discontinuities or sharp spikes (Graps, 1995). However, in many applications, the signal of concern is both time and frequency dependent, and as such, the Fourier theory is "incapable" of providing a global and complete analysis. The shortcomings of the Fourier transform, in addition to its failure to deal with non-periodic signals led to the adoption by the scientific community of a windowed version of this transform known as the STFT. The STFT transform of a signal f(t) is defined around a time through the usage of a sliding window w (centred at time ) and a frequency as (Wickerhauser, 1994; Graps, 1995; Burrus et al., 1998; David, 2002 & Oppenheim & Schafer, 2010):

1995). Since then, different works have been carried out either in the development of the theory related to the wavelet, or towards its application in different fields. In the field of signal processing, the great achievements reached in different studies by Mallat, Meyer and Daubechies have allowed the emergence of a wide range of wavelet-based applications. In fact, inspired by the work developed by Mallat on the relationships between the Quadrature Mirror Filters (QMF), pyramid algorithms and orthonormal wavelet bases (Mallat, 1989), Meyer constructed the first non-trivial wavelets (Meyer, 1989). However, the most important work was carried out by Ingrid Daubechies. Based on Mallat's work, Daubechies succeeded to construct a set of wavelet orthonormal basis functions, which have become the cornerstone of many applications (Daubechies, 1988). Few years later, the same author, in collaboration with others (Cody, 1994), presented a set of wavelet biorthogonal basis function, which later found their use in different applications, especially in image coding. Recently, JPEG2000, a biorthogonal wavelet-based compression has been adopted as the

Different ways to introduce the wavelet transform can be envisaged (Starck et al., 1998). However, the traditional method to achieve this goal remains the use of the Fourier theory (more precisely, STFT). The Fourier theory uses sine and cosine as basis functions to analyse a particular signal. Due to the infinite expansion of the basis functions, the FT is more appropriate for signals of the same nature, which generally are assumed to be periodic. Hence, the Fourier theory is purely a frequency domain approach, which means that a

<sup>j</sup>ω<sup>t</sup> <sup>F</sup> <sup>ω</sup> f(t)e dt

The original signal can be recovered, under certain conditions, by the inverse Fourier

  

*<sup>f</sup> <sup>t</sup> <sup>F</sup> <sup>e</sup> <sup>d</sup> <sup>j</sup> <sup>t</sup>* ( ) ( ) <sup>2</sup>

Obviously, discrete-time versions of both direct and inverse forms of the Fourier transform

Due to the non-locality and the time-independence of the basis functions in the Fourier analysis, as represented by the exponential factor of equation (1), the FT can only suit signals with "time-independent" statistical properties. In other words, the FT can only provide global information of a signal and fails in dealing with local patterns like discontinuities or sharp spikes (Graps, 1995). However, in many applications, the signal of concern is both time and frequency dependent, and as such, the Fourier theory is "incapable" of providing a global and complete analysis. The shortcomings of the Fourier transform, in addition to its failure to deal with non-periodic signals led to the adoption by the scientific community of a windowed version of this transform known as the STFT. The STFT transform of a signal f(t) is defined around a time through the usage of a sliding window w (centred at time ) and a frequency as (Wickerhauser, 1994; Graps, 1995; Burrus et al., 1998; David, 2002 &

(1)

<sup>1</sup> (2)

particular signal f(t) can be represented by the frequency spectrum F(w), as follows:

new compression standard (Ebrahimi et al., 2002).

**2. Continuous Wavelet Transform** 

Transform as follows:

Oppenheim & Schafer, 2010):

are possible.

$$\text{STFT}(\theta, \mathbf{w}) = \int\_{-\infty}^{+\infty} \mathbf{f}(\mathbf{t}) \mathbf{w}(\mathbf{t} - \theta) \mathbf{e}^{-\text{jwt}} \mathbf{d}\mathbf{t} \tag{3}$$

As it is apparent from equation (3), even if the integral limits are infinite, the analysis is always limited to a portion of the signal, bounded by the limits [-, ] of the sliding window. The time-frequency plane of a fixed window STFT transform is illustrated in Figure 1.

Fig. 1. Fourier time-frequency plane (Graps, 1995)

Although, this approach (using STFT transform) succeeds well in giving both time and frequency information about a portion of the signal, however, as its predecessor, it has a major drawback. The fact is that the choice of the window size is crucial. As stated by Starck and al (Starck et al., 1998): " The smaller the window size, the better the time-resolution. However, the smaller the window size also, the more the number of discrete frequencies which can be represented in the frequency domain will be reduced, and therefore the more weakened will be the discrimination potential among frequencies". This problem is closely linked to the Heisenberg's uncertainty principle, which states that a signal (e.g. a very short portion of the signal) cannot be represented as a point in the time-frequency domain.

This shortcoming brings us to rise the fundamental question: how to size then the sliding window? Not surprisingly, the answer to this question leads us by means of certain transformations to the wavelet transform. In fact, by considering the convolution of the sliding window with the time-dependant exponential *e-jwt* within the integral of equation (3):

$$\mathbf{K}\_{\theta, \alpha}(\mathbf{t}) = \mathbf{w}(\mathbf{t} - \theta) \mathbf{e}^{-\text{jwt}} \tag{4}$$

And replacing the frequency by a scaling factor a, and the window bound by a shifting factor b, leads us to the first step leading to the Continuous Wavelet Transform (CWT), as represented in equation (5):

$$\mathbf{K}\_{\mathbf{a},\mathbf{b}}(\mathbf{t}) = \frac{1}{\sqrt{\mathbf{a}}} \boldsymbol{\upmu}^\*(\frac{\mathbf{t} - \mathbf{b}}{\mathbf{a}}) \qquad \mathbf{a} \in \mathbb{R}^+, \,\mathbf{b} \in \mathbb{R} \tag{5}$$

The combination of equation (5) with equation (3), leads to the CWT as defined by Morlet and Grossman (Grossman & Morlet, 1984).

$$\text{BW}(\mathbf{a}, \mathbf{b}) = \frac{1}{\sqrt{\mathbf{a}}} \int\_{-\infty}^{+\infty} \mathbf{f}(\mathbf{t}) \boldsymbol{\upmu}^\*(\frac{\mathbf{t} - \mathbf{b}}{\mathbf{a}}) \, \mathrm{d}\mathbf{t} \tag{6}$$

The Wavelet Transform for Image Processing Applications 399

 Wavelets are building blocks for general functions: They are used to represent signals and more generally functions. In other words, a function is represented in the wavelet

 Wavelets have space – frequency localisation: Which means that most of the energy of a wavelet is confined in a finite interval and that the transform contains only frequencies

 Wavelets support fast and efficient transform algorithms: This requirement is needed when implementing the transform. Often wavelet transforms need O(n) operations, which means that the number of multiplications and additions follows linearly the length of the signal. This is a direct implication of the compactness property of the transform. However, more general wavelet transforms require O(nlog(n)) operations

To refine the wavelet definition, the three following characteristics have been added by Sweldens and Daubechies (Sweldens, 1996 & Daubechies, 1992, 1993) as reported in (Burrus

 Oneness of the generating function: Refers to the ability of generating a wavelet system from a single scaling function or wavelet function just by scaling and translating. Multiresolution ability: This concept, which has first been introduced by Mallat, states the ability of the transform to represent a signal or function at different level, by

 Ability of generating lower level coefficients from the higher level coefficients. This can be achieved through the use of tree-like structured chain of filters called Filter Banks.

The multiresolution concept has been introduced first by Mallat (Mallat, 1989). It defines clearly the relationships between the QMF, pyramid algorithms and orthonormal wavelet bases through basically, the definition of a set of nested subspaces and a so-called scaling function. The strength of multiresolution lies in its ability to decompose a signal in finer and finer details. Most importantly, it allows the description of a signal in terms of time-

The basic requirement for multiresolution analysis is the existence of a set of approximation subspaces of L2(R) (square integrable function space) with different resolutions, as represented schematically for the three intermediate subspaces in Figure 3 and stated by

In such a way that, if Vf(t) j then Vf(2t) 1j . Which means that the subspace containing high resolution will automatically contains those of lower resolution. In a more general case,

<sup>k</sup> Vt)f(2 . This implication is known as the scale invariance property.

2 <sup>101</sup> *V V V V V LR* .... .... ( ) (8)

different weighted sums, derived from the original one.

space by mean of infinite series of wavelets.

from a certain frequency band.

(e.g. undecimated wavelet).

et al., 1998):

**3. Multiresolution** 

frequency or time-scale analysis.

**3.1 Nested subspaces** 

if Vf(t) <sup>0</sup> , then k

equation (8):

Where f(t) belongs to the square integrable functions space, L2(R). In the same way, the inverse CWT can be defined as (Grossman & Morlet, 1984):

$$\mathbf{f}(\mathbf{t}) = \frac{1}{\mathbf{C}\_{\Psi}} \int\_{0}^{+\alpha} \int\_{-\alpha}^{+\alpha} \frac{1}{\sqrt{\mathbf{a}}} \mathbf{W}(\mathbf{a}, \mathbf{b}) \boldsymbol{\nu}(\frac{\mathbf{t} - \mathbf{b}}{\mathbf{a}}) \frac{\mathbf{da} \, \mathbf{d} \mathbf{b}}{\mathbf{a}^{2}} \tag{7}$$

The *C<sup>ψ</sup>* factor is needed for reconstruction purposes. In fact, the reconstruction is only possible if this factor is defined. This requirement is known as the admissibility condition. In a more general way, *ψ(t)* is replaced by χ(t), allowing a variety of choices, which can enhance certain features for some particular applications (Starck et al., 1998; Stromme, 1999 & Hankerson et al., 2005). However, the CWT in the form defined by equation (6) is highly redundant, which makes its direct implementation of minor interest. The time-frequency plane of a wavelet transformation is illustrated in Figure 2. The differences with the STFT transform are visually clear.

Fig. 2. Wavelet time-frequency plane ((Graps, 1995) with minor modifications)

At this stage and after this brief introduction, it is natural to ask the question: therefore what are wavelet Transforms*?*

Although wavelet transforms are defined as a mathematical tool or technique, there is no consensus within the scientific community on a particular definition. This "embarrassment" has been stated by Sweldens as (Sweldens, 1996): "Giving that the wavelet field keeps growing, the definition of a wavelet continuously changes. Therefore it is impossible to rigorously define a wavelet". According to the same author, to call a particular function a wavelet system, it has to fulfil the three following properties:

The combination of equation (5) with equation (3), leads to the CWT as defined by Morlet

W(a,b) f(t) 1 tb \* <sup>ψ</sup> ( ) dt a a

Where f(t) belongs to the square integrable functions space, L2(R). In the same way, the

1 1 tb dadb f(t) W(a,b) ( ) C a a a

The *C<sup>ψ</sup>* factor is needed for reconstruction purposes. In fact, the reconstruction is only possible if this factor is defined. This requirement is known as the admissibility condition. In a more general way, *ψ(t)* is replaced by χ(t), allowing a variety of choices, which can enhance certain features for some particular applications (Starck et al., 1998; Stromme, 1999 & Hankerson et al., 2005). However, the CWT in the form defined by equation (6) is highly redundant, which makes its direct implementation of minor interest. The time-frequency plane of a wavelet transformation is illustrated in Figure 2. The differences with the STFT

 

0 2

(6)

(7)

 

and Grossman (Grossman & Morlet, 1984).

transform are visually clear.

are wavelet Transforms*?*

inverse CWT can be defined as (Grossman & Morlet, 1984):

ψ

Fig. 2. Wavelet time-frequency plane ((Graps, 1995) with minor modifications)

wavelet system, it has to fulfil the three following properties:

At this stage and after this brief introduction, it is natural to ask the question: therefore what

Although wavelet transforms are defined as a mathematical tool or technique, there is no consensus within the scientific community on a particular definition. This "embarrassment" has been stated by Sweldens as (Sweldens, 1996): "Giving that the wavelet field keeps growing, the definition of a wavelet continuously changes. Therefore it is impossible to rigorously define a wavelet". According to the same author, to call a particular function a


To refine the wavelet definition, the three following characteristics have been added by Sweldens and Daubechies (Sweldens, 1996 & Daubechies, 1992, 1993) as reported in (Burrus et al., 1998):

