a) It obeys the admissibility condition which is given as C ψ = ψ (ω)

Chapter 2 Introduction to Wavelets Wavelets were shown in 1987 to be the foundation of a powerful new approach to signal processing and analysis called multi-resolution theory by S. Mallat. As its name implies, multi-resolution theory is concerned with the representation and analysis of signals (or images) at more than one resolution. The motivation for such an approach is that features that might go undetected at one resolution may be easy to spot at another [Gon2002]. Wavelets enable analysis on several timescales of the local properties of complex signals that can present non-stationary zones. They are the foundation for new techniques of signal analysis and synthesis and find beautiful applications to general problems such as compression and denoising. In the last two decades, wavelets have essentially emerged as a fruitful mathematical theory and a tool for signal and image processing. They lead to a huge number of applications in various fields, such as geophysics, astrophysics, telecommunications, imagery and video coding [Mis2007]. 2.1 What is a Wavelet? A Wavelet ψ(t) is a waveform of effectively limited time duration satisfying following conditions. a) It obeys the admissibility condition which is given as C ψ = ψ (ω) dω <. Where, ψ (ω) is Fourier Transform of wavelet function ψ(t). This condition ensures that ψ (ω) goes to zero quickly as ω 0. b) If C ψ <, then condition of ψ (0) = 0 is imposed on ψ (ω), which is equivalent + to zero average value i.e. ψ(t)dt = 0. + c) The energy of wavelet function ψ(t) is unity i.e. ψ(t) 2 dt = 1. An example Wavelet (A Morlet Wavelet) expressed by ψ(t) = e jω 0t e t2 /2 is shown in figure 2.1 [Dau2006]. More about wavelets such as its evolution, comparison 0 ω 40

with Fourier Transform, different types of wavelets and their properties are explained in appendix A. Re[ψ(t)] t Fig. 2.1: A Morlet Wavelet 2.2 Wavelet Transform Wavelet transforms are of mainly two types, Continuous Wavelet Transform (CWT) and Discrete Wavelet Transform (DWT). The DWT is further divided into Redundant Discrete Systems (Frames) and Orthonormal (and other) bases for wavelets or Multi Resolution Analysis (MRA) [Dau2006]. 2.2.1 Continuous Wavelet Transform (CWT) The wavelet basis functions can be obtained by dilating and translating the mother wavelet function ψ(t) as given in equation 2.1. ψ a,b (t) = 1 a b ψ (t ) a, b R (2.1) a Here a and b are called Dilation and Translation parameters respectively. The parameters a and b are varied continuously over R (with the constraint a 0). The Continuous Wavelet Transform (CWT) of a signal f(t) is then given by equation 2.2. + CWT(a, b) = f, ψ a,b = 1 a f(t). ψ t b ( ) dt (2.2) a Here, f, ψ a,b is the L 2 inner product. The results of the CWT are many wavelet coefficients, which are a function of a (scale) and b (position). These wavelet coefficients at several scale values are shown in figure 2.2 [Dau2006]. 41

The representation of wavelet coefficients as shown in figure 2.2 is called Scalogram. From figure 2.2, it can be seen that larger peaks (higher wavelet coefficients) are obtained at higher scales. Fig. 2.2: Wavelets coefficients at various scales (Scalogram) The top view representation of this Scalogram is also shown in figure 2.3, which is a common way of representing a Scalogram. Fig. 2.3: CWT Scalogram: A common representation view The higher scales correspond to the most stretched wavelets and the more stretched the wavelet, the longer the portion of the signal with which it is being compared. Thus the coarser signal features are measured by the wavelet coefficients. Similarly at lower scales, the wavelet function is compressed and therefore finer signal features are measured. Therefore, there is a correspondence between wavelet scales and frequency as revealed by wavelet analysis. i) Low scale Compressed wavelet Rapidly changing fine details High frequency. ii) High scale Stretched wavelet Slowly changing coarse features Low frequency. 42

The wavelet transform computation is done in the following steps [Doc2013]. i) Take a wavelet and compare it to a section at the start of the original signal. ii) Calculate a number C, that represents how closely correlated the wavelet is with this section of the signal. The larger the number C is in absolute value, the more is the similarity. If the signal energy and the wavelet energy are equal to one, C may be interpreted as a correlation coefficient. iii) Shift the wavelet to the right and repeat steps 1 and 2 until whole signal is covered. iv) Scale (stretch) the wavelet and repeat steps i) through iii). v) Repeat steps i) through iv) for all scales. Arranging all these C values on time axis for different scales, the Scalogram is obtained. 43

2.2.2 Discrete Wavelet Transform (DWT) In CWT, calculating wavelet coefficients at every possible scale is a fair amount of work, and it generates an awful lot of data. If scales and positions are chosen to be discrete then analysis will be much easier and will not generate the huge data. This idea of choosing discrete values of translation and dilation parameter is implemented in Redundant Wavelet Transform (Frames) and Orthonormal bases for wavelets or Multi Resolution Analysis (MRA) [Dau2006]. 2.2.2.1 Redundant Wavelet Transform (Frames) In this case, the dilation parameter a and the translation parameter b both take only discrete values. The a is chosen to be an integer (positive and negative) powers of one fixed dilation parameter a 0 > 1, i.e. a = a m 0. The different values of m correspond to wavelets of different widths. It follows that the discretization of the translation parameter b should depend on m. The narrow (high frequency) wavelets are translated by small steps in order to cover the whole time range, while wider (lower frequency) wavelets are translated by larger steps. Since the width of ψ(a m 0 k) is proportional to a m 0, therefore, b is discretized by b = nb 0 a m 0, where b 0 > 0 is fixed and n Z. The corresponding discretely labeled wavelets are therefore, m2 ψ m,n (k) = a 0 ψ(a m 0 (k nb 0 a m 0 )) m, n Z (2.3) For a given function f(k), the inner product f, ψ m,n then gives the discrete wavelet transform as given in equation 2.4 [Dau2006]. DWT(m, n) = f, ψ m,n = a 0 m2 f(k). ψ m (a 0 k nb 0 ) (2.4) k= 2.2.2.2 Orthonormal Wavelet Bases (Multi Resolution Analysis (MRA)) If scales and positions are chosen based on powers of two, so-called Dyadic scales and positions, then analysis becomes much more efficient and just as 44

accurate. It was developed in 1988 by S. Mallat. For some very special choice of ψ and a 0, b 0, the ψ m,n constitutes an orthonormal basis for L 2 (R). In particular, if a 0 = 2, b 0 = 1, then there exist ψ with good time-frequency localization properties, such that the, ψ m,n (k) = 2 m 2 ψ(2 m k n) m, n Z (2.5) Constitute an orthonormal basis for L 2 (R). For a given function f(k), the inner product f, ψ m,n then gives the discrete wavelet transform as given in equation 2.6 [Dau2006]. DWT(m, n) = f, ψ m,n = 2 m 2 f(k). ψ (2 m k n) (2.6) k= Since the scaling and translation are discrete in DWT, the obtained Scalogram also looks discrete as shown in figure 2.4. Fig. 2.4: DWT Scalogram: A common representation view 2.3 Wavelet Transform and Filter Banks The multi resolution theory given by S. Mallat and Meyer [Mal2008] proves that any conjugate mirror filter characterizes a wavelet ψ(t) that generates an orthonormal basis of L 2 (R), and that a fast discrete wavelet transform is implemented by cascading these conjugate mirror filters. The equivalence between this continuous time wavelet theory and discrete filter banks led to a new fruitful interface between digital signal processing and harmonic analysis [Mal2008]. The wavelet decomposition of a signal s(t) based on the multi resolution theory given by S. Mallat and Meyer can done using digital FIR filters as shown in figure 2.5 [Doc2013]. 45

The arrangement shown above has used two wavelet decomposition (Analysis) filters which are High Pass and Low Pass respectively followed by down sampling by 2 producing half of input data point of High and Low frequency. The High frequency coefficients are called Detailed Coefficients (cd) and Low frequency coefficients are called Approximation Coefficients (ca) [Doc2013]. The scheme shown above represents one level of decomposition. The approximation coefficients (ca) can further be decomposed in another set of wavelet coefficients as shown in figure 2.6. The structure of figure 2.6 is called a wavelet decomposition tree. After decomposition, the signal can be reconstructed back by Inverse Wavelet Transform. The corresponding Filter Bank structure for reconstruction is shown in figure 2.7. Fig. 2.5: One level wavelet decomposition (Analysis) Fig. 2.6: Multi level wavelet decomposition 46

Fig. 2.7: One level wavelet reconstruction (Synthesis) In figure 2.7, the wavelet coefficients are up sampled by 2 and then filtered by reconstruction (Synthesis) wavelet filters [Doc2013]. As an example, a noisy sinusoid (s) and its 3 levels of decomposition showing approximation coefficients (a 1, a 2, a 3 ), detailed coefficients (d 1, d 2, d 3 ), Scalogram and reconstructed signal (s ) are shown in figure 2.8. Similar to the 1-D signals, wavelet transform can also be applied to 2-D signals i.e. images. In case of 2-D, wavelet functions become two dimensional also. The wavelet analysis for images is explained in next sections. 2.4 Wavelet Transform for 2-D signals (Images) Wavelet orthonormal bases of images can be constructed from wavelet orthonormal basis of one dimensional signal. Three mother wavelets ψ 1 (x), ψ 2 (x) and ψ 3 (x) with x = (x 1, x 2 ) R 2, are dilated by 2 j and translated by 2 j n with n = (n 1, n 2 ) Z 2. This yields an orthonormal basis of the space L 2 (R 2 ) of finite energy functions f(x) = f (x 1, x 2 ): {ψ k j,n (x) = 1 2 j ψk ( x 2j n 2 j )} j Z,n Z 2,1 k 3 (2.7) k The support of a wavelet ψ j,n is a square of width proportional to the scale 2 j. Two dimensional wavelet bases are discretized to define orthonormal bases of images including N pixels. Wavelet coefficients are calculated with a fast O(N) algorithm using multi-rate filter banks [Mal2008]. 47

The wavelet decomposition of an image based on the multi resolution theory can done using digital FIR filters [Doc2013] as shown in figure 2.9. Fig. 2.8: A noisy sinusoid and its wavelet coefficients Fig. 2.9: One level wavelet decomposition of an Image 48

In the figure 2.9, Lo_D represents a Low Pass FIR filter and Hi_D represents a High Pass FIR filter. The input image of size M M is converted into four coefficients matrices ca, ch, cv and cd of size M M. The coefficients represented by ca are 2 2 called approximation coefficients and contain low frequency details of the image while coefficients ch, cv and cd are called Detailed Coefficients and contain horizontal, vertical and diagonal high frequency details of the image. The wavelet decomposed image can be reconstructed back by these coefficients using Inverse DWT as shown in figure 2.10. Fig. 2.10: One level wavelet Reconstruction of an Image Figures 2.9 and 2.10 show, one level wavelet decomposition and reconstruction steps. Multi-level decompositions can also be achieved by further decomposing approximation coefficient matrix ca similar to the scheme of figure 2.9. An example image and its wavelet coefficients for Bior6.8 wavelet are shown in figure 2.11 for 3-level decomposition. From figure 2.11, it can be seen clearly how well wavelets are capable of capturing the horizontal, vertical and diagonal details of the images. This capability of wavelets leads to various image processing applications such as edge detection, edge enhancement, feature extraction for pattern recognition, image retrieval etc. 49

Fig. 2.11: 3-level wavelet decomposition of an image There is another way of representing these wavelet coefficients, it is called pyramidal structure. For example image Lena, this pyramidal structure is shown for Coif5 wavelet and 2 decomposition levels in figure 2.12. Fig. 2.12: 2-level wavelet decomposition of an image Lena (Pyramidal View) In pyramidal view of figure 2.12, the LL represents approximation coefficients, LH represents horizontal detailed coefficients, HL represents vertical detailed coefficients and HH represents diagonal detailed coefficients. 50

2.5 Working with Wavelets Of course, there is no point in breaking up a signal merely to have the satisfaction of immediately reconstructing it. The wavelet coefficients, before performing the reconstruction step can be modified to achieve certain objectives such as denoising, compression etc. as shown in figure 2.13. Fig. 2.13: Scheme of wavelet coefficients modification For example, to achieve denoising, the high frequency wavelet coefficients are thresholded non-linearly. Similarly, to achieve compression, the insignificant wavelets coefficients below a particular threshold are neglected. 2.6 Limitations of Wavelet Analysis Although the standard DWT is a powerful tool, it has three major limitations that has undermined its application for certain signal and image processing tasks [Fer2002]. These are as follows. 2.6.1 Shift Sensitivity A transform is shift sensitive if the shifting in time for input-signal causes an unpredictable change in the transform coefficients. It has been observed that the standard DWT is seriously disadvantaged by the shift sensitivity that arises from down samplers in the DWT implementation [Sel2005]. Shift sensitivity is an undesirable property because it implies that DWT coefficients fail to distinguish 51

between input-signal shifts. The shift variant nature of DWT is demonstrated with three shifted step-inputs in figure 2.14. In figure 2.14 input shifted signals are decomposed up to 4 levels using db5. It shows the unpredictable variations in the reconstructed detail signal at various levels and in final approximation. Wavelet packets have also been investigated for shift sensitivity. Wavelet Packet (WP) gives better results than standard DWT implementation at the cost of additional complexity. Fig. 2.14: Shift-sensitivity of standard 1-D DWT 2.6.2 Poor Directionality An m-dimensional transform (m>1) suffers poor directionality when the transform coefficients reveal only a few feature orientations in the spatial domain. While Fourier sinusoids in higher dimensions correspond to highly directional plane waves, the standard tensor product construction of M-D wavelets produces a checkerboard pattern that is simultaneously oriented along several directions. This lack of directional selectivity greatly complicates modeling and processing of geometric image features like ridges and edges [Guo1992]. 52

2.6.3 Absence of Phase Information For a complex valued signal or vector, its phase can be computed by its real and imaginary projections. Most DWT implementations (including standard DWT, WPT and Stationary Wavelet Transform (SWT)) use separable filtering with real coefficient filters associated with real wavelets resulting in real-valued approximations and details. Such DWT implementations cannot provide the local phase information. All natural signals are basically real-valued, hence to avail the local phase information, complex-valued filtering is required. The difference between real and analytic wavelets is shown in figure 2.15. Fig. 2.15 Presentation of (a) real and (b) analytic wavelets 2.6.4 Aliasing The wide spacing of the wavelet coefficient samples, or equivalently, the fact that the wavelet coefficients are computed via iterated discrete-time down sampling operations interspersed with non-ideal low-pass and high-pass filters, results in substantial aliasing. The inverse DWT cancels this aliasing, of course, but only if the wavelet and scaling coefficients are not changed. Any wavelet coefficient processing (thresholding, filtering, and quantization) upsets the delicate balance between the forward and inverse transforms, leading to artifacts in the reconstructed signal [Sel2005]. 53

2.7 Wavelet Packet Analysis of images Wavelet packets are used to get the advantage of better frequency resolution representation. Wavelet packets analysis is a generalization of orthogonal wavelets that allow richer signal analysis by breaking up detail (high frequency) spaces, which are never decomposed in the case of wavelets. Wavelet packets were introduced by Coifman and Wickerhauser [Coi1992] in early 1990s in order to mitigate the lack of frequency resolution of wavelet analysis. The basic idea is to cut up detail spaces into frequency sections. In wavelet analysis, a signal is split into an approximation part and a detail part. The approximation is then itself split into a second-level approximation and detail, and the process is repeated. For n-level decomposition, there are n + 1 possible ways to decompose or encode the signal. In wavelet packet analysis, the details as well as the approximations can be split. This yields more than 2 2n 1 different ways to encode the signal. The set of functions w j,n = ( w j,n,k (x), k Z) is the (j, n) wavelet packet. For positive values of integers j and n, wavelet packets are organized in binary trees. Here scale j defines depth and frequency n defines position in the tree. The notation w j,n where j denotes scale parameter and n the frequency parameter, is consistent with the usual depth-position tree labeling [Mis2007]. The two-level wavelet packet decomposition is shown in figure 2.16. Fig. 2.16: Three level wavelet packet decomposition binary tree In the figure 2.16, w 1,0 is the outcome of a low pass wavelet filter having low frequency details and known as Approximation Coefficient band. The w 1,1 is the 54

output of high pass wavelet filter which contains high frequency details and known as Detailed Coefficient band. The coefficients w 1,0 and w 1,1 are further split into high and low frequency bands at every scale. The same theory can be applied to two dimensional signals (images). The binary tree of figure 2.16 is extended to quad tree as shown in figure 2.17 for depth 2. Fig. 2.17: Two-level wavelet packet decomposition quad-tree for images In figure 2.17, ca represents low frequency coefficients known as Approximation Coefficients and ch, cv and cd represent high frequency coefficients known as Horizontal, Vertical and Diagonal Coefficients respectively. As an example, 2-level wavelet packet decomposition is applied on Lena image using Haar wavelet basis and is shown in figure 2.18. Fig. 2.18: 2 level wavelet packet decomposition of image using Haar wavelet 55

2.8 Discrete Curvelet Transform of images Ridgelets were introduced in 1999 by Candes and Donoho [Can1999] to address the edge representation problem. Later they were modified and fundamental curvelets came into existence in year 2000 [Can2000] based on windowed Ridgelets. In 2004, the definition was changed and the curvelets form a new tight frame [Can2004]. Moreover, curvelet was claimed to be optimal in representing objects with smooth singularities. This is the so-called second generation of curvelet. In 2006, a fast algorithm Fast Discrete Curvelet Transform (FDCT) was developed for both 2D and 3D [Can2006]. In 2007, a new variant of the FDCT was developed [Dem2007]. It extended the ideas of 'wavelets on an interval' to curvelets and wave atoms. Wavelet transform is widely used for denoising but it suffers from shift and rotation sensitivity as well as shows poor directionality. Curvelet transform is more suitable for detection of directional properties as it provides optimally sparse representation of objects giving maximum energy concentration along the edges. Curvelets are the basis elements which show high directional sensitivity and are highly anisotropic. Curvelets have variable width and variable length and so a variable anisotropy. The length and width at fine scales are related by a scaling law (width length 2 ) and so the anisotropy increases with decreasing scale like a power law. In two dimensional plane, curvelets are localized not only in position (the spatial domain) and scale (the frequency domain), but also in orientation. The curvelet transform, like the wavelet transform, is a multi-scale transform. In addition, the curvelet transform is based on a certain anisotropic scaling principle which is quite different from the isotropic scaling of wavelets. The discrete curvelet transform of 2-D function f(x 1, x 2 ) makes use of dyadic sequence of scales and a bank of filters (P 0 f, 1 f, 2 f,.. ) with the property that the bandpass filter s is concentrated near the frequencies [2 2s, 2 2s+2 ], e.g. s = ψ 2s f, ψ 2s (ξ) = ψ (2 2s ξ). In wavelet theory, one uses decomposition into dyadic subbands [2 s, 2 s+1 ]. In contrast, the sub-bands used in the discrete curvelet transform of continuum function have the nonstandard form [2 2s, 2 2s+2 ]. The basic process 56

of the digital realization for curvelet transform of a 2-D function f(x 1, x 2 ) is given as follows [Sta2002]. Sub-band Decomposition: The function f(x 1, x 2 ) is decomposed into sub-bands using a trous algorithm as, f (P 0 f, 1 f, 2 f,.. ) The different sub-bands s f contain details about 2 2s wide. Smooth Partitioning: Each sub-band is smoothly windowed into squares of an appropriate scale (of side length ~2 s ), s f (w Q s f) Q Qs Where w Q is a collection of smooth window localized around dyadic squares, Q = [k 1 /2 s, (k 1 + 1)/2 s ] [k 2 /2 s, (k 2 + 1)/2 s ] Renormalization: Each resulting square is renormalized to unit scale, g Q = (T Q ) 1 (w Q s f), Q Q s Where, (T Q f)(x 1, x 2 ) = 2 s f(2 s x 1 k 1, 2 s x 2 k 2 ) is a renormalization operator. Ridgelet Analysis: Each square is analyzed via the orthonormal discrete ridgelet transform. This is a system of basis elements p λ making an orthonormal basis for L 2 (R 2 ): α μ = g Q, p λ. In this definition, the two dyadic sub-bands [2 2s, 2 2s+1 ] and [2 2s+1, 2 2s+2 ] are merged before applying ridgelet transform. All these steps of computing Discrete Curvelet Transform (DCT) are shown in figure 2.19. Curvelets at different scales and orientations are shown in figure 2.20. Wavelet transform has been deployed a countless number of times in many fields of science and technology. Similarly, digital curvelet transform is also very useful in various applications, especially in the fields of image processing and scientific computing. In image analysis for example, the curvelet transform may be used for the compression of image data, for the enhancement and restoration of images as acquired by many common data acquisition devices (e.g., CT scanners), and for post-processing applications such as extracting patterns from large digital images, detecting features embedded in very noisy images, enhancing low contrast 57

images, or registering a series of images acquired with very different types of sensors. Curvelet-based seismic imaging already is a very active field of research [Can2006]. Fig. 2.19: Curvelet transform flowgraph (The figure illustrates the decomposition of the original image into subbands followed by the spatial partitioning of each subband. The ridgelet transform is then applied to each block) Fig. 2.20: Curvelets at different scales and orientations 58

In scientific computation, curvelets may be used for speeding up fundamental computations; numerical propagation of waves in inhomogeneous media is of special interest. Promising applications include seismic migration and computational geophysics [Can2006]. The detailed theory of curvelet transform and various other applications of curvelets are reported in [Maj2010]. 59