# - Acharya T.

ISBN 0-471-48422-9

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m= â€” oo n = â€” oo

The transform shown in Eq. 4.9 is called the wavelet series, which is analogous to the Fourier series because the input function /(f) is still a continuous function whereas the transform coefficients are discrete. This is often called the discrete time wavelet transform (DTWT). For digital signal or image processing applications executed by a digital computer, the input signal /(f)

WAVELET TRANSFORMS 83

needs to be discrete in nature because of the digital sampling of the original data, which is represented by a finite number of bits. When the input function f(t) as well as the wavelet parameters a and b are represented in discrete form, the transformation is commonly referred to as the discrete wavelet transform (DWT) of signal f(t) [6, 11].

The discrete wavelet transform (DWT) became a very versatile signal processing tool after Mallat [6] proposed the multiresolution representation of signals based on wavelet decomposition. The method of multiresolution is to represent a function (signal) with a collection of coefficients, each of which provides information about the position as well as the frequency of the signal (function). The advantage of the DWT over Fourier transformation is that it performs multiresolution analysis of signals with localization both in time and frequency, popularly known as time-frequency localization. As a result, the DWT decomposes a digital signal into different subbands so that the lower frequency subbands have finer frequency resolution and coarser time resolution compared to the higher frequency subbands. The DWT is being increasingly used for image compression due to the fact that the DWT supports features like progressive image transmission (by quality, by resolution), ease of compressed image manipulation, region of interest coding, etc. Because of these characteristics, the DWT is the basis of the new JPEG2000 image compression standard [8].

4.2.2 Concept of Multiresolution Analysis

There were a number of orthonormal wavelet basis functions of the form ipm,n{t) = 2~m/2ip (2~mt â€” n) discovered in 1980s. The theory of multiresolution analysis presented a systematic approach to generate the wavelets [6, 9, 10]. The idea of multiresolution analysis is to approximate a function f(t) at different levels of resolution.

In multiresolution analysis, we consider two functions: the mother wavelet ip(t) and the scaling function <j>(t). The dilated (scaled) and translated (shifted) version of the scaling function is given by <t>m<n(t) â€” 2~Tnl2(f)(2~rnt â€” n). For fixed m, the set of scaling functions <j>m,n(t) are orthonormal. By the linear combinations of the scaling function and its translations we can generate a set of functions

f(t) = X an^mA1)- (4-12)

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The set of all such functions generated by linear combination of the set {<t>m,n(t)} is called the span of the set denoted by Span{0m n(f)}.

Now consider Vm to be a vector space corresponding to Spanâ€ž(Â£)}â– Assuming that the resolution increases with decreasing m, these vector spaces describe successive approximation vector spaces, â€¢ â€¢ â€¢ C V2 C Vi C Vo C V-\ C V-2 C â€¢ â– each with resolution 2m (i.e., each space Vj+i is contained in the next resolution space Vj). In multiresolution analysis, the set of subspaces satisfies the following properties:

84

INTRODUCTION TO DISCRETE WAVELET TRANSFORM

1. Vm+i C Vm, for all m: This property states that each subspace is contained in the next resolution subspace.

2- \JVm = L2(1Z): This property indicates that the union of subspaces is dense in the space of square integrable functions L2(JV)\ 1Z indicates a set of real numbers (upward completeness property).

3. HKn = 0 (an empty set): This property is called downward completeness property.

4. f(t) â‚ Vo *-> /(2~mt) Â£ Vm: Dilating a function from resolution space Vo by a factor of 2m results in the lower resolution space Vm (scale or dilation invariance property).

5. f(t) â‚ Vo <-> f(t â€” n) â‚ Vo: Combining this with the scale invariance property above, this property states that translating a function in a resolution space does not change the resolution (translation invariance property).

6. There exists a set {<fi(t â€” n) â‚ Vo: n is an integer} that forms an orthonormal basis of Vo-

The basic tenet of multiresolution analysis is that whenever the above properties are satisfied, there exists an orthonormal wavelet basis ipm<n(t) = 2â€œm/20(2~mt - n) such that â€™

where Pj is the orthogonal projection of ip onto Vj. For each m, consider the wavelet functions ipm,n{t) span a vector space Wm. It is clear from Eq. 4.13 that the wavelet that generates the space Wm and the scaling function that generates the space Vm are not independent. Wm is exactly the orthogonal complement of Vm in V^-i. Thus, any function in V"m_i can be expressed as the sum of a function in Vm and a function in the wavelet space Wm. Symbolically, we can express this as

(4.13)

Vrn-1 -KnÂ®

(4.14)

(4.15)

(4.16)

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