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- Acharya T.

Acharya T. - John Wiley & Sons, 2000. - 292 p.
ISBN 0-471-48422-9
Download (direct link): standardforImagecompressioncon2000.pdf
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79
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INTRODUCTION TO DISCRETE WAVELET TRANSFORM
may not be statistically predictable especially at the region of discontinuities — a special feature that is typical of images having discontinuities at the edges.
In 1989, Mallat proposed the theory of signal analysis based on “multiresolution decomposition” of signals using wavelets in time-scale space and proposed the popular pyramid algorithm [6]. For historical perspectives of wavelets and the underlying mathematical foundation of wavelet transform, the reader is referred to the treatise by Ives Meyer [7].
4.2 WAVELET TRANSFORMS
Wavelets are functions generated from one single function (basis function) called the prototype or mother wavelet by dilations (scalings) and translations (shifts) in time (frequency) domain. If the mother wavelet is denoted by ip(t), the other wavelets ipa,b{t) can be represented as
Mt) = Tti4, (^r) (41)
where a and 6 are two arbitrary real numbers. The variables a and b represent the parameters for dilations and translations respectively in the time axis. From Eq. 4.1, it is obvious that the mother wavelet can be essentially represented as
ip{t) = ipit0{t)- (4.2)
For any arbitrary a ^ 1 and 6 = 0, we can derive that
Mt} = W? (h) ' (4-3)
As shown in Eq. 4.3, is nothing but a time-scaled (by a) and
amplitude-scaled (by ^/\a\) version of the mother wavelet function ip(t) in Eq. 4.2. The parameter a causes contraction of in the time axis when a < 1 and expansion or stretching when a > 1. That’s why the parameter a is called the dilation (scaling) parameter. For a < 0, the function ipa,b(t) results in time reversal with dilation.
Mathematically, we can substitute t in Eq. 4.3 by t — b to cause a translation or shift in the time axis resulting in the wavelet function ipa,b(t) as shown in Eq. 4.1. The function ipa,b(t) is a shift of ipa,o{t) in right along the time axis by an amount 6 when b > 0 whereas it is a shift in left along the time axis by an amount 6 when 6 < 0. That’s why the variable b represents the translation in time (shift in frequency) domain.
In Figure 4.1, we have shown an illustration of a mother wavelet and its dilations in the time domain with the dilation parameter a = a. For the mother wavelet ip(t) shown in Figure 4.1(a), a contraction of the signal in
WAVELET TRANSFORMS 81
(a)
(b)
(c)
Fig. 4.1 (a) A mother wavelet ip(t), (b) ip(t/a): 0 < a < 1, (c) ip(t/a)\ a > 1.
the time axis when a < 1 is shown in Figure 4.1(b) and expansion of the signal in the time axis when a > 1 is shown in Figure 4.1(c). Based on this definition of wavelets, the wavelet transform (WT) of a function (signal) f{t) is mathematically represented by
The inverse transform to reconstruct /(f) from W(a,b) is mathematically represented by
and ^(w) is the Fourier transform of the mother wavelet ip(t).
If a and b are two continuous (nondiscrete) variables and /(f) is also a continuous function, W(a,b) is called the continuous wavelet transform (CWT). Hence the CWT maps a one-dimensional function /(f) to a function W(a,b) of two continuous real variables a (dilation) and b (translation).
(4.4)
(4.5)
where
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INTRODUCTION TO DISCRETE WAVELET TRANSFORM
4.2.1 Discrete Wavelet Transforms
Since the input signal (e.g., a digital image) is processed by a digital computing machine, it is prudent to define the discrete version of the wavelet transform. Before we define the discrete wavelet transform, it is essential to define the wavelets in terms of discrete values of the dilation and translation parameters a and b instead of being continuous. There are many ways we can discretize a and b and then represent the discrete wavelets accordingly. The most popular approach of discretizing a and b is using Eq. 4.6,
a = a™, b^nboa^ (4.6)
where m and n are integers. Substituting a and b in Eq. 4.1 by Eq. 4.6, the discrete wavelets can be represented by Eq. 4.7.
VVnW = aom/2V; (aomt - nbo) • (4-7)
There are many choices to select the values of ao and 6o- We select the most common choice here: Oo = 2 and b0 = 1; hence, a = 2m and b — n2m. This corresponds to sampling (discretization) of a and b in such a way that the consecutive discrete values of a and b as well as the sampling intervals differ by a factor of two. This way of sampling is popularly known as dyadic sampling and the corresponding decomposition of the signals is called the dyadic decomposition. Using these values, we can represent the discrete wavelets as in Eq. 4.8, which constitutes a family of orthonormal basis functions.
rl>m,n(t) = 2-"^ (2“m< - n) • (4-8)
In general, the wavelet coefficients for function /(f) are given by
cm,n (/) = «(7m/2 J fW'tP (aomt - nM dt (4.9)
and hence for dyadic decomposition, the wavelet coefficients can be derived accordingly as
Cm,n(f) = 2~m'2 J f(t)rl> {2~mt - n) dt. (4.10)
This allows us to reconstruct the signal /(f) in from the discrete wavelet coefficients as
OO OO
/(*) = cmt„(/)v>TO,n(f)- (4.11)
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