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8.6 Discrete Wavelet Transform

Wavelet-based analysis of signals is an interesting, and relatively recent, new tool [Rio91, Str96, Vai93]. Similar to Fourier series analysis, where sinusoids are chosen as the basis function, wavelet analysis is also based on a decomposition of a signal using an orthonormal (typically, although not necessarily) family of basis functions. Unlike a sine wave, a wavelet has its energy concentrated in time. Sinusoids are useful in analyzing periodic and time-invariant phenomena, while wavelets are well suited for the analysis of transient, time-varying signals.
A wavelet expansion is similar in form to the well-known Fourier series expansion, but is defined by a two-parameter family of functions
where j and k are integers and the functions j,k(t) are the wavelet expansion functions. As indicated earlier, they usually form an orthogonal basis. The two-parameter expansion coefficients aj,k are called the discrete wavelet transform (DWT) coefficients of f(t) and Equation (8.6.1) is known as the synthesis formula (i.e., inverse transformation). The coefficients are given by
The wavelet basis functions are a two-parameter family of functions that are related to a function (t) called the generating or mother wavelet by
where k is the translation and j the dilation or compression parameter. Therefore, wavelet basis functions are obtained from a single wavelet by translation and scaling. There is, however, no single and universal mother wavelet function. The mother wavelet must simply satisfy a small set of conditions and is typically selected based on the signal processing problem domain. Almost all useful wavelet systems satisfy the multi-resolution condition. This means that given an approximation of a signal f(t) using translations of a mother wavelet up to some chosen scale, we can achieve a better approximation by using expansion signals with half the width and half as wide translation steps. This is conceptually similar to improving frequency resolution by doubling the number of harmonics (i.e., halving the fundamental harmonic) in a Fourier series expansion. It is important to note that the wavelet functions never actually enter into the calculation of the discrete wavelet transform. The computation of the transform may be formulated as a filtering operation with two related FIR filters.

Image transforms.

The 1D discrete wavelet transform is calculated using Mallat's algorithm [Mal89]. The transform coefficients, ck and dk at different scales, are calculated using the following convolution-like expressions:
where j denotes the resolution and k is the index for the samples. The operation defined in Equation (8.6.4) is a linear digital filtering operation using filters h and g, followed by down-sampling. The coefficients ckj and dkj are known respectively as the level j scaling and wavelet coefficients. The top-level coefficients cJ represent the original signal. For a signal of length N, where N is a power of two, we get J=Log [2, N]. In such a case, the iteration may be repeated J times with the last stage being of length one, one scaling coefficient, and one wavelet coefficient. Note that the iteration is performed over the scaling coefficients only. Here is an illustration for N=8.
Filters h and g are FIR quadrature-mirror filters known as the scaling and wavelet filters, respectively. The scaling filter is a lowpass filter, while the wavelet filter is highpass. For an even-length scaling filter, the two are related by the following formula:
The ImageProcessing package includes two well-known families of scaling/wavelet filters.

Scaling filters.

This loads the packages.
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This shows the magnitude response of Daubechies length-4 scaling (red) and wavelet (blue) filters [Dau92].
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The 1D series form of the inverse discrete wavelet transform (IDWT) is given by the following formula:
where j denotes the scale and k is the vector index. The operation defined in Equation (8.6.7) is an up-sampling operation followed by linear digital filtering. At scale j, the scaling and wavelet coefficients are combined to form the level j+1 scaling coefficients. The top-level sequence cJ represents the original signal. For a signal of length N, where N is a power of two, we get J=Log [2, N]. In such a case, the iteration may be repeated J times, beginning with two one-sample long sequences and ending with a sequence N samples long.
Similarly, the 2D discrete wavelet transform is computed using 2D filtering and downsampling operations. At each scale, the coefficients are given by the following formulas:
where the filters h0, h1, h2, and h3 are formed by vector outer products of the quadrature mirror filters h and g.
For example, the filter h1 has the form
Here is an illustration of the decomposition of an 8 × 8 array.
At each scale j, the array cj contains the low-frequency information from the previous stage, while dj, ej, and fj contain the horizontal, vertical, and bi-directional edge information (high-frequency or detail), respectively. As in the 1D transform, the iteration is performed over the lowpass coefficients only.
The 2D series formulation of the inverse discrete wavelet transform is given by
where the filters h0, h1, h2, and h3 are formed by vector outer products of the quadrature mirror filters h and g (see Equation (8.6.9)).
The wavelet expansion in Equation (8.6.1) is a multi-resolution representation of f(t). The multi-resolution formulation needs two closely related functions: the wavelet function (t) and the so-called scaling function (t). These are defined by two-scale difference equations with coefficients coming from filters g and h, respectively. The scaling function (t) has a recursive formulation known as the dilation equation, which involves the scaling coefficients hn.
The wavelet function (t) is related to the scaling function via the following difference equation:
Given a finite set of real (or possibly complex) scaling and wavelet coefficients and a unit pulse for the lowest resolution scaling function, Equations (8.6.13) and (8.6.14) may be iterated to obtain these functions at higher resolutions.

Scaling and wavelet sequences.

Here we calculate and display the scaling and wavelet sequences corresponding to a Daubechies filter of length 4.
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In the first example, we evaluate the discrete wavelet transform of a 1D sequence, a piecewise constant signal with some uniformly distributed additive noise.
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Here we compute a three-level decomposition. This returns a list of lists of scaling and wavelet transform coefficients.
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Here we plot the transform coefficients, beginning with the coarsest scale or equivalently the lowest computed decomposition level (j=3). For visualization purposes, each level was upsampled to the scale of the original signal (i.e., j=6).
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DiscreteWaveletTransform returns transform coefficients in a nested list. This structure, while convenient in computing the inverse transformation and in accessing and manipulating the transform coefficients, makes it difficult to plot the coefficients with standard built-in graphics functions. For this reason two structure manipulation functions and a new density plot function are available.
WaveletFlatten[coefs]returns a simple list or a list of lists given wavelet coefficients coefs
WaveletPartition[list, lev]
partitions list into wavelet transform format with number of levels given by lev
WaveletDensityPlot[coefs]
generates a density plot of the 2D discrete wavelet transform coefficients coefs

Wavelet transform structure manipulation and graphics functions.

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In the next example, we take the 2D discrete wavelet transform of an example image. We use a 2-level decomposition.
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