4.5 shows respectively the real and imaginary parts.
The Mexican hat defined by Murenzi [17] is:
(4.28)
which is the second derivative of a Gaussian waveform (see Figure 4.6).
4.5.1.3 Discrete‐Time Wavelet Transform
In order to process digital signals a discrete approximation of the wavelet coefficients is required. The discrete wavelet transform (DWT) can be derived in accordance with the sampling theorem if we process a frequency band‐limited signal.
The continuous form of the WT may be discretized with some simple considerations on the modification of the wavelet pattern by dilation. Since generally the wavelet function is not band limited, it is necessary to suppress the values outside the frequency components above half the sampling frequency to avoid aliasing (overlapping in frequency) effects.
Figure 4.5 Morlet's wavelet: real and imaginary parts shown respectively in (a) and (b).
Figure 4.6 Mexican hat wavelet.
A Fourier space may be used to compute the transform scale‐by‐scale. The number of elements for a scale can be reduced if the frequency bandwidth is also reduced. This requires a band‐limited wavelet. The decomposition proposed by Littlewood and Paley [21] provides a very nice illustration of the reduction of elements scale‐by‐scale. This decomposition is based on an iterative dichotomy of the frequency band. The associated wavelet is well localized in Fourier space where it allows a reasonable analysis to be made although not in the original space. The search for a discrete transform, which is well localized in both spaces leads to multiresolution analysis.
4.5.1.4 Multiresolution Analysis
Multiresolution analysis results from the embedded subsets generated by the interpolations (or down‐sampling and filtering) of the signal at different scales. A function f(t) is projected at each step j onto the subset Vj . This projection is defined by the scalar product cj (k) of f(t) with the scaling function φ(t), which is dilated and translated:
(4.29)
where 〈·, ·〉 denotes an inner product and φ(t) has the property:
(4.30)
where the right side is convolution of h and ϕ. By taking the Fourier transform of both sides:
where and
are the Fourier transforms of h(t) and ϕ(t) respectively. For a discrete frequency space (i.e. using the DFT) the above equation (Eq. 4.31) permits the computation of the wavelet coefficient Cj + 1(k) from Cj (k) directly. If we start from C 0(k) we compute all Cj (k), with j > 0, without directly computing any other scalar product:
(4.32)
where k is the discrete frequency index.
At each step, the number of scalar products is divided by two and consequently the signal is smoothed. Using this procedure, the first part of a filter bank is built up. In order to restore the original data, Mallat uses the properties of orthogonal wavelets, but the theory has been generalized to a large class of filters by introducing two other filters and
, also called conjugate filters. The restoration is performed with:
(4.33)
where wj + 1(∙) are the wavelet coefficients at the scale j + 1 defined later in this section. For an exact restoration, two conditions have to be satisfied for the conjugate filters:
Anti‐aliasing condition:
Exact restoration:
In the decomposition, the input is successively convolved with the two filters H (low frequencies) and G (high frequencies). Each resulting function is decimated by suppression of one sample out of two. The high frequency signal is left, and we iterate with the low frequency signal (left side of Figure 4.7). In the reconstruction, we restore the sampling by inserting a zero between each sample, then we convolve with the conjugate filters and
, we add the resulting outputs and we multiply the result by 2. We iterate up to the smallest scale (right side of Figure 4.7). Orthogonal wavelets correspond to the restricted case where:
(4.36)
Figure 4.7 The filter bank associated with the multiresolution