High-pass filter coefficients

By iterative application of the FWT to the high-pass filter coefficients, a shape emerges that is an approximation of the wavelet function. The same applies to the iterative convolution of the low-pass filter that produces a shape approximating the scaling function. Figure 4.6 and Figure 4.7 display the construction of the scaling and wavelet functions, respectively ... 

FIGURE 4.7 Construction of wavelet function / with high-pass D4 filter coefficients (above). Below the functional representation of the high-pass filter coefficients (left) and their refinement by iterative calculation (increasing resolution level j) leading to an approximation of the wavelet function / (right). 

The first row in the wavelet matrix A, simply eontains the low-pass filter coefficients. The second row in the wavelet matrix A, contains the high-pass filter coefficients. As shown above, it is sometimes more convenient to store the filter eoefficients in the matrix A as a sequence of sub-blocks Aj j —. ..,-2,-1,0,1,2,...). The sub-blocks are simply the filter coefficients found in the lattice decomposition and reconstruction equations ar-... 

For reasons of convenience, and as will be shown in Chapter 6. the decomposition and reconstruction lattice equations can also be written as a pair of equations of recursive matrix products where we separate out the low-pass and high-pass filter coefficients in the form of infinite matrices Cj and Dy. That is. 

Using the notation of Chapter 4, where the low- and high-pass filter coefficients are combined into one matrix, the 3-band DWT outlined in Fig. 1. Going from level 2 to level 1 is written... 

If we let A denote the matrix of filter coefficients with the first row containing the low-pass filter coefficients and the remaining m - 1 rows the sets of high pass filter coefficients and if Nf is the number of filter coefficients contained in each filter, then A will be an m x Nf matrix. A can be partitioned into m x m sub-matrices as follows... 

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