Abstract
The class of digital filter banks (DFB's) and wavelets composed of zero-phase filters is particularly useful for image processing because of the desirable filter responses and the possibility of using the McClellan transformation for two-dimensional design. In this paper unimodular polyphase matrices with the identity matrix for Smith canonical form are introduced, and then decomposed to a product of unit upper and lower triangular or block-triangular matrices which define ladder structures. A fundamental approach to obtaining suitable unimodular matrices for one and two dimensions is to focus on the shift (translation) operators, as is done in the harmonic analysis discipline. Several matrix shift operators of different dimensions are introduced and their properties and applications are presented, most notable of which is that the McClellan transformation can be effected by a simple substitution of a 2 × 2 circulant matrix for the polynomial variable, w = (z + z -1)/2. Unimodular matrix groups and pertinent subgroups are identified, and these are observed to be subgroups of the special linear group over polynomials, SL(k[w]) . A class of coiflet-like wavelets containing the well-known wavelet, based on the Burt and Adelson filter, is decomposed by these methods and is seen to require only 3/2 multiplications/sample if a scaling property, introduced herein, is satisfied. Making use of certain paraunitary wavelets, coiflets, that are closely comparable to the zero-phase wavelets of this class, it is seen that, in these cases, the zero-phase ladder algorithm is twice as fast as the paraunitary lattice algorithm.
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Marshall, T.G. Zero-Phase Filter Bank and Wavelet Code r Matrices: Properties, Triangular Decompositions, and a Fast Algorithm. Multidimensional Systems and Signal Processing 8, 71–88 (1997). https://doi.org/10.1023/A:1008216922668
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DOI: https://doi.org/10.1023/A:1008216922668