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Abstract
In recent years multifrequency channel decompositions have received a great amount of attention in signal processing, especially digital image processing. The expansion of a function into several frequency channels provides a representation which is well adapted for quantization and also coding based on psychophysiology of the human visual system. Moreover, data compression techniques based on multifrequency channel decompositions have been proven to be very efficient; they are robust and can be implemented on a digital computer using fast algorithms.
The work of this thesis is to describe a mathematical model for the computation and interpretation of the concept of multiresolution analysis and wavelet theory. The relations between the multiresolution analysis, wavelet transform and halfband filters are firmly established. We explain how the approximations of a signal at different resolutions are obtained, and how the difference of information between two successive approximations is calculated. The wavelet representation is obtained by decomposing the signal using wavelet orthonormal basis functions. A complete constructive characterization of orthogonal FIR filter bank is explained which generates orthonormal compactly supported wavelet bases.
We present a method of constructing compactly supported wavelets based on maximally flat halfband FIR filters generated by normalized incomplete beta functions. The halfband beta filters are shown to be factorable such that they yield linear phase and conjugate quadrature filters for use in exact reconstruction subband coders. The CQF solutions are shown to be the same as wavelet solutions obtained by Daubechies.
