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Filter Banks and Wavelets: Extensions and Applications
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| Columbia University, Center for Telecommunications Research Technical Report 257-91-38 |
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The approach researchers often take when analyzing a signal is to identify its nature and most notable characteristics, and subsequently use that knowledge to process it efficiently. This involves observing the signal's long-term behavior as well as its short-term variations. The basic toolbox used to accomplish this task consists of various transform-domain techniques, the Fourier transform being one of them. Although in wide use for quite some time, the use of the Fourier transform entails a difficulty, namely any time-local information about the signal is lost. The short-time Fourier transform, meant to alleviate this problem, allows the extraction of some finer time or frequency information but still results in a fixed resolution time-frequency analysis. The recently introduced wavelet transform, its natural framework of multiresolution analysis and its discrete-time counterpart--the discrete wavelet transform performed by multirate filter banks--allow for better analysis of signals, by looking at them at various scales or resolutions, much like using a magnifying glass. The three techniques, wavelet transform, multiresolution analysis and multirate filter banks, although stemming from different fields (applied mathematics, computer vision and digital signal processing) have recently converged to form a single theory. The scope of this thesis is to extend some of the concepts pertaining to the theory of multirate filter banks and wavelets. These include: *solving the open problem of constructing perfect reconstruction filter banks (discrete wavelet transform) with rational scaling factors, *extending some of the one-dimensional filter bank results to the multi-dimensional case using arbitrary sampling lattices, *using the iterated filter bank method to construct irreducible multidimensional wavelet bases and *applying some of the above concept to image and video representation and coding (experimental results are given).
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