An optimally well-localized multi-channel parallel perfect reconstruction filter bank

This dissertation defines a measure of uncertainty for finite length discrete-time signals. Using this uncertainty measure, a relationship analogous to the well known continuous-time Heisenberg-Weyl inequality is developed. This uncertainty measure is applied to quantify the joint discrete time-discrete frequency localization of finite impulse response filters, which are used in a quadrature mirror filter bank (QMF). A formulation of a biorthogonal QMF where the low pass analysis filter minimizes the newly defined measure of uncertainty is presented. The search algorithm used in the design of the length-N linear phase low pass analysis FIR filter is given for N = 6 and 8. In each case, the other three filters, which constitute a perfect reconstruction QMF, are determined by adapting a method due to Vetterli and Le Gall. From a set of well known QMFs comprised of length six filters, L-channel perfect reconstruction parallel filter banks (PRPFB) are constructed. The Noble identities are used to show that the L-channel PRPFB is equivalent to a L − 1 level discrete wavelet filter bank. Several five-channel PRPFBs are implemented. A separable implementation of a five-channel, one-dimensional filter bank produces twenty-five channel, two-dimensional filter bank. Each non-low pass, two-dimensional filter is decomposed in a novel, nonseparable way to obtain equivalent channel filters that possess orientation selectivity. This results in a forty-one channel, two-dimensional, orientation selective, PRPFB.; Joint uncertainty for the overall L-channel, one-dimensional, parallel filter bank is quantified by a metric which is a weighted sum of the time and frequency localizations of the individual filters. Evidence is presented to show that a filter bank possessing a lower joint filter bank uncertainty with respect to this metric results in a computed multicomponent AM-FM image model that yields lower reconstruction errors. This strongly supports the theory that there is a direct relationship between joint uncertainty as quantified by the measures developed and the degree of local smoothness or “local coherency” that may be expected in the filter bank channel responses. Thus, as demonstrated by the examples, these new measures may be used to construct new filter banks that offer excellent localization properties on par with those of Gabor filter banks.