Multi-dimensional Filter Banks and Wavelets - A System Theoretic Perspective

Abstract

We review the current status of multi-dimensional filters bank and wavelet design from the perspective of signal and system theory. The study of wavelets and perfect reconstruction filter banks are known to have roots in traditional filter design techniques. On the other hand, the field of multi-dimensional systems and signal processing has developed a set of tools intrinsic to itself, and has attained a certain level of maturity over the last two decades. We have recently noted a degree of synergy between the two fields of wavelets and multi-dimensional systems. This arises from the fact that many ideas crucial to wavelet design are inherently system theoretic in nature. While there are many examples of this synergy manifested in recent publications, we provide a flavor of techniques germane to this development by considering a few specific problems in detail. The construction of orthogonal wavelets can be essentially viewed as a circuit and system theoretic problem of design of energy dissipative (passive) filters, the multi-dimensional version of which has very close ties with a classic problem of lumped-distributed passive network synthesis. Groebner basis techniques, matrix completion problems over rings of polynomials or rings of stable rational functions, i.e., Quillen-Suslin (31) type problems are still other examples, which feature in our discussion in an important manner. A number of open problems are also cited. © 1998 The Franklin Institute. Published by Elsevier Science Ltd.