Multidimensional filter banks and wavelets - A system theoretic perspective

We review the current status of multidimensional 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 otherhand, the field of multidimensional 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 multidimensional systems. This arises from the fact that many ideas crucial to the wavelet design are inherently system, theoretic in nature. While there are many examples of this synergy manifested in recent publications, we provide a flavour 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 multidimensional version of which has very close ties with 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 important manner. A number of open problems are also cited.