WAVELETS AND FILTER BANKS - THEORY AND DESIGN

Wavelets, filter banks, and multiresolution signal analysis, which have been used independently in the fields of applied mathematics, signal processing, and computer vision, respectively, have recently converged to form a single theory. In this paper, we first compare the wavelet transform with the more classical short-time Fourier transform approach to signal analysis. Then we explore the relations between wavelets, filter banks, and multiresolution signal processing. We briefly review perfect reconstruction filter banks, which can be used both for computing the discrete wavelet transform, and for deriving continuous wavelet bases, provided that the filters meet a constraint known as regularity. Given a low-pass filter, we derive necessary and sufficient conditions for the existence of a complementary high-pass filter that will permit perfect reconstruction. Posing the perfect reconstruction condition as a Bezout identity, we then show how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations. An alternative approach based on the theory of continued fractions is also given. We use these results to design highly regular filter banks, which generate biorthogonal continuous wavelet bases with symmetries.