A family of polynomial spline wavelet transforms

Abstract

This paper presents an extension of the family of orthogonal Battle/Lemarié spline wavelet transforms with emphasis on filter bank implementation. Spline wavelets that are not necessarily orthogonal within the same resoluton level, are constructed by linear combination of polynomial spline wavelets of compact support, the natural counterpart of classical B-spline functions. Mallat's fast wavelet transform algorithm is extended to deal with these non-orthogonal basis functions. The impulse and frequency responses of the corresponding analysis and synthesis filters are derived explicitly for polynomial splines of any order n ( n odd). The link with the general framework of biorthogonal wavelet transforms is also made explicit. The special cases of orthogonal, B-spline, cardinal and dual wavelets are considered in greater detail. The B-spline (respectively dual) representation is associated with simple FIR binomial synthesis (respectively analysis) filters and recursive analysis (respectively synthesis) filters. The cardinal representation provides a sampled representation of the underlying continuous functions (interpolation property). The distinction between cardinal and orthogonal representation vanishes as the order of the spline is increased; both wavelets tend asymptotically to the bandlimited sinc-wavelet. The distinctive features of these various representations are discussed and illustrated with a texture analysis example.