Exponential-Spline Wavelet Bases
Michael Unser, BIG, EPFL

Test Run • 16 March 2005 • BM 4.235

Abstract
We build a multiresolution analysis based on shift-invariant exponential B-spline spaces. We construct the basis functions for these spaces and for their orthogonal complements. This yields a new family of wavelet-like basis functions of L2, with some remarkable properties. The wavelets, which are characterized by a set of poles and zeros, have an explicit analytical form (exponential spline). They are non-stationary is the sense that they are scale-dependent and that they are not necessarily the dilates of one another. They behave like multi-scale versions of some underlying differential operator $\Lop$; in particular, they are orthogonal to the exponentials that are in the null space of $\Lop$. The corresponding wavelet transforms are implemented efficiently using an adaptation of Mallat's filterbank algorithm.