Wavelets and B-Splines Multiresolution Analysis

Abstract

this paper is the development of cardinal B-splines and their properties from the perspective of wavelet analysis. Although B-splines were known already to Laplace, the relatively recent development of "spline-wavelets" (cf. [1]) has proven extremely useful in image processing, particularly in data compression and image reconstruction. Using a multiresolution analysis approach, these wavelets occur in a natural way from the more classical B-splines. In section 1, we give some background regarding the Haar wavelet, and give a formal presentation of the multiresolution analysis framework on which the rest of the development is based. In section 2, we define Cardinal and B-splines, and prove that the B-splines form a basis for the cardinal splines of a given order. We also introduce the notation of a Riesz basis, and show that B-splines form a multiresolution analysis. We show that B-spline