Conventional digital signal processing techniques consider signals that are represented by a set of uniformly spaced sample values. Although most processing algorithms are derived within a purely discrete framework, there are a variety of biomedical problems (e.g., detection of anatomical structures in images, registration) that would be better formulated by considering a signal as a continuous real-valued function defined over some domain. This study is concerned with the development of new processing techniques that represent signals by continuous polynomial spline functions. Traditionally, polynomial spline interpolation or approximation problems are approached using a matrix formulation. Our initial contribution has been to derive efficient computational algorithms using recursive digital filters. These techniques have been applied to the design of fast algorithms for image interpolation and compression. We have also derived a sampling theory for polynomial splines that generalizes Shannon's sampling theorem for band limited functions. In addition, we have defined a whole family of polynomial spline wavelet transforms that allows the expansion of continuous image functions in terms of basis functions obtained by dilatation and expansion of a single prototype. Such wavelet transforms may provide space/frequency signal decompositions that are close to optimal, according to the uncertainty principle.
