B-spline signal processing. I. Theory

Abstract

This paper describes a set of efficient filtering techniques for the processing and representation of singals in terms of continuous B-spline basis functions. We first consider the roblem of determining the spline coefficients for an exact signal interpolation (direct B-spline transform). The reverse operation is the signal reconstruction from its spline coefficients with an optional zoming factor m (indirect B-spline transform). We derive general expressions for the z transforms and the equivalent continuous impulse responses of B-spline interpolators of order n. We present simple techniques for signal differentiation and filtering in the transformed domain. We then derive recursive filters that efficiently solve the problems of smoothing spline and least squares approximations. The smoothing spline technique approximates a signal with a complete set of coefficients subject to certain regularization or smoothness constraints. The least squares approach, on the other hand, uses a reduced number of B-spline coefficients with equally spaced nodes; this techniques is in many ways analogous to the application of antialiasing low-pass filter prior to decimation in order to represent a signal correctly with a reduced number of samples.