Local cubic splines on non-uniform grids and real-time computation of wavelet transform

Abstract

In this paper, local cubic quasi-interpolating splines on non-uniform grids are described. The splines are designed by fast computational algorithms that utilize the relation between splines and cubic interpolation polynomials. These splines provide an efficient tool for real-time signal processing. As an input, the splines use either clean or noised arbitrarily-spaced samples. Formulas for the spline’s extrapolation beyond the sampling interval are established. Sharp estimations of the approximation errors are presented. The capability to adapt the grid to the structure of an object and to have minimal requirements to the operating memory are of great advantages for offline processing of signals and multidimensional data arrays. The designed splines serve as a source for generating real-time wavelet transforms to apply to signals in scenarios where the signal’s samples subsequently arrive one after the other at random times. The wavelet transforms are executed by six-tap weighted moving averages of the signal’s samples without delay. On arrival of new samples, only a couple of adjacent transform coefficients are updated in a way that no boundary effects arise.