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

In this paper, local cubic quasi-interpolating splines on non-uniform grids are described. The splines are computed 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, they use either clean or noised arbitrarily-spaced samples. Exact estimations of the approximation errors are established. 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 for signals in scenarios where signal’s samples subsequently arrive one after another at randomized times. The wavelet transforms are executed without delay. On arrival of samples, only a couple of adjacent wavelet transform coefficients are updated.