The Discrete Pulse Transform

We investigate a recent algorithm, here called a discrete pulse transform (DPT), for the multiresolution analysis of a sequence. A DPT represents a sequence as a sum of pulses, where a pulse is a sequence which is zero everywhere except for a certain number of consecutive elements which have a constant nonzero value. Unlike the discrete Fourier and wavelet transforms, the DPT is not a discretization of an underlying continuous model, but is inherently discrete. The DPT is composed of nonlinear morphological filters based only on the order relations between elements of the sequence. It is comparable to, but computationally more efficient than, the median transform, and more amenable to theoretical analysis. In particular, we show that a DPT has remarkable shape preserving and consistency properties.