Hermite sieve as a wavelet-like array for 1D and 2D signal decomposition

Abstract

A new class of an array of wavelet-like functions, derived from generalised Hermite polynomials and controlled by a scale parameter, has been used to create a multilayered representation for one- and two-dimensional signals. This representation, which is explicitly in terms of an array of coefficients, reminiscent of Fourier series, is stable. Among its other properties, (a) the shape of the resolution cell in the 'phase-space' is variable even at a specified scale, depending on the nature of the signal under consideration; and (b) zero crossings at the various scales can be extracted directly from the coefficients. The new representation is illustrated by examples. However, there do remain some basic problems with respect to the new representation.