Analytical Footprints: Compact Representation of Elementary Singularities in Wavelet Bases

Abstract

Abstract-We introduce a family of elementary singularities that are point-Hölder -regular. These singularities are self-similar and are the Green functions of fractional derivative operators; i.e., by suitable fractional differentiation, one retrieves a Dirac function at the exact location of the singularity. We propose to use fractional operator-like wavelets that act as a multiscale version of the derivative in order to characterize and localize singularities in the wavelet domain. We show that the characteristic signature when the wavelet interacts with an elementary singularity has an asymptotic closed-form expression, termed the analytical footprint. Practically, this means that the dictionary of wavelet footprints is embodied in a single analytical form. We show that the wavelet coefficients of the (nonredundant) decomposition can be fitted in a multiscale fashion to retrieve the parameters of the underlying singularity. We propose an algorithm based on stepwise parametric fitting and the feasibility of the approach to recover singular signal representations.

Stochastic Models for Sparse and Piecewise-Smooth Signals

“…Remarkably, this concept carries over to more general classes of operators provided that there is a corresponding spline construction available. In particular, there exist wavelet bases that are perfectly matched to the complete range of fractional derivative operators in Table I: -the fractional spline wavelets which are linked to the operator [42], and well as [43], [44]; -the multidimensional polyharmonic spline wavelets associated with the fractional Laplacian [45]. The latter are thin-plate spline functions that live in the span of the radial basis functions (cf.…”

“…Green's function in Table I). The spline operator-like wavelets discussed earlier turn out to be ideal for this task because the calculations of in (18) can be carried out analytically [44,Theorem 1]. Such a behavior of the wavelet transform of a piecewise-smooth signal is well documented in the literature, but it has not been made as explicit before, to the best of our knowledge.…”

Operator-Like Wavelet Bases of $L_{2}(\mathbb{R}^{d})$

“…What makes the approach even more attractive is that, at each scale, the wavelet space is generated by the shifts of a single function. Our work provides a generalization of some known constructions including: cardinal spline wavelets [5], elliptic wavelets [19], polyharmonic spline wavelets [25,26], Wirtinger-Laplace operator-like wavelets [27], and exponential-spline wavelets [15].…”