Analytical Footprints: Compact Representation of Elementary Singularities in Wavelet Bases

We introduce a family of elementary singularities that are point-Hölder α-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 δ 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.