A Family of Complex Wavelets for the Characterization of Singularities
Maria Haase · Defense Technical Information Center (DTIC) · 2000
In the presence of oscillating singularities the standard Wavelet Transfrom Modulus Maxima (WTMM) method gives irrelevant information on the Hoelder regularity of the function. In general, two exponents h,beta are necessary to describe the singular behavior of a function f(x), namely the Hoelder exponent h and the oscillation exponent beta describing the local power law divergence of the instantaneous frequency. If f(x) contains oscillating singularities the regularity of the primitive of f(x) depends on beta. In this case the Hoelder exponent does not increase by 1 as in the case of a cusp singularity but by beta + 1. Thus, the singularity spectrum of general functions depends on both exponents, D(h,beta). In order to extract Hoelder exponents and to quantify at the same time the oscillating behavior we propose to use a family of complex progressive wavelets ?psi sub n! with an increasing number of vanishing moments.