Singularity detection and processing with wavelets

Most of a signal information is often found in irregular structures and transient phenomena. We review the mathematical characterization of singularities with Lipschitz exponents. The main theorems that estimate local Lipschitz exponents of functions, from the evolution across scales of their wavelet transform are explained. We then prove that the local maxima of a wavelet transform detect the location of irregular structures and provide numerical procedures to compute their Lipschitz exponents. The wavelet transform of singularities with fast oscillations have a different behavior that we study separately. We show that the size of the oscillations can be measured from the wavelet transform local maxima. It has been shown that one and two-dimensional signals can be reconstructed from the local maxima of their wavelet transform [14]. As an application, we develop an algorithm that removes white noises by discriminating the noise and the signal singularities through an analysis of their wavelet transform maxima. In two-dimensions, the wavelet transform maxima indicate the location of edges in images. We extend the denoising algorithm for image enhancement applications. hhhhhhhhhhhhhhhh This research was supported by the NSF grant IRI-890331 and Airforce grant AFOSR-90-0040.