SINGULARITY DETECTION AND PROCESSING WITH WAVELETS

Most of a signal information is often carried by irregular structures and transient phenomena. The mathematical characterization of singularities with Lipschitz exponents is explained. Theorems are reviewed that estimate local Lipschitz exponents of functions from the evolution across scales of their wavelet transform. It is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their Lipschitz exponents. The wavelet transform of singularities with fast oscillations have a particular behavior that is studied separately. The local frequency of such oscillations are measured from the wavelet transform modulus maxima. It has been shown numerically that one- and two-dimensional signals can be reconstructed, with a good approximation, from the local maxima of their wavelet transform modulus. As an application, an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales. In two-dimensions, the wavelet transform maxima indicate the location of edges in images. The denoising algorithm is extended for image enhancement.