Singularity detection and processing with wavelets
Information Theory, IEEE Transactions on, 38 (2):
617--643 (March 1992)DOI: 10.1109/18.119727
Abstract
The mathematical characterization of singularities with Lipschitz
exponents is reviewed. Theorems that estimate local Lipschitz exponents
of functions from the evolution across scales of their wavelet transform
are reviewed. 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 has a
particular behavior that is studied separately. The local frequency of
such oscillations is 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
