Images, Numerical Analysis of Singularities and Shock Filters

Abstract

This work is concerned primarily with establishing a natural mathematical framework for the Numerical Analysis of Singularities, a term which we coined for this new evolving branch of numerical analysis. The problem of analyzing singular behavior of nonsmooth functions is implicitly or explicitly ingrained in any successful attempt to extract information from images. The abundance of papers on the so called Edge Detection testifies to this statement. We attempt to make a fresh start by reformulating this old problem in the rigorous context of the Theory of Generalized Functions of several variables with stress put on the computational aspects of essential singularities. We state and prove a variant of the Divergence Theorem for discontinuous functions which we call Fundamental Theorem of Edge Detection, for it is the backbone of the advocated here numerical analysis based on the estimates of contribution5 furnished by the essential singularities of functions. We further extend this analysis to arbitrary order singularities by utilization of the Miranda's calculus of tangential derivatives. With this machinery we are able to explore computationally the internal geometry of singularities including singular, i.e., nonsmooth, singularity boundaries. This theory give5 rise to singularity detection scheme called "rotating thin masks" which is applicable to arbitrary order n-dimensional essential singularities. In the particular implementation we combined first-order detector with derived here various curvature detectors. Preliminary experimental results are presented. We also derive a new class of nonlinear singularity detection schemes based on tensor products of distributions. Finally, a novel computational approach to the problem of image enhancement is presented. We call this construction the Shock Filters, since it is founded on the nonlinear PDE's whose solutions exhibit formation of discontinuous profiles, corresponding to shock waves in gas dynamics. An algorithm for experimental Shock Filter, based on the upwind finite difference scheme is presented and tested on the one and two dimensional data.

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