Analytic Functions, Singularities and Edges: A New Formalism

Abstract

Methods for detecting edges, be they multidimensional or multiresolution, ultimately reduce to finding extremal points, first derivatives or zeros of second derivatives. However, problems such as missing edges, weak edges due to thresholding, derivatives not existing and false edge generation, are some of the consequences. We adopt a new formalism: Edges are singularities of the mathematically smoothest function possible - the complex analytic function. We embed a real image into the real part of an analytic function. After solving the conjugate harmonic problem, edges in discrete images are identified from the imaginary part. The analytic function model is inherently two-dimensional and an invariant measure. Comparisons are made with other standard edge detection methods. We outline issues that need to be considered for establishing analytic functions for edge detection.