The analytic edge - image reconstruction from edge data via the Cauchy Integral

Abstract

A novel image reconstruction algorithm from edges (image gradients) follows from the Sokhostki-Plemelj Theorem of complex analysis, an elaboration of the standard Cauchy (Singular) Integral. This algorithm demonstrates the use of Singular Integral Equation methods to image processing, extending the more common use of Partial Differential Equations (e.g. based on variants of the Diffusion or Poisson equations). The Cauchy Integral approach has a deep connection to and sheds light on the (linear and non-linear) diffusion equation, the retinex algorithm and energy-based image regularization. It extends the commonly understood local definition of an edge to a global, complex analytic structure - the analytic edge - the contrast weighted kernel of the Cauchy Integral. Superposition of the set of analytic edges provides a "filled-in" image which is the piece-wise analytic image corresponding to the edge (gradient data) supplied. This is a fully parallel operation which avoids the time penalty associated with iterative solutions and thus is compatible with the short time (about 150 milliseconds) that is biologically available for the brain to construct a perceptual image from edge data. Although this algorithm produces an exact reconstruction of a filled-in image from the gradients of that image, slight modifications of it produce images which correspond to perceptual reports of human observers when presented with a wide range of "visual contrast illusion" images.