IRMACS: The Interdisciplinary Colloquium: "Geometry of Image Processing"

Abstract

General relativity is often thought of as "rocket science". Indeed, engineers have to worry about relativistic effects pretty much only when building rockets. Yet ideas and mathematical techniques of general relativity are much more useful in everyday life than one might think. In this colloquium, I will discuss how differential geometry can be used in image processing and computer vision. The idea of a metric appears early in color theory, as perceptual distance measures in many color spaces are not uniform. The real fun begins when one thinks about merging spatial and feature space information into a single curved manifold. This idea results in a broad class of image enhancement and segmentation algorithms, which can have very good performance. For example, covariant formulation of diffusion equation leads to a state-of-the-art noise removal algorithm using non-linear PDEs, and variational methods minimizing geometrical "energy" functionals can be used to segment an image.