Geometric and Topological Variational Methods for Imaging and Computer Vision

Abstract

The great challenge in signal/image processing is to devise computationally efficient and optimal algorithms for estimating signals/images contaminated by noise and preserving their geometrical structure. The first problem addressed is this thesis is image denoising formulated in the calculus of variations framework. We propose robust variational models for image denoising by numerically solving partial differential equations. The core idea behind our proposed approaches is to use geometric insight in helping construct regularizing functionals and avoiding a subjective choice of a prior in maximum a posteriori estimation. Using tools from robust statistics and information theory, we show that we can extend this strategy and develop two gradient descent flows for image denoising with a demonstrated performance through illustrating experimental results.

The rest of the thesis is devoted to a joint exploitation of geometry and topology of objects for as parsimonious as possible representation of objects and its subsequent application in object classification and recognition problems. Attempting to extend current approaches to image registration which have generally relied on the assumption of 2D images, we propose a novel technique for 3D object matching using a joint exploitation of geometry and topology. The key idea consists of capturing geometry along all topologically homogeneous parts of an object by way of level curves superimposed on a Reeb graph usually extracted by way of the object critical points. This resulting skeletal representation, however, is not rotationally invariant. We propose a new methodology called {em geodesic shape distribution} that lifts this limitation and which we apply to 3D object matching. The central idea is to encode a 3D shape into a 1D geodesic shape distribution. Object matching is then achieved by calculating an information-theoretic measure of dissimilarity between the resulting geodesic shape distributions in a lower dimensional space. Illustrating numerical experiments with synthetic and real data are provided to demonstrate the potential and the much improved performance of the proposed methodology in 3D object matching.