APPLICATIONS OF VARIATIONAL PDE ACCELERATION TO COMPUTER VISION PROBLEMS

Abstract

This dissertation addresses general optimization in the field of computer vision. In this manuscript we derive a new mathematical framework, Partial Differential Equation (PDE) acceleration, for addressing problems in optimization and image processing. We demonstrate the strength of our framework by applying it to problems in image restoration, object tracking, segmentation, and 3D reconstruction. We address these image processing problems using a class of optimization methods known as variational PDEs. First employed in computer vision in the late 1980s, variational PDE methods are an iterative model-based approach that do not rely on extensive training data or model tuning. We also demonstrate for this class of optimization problems how PDE acceleration offers robust performance against classical optimization methods. Beginning with the most straightforward application, image restoration, we then show how to extend PDE acceleration to object tracking, segmentation and a highly non-convex formulation for 3D reconstruction. We also compare across a wide class of optimization methods for functions, curves, and surfaces and demonstrate that not only is PDE acceleration easy to implement, but that it remains competitive in a variety of both convex and non-convex computer vision applications.