Geometric optimization for computer vision

Abstract

Drawing ideas from differential geometry and optimization, this thesis presents novel parameterization-based framework to address optimization problems formulated on a differentiable manifold. The framework views the manifold as a collection of local coordinate charts. It involves successive parameterizations of a manifold, carrying out optimization of the local cost function in parameter space and then projecting the optimal vector back to the manifold. Several algorithms based on this approach are devised and applied to four computer vision tasks involving recovering pose information from images. First, we cast 2D-3D pose estimation as an optimization problem on the intersection of the special orthogonal group and a cone. We move on to estimate the essential matrix by minimizing a smooth function over the essential manifold. This is followed by formulating the problem of locating quadratic surfaces as an optimization problem cast on the special Euclidean group. Last, we demonstrate how one could simultaneously register multiple partial views of a 3D object within a common coordinate frame by solving an optimization problem involving the N -fold product of the special orthogonal group with itself. A mathematical proof establishes the local quadratic convergent rate of the Newton-like algorithms. Simulation results demonstrate the robustness of techniques against measurement noise and / or occlusion. New closed form calculations for the problems serve as a good initial estimate for any iterative algorithm presented, and give exact solution in the noise free case. The algorithmic technique and mathematical insights developed appear applicable to many problems in computer vision, as well as in other areas.