Computational Methods for Computer Vision: Minimal Solvers and Convex Relaxations

Robust fitting of geometric models is a core problem in computer vision. The most common approach is to use a hypothesize-and-test framework, such as RANSAC. In these frameworks the model is estimated from as few measurements as possible, which minimizes the risk of selecting corrupted measurements. These estimation problems are called minimal problems, and they can often be formulated as systems of polynomial equations. In this thesis we present new methods for building so-called minimal solvers or polynomial solvers, which are specialized code for solving such systems. On several minimal problems we improve on the state-of-the-art both with respect to numerical stability and execution time.

In many computer vision problems low rank matrices naturally occur. The rank can serve as a measure of model complexity and typically a low rank is desired. Optimization problems containing rank penalties or constraints are in general difficult. Recently convex relaxations, such as the nuclear norm, have been used to make these problems tractable. In this thesis we present new convex relaxations for rank-based optimization which avoid drawbacks of previous approaches and provide tighter relaxations. We evaluate our methods on a number of real and synthetic datasets and show state-of-the-art results.