Higher-Order Regularization in Computer Vision

Sammanfattning: At the core of many computer vision models lies the minimization of an objective function consisting of a sum of functions with few arguments. The order of the objective function is defined as the highest number of arguments of any summand. To reduce ambiguity and noise in the solution, regularization terms are included into the objective function, enforcing different properties of the solution. The most commonly used regularization is penalization of boundary length, which requires a second-order objective function. Most of this thesis is devoted to introducing higher-order regularization terms and presenting efficient minimization schemes.

One of the topics of the thesis covers a reformulation of a large class of discrete functions into an equivalent form. The reformulation is shown, both in theory and practical experiments, to be advantageous for higher-order regularization models based on curvature and second-order derivatives. Another topic is the parametric max-flow problem. An analysis is given, showing its inherent limitations for large-scale problems which are common in computer vision. The thesis also introduces a segmentation approach for finding thin and elongated structures in 3D volumes. Using a line-graph formulation, it is shown how to efficiently regularize with respect to higher-order differential geometric properties such as curvature and torsion. Furthermore, an efficient optimization approach for a multi-region model is presented which, in addition to standard regularization, is able to enforce geometric constraints such as inclusion or exclusion of different regions. The final part of the thesis deals with dense stereo estimation. A new regularization model is introduced, penalizing the second-order derivatives of a depth or disparity map. Compared to previous second-order approaches to dense stereo estimation, the new regularization model is shown to be more easily optimized.