Early vision is the process occurring before any semantic interpretation of an image takes place. Motion estimation, object segmentation and detection are all parts of early vision, but recognition is not. Some models in early vision are easy to perform inference with—they are tractable. Others describe the reality well—they have high fidelity. This thesis improves the tractability-fidelity trade-off of the current state of the art by introducing new discrete methods for image segmentation and other problems of early vision.
The first part studies pseudo-boolean optimization, both from a theoretical perspective as well as a practical one by introducing new algorithms. The main result is the generalization of the roof duality concept to polynomials of higher degree than two. Another focus is parallelization; discrete optimization methods for multi-core processors, computer clusters, and graphical processing units are presented.
Remaining in an image segmentation context, the second part studies parametric problems where a set of model parameters and a segmentation are estimated simultaneously. For a small number of parameters these problems can still be optimally solved. One application is an optimal method for solving the two-phase Mumford-Shah functional.
The third part shifts the focus to curvature regularization—where the commonly used length and area penalization is replaced by curvature in two and three dimensions. These problems can be discretized over a mesh and special attention is given to the mesh geometry. Specifically, hexagonal meshes in the plane are compared to square ones and a method for generating adaptive meshes is introduced and evaluated. The framework is then extended to curvature regularization of surfaces.
Finally, the thesis is concluded by three applications to early vision problems: cardiac MRI segmentation, image registration, and cell classification.