Applying Convex Optimization Techniques to Energy Minimization Problems in Computer Vision

APPLYING CONVEX OPTIMIZATION TECHNIQUES TO ENERGY MINIMIZATION PROBLEMS IN COMPUTER VISION Arvind Bhusnurmath C.J.Taylor Energy minimization is an important technique in computer vision that has been applied to practically all early vision problems. Several methods have been proposed to solve the problem and to date, the most popular ones have been discrete optimization approaches such as graph cuts and belief propagation. While convex programming approaches have been proposed, the resulting problems were practically unsolvable due to their size. This thesis describes a novel approach to applying convex optimization, in particular interior point barrier methods, to solve some energy minimization problems in an efficient manner. The methods exploit the fact that computer vision problems usually have specific structure to them. Two problems are handled in this thesis. In the first, a popular energy minimization technique, graph cuts, is reduced to an unconstrained `1 norm minimization problem. This problem is then solved using linear programming. The resultant algorithm is parallelizable and the thesis provides one possible implementation using a graphical processing unit. Results of using this approach on interactive foregroundbackground segmentation problems are provided. In the second problem, the energy function associated with stereo matching is directly converted into a convex function via an approximation of the data penalties. The resultant linear program is again solved using interior point methods and exhibits very similar structure to the first approach. Results on the standard Middlebury stereo matching dataset will be discussed.