# American Institute of Mathematical Sciences

November  2012, 6(4): 623-644. doi: 10.3934/ipi.2012.6.623

## Global minimization of Markov random fields with applications to optical flow

 1 Rice University, Department of Electrical and Computer Engineering, Houston, 77251, United States 2 City University of Hong Kong, Department of Computer Science, Hong Kong, China 3 UCLA, Department of Mathematics, Los Angeles, 90095, United States

Received  September 2011 Revised  June 2012 Published  November 2012

Many problems in image processing can be posed as non-convex minimization problems. For certain classes of non-convex problems involving scalar-valued functions, it is possible to recast the problem in a convex form using a functional lifting'' technique. In this paper, we present a variational functional lifting technique that can be viewed as a generalization of previous works by Pock et. al and Ishikawa. We then generalize this technique to the case of minimization over vector-valued problems, and discuss a condition which allows us to determine when the solution to the convex problem corresponds to a global minimizer. This generalization allows functional lifting to be applied to a wider range of problems then previously considered. Finally, we present a numerical method for solving the convexified problems, and apply the technique to find global minimizers for optical flow image registration.
Citation: Tom Goldstein, Xavier Bresson, Stan Osher. Global minimization of Markov random fields with applications to optical flow. Inverse Problems & Imaging, 2012, 6 (4) : 623-644. doi: 10.3934/ipi.2012.6.623
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