Minimizing energy functions on 4-connected lattices using elimination

We describe an energy minimization algorithm for functions defined on 4-connected lattices, of the type usually encountered in problems involving images. Such functions are often minimized using graph-cuts/max-flow, but this method is only applicable to submodular problems. In this paper, we describe an algorithm that will solve any binary problem, irrespective of whether it is submodular or not, and for multilabel problems we use alpha-expansion. The method is based on the elimination algorithm, which eliminates nodes from the graph until the remaining function is submodular. It can then be solved using max-flow. Values of eliminated variables are recovered using back-substitution. We compare the algorithm's performance against alternative methods for solving non-submodular problems, with favourable results.

[1]  R. Szeliski.  Locally adapted hierarchical basis preconditioning , 2006, SIGGRAPH 2006.

[2]  Vladimir Kolmogorov,et al.  Convergent Tree-Reweighted Message Passing for Energy Minimization , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[3]  Vladimir Kolmogorov,et al.  Optimizing Binary MRFs via Extended Roof Duality , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[4]  Endre Boros,et al.  Pseudo-Boolean optimization , 2002, Discret. Appl. Math..

[5]  Nikos Komodakis,et al.  Performance vs computational efficiency for optimizing single and dynamic MRFs: Setting the state of the art with primal-dual strategies , 2008, Comput. Vis. Image Underst..

[6]  Richard Szeliski,et al.  Locally adapted hierarchical basis preconditioning , 2006, SIGGRAPH '06.

[7]  Olga Veksler,et al.  Graph cut with ordering constraints on labels and its applications , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[8]  Olga Veksler,et al.  Fast approximate energy minimization via graph cuts , 2001, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[9]  Pushmeet Kohli,et al.  Dynamic Graph Cuts for Efficient Inference in Markov Random Fields , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[10]  Vladimir Kolmogorov,et al.  What energy functions can be minimized via graph cuts? , 2002, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[11]  Nikos Komodakis,et al.  Approximate Labeling via Graph Cuts Based on Linear Programming , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[12]  Richard Szeliski,et al.  A Comparative Study of Energy Minimization Methods for Markov Random Fields with Smoothness-Based Priors , 2008, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[13]  Daniel Freedman,et al.  Energy minimization via graph cuts: settling what is possible , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[14]  Vladimir Kolmogorov,et al.  Minimizing Nonsubmodular Functions with Graph Cuts-A Review , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.