A graph cut algorithm for higher-order Markov Random Fields

Abstract

Higher-order Markov Random Fields, which can capture important properties of natural images, have become increasingly important in computer vision. While graph cuts work well for first-order MRF's, until recently they have rarely been effective for higher-order MRF's. Ishikawa's graph cut technique [8,9] shows great promise for many higher-order MRF's. His method transforms an arbitrary higher-order MRF with binary labels into a first-order one with the same minima. If all the terms are submodular the exact solution can be easily found; otherwise, pseudoboolean optimization techniques can produce an optimal labeling for a subset of the variables. We present a new transformation with better performance than [8,9], both theoretically and experimentally. While [8,9] transforms each higher-order term independently, we transform a group of terms at once. For n binary variables, each of which appears in terms with k other variables, at worst we produce n non-submodular terms, while [8,9] produces O(nk). We identify a local completeness property that makes our method perform even better, and show that under certain assumptions several important vision problems (including common variants of fusion moves) have this property. Running on the same field of experts dataset used in [8,9] we optimally label significantly more variables (96% versus 80%) and converge more rapidly to a lower energy. Preliminary experiments suggest that some other higher-order MRF's used in stereo [20] and segmentation [1] are also locally complete and would thus benefit from our work.

Tighter Relaxations for Higher-Order Models Based on Generalized Roof Duality

“…We will use the methods listed in Table 2. [12] GRD-gen GRD using generators (Section 3) GRD-ext GRD-gen in combination with symmetric extension (Section 4) Fix et al The reductions proposed in [25] HOCR The reductions proposed in [22] …”

Higher-Order Clique Reduction without Auxiliary Variables

“…While this does not have the guarantee that the minimization problem stays the same, used with the fusionmove algorithm it minimizes the energy much faster than the state of the art such as Fix et al [6].…”

“…Although their importance was long understood and there were earlier attempts on utilizing them [5,18,21], their use [9,27] and research into minimization of various classes of higherorder MRF energies [1,6,8,11,12,17,20,24,25,26] have intensified significantly in the past few years. While there are useful higher-order MRF energies with specific forms that can be efficiently minimized [15,16], minimization of general higher-order energy is needed to utilize sophisticated priors, especially if it is learned from data as in the case of Fields of Experts (FoEs) [23].…”

“…It can be seen that the proposed ELC reduction produces an energy with much less variables. From left to right: ELC reduction followed by the reduction in [10,11], ELC reduction followed by the reduction in Fix et al [6], approximate ELC reduction, the reduction in [10,11], and the reduction in [6]. The numbers are the average over about 180 fusionmove iterations while denoising 10 test images.…”

“…To extend these methods to the higher-order case, reduction of higher-order potentials with binary labels into firstorder cases have been proposed [6,7,8,11,13]. The idea is to add new variables to reduce the order of the energy, as discussed in the next section.…”