Many recent advances in computer vision have demonstrated the impressive power of dense and nonsubmodular energy functions in solving visual labeling problems. However, minimizing such energies is challenging. None of existing techniques (such as s-t graph cut, QPBO, BP and TRW-S) can individually do this well. In this paper, we present an efficient method, namely ESSP, to optimize binary MRFs with arbitrary pairwise potentials, which could be nonsubmodular and with dense connectivity. We also provide a comparative study of our approach and several recent promising methods. From our study, we make some reasonable recommendations of combining existing methods that perform the best in different situations for this challenging problem. Experimental results validate that for dense and nonsubmodular energy functions, the proposed approach can usually obtain lower energies than the best combination of other techniques using comparably reasonable time.
QPBF is also a general form of the energy functions of binary MRFs. State-of-the-art methods for binary MRF labeling include BP @cite_0 , TRW @cite_23 @cite_21 and graph cuts @cite_27 . An important conclusion is that any submodular QPBF can be exactly minimized by solving an st-mincut problem on a directed graph @cite_27 . For nonsubmodular QPBFs, the roof duality @cite_19 is used to find an optimal partial labeling, which is called QPBO and two variants P and I @cite_10 @cite_26 . Our algorithm is in contrast to the directed graph formulation @cite_27 and QPBO(P,I) @cite_26 . First, the undirected graph characterization maps a QPBF to a symmetric set function, thus making us easily suppress the supermodular part, which is important for the optimality of our algorithm. Second, we use extended SSP to handle nonsubmodular terms. As a result, our algorithm solves a minimum cut problem of an undirected graph, which needs only half the number of nodes of QPBO(P,I).
@cite_26 @cite_21 @cite_0 @cite_19 @cite_27 @cite_23 @cite_10
2110306576 1582030780 2004559848 2137117160
This paper addresses the problem of exactly inferring the maximum a posteriori solutions of discrete multi-label MRFs or CRFs with higher order cliques. We present a framework to transform special classes of multi-label higher order functions to submodular second order Boolean functions (referred to as Fs 2), which can be minimized exactly using graph cuts and we characterize those classes. The basic idea is to use two or more Boolean variables to encode the states of a single multi-label variable. There are many ways in which this can be done and much interesting research lies in finding ways which are optimal or minimal in some sense. We study the space of possible encodings and find the ones that can transform the most general class of functions to Fs 2. Our main contributions are two-fold. First, we extend the subclass of submodular energy functions that can be minimized exactly using graph cuts. Second, we show how higher order potentials can be used to improve single view 3D reconstruction results. We believe that our work on exact minimization of higher order energy functions will lead to similar improvements in solutions of other labelling problems. Graph cut algorithms [9], commonly used in computer vision, solve a first-order MRF over binary variables. The state of the art for this NP-hard problem is QPBO [1,2], which finds the values for a subset of the variables in the global minimum. While QPBO is very effective overall there are still many difficult problems where it can only label a small subset of the variables. We propose a new approach that, instead of optimizing the original graphical model, instead optimizes a tractable sub-model, defined as an energy function that uses a subset of the pairwise interactions of the original, but for which exact inference can be done efficiently. Our Bounded Treewidth Subgraph (k-BTS) algorithm greedily computes a large weight treewidth-k subgraph of the signed graph, then solves the energy minimization problem for this subgraph by dynamic programming. The edges omitted by our greedy method provide a per-instance lower bound. We demonstrate promising experimental results for binary deconvolution, a challenging problem used to benchmark QPBO [2]: our algorithm performs an order of magnitude better than QPBO or its common variants [4], both in terms of energy and accuracy, and the visual quality of our output is strikingly better as well. We also obtain a significant improvement in energy and accuracy on a stereo benchmark with 2nd order priors [5], although the improvement in visual quality is more modest. Our method's running time is comparable to QPBO. In this work, we propose the construction of two-channel wavelet filter banks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graph-signals as we build a framework in which many concepts from the classical signal processing domain, such as Fourier decomposition, signal filtering and downsampling can be extended to graph domain. Especially, we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals. This property of bipartite graphs, allows us to design critically sampled two-channel filter banks, and we propose quadrature mirror filters (referred to as graph-QMF) for bipartite graph which cancel aliasing and lead to perfect reconstruction. For arbitrary graphs we present a bipartite subgraph decomposition which produces an edge-disjoint collection of bipartite subgraphs. Graph-QMFs are then constructed on each bipartite subgraph leading to “multi-dimensional” separable wavelet filter banks on graphs. Our proposed filter banks are critically sampled and we state necessary and sufficient conditions for orthogonality, aliasing cancellation and perfect reconstruction. The filter banks are realized by Chebychev polynomial approximations. Many computer vision applications rely on the efficient optimization of challenging, so-called non-submodular, binary pairwise MRFs. A promising graph cut based approach for optimizing such MRFs known as "roof duality" was recently introduced into computer vision. We study two methods which extend this approach. First, we discuss an efficient implementation of the "probing" technique introduced recently by (2006). It simplifies the MRF while preserving the global optimum. Our code is 400-700 faster on some graphs than the implementation of the work of (2006). Second, we present a new technique which takes an arbitrary input labeling and tries to improve its energy. We give theoretical characterizations of local minima of this procedure. We applied both techniques to many applications, including image segmentation, new view synthesis, super-resolution, diagram recognition, parameter learning, texture restoration, and image deconvolution. For several applications we see that we are able to find the global minimum very efficiently, and considerably outperform the original roof duality approach. In comparison to existing techniques, such as graph cut, TRW, BP, ICM, and simulated annealing, we nearly always find a lower energy.