While Markov Random Fields (MRFs) are widely used in computer vision, they present a quite challenging inference problem. MRF inference can be accelerated by pre-processing techniques like Dead End Elimination (DEE) or QPBO-based approaches which compute the optimal labeling of a subset of variables. These techniques are guaranteed to never wrongly label a variable but they often leave a large number of variables unlabeled. We address this shortcoming by interpreting pre-processing as a classification problem, which allows us to trade off false positives (i.e., giving a variable an incorrect label) versus false negatives (i.e., failing to label a variable). We describe an efficient discriminative rule that finds optimal solutions for a subset of variables. Our technique provides both per-instance and worst-case guarantees concerning the quality of the solution. Empirical studies were conducted over several benchmark datasets. We obtain a speedup factor of 2 to 12 over expansion moves without preprocessing, and on difficult non-submodular energy functions produce slightly lower energy.
A popular approach to the inference problem is to find the optimal labeling for a subset of the variables @cite_17 @cite_3 @cite_24 @cite_41 @cite_23 @cite_40 @cite_2 @cite_13 @cite_4 . A partial labeling that holds in every global minimizer is said to be @cite_32 . Techniques like QPBO @cite_32 @cite_18 find an optimal partial labeling by enforcing an even stronger condition: a partial labeling that will decrease the energy if it is substituted into any complete labeling. QPBO is naturally viewed as a pre-processing method since it finds persistent partial labelings, and leaves the task of labeling the remaining variables to some other algorithm. This stronger property is sometimes called an @cite_32 , which was generalized by @cite_40 . QPBO in particular is widely used in computer vision since it often finds the correct label for the majority of the variables.
@cite_18 @cite_4 @cite_41 @cite_32 @cite_3 @cite_24 @cite_40 @cite_23 @cite_2 @cite_13 @cite_17
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Optimization techniques based on graph cuts have become a standard tool for many vision applications. These techniques allow to minimize efficiently certain energy functions corresponding to pairwise Markov random fields (MRFs). Currently, there is an accepted view within the computer vision community that graph cuts can only be used for optimizing a limited class of MRF energies (e.g., submodular functions). In this survey, we review some results that show that graph cuts can be applied to a much larger class of energy functions (in particular, nonsubmodular functions). While these results are well-known in the optimization community, to our knowledge they were not used in the context of computer vision and MRF optimization. We demonstrate the relevance of these results to vision on the problem of binary texture restoration. We extend the concept of generalized roof duality from pseudo-boolean functions to real-valued functions over multi-label variables. In particular, we prove that an analogue of the persistency property holds for energies of any order with any number of linearly ordered labels. Moreover, we show how the optimal submodular relaxation can be constructed in the first-order case. Consider a convex relaxation @math of a pseudo-boolean function @math . We say that the relaxation is totally half-integral if @math is a polyhedral function with half-integral extreme points @math , and this property is preserved after adding an arbitrary combination of constraints of the form @math , @math , and @math where @math is a constant. A well-known example is the roof duality relaxation for quadratic pseudo-boolean functions @math . We argue that total half-integrality is a natural requirement for generalizations of roof duality to arbitrary pseudo-boolean functions. Our contributions are as follows. First, we provide a complete characterization of totally half-integral relaxations @math by establishing a one-to-one correspondence with bisubmodular functions . Second, we give a new characterization of bisubmodular functions. Finally, we show some relationships between general totally half-integral relaxations and relaxations based on the roof duality. We consider the problem of optimizing multilabel MRFs, which is in general NP-hard and ubiquitous in low-level computer vision. One approach for its solution is to formulate it as an integer linear programming and relax the integrality constraints. The approach we consider in this paper is to first convert the multi-label MRF into an equivalent binary-label MRF and then to relax it. The resulting relaxation can be efficiently solved using a maximum flow algorithm. Its solution provides us with a partially optimal labelling of the binary variables. This partial labelling is then easily transferred to the multi-label problem. We study the theoretical properties of the new relaxation and compare it with the standard one. Specifically, we compare tightness, and characterize a subclass of problems where the two relaxations coincide. We propose several combined algorithms based on the technique and demonstrate their performance on challenging computer vision problems. We consider discrete pairwise energy minimization problem (weighted constraint satisfaction, max-sum labeling) and methods that identify a globally optimal partial assignment of variables. When finding a complete optimal assignment is intractable, determining optimal values for a part of variables is an interesting possibility. Existing methods are based on different sufficient conditions. We propose a new sufficient condition for partial optimality which is: (1) verifiable in polynomial time (2) invariant to reparametrization of the problem and permutation of labels and (3) includes many existing sufficient conditions as special cases. We pose the problem of finding the maximum optimal partial assignment identifiable by the new sufficient condition. A polynomial method is proposed which is guaranteed to assign same or larger part of variables than several existing approaches. The core of the method is a specially constructed linear program that identifies persistent assignments in an arbitrary multi-label setting. We consider energy minimization for undirected graphical models, also known as the MAP-inference problem for Markov random fields. Although combinatorial methods, which return a provably optimal integral solution of the problem, made a significant progress in the past decade, they are still typically unable to cope with large-scale datasets. On the other hand, large scale datasets are often defined on sparse graphs and convex relaxation methods, such as linear programming relaxations then provide good approximations to integral solutions. We propose a novel method of combining combinatorial and convex programming techniques to obtain a global solution of the initial combinatorial problem. Based on the information obtained from the solution of the convex relaxation, our method confines application of the combinatorial solver to a small fraction of the initial graphical model, which allows to optimally solve much larger problems. We demonstrate the efficacy of our approach on a computer vision energy minimization benchmark. We consider the NP-hard problem of MAP-inference for undirected discrete graphical models. We propose a polynomial time and practically efficient algorithm for finding a part of its optimal solution. Specifically, our algorithm marks some labels of the considered graphical model either as (i) optimal, meaning that they belong to all optimal solutions of the inference problem; (ii) non-optimal if they provably do not belong to any solution. With access to an exact solver of a linear programming relaxation to the MAP-inference problem, our algorithm marks the maximal possible (in a specified sense) number of labels. We also present a version of the algorithm, which has access to a suboptimal dual solver only and still can ensure the (non-)optimality for the marked labels, although the overall number of the marked labels may decrease. We propose an efficient implementation, which runs in time comparable to a single run of a suboptimal dual solver. Our method is well-scalable and shows state-of-the-art results on computational benchmarks from machine learning and computer vision. We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal non-relaxed integral solution. Our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the MAP-inference problem and iteratively prunes those, which do not satisfy our criterion for partial optimality. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous approaches in terms of the number of persistently labelled variables. The method is very general, as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem. Our method's runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem. THE prediction of a protein's tertiary structure is still a considerable problem because the huge amount of possible conformational space1 makes it computationally difficult. With regard to side-chain modelling, a solution has been attempted by the grouping of side-chain conformations into representative sets of rotamers2–5. Nonetheless, an exhaustive combinatorial search is still limited to carefully identified packing units5,6containing a limited number of residues. For larger systems other strategies had to be develop-ped, such as the Monte Carlo Procedure6,7 and the genetic algorithm and clustering approach8. Here we present a theorem, referred to as the 'dead-end elimination' theorem, which imposes a suitable condition to identify rotamers that cannot be members of the global minimum energy conformation. Application of this theorem effectively controls the computational explosion of the rotamer combinatorial problem, thereby allowing the determination of the global minimum energy conformation of a large collection of side chains.