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29
A graph cut algorithm for higher-order markov random fields
- IN: INT. CONF. COMPUTER VISION
, 2011
"... 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 ..."
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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.
Markov Random Field Modeling, Inference & Learning in Computer Vision & Image Understanding: A Survey
, 2013
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Generalized Roof Duality for Pseudo-Boolean Optimization
"... The number of applications in computer vision that model higher-order interactions has exploded over the last few years. The standard technique for solving such problems is to reduce the higher-order objective function to a quadratic pseudo-boolean function, and then use roof duality for obtaining a ..."
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Cited by 7 (1 self)
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The number of applications in computer vision that model higher-order interactions has exploded over the last few years. The standard technique for solving such problems is to reduce the higher-order objective function to a quadratic pseudo-boolean function, and then use roof duality for obtaining a lower bound. Roof duality works by constructing the tightest possible lower-bounding submodular function, and instead of optimizing the original objective function, the relaxation is minimized. We generalize this idea to polynomials of higher degree, where quadratic roof duality appears as a special case. Optimal relaxations are defined to be the ones that give the maximum lower bound. We demonstrate that important properties such as persistency still hold and how the relaxations can be efficiently constructed for general cubic and quartic pseudo-boolean functions. From a practical point of view, we show that our relaxations perform better than state-ofthe-art for a wide range of problems, both in terms of lower bounds and in the number of assigned variables. 1.
Landmark/Image-based Deformable Registration of Gene Expression Data
"... Analysis of gene expression patterns in brain images obtained from high-throughput in situ hybridization requires accurate and consistent annotations of anatomical regions/subregions. Such annotations are obtained by mapping an anatomical atlas onto the gene expression images through intensity- and/ ..."
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Cited by 6 (4 self)
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Analysis of gene expression patterns in brain images obtained from high-throughput in situ hybridization requires accurate and consistent annotations of anatomical regions/subregions. Such annotations are obtained by mapping an anatomical atlas onto the gene expression images through intensity- and/or landmark-based registration methods or deformable model-based segmentation methods. Due to the complex appearance of the gene expression images, these approaches require a pre-processing step to determine landmark correspondences in order to incorporate landmark-based geometric constraints. In this paper, we propose a novel method for landmark-constrained, intensity-based registration without determining landmark correspondences a priori. The proposed method performs dense image registration and identifies the landmark correspondences, simultaneously, using a single higher-order Markov Random Field model. In addition, a machine learning technique is used to improve the discriminating properties of local descriptors for landmark matching by projecting them in a Hamming space of lower dimension. We qualitatively show that our method achieves promising results and also compares well, quantitatively, with the expert’s annotations, outperforming previous methods. 1.
Generalized Roof Duality for Multi-Label Optimization: Optimal
"... Abstract. 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 ..."
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Cited by 5 (1 self)
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Abstract. 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.
Maximum Persistency in Energy Minimization
"... We consider discrete pairwise energy minimization prob-lem (weighted constraint satisfaction, max-sum labeling) and methods that identify a globally optimal partial assign-ment of variables. When finding a complete optimal assign-ment is intractable, determining optimal values for a part of variable ..."
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Cited by 4 (1 self)
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We consider discrete pairwise energy minimization prob-lem (weighted constraint satisfaction, max-sum labeling) and methods that identify a globally optimal partial assign-ment of variables. When finding a complete optimal assign-ment 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) ver-ifiable in polynomial time (2) invariant to reparametriza-tion of the problem and permutation of labels and (3) in-cludes many existing sufficient conditions as special cases. We pose the problem of finding the maximum optimal par-tial assignment identifiable by the new sufficient condition. A polynomial method is proposed which is guaranteed to
Partial optimality by pruning for MAP-inference with general graphical models
- In CVPR
, 2014
"... 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 initiali ..."
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Cited by 4 (1 self)
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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 cri-terion for partial optimality. We show that our prun-ing strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous ap-proaches in terms of the number of persistently labelled variables. The method is very general, as it is appli-cable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered re-laxed problem. Our method’s runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem. 1.
Non-rigid 2D-3D Medical Image Registration using Markov Random Fields
, 2013
"... Abstract. The aim of this paper is to propose a novel mapping algorithm between 2D images and a 3D volume seeking simultaneously a linear plane transformation and an in-plane dense deformation. We adopt a metric free locally over-parametrized graphical model that combines linear and deformable param ..."
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Cited by 3 (1 self)
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Abstract. The aim of this paper is to propose a novel mapping algorithm between 2D images and a 3D volume seeking simultaneously a linear plane transformation and an in-plane dense deformation. We adopt a metric free locally over-parametrized graphical model that combines linear and deformable parameters within a coupled formulation on a 5-dimensional space. Image similarity is encoded in singleton terms, while geometric linear consistency of the solution (common/single plane) and in-plane deformations smoothness are modeled in a pair-wise term. The robustness of the method and its promising results with respect to the state of the art demonstrate the extreme potential of this approach.
A Primal-Dual Algorithm for Higher-Order Multilabel Markov Random Fields
"... Graph cuts method such as α-expansion [4] and fu-sion moves [22] have been successful at solving many optimization problems in computer vision. Higher-order Markov Random Fields (MRF’s), which are important for numerous applications, have proven to be very difficult, es-pecially for multilabel MRF’s ..."
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Cited by 3 (0 self)
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Graph cuts method such as α-expansion [4] and fu-sion moves [22] have been successful at solving many optimization problems in computer vision. Higher-order Markov Random Fields (MRF’s), which are important for numerous applications, have proven to be very difficult, es-pecially for multilabel MRF’s (i.e. more than 2 labels). In this paper we propose a new primal-dual energy minimiza-tion method for arbitrary higher-order multilabel MRF’s. Primal-dual methods provide guaranteed approximation bounds, and can exploit information in the dual variables to improve their efficiency. Our algorithm generalizes the PD3 [19] technique for first-order MRFs, and relies on a variant of max-flow that can exactly optimize certain higher-order binary MRF’s [14]. We provide approximation bounds sim-ilar to PD3 [19], and the method is fast in practice. It can optimize non-submodular MRF’s, and additionally can in-corporate problem-specific knowledge in the form of fusion proposals. We compare experimentally against the exist-ing approaches that can efficiently handle these difficult en-ergy functions [6, 10, 11]. For higher-order denoising and stereo MRF’s, we produce lower energy while running sig-nificantly faster. 1. Higher-order MRFs There is widespread interest in higher-order MRF’s for problems like denoising [23]and stereo [30], yet the result-ing energy functions have proven to be very difficult to min-imize. The optimization problem for a higher-order MRF is defined over a hypergraph with vertices V and cliques C plus a label set L. We minimize the cost of the labeling f: L|V | → < defined by f(x) =
Structured learning of sum-of-submodular higher order energy functions
, 1309
"... Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow [18] has had significant impact in computer vision [5, 20, 27]. In this paper we address the important class of sum-of-submodular (SoS) functions [2, 17], which can be efficient ..."
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Cited by 3 (1 self)
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Submodular functions can be exactly minimized in polynomial time, and the special case that graph cuts solve with max flow [18] has had significant impact in computer vision [5, 20, 27]. In this paper we address the important class of sum-of-submodular (SoS) functions [2, 17], which can be efficiently minimized via a variant of max flow called submodular flow [6]. SoS functions can naturally express higher order priors involving, e.g., local image patches; however, it is difficult to fully exploit their expressive power because they have so many parameters. Rather than trying to formulate existing higher order priors as an SoS function, we take a discriminative learning approach, effectively searching the space of SoS functions for a higher order prior that performs well on our training set. We adopt a structural SVM approach [14, 33] and formulate the training problem in terms of quadratic programming; as a result we can efficiently search the space of SoS priors via an extended cutting-plane algorithm. We also show how the state-of-the-art max flow method for vision problems [10] can be modified to efficiently solve the submodular flow problem. Experimental comparisons are made against the OpenCV implementation of the GrabCut interactive segmentation technique [27], which uses hand-tuned parameters instead of machine learning. On a standard dataset [11] our method learns higher order priors with hundreds of parameter values, and produces significantly better segmentations. While our focus is on binary labeling problems, we show that our techniques can be naturally generalized to handle more than two labels. 1.
