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TriangleFlow: Optical Flow with Triangulationbased HigherOrder Likelihoods
"... Abstract. We use a simple yet powerful higherorder conditional random field (CRF) to model optical flow. It consists of a standard photoconsistency cost and a prior on affine motions both modeled in terms of higherorder potential functions. Reasoning jointly over a large set of unknown variables p ..."
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Abstract. We use a simple yet powerful higherorder conditional random field (CRF) to model optical flow. It consists of a standard photoconsistency cost and a prior on affine motions both modeled in terms of higherorder potential functions. Reasoning jointly over a large set of unknown variables provides more reliable motion estimates and a robust matching criterion. One of the main contributions is that unlike previous regionbased methods, we omit the assumption of constant flow. Instead, we consider local affine warps whose likelihood energy can be computed exactly without approximations. This results in a tractable, socalled, higherorder likelihood function. We realize this idea by employing triangulation meshes which immensely reduce the complexity of the problem. Optimization is performed by hierarchical QPBO moves and an adaptive mesh refinement strategy. Experiments show that we achieve highquality motion fields on several data sets including the Middlebury optical flow database. 1
Markov Random Field Modeling, Inference & Learning in Computer Vision & Image Understanding: A Survey
, 2013
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Classes of submodular constraints expressible by graph cuts
 IN: PROCEEDINGS OF THE 14TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONTRAINT PROGRAMMING (CP’08). VOLUME 5202 OF LNCS. (2008
, 2008
"... Submodular constraints play an important role both in theory and practice of valued constraint satisfaction problems (VCSPs). It has previously been shown, using results from the theory of combinatorial optimisation, that instances of VCSPs with submodular constraints can be minimised in polynomial ..."
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Cited by 13 (2 self)
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Submodular constraints play an important role both in theory and practice of valued constraint satisfaction problems (VCSPs). It has previously been shown, using results from the theory of combinatorial optimisation, that instances of VCSPs with submodular constraints can be minimised in polynomial time. However, the general algorithm is of order O(n 6) and hence rather impractical. In this paper, by using results from the theory of pseudoBoolean optimisation, we identify several broad classes of submodular constraints over a Boolean domain which are expressible using binary submodular constraints, and hence can be minimised in cubic time. Furthermore, we describe how our results translate to certain optimisation problems arising in computer vision.
Global interactions in random field models: A potential function ensuring connectedness
 SIAM J. Img. Sci
"... Abstract. Markov random field (MRF) models, including conditional random field models, are popular in computer vision. However, in order to be computationally tractable, they are limited to incorporating only local interactions and cannot model global properties such as connectedness, which is a pot ..."
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Abstract. Markov random field (MRF) models, including conditional random field models, are popular in computer vision. However, in order to be computationally tractable, they are limited to incorporating only local interactions and cannot model global properties such as connectedness, which is a potentially useful highlevel prior for object segmentation. In this work, we overcome this limitation by deriving a potential function that forces the output labeling to be connected and that can naturally be used in the framework of recent maximum a posteriori (MAP)MRF linear program (LP) relaxations. Using techniques from polyhedral combinatorics, we show that a provably strong approximation to the MAP solution of the resulting MRF can still be found efficiently by solving a sequence of maxflow problems. The efficiency of the inference procedure also allows us to learn the parameters of an MRF with global connectivity potentials by means of a cutting plane algorithm. We experimentally evaluate our algorithm on both synthetic data and on the challenging image segmentation task of the PASCAL Visual Object Classes 2008 data set. We show that in both cases the addition of a connectedness prior significantly reduces the segmentation error. Key words. Markov random fields, potential functions, large cliques, higharity interactions
Curvature Prior for MRFbased Segmentation and Shape
, 2011
"... Most image labeling problems such as segmentation and image reconstruction are fundamentally illposed and suffer from ambiguities and noise. Higher order image priors encode high level structural dependencies between pixels and are key to overcoming these problems. However, these priors in general ..."
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Cited by 10 (1 self)
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Most image labeling problems such as segmentation and image reconstruction are fundamentally illposed and suffer from ambiguities and noise. Higher order image priors encode high level structural dependencies between pixels and are key to overcoming these problems. However, these priors in general lead to computationally intractable models. This paper addresses the problem of discovering compact representations of higher order priors which allow efficient inference. We propose a framework for solving this problem which uses a recently proposed representation of higher order functions where they are encoded as lower envelopes of linear functions. Maximum a Posterior inference on our learned models reduces to minimizing a pairwise function of discrete variables, which can be done approximately using standard methods. Although this is a primarily theoretical paper, we also demonstrate the practical effectiveness of our framework on the problem of learning a shape prior for image segmentation and reconstruction. We show that our framework can learn a compact representation that approximates a prior that encourages low curvature shapes. We evaluate the approximation accuracy, discuss properties of the trained model, and show various results for shape inpainting and image segmentation. 1
Linear Intensitybased Image Registration by Markov Random Fields and Discrete Optimization
, 2010
"... We propose a framework for intensitybased registration of images by linear transformations, based on a discrete Markov Random Field (MRF) formulation. Here, the challenge arises from the fact that optimizing the energy associated with this problem requires a highorder MRF model. Currently, methods ..."
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We propose a framework for intensitybased registration of images by linear transformations, based on a discrete Markov Random Field (MRF) formulation. Here, the challenge arises from the fact that optimizing the energy associated with this problem requires a highorder MRF model. Currently, methods for optimizing such highorder models are less general, easy to use, and efficient, than methods for the popular secondorder models. Therefore, we propose an approximation to the original energy by an MRF with tractable secondorder terms. The approximation at a certain point p in the parameter space is the normalized sum of evaluations of the original energy at projections of p to twodimensional subspaces. We demonstrate the quality of the proposed approximation by computing the correlation with the original energy, and show that registration can be performed by discrete optimization of the approximated energy in an iteration loop. A search space refinement strategy is employed over iterations to achieve subpixel accuracy, while keeping the number of labels small for efficiency. The proposed framework can encode any similarity measure, is robust to the settings of the internal parameters, and allows an intuitive control of the parameter ranges. We demonstrate the applicability of the framework by intensitybased registration, and 2D3D registration of medical images. The evaluation is performed by random studies and real registration tasks. The tests indicate increased robustness and precision compared to corresponding standard optimization of the original energy, and demonstrate robustness to noise. Finally, the proposed framework
Exploring Compositional High Order Pattern Potentials for Structured Output Learning
"... When modeling structured outputs such as image segmentations, prediction can be improved by accurately modeling structure present in the labels. A key challenge is developing tractable models that are able to capture complex high level structure like shape. In this work, we study the learning of a g ..."
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Cited by 8 (0 self)
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When modeling structured outputs such as image segmentations, prediction can be improved by accurately modeling structure present in the labels. A key challenge is developing tractable models that are able to capture complex high level structure like shape. In this work, we study the learning of a general class of patternlike high order potential, which we call Compositional High Order Pattern Potentials (CHOPPs). We show that CHOPPs include the linear deviation pattern potentials of Rother et al. [26] and also Restricted Boltzmann Machines (RBMs); we also establish the near equivalence of these two models. Experimentally, we show that performance is affected significantly by the degree of variability present in the datasets, and we define a quantitative variability measure to aid in studying this. We then improve CHOPPs performance in high variability datasets with two primary contributions: (a) developing a losssensitive joint learning procedure, so that internal pattern parameters can be learned in conjunction with other model potentials to minimize expected loss;and (b) learning an imagedependent mapping that encourages or inhibits patterns depending on image features. We also explore varying how multiple patterns are composed, and learning convolutional patterns. Quantitative results on challenging highly variable datasets show that the joint learning and imagedependent high order potentials can improve performance. 1.
Fast exact inference for recursive cardinality models
 In Uncertainty in Artificial Intelligence
, 2012
"... Cardinality potentials are a generally useful class of high order potential that affect probabilities based on how many of D binary variables are active. Maximum a posteriori (MAP) inference for cardinality potential models is wellunderstood, with efficient computations taking O(D log D) time. Yet ..."
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Cited by 8 (4 self)
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Cardinality potentials are a generally useful class of high order potential that affect probabilities based on how many of D binary variables are active. Maximum a posteriori (MAP) inference for cardinality potential models is wellunderstood, with efficient computations taking O(D log D) time. Yet efficient marginalization and sampling have not been addressed as thoroughly in the machine learning community. We show that there exists a simple algorithm for computing marginal probabilities and drawing exact joint samples that runs in O(D log 2 D) time, and we show how to frame the algorithm as efficient belief propagation in a low order treestructured model that includes additional auxiliary variables. We then develop a new, more general class of models, termed Recursive Cardinality models, which take advantage of this efficiency. Finally, we show how to do efficient exact inference in models composed of a tree structure and a cardinality potential. We explore the expressive power of Recursive Cardinality models and empirically demonstrate their utility. 1
Generalized Roof Duality for PseudoBoolean Optimization
"... The number of applications in computer vision that model higherorder interactions has exploded over the last few years. The standard technique for solving such problems is to reduce the higherorder objective function to a quadratic pseudoboolean 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 higherorder interactions has exploded over the last few years. The standard technique for solving such problems is to reduce the higherorder objective function to a quadratic pseudoboolean function, and then use roof duality for obtaining a lower bound. Roof duality works by constructing the tightest possible lowerbounding 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 pseudoboolean functions. From a practical point of view, we show that our relaxations perform better than stateoftheart for a wide range of problems, both in terms of lower bounds and in the number of assigned variables. 1.
Higherorder segmentation via multicuts
 CORR ABS/1305.6387
"... Multicuts enable to conveniently represent discrete graphical models for unsupervised and supervised image segmentation, based on local energy functions that exhibit symmetries. The basic Potts model and natural extensions thereof to higherorder models provide a prominent class of representatives, ..."
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Multicuts enable to conveniently represent discrete graphical models for unsupervised and supervised image segmentation, based on local energy functions that exhibit symmetries. The basic Potts model and natural extensions thereof to higherorder models provide a prominent class of representatives, that cover a broad range of segmentation problems relevant to image analysis and computer vision. We show how to take into account such higherorder terms systematically in view of computational inference, and present results of a comprehensive and competitive numerical evaluation of a variety of dedicated cuttingplane algorithms. Our results reveal ways to evaluate a significant subset of models globally optimal, without compromising runtime. Polynomially solvable relaxations are studied as well, along with advanced rounding schemes for postprocessing.