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512
Fast approximate energy minimization via graph cuts
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2001
"... In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when v ..."
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Cited by 2127 (61 self)
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In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when very large moves are allowed. The first move we consider is an αβswap: for a pair of labels α, β, this move exchanges the labels between an arbitrary set of pixels labeled α and another arbitrary set labeled β. Our first algorithm generates a labeling such that there is no swap move that decreases the energy. The second move we consider is an αexpansion: for a label α, this move assigns an arbitrary set of pixels the label α. Our second
What energy functions can be minimized via graph cuts?
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2004
"... In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are co ..."
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Cited by 1047 (23 self)
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In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a generalpurpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to create the appropriate graph. A software implementation is freely available.
Segmentation of brain MR images through a hidden Markov random field model and the expectationmaximization algorithm
 IEEE TRANSACTIONS ON MEDICAL. IMAGING
, 2001
"... The finite mixture (FM) model is the most commonly used model for statistical segmentation of brain magnetic resonance (MR) images because of its simple mathematical form and the piecewise constant nature of ideal brain MR images. However, being a histogrambased model, the FM has an intrinsic limi ..."
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Cited by 619 (14 self)
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The finite mixture (FM) model is the most commonly used model for statistical segmentation of brain magnetic resonance (MR) images because of its simple mathematical form and the piecewise constant nature of ideal brain MR images. However, being a histogrambased model, the FM has an intrinsic limitation—no spatial information is taken into account. This causes the FM model to work only on welldefined images with low levels of noise; unfortunately, this is often not the the case due to artifacts such as partial volume effect and bias field distortion. Under these conditions, FM modelbased methods produce unreliable results. In this paper, we propose a novel hidden Markov random field (HMRF) model, which is a stochastic process generated by a MRF whose state sequence cannot be observed directly but which can be indirectly estimated through observations. Mathematically, it can be shown that the FM model is a degenerate version of the HMRF model. The advantage of the HMRF model derives from the way in which the spatial information is encoded through the mutual influences of neighboring sites. Although MRF modeling has been employed in MR image segmentation by other researchers, most reported methods are limited to using MRF as a general prior in an FM modelbased approach. To fit the HMRF model, an EM algorithm is used. We show that by incorporating both the HMRF model and the EM algorithm into a HMRFEM framework, an accurate and robust segmentation can be achieved. More importantly, the HMRFEM framework can easily be combined with other techniques. As an example, we show how the bias field correction algorithm of Guillemaud and Brady (1997) can be incorporated into this framework to achieve a threedimensional fully automated approach for brain MR image segmentation.
Graphcut textures: Image and video synthesis using graph cuts
 ACM Transactions on Graphics, SIGGRAPH 2003
, 2003
"... This banner was generated by merging the source images in Figure 6 using our interactive texture merging technique. In this paper we introduce a new algorithm for image and video texture synthesis. In our approach, patch regions from a sample image or video are transformed and copied to the output a ..."
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Cited by 492 (9 self)
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This banner was generated by merging the source images in Figure 6 using our interactive texture merging technique. In this paper we introduce a new algorithm for image and video texture synthesis. In our approach, patch regions from a sample image or video are transformed and copied to the output and then stitched together along optimal seams to generate a new (and typically larger) output. In contrast to other techniques, the size of the patch is not chosen apriori, but instead a graph cut technique is used to determine the optimal patch region for any given offset between the input and output texture. Unlike dynamic programming, our graph cut technique for seam optimization is applicable in any dimension. We specifically explore it in 2D and 3D to perform video texture synthesis in addition to regular image synthesis. We present approximative offset search techniques that work well in conjunction with the presented patch size optimization. We show results for synthesizing regular, random, and natural images and videos. We also demonstrate how this method can be used to interactively merge different images to generate new scenes.
A comparative study of energy minimization methods for Markov random fields
 IN ECCV
, 2006
"... One of the most exciting advances in early vision has been the development of efficient energy minimization algorithms. Many early vision tasks require labeling each pixel with some quantity such as depth or texture. While many such problems can be elegantly expressed in the language of Markov Ran ..."
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Cited by 414 (36 self)
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One of the most exciting advances in early vision has been the development of efficient energy minimization algorithms. Many early vision tasks require labeling each pixel with some quantity such as depth or texture. While many such problems can be elegantly expressed in the language of Markov Random Fields (MRF’s), the resulting energy minimization problems were widely viewed as intractable. Recently, algorithms such as graph cuts and loopy belief propagation (LBP) have proven to be very powerful: for example, such methods form the basis for almost all the topperforming stereo methods. Unfortunately, most papers define their own energy function, which is minimized with a specific algorithm of their choice. As a result, the tradeoffs among different energy minimization algorithms are not well understood. In this paper we describe a set of energy minimization benchmarks, which we use to compare the solution quality and running time of several common energy minimization algorithms. We investigate three promising recent methods—graph cuts, LBP, and treereweighted message passing—as well as the wellknown older iterated conditional modes (ICM) algorithm. Our benchmark problems are drawn from published energy functions used for stereo, image stitching and interactive segmentation. We also provide a generalpurpose software interface that allows vision researchers to easily switch between optimization methods with minimal overhead. We expect that the availability of our benchmarks and interface will make it significantly easier for vision researchers to adopt the best method for their specific problems. Benchmarks, code, results and images are available at
Markov random fields with efficient approximations
 In IEEE Conference on Computer Vision and Pattern Recognition
, 1998
"... Markov Random Fields (MRF’s) can be used for a wide variety of vision problems. In this paper we focus on MRF’s with twovalued clique potentials, which form a generalized Potts model. We show that the maximum a posteriori estimate of such an MRF can be obtained by solving a multiway minimum cut pro ..."
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Cited by 213 (24 self)
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Markov Random Fields (MRF’s) can be used for a wide variety of vision problems. In this paper we focus on MRF’s with twovalued clique potentials, which form a generalized Potts model. We show that the maximum a posteriori estimate of such an MRF can be obtained by solving a multiway minimum cut problem on a graph. We develop efficient algorithms for computing good approximations to the minimum multiway cut. The visual correspondence problem can be formulated as an MRF in our framework; this yields quite promising results on real data with ground truth. We also apply our techniques to MRF’s with linear clique potentials. 1
Automated modelbased tissue classification of MR images of the brain
, 1999
"... We describe a fully automated method for modelbased tissue classification of Magnetic Resonance (MR) images of the brain. The method interleaves classification with estimation of the model parameters, improving the classification at each iteration. The algorithm is able to segment single and multi ..."
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Cited by 212 (14 self)
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We describe a fully automated method for modelbased tissue classification of Magnetic Resonance (MR) images of the brain. The method interleaves classification with estimation of the model parameters, improving the classification at each iteration. The algorithm is able to segment single and multispectral MR images, corrects for MR signal inhomogeneities and incorporates contextual information by means of Markov Random Fields. A digital brain atlas containing prior expectations about the spatial location of tissue classes is used to initialize the algorithm. This makes the method fully automated and therefore provides objective and reproducible segmentations. We have validated the technique on simulated as well as on real MR images of the brain.
MCMCbased particle filtering for tracking a variable number of interacting targets
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2005
"... We describe a particle filter that effectively deals with interacting targets targets that are influenced by the proximity and/or behavior of other targets. The particle filter includes a Markov random field (MRF) motion prior that helps maintain the identity of targets throughout an interaction, s ..."
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Cited by 204 (6 self)
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We describe a particle filter that effectively deals with interacting targets targets that are influenced by the proximity and/or behavior of other targets. The particle filter includes a Markov random field (MRF) motion prior that helps maintain the identity of targets throughout an interaction, significantly reducing tracker failures. We show that this MRF prior can be easily implemented by including an additional interaction factor in the importance weights of the particle filter. However, the computational requirements of the resulting multitarget filter render it unusable for large numbers of targets. Consequently, we replace the traditional importance sampling step in the particle filter with a novel Markov chain Monte Carlo (MCMC) sampling step to obtain a more efficient MCMCbased multitarget filter. We also show how to extend this MCMCbased filter to address a variable number of interacting targets. Finally, we present both qualitative and quantitative experimental results, demonstrating that the resulting particle filters deal efficiently and effectively with complicated target interactions.
Approximation Algorithms for Classification Problems with Pairwise Relationships: Metric Labeling and Markov Random Fields
 IN IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1999
"... In a traditional classification problem, we wish to assign one of k labels (or classes) to each of n objects, in a way that is consistent with some observed data that we have about the problem. An active line of research in this area is concerned with classification when one has information about pa ..."
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Cited by 195 (2 self)
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In a traditional classification problem, we wish to assign one of k labels (or classes) to each of n objects, in a way that is consistent with some observed data that we have about the problem. An active line of research in this area is concerned with classification when one has information about pairwise relationships among the objects to be classified; this issue is one of the principal motivations for the framework of Markov random fields, and it arises in areas such as image processing, biometry, and document analysis. In its most basic form, this style of analysis seeks a classification that optimizes a combinatorial function consisting of assignment costs  based on the individual choice of label we make for each object  and separation costs  based on the pair of choices we make for two "related" objects. We formulate a general classification problem of this type, the metric labeling problem; we show that it contains as special cases a number of standard classification f...