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13
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 1377 (51 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
An Experimental Comparison of MinCut/MaxFlow Algorithms for Energy Minimization in Vision
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2001
"... After [10, 15, 12, 2, 4] minimum cut/maximum flow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in lowlevel vision. The combinatorial optimization literature provides many mincut/maxflow algorithms with different polynomial time compl ..."
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Cited by 791 (48 self)
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After [10, 15, 12, 2, 4] minimum cut/maximum flow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in lowlevel vision. The combinatorial optimization literature provides many mincut/maxflow algorithms with different polynomial time complexity. Their practical efficiency, however, has to date been studied mainly outside the scope of computer vision. The goal of this paper
Learning from Labeled and Unlabeled Data using Graph Mincuts
, 2001
"... Many application domains suffer from not having enough labeled training data for learning. However, large amounts of unlabeled examples can often be gathered cheaply. As a result, there has been a great deal of work in recent years on how unlabeled data can be used to aid classification. We consi ..."
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Cited by 266 (5 self)
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Many application domains suffer from not having enough labeled training data for learning. However, large amounts of unlabeled examples can often be gathered cheaply. As a result, there has been a great deal of work in recent years on how unlabeled data can be used to aid classification. We consider an algorithm based on finding minimum cuts in graphs, that uses pairwise relationships among the examples in order to learn from both labeled and unlabeled data. Our algorithm
Efficient GraphBased Energy Minimization Methods In Computer Vision
, 1999
"... ms (we show that exact minimization in NPhard in these cases). These algorithms produce a local minimum in interesting large move spaces. Furthermore, one of them nds a solution within a known factor from the optimum. The algorithms are iterative and compute several graph cuts at each iteration. Th ..."
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Cited by 82 (5 self)
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ms (we show that exact minimization in NPhard in these cases). These algorithms produce a local minimum in interesting large move spaces. Furthermore, one of them nds a solution within a known factor from the optimum. The algorithms are iterative and compute several graph cuts at each iteration. The running time at each iteration is eectively linear due to the special graph structure. In practice it takes just a few iterations to converge. Moreover most of the progress happens during the rst iteration. For a certain piecewise constant prior we adapt the algorithms developed for the piecewise smooth prior. One of them nds a solution within a factor of two from the optimum. In addition we develop a third algorithm which nds a local minimum in yet another move space. We demonstrate the eectiveness of our approach on image restoration, stereo, and motion. For the data with ground truth, our methods signicantly outperform standard methods. Biographical Sketch Olga
Efficiently Computing a Good Segmentation
, 1998
"... This paper addresses the problem of segmenting an image into regions. We develop a framework for image segmentation based on the intuition that there should be evidence for a boundary between each pair of neighboring regions. This framework provides precise definitions of what it means for a segment ..."
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Cited by 26 (0 self)
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This paper addresses the problem of segmenting an image into regions. We develop a framework for image segmentation based on the intuition that there should be evidence for a boundary between each pair of neighboring regions. This framework provides precise definitions of what it means for a segmentation to be too coarse or too fine, in terms of boundaries between pairs of regions. Within this framework, we de ne a pairwise region comparison function using standard graphbased representations of an image. Then we present an efficient algorithm for computing a segmentation based on this comparison function, and prove that it produces segmentations that satisfy the global properties of being neither too coarse nor too fine according to the framework. We apply this algorithm to image segmentation using two different graphbased representations of an image, and illustrate the results with both real and synthetic images. The algorithm runs in time nearly linear in the number of graph edges and is also fast in practice. An important characteristic of the method is its ability to preserve detail in lowvariability image regions while ignoring detail in highvariability regions.
Comparing and unifying searchbased and similaritybased approaches to semisupervised clustering
 In Proceedings of the ICML2003 Workshop on the Continuum from Labeled to Unlabeled Data in Machine Learning and Data Mining
, 2003
"... Semisupervised clustering employs a small amount of labeled data to aid unsupervised learning. Previous work in the area has employed one of two approaches: 1) Searchbased methods that utilize supervised data to guide the search for the best clustering, and 2) Similaritybased methods that use supe ..."
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Cited by 19 (3 self)
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Semisupervised clustering employs a small amount of labeled data to aid unsupervised learning. Previous work in the area has employed one of two approaches: 1) Searchbased methods that utilize supervised data to guide the search for the best clustering, and 2) Similaritybased methods that use supervised data to adapt the underlying similarity metric used by the clustering algorithm. This paper presents a unified approach based on the KMeans clustering algorithm that incorporates both of these techniques. Experimental results demonstrate that the combined approach generally produces better clusters than either of the individual approaches. 1.
Approximate Classification via Earthmover Metrics
 In SODA ’04: Proceedings of the fifteenth annual ACMSIAM symposium on Discrete algorithms
, 2004
"... Given a metric space (X, d), a natural distance measure on probability distributions over X is the earthmover metric. We use randomized rounding of earthmover metrics to devise new approximation algorithms for two wellknown classification problems, namely, metric labeling and 0extension. ..."
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Cited by 17 (3 self)
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Given a metric space (X, d), a natural distance measure on probability distributions over X is the earthmover metric. We use randomized rounding of earthmover metrics to devise new approximation algorithms for two wellknown classification problems, namely, metric labeling and 0extension.
Object Recognition with Pictorial Structures
, 2001
"... This thesis presents a statistical framework for object recognition. The framework is motivated by the pictorial structure models introduced by Fischler and Elschlager nearly 30 years ago. The basic idea is to model an object by a collection of parts arranged in a deformable con guration. The appear ..."
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Cited by 4 (0 self)
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This thesis presents a statistical framework for object recognition. The framework is motivated by the pictorial structure models introduced by Fischler and Elschlager nearly 30 years ago. The basic idea is to model an object by a collection of parts arranged in a deformable con guration. The appearance of each part is modeled separately, and the deformable configuration is represented by springlike connections between pairs of parts. These models allow for qualitative descriptions of visual appearance, and are suitable for generic recognition problems. The problem of detecting an object in an image and the problem of learning an object model using training examples are naturally formulated under a statistical approach. We present efficient algorithms to solve these problems in our framework. We demonstrate our techniques by training models to represent faces and human bodies. The models are then used to locate the corresponding objects in novel images.
Image Labeling and Grouping by Minimizing Linear Functionals over Cones
 Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition
, 2001
"... We consider energy minimization problems related to image labeling, partitioning, and grouping, which typically show up at midlevel stages of computer vision systems. A common feature of these problems is their intrinsic combinatorial complexity from an optimization pointofview. Rather than trying ..."
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Cited by 3 (2 self)
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We consider energy minimization problems related to image labeling, partitioning, and grouping, which typically show up at midlevel stages of computer vision systems. A common feature of these problems is their intrinsic combinatorial complexity from an optimization pointofview. Rather than trying to compute the global minimum { a goal we consider as elusive in these cases { we wish to design optimization approaches which exhibit two relevant properties: First, in each application a solution with guaranteed degree of suboptimality can be computed. Secondly, the computations are based on clearly de ned algorithms which do not comprise any (hidden) tuning parameters. In this paper, we focus on the second property and introduce a novel and general optimization technique to the eld of computer vision which amounts to compute a suboptimal solution by just solving a convex optimization problem. As representative examples, we consider two binary quadratic energy functionals related to image labeling and perceptual grouping. Both problems can be considered as instances of a general quadratic functional in binary variables, which is embedded into a higher{dimensional space such that suboptimal solutions can be computed as minima of linear functionals over cones in that space (semidefinite programs). Extensive numerical results reveal that, on the average, suboptimal solutions can be computed which yield a gap below 5% with respect to the global optimum in case where this is known.
Quadratic Minimization for Labeling Problems
, 2002
"... Many tasks in Computer Vision can be formulated in the framework of Labeling Problems. Thereby we are asked to assign labels to objects. The assignment is based on a prior model for observationals in the sehensfeld and posteriori data. The labeling is to compute which minimizes ambiguities in the me ..."
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Cited by 1 (0 self)
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Many tasks in Computer Vision can be formulated in the framework of Labeling Problems. Thereby we are asked to assign labels to objects. The assignment is based on a prior model for observationals in the sehensfeld and posteriori data. The labeling is to compute which minimizes ambiguities in the measurements. This computation involves an appropriate functional over objects and labels, which defines a notion of consistency.