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64
A convex relaxation approach for computing minimal partitions
 In Proc. of CVPR
, 2009
"... In this work we propose a convex relaxation approach for computing minimal partitions. Our approach is based on rewriting the minimal partition problem (also known as Potts model) in terms of a primal dual Total Variation functional. We show that the Potts prior can be incorporated by means of conve ..."
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Cited by 70 (16 self)
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In this work we propose a convex relaxation approach for computing minimal partitions. Our approach is based on rewriting the minimal partition problem (also known as Potts model) in terms of a primal dual Total Variation functional. We show that the Potts prior can be incorporated by means of convex constraints on the dual variables. For minimization we propose an efficient primal dual projected gradient algorithm which also allows a fast implementation on parallel hardware. Although our approach does not guarantee to find global minimizers of the Potts model we can give a tight bound on the energy between the computed solution and the true minimizer. Furthermore we show that our relaxation approach dominates recently proposed relaxations. As a consequence, our approach allows to compute solutions closer to the true minimizer. For many practical problems we even find the global minimizer. We demonstrate the excellent performance of our approach on several multilabel image segmentation and stereo problems. 1.
Fast Global Labeling for RealTime Stereo Using Multiple Plane Sweeps
, 2008
"... This work presents a realtime, dataparallel approach for global label assignment on regular grids. The labels are selected according to a Markov random field energy with a Potts prior term for binary interactions. We apply the proposed method to accelerate the cleanup step of a realtime dense ste ..."
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Cited by 48 (2 self)
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This work presents a realtime, dataparallel approach for global label assignment on regular grids. The labels are selected according to a Markov random field energy with a Potts prior term for binary interactions. We apply the proposed method to accelerate the cleanup step of a realtime dense stereo method based on plane sweeping with multiple sweeping directions, where the label set directly corresponds to the employed directions. In this setting the Potts smoothness model is suitable, since the set of labels does not possess an intrinsic metric or total order. The observed runtimes are approximately 30 times faster than the ones obtained by graph cut approaches. 1
GLOBAL SOLUTIONS OF VARIATIONAL MODELS WITH CONVEX REGULARIZATION
"... Abstract. We propose an algorithmic framework to compute global solutions of variational models with convex regularity terms that permit quite arbitrary data terms. While the minimization of variational problems with convex data and regularity terms is straight forward (using for example gradient de ..."
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Cited by 32 (9 self)
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Abstract. We propose an algorithmic framework to compute global solutions of variational models with convex regularity terms that permit quite arbitrary data terms. While the minimization of variational problems with convex data and regularity terms is straight forward (using for example gradient descent), this is no longer trivial for functionals with nonconvex data terms. Using the theoretical framework of calibrations the original variational problem can be written as the maximum flux of a particular vector field going through the boundary of the subgraph of the unknown function. Upon relaxation this formulation turns the problem into a convex problem, however, in higher dimension. In order to solve this problem, we propose a fast primal dual algorithm which significantly outperforms existing algorithms. In experimental results we show the application of our method to outlier filtering of range images and disparity estimation in stereo images using a variety of convex regularity terms. Key words. Variational methods, calibrations, total variation, convex optimization. AMS subject classifications. 49M20, 49M29, 65K15, 68U10. 1. Introduction. Energy
Globally optimal segmentation of multiregion objects
 In ICCV
, 2009
"... colours are hard to separate. In the absence of user localization, above at center is the best result we can expect from such models. Now we can design multiregion models with geometric interactions to segment such objects more robustly in a single graph cut. Many objects contain spatially distinct ..."
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Cited by 32 (2 self)
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colours are hard to separate. In the absence of user localization, above at center is the best result we can expect from such models. Now we can design multiregion models with geometric interactions to segment such objects more robustly in a single graph cut. Many objects contain spatially distinct regions, each with a unique colour/texture model. Mixture models ignore the spatial distribution of colours within an object, and thus cannot distinguish between coherent parts versus randomly distributed colours. We show how to encode geometric interactions between distinct region+boundary models, such as regions being interior/exterior to each other along with preferred distances between their boundaries. With a single graph cut, our method extracts only those multiregion objects that satisfy such a combined model. We show applications in medical segmentation and scene layout estimation. Unlike Li et al. [17] we do not need “domain unwrapping” nor do we have topological limits on shapes. 1.
RealTime Dense Geometry from a Handheld Camera
"... Abstract. We present a novel variational approach to estimate dense depth maps from multiple images in realtime. By using robust penalizers for both data term and regularizer, our method preserves discontinuities in the depth map. We demonstrate that the integration of multiple images substantially ..."
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Cited by 29 (4 self)
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Abstract. We present a novel variational approach to estimate dense depth maps from multiple images in realtime. By using robust penalizers for both data term and regularizer, our method preserves discontinuities in the depth map. We demonstrate that the integration of multiple images substantially increases the robustness of estimated depth maps to noise in the input images. The integration of our method into recently published algorithms for camera tracking allows dense geometry reconstruction in realtime using a single handheld camera. We demonstrate the performance of our algorithm with realworld data. 1
A convex approach to minimal partitions
 J. IMAGING SCI
, 2012
"... We describe a convex relaxation for a family of problems of minimal perimeter partitions. The minimization of the relaxed problem can be tackled numerically, we describe an algorithm and show some results. In most cases, our relaxed problem finds a correct numerical approximation of the optimal solu ..."
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Cited by 24 (9 self)
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We describe a convex relaxation for a family of problems of minimal perimeter partitions. The minimization of the relaxed problem can be tackled numerically, we describe an algorithm and show some results. In most cases, our relaxed problem finds a correct numerical approximation of the optimal solution: we give some arguments to explain why it should be so, and also discuss some situation where it fails.
A continuous maxflow approach to Potts model
 In European Conference on Computer Vision (ECCV), Iraklion
, 2010
"... Abstract. We address the continuous problem of assigning multiple (unordered) labels with the minimum perimeter. The corresponding discrete Potts model is typically addressed with aexpansion which can generate metrication artifacts. Existing convex continuous formulations of the Potts model use TV ..."
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Cited by 19 (3 self)
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Abstract. We address the continuous problem of assigning multiple (unordered) labels with the minimum perimeter. The corresponding discrete Potts model is typically addressed with aexpansion which can generate metrication artifacts. Existing convex continuous formulations of the Potts model use TVbased functionals directly encoding perimeter costs. Such formulations are analogous to ’mincut ’ problems on graphs. We propose a novel convex formulation with a continous ’maxflow ’ functional. This approach is dual to the standard TVbased formulations of the Potts model. Our continous maxflow approach has significant numerical advantages; it avoids extra computational load in enforcing the simplex constraints and naturally allows parallel computations over different labels. Numerical experiments show competitive performance in terms of quality and significantly reduced number of iterations compared to the previous state of the art convex methods for the continuous Potts model. 1
Multilabel linear discriminant analysis
 In ECCV
"... Abstract. Multilabel problems arise frequently in image and video annotations, and many other related applications such as multitopic text categorization, music classification, etc. Like other computer vision tasks, multilabel image and video annotations also suffer from the difficulty of high d ..."
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Cited by 16 (10 self)
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Abstract. Multilabel problems arise frequently in image and video annotations, and many other related applications such as multitopic text categorization, music classification, etc. Like other computer vision tasks, multilabel image and video annotations also suffer from the difficulty of high dimensionality because images often have a large number of features. Linear discriminant analysis (LDA) is a wellknown method for dimensionality reduction. However, the classical Linear Discriminant Analysis (LDA) only works for singlelabel multiclass classifications and cannot be directly applied to multilabel multiclass classifications. It is desirable to naturally generalize the classical LDA to multilabel formulations. At the same time, multilabel data present a new opportunity to improve classification accuracy through label correlations, which are absent in singlelabel data. In this work, we propose a novel Multilabel Linear Discriminant Analysis (MLDA) method to take advantage of label correlations and explore the powerful classification capability of the classical LDA to deal with multilabel multiclass problems. Extensive experimental evaluations on five public multilabel data sets demonstrate excellent performance of our method.
TIGHT CONVEX RELAXATIONS FOR VECTORVALUED LABELING
"... Abstract. Multilabel problems are of fundamental importance in computer vision and image analysis. Yet, finding global minima of the associated energies is typically a hard computational challenge. Recently, progress has been made by reverting to spatially continuous formulations of respective prob ..."
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Cited by 14 (6 self)
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Abstract. Multilabel problems are of fundamental importance in computer vision and image analysis. Yet, finding global minima of the associated energies is typically a hard computational challenge. Recently, progress has been made by reverting to spatially continuous formulations of respective problems and solving the arising convex relaxation globally. In practice this leads to solutions which are either optimal or within an a posteriori bound of the optimum. Unfortunately, in previous methods, both run time and memory requirements scale linearly in the total number of labels, making them very inefficient and often not applicable to problems with higher dimensional label spaces. In this paper, we propose a reduction technique for the case that the label space is a continuous product space and the regularizer is separable, i.e. a sum of regularizers for each dimension of the label space. On typical realworld labeling problems, the resulting convex relaxation requires orders of magnitude less memory and computation time than previous methods. This enables us to apply it to largescale problems like optic flow, stereo with occlusion detection, segmentation into a very large number of regions, and joint denoising and local noise estimation. Experiments show that despite the drastic gain in performance, we do not arrive at less accurate solutions than the original