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152
Fast Global Minimization of the Active Contour/Snake Model
"... The active contour/snake model is one of the most successful variational models in image segmentation. It consists of evolving a contour in images toward the boundaries of objects. Its success is based on strong mathematical properties and efficient numerical schemes based on the level set method. ..."
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Cited by 155 (9 self)
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The active contour/snake model is one of the most successful variational models in image segmentation. It consists of evolving a contour in images toward the boundaries of objects. Its success is based on strong mathematical properties and efficient numerical schemes based on the level set method. The only drawback of this model is the existence of local minima in the active contour energy, which makes the initial guess critical to get satisfactory results. In this paper, we propose to solve this problem by determining a global minimum of the active contour model. Our approach is based on the unification of image segmentation and image denoising tasks into a global minimization framework. More precisely, we propose to unify three wellknown image variational models, namely the snake model, the RudinOsherFatemi denoising model and the MumfordShah segmentation model. We will establish theorems with proofs to determine the existence of a global minimum of the active contour model. From a numerical point of view, we propose a new practical way to solve the active contour propagation problem toward object boundaries through a dual formulation of the minimization problem. The dual formulation, easy to implement, allows us a fast global minimization of the snake energy. It avoids the usual drawback in the level set approach that consists of initializing the active contour in a distance function and reinitializing it periodically during the evolution, which is timeconsuming. We apply our segmentation algorithms on synthetic and realworld images, such as texture images and medical images, to emphasize the performances of our model compared with other segmentation models.
Image Restoration with Discrete Constrained Total Variation Part I: Fast and Exact Optimization
, 2006
"... This paper deals with the total variation minimization problem in image restoration for convex data fidelity functionals. We propose a new and fast algorithm which computes an exact solution in the discrete framework. Our method relies on the decomposition of an image into its level sets. It maps ..."
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Cited by 94 (10 self)
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This paper deals with the total variation minimization problem in image restoration for convex data fidelity functionals. We propose a new and fast algorithm which computes an exact solution in the discrete framework. Our method relies on the decomposition of an image into its level sets. It maps the original problems into independent binary Markov Random Field optimization problems at each level. Exact solutions of these binary problems are found thanks to minimum cost cut techniques in graphs. These binary solutions are proved to be monotone increasing with levels and yield thus an exact solution of the discrete original problem. Furthermore we show that minimization of total variation under L1 data fidelity term yields a selfdual contrast invariant filter. Finally we present some results.
A Globally Optimal Algorithm for Robust TVL 1 Range Image Integration
"... Robust integration of range images is an important task for building highquality 3D models. Since range images, and in particular range maps from stereo vision, may have a substantial amount of outliers, any integration approach aiming at highquality models needs an increased level of robustness. ..."
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Cited by 77 (8 self)
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Robust integration of range images is an important task for building highquality 3D models. Since range images, and in particular range maps from stereo vision, may have a substantial amount of outliers, any integration approach aiming at highquality models needs an increased level of robustness. Additionally, a certain level of regularization is required to obtain smooth surfaces. Computational efficiency and global convergence are further preferable properties. The contribution of this paper is a unified framework to solve all these issues. Our method is based on minimizing an energy functional consisting of a total variation (TV) regularization force and an L 1 data fidelity term. We present a novel and efficient numerical scheme, which combines the duality principle for the TV term with a pointwise optimization step. We demonstrate the superior performance of our algorithm on the wellknown Middlebury multiview database and additionally on realworld multiview images. 1.
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.
A convex formulation of continuous multilabel problems
 In ECCV, pages III: 792–805
, 2008
"... Abstract. We propose a spatially continuous formulation of Ishikawa’s discrete multilabel problem. We show that the resulting nonconvex variational problem can be reformulated as a convex variational problem via embedding in a higher dimensional space. This variational problem can be interpreted a ..."
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Cited by 65 (13 self)
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Abstract. We propose a spatially continuous formulation of Ishikawa’s discrete multilabel problem. We show that the resulting nonconvex variational problem can be reformulated as a convex variational problem via embedding in a higher dimensional space. This variational problem can be interpreted as a minimal surface problem in an anisotropic Riemannian space. In several stereo experiments we show that the proposed continuous formulation is superior to its discrete counterpart in terms of computing time, memory efficiency and metrication errors. 1
Geometric Applications of the Split Bregman Method: Segmentation and Surface Reconstruction
, 2009
"... Variational models for image segmentation have many applications, but can be slow to compute. Recently, globally convex segmentation models have been introduced which are very reliable, but contain TVregularizers, making them difficult to compute. The previously introduced Split Bregman method is a ..."
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Cited by 58 (7 self)
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Variational models for image segmentation have many applications, but can be slow to compute. Recently, globally convex segmentation models have been introduced which are very reliable, but contain TVregularizers, making them difficult to compute. The previously introduced Split Bregman method is a technique for fast minimization of L1 regularized functionals, and has been applied to denoising and compressed sensing problems. By applying the Split Bregman concept to image segmentation problems, we build fast solvers which can outperform more conventional schemes, such as duality based methods and graphcuts. We also consider the related problem of surface reconstruction from unorganized data points, which is used for constructing level set representations in 3 dimensions.
An algorithm for minimizing the mumfordshah functional
 in Proc. International Conference on Computer Vision, 2009
"... In this work we revisit the MumfordShah functional, one of the most studied variational approaches to image segmentation. The contribution of this paper is to propose an algorithm which allows to minimize a convex relaxation of the MumfordShah functional obtained by functional lifting. The algorit ..."
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Cited by 49 (14 self)
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In this work we revisit the MumfordShah functional, one of the most studied variational approaches to image segmentation. The contribution of this paper is to propose an algorithm which allows to minimize a convex relaxation of the MumfordShah functional obtained by functional lifting. The algorithm is an efficient primaldual projection algorithm for which we prove convergence. In contrast to existing algorithms for minimizing the full MumfordShah this is the first one which is based on a convex relaxation. As a consequence the computed solutions are independent of the initialization. Experimental results confirm that the proposed algorithm determines smooth approximations while preserving discontinuities of the underlying signal. 1.
Continuous global optimization in multiview 3d reconstruction
 In International Conference on Energy Minimization Methods in Computer Vision and Pattern Recognition
, 2007
"... Abstract. In this work, we introduce a robust energy model for multiview 3D reconstruction that fuses silhouette and stereobased image information. It allows to cope with significant amounts of noise without manual presegmentation of the input images. Moreover, we suggest a method that can global ..."
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Cited by 46 (11 self)
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Abstract. In this work, we introduce a robust energy model for multiview 3D reconstruction that fuses silhouette and stereobased image information. It allows to cope with significant amounts of noise without manual presegmentation of the input images. Moreover, we suggest a method that can globally optimize this energy up to the visibility constraint. While similar global optimization has been presented in the discrete context in form of the maxflowmincut framework, we suggest the use of a continuous counterpart. In contrast to graph cut methods, discretizations of the continuous optimization technique are consistent and independent of the choice of the grid connectivity. Our experiments demonstrate that this leads to visible improvements. Moreover, memory requirements are reduced, allowing for global reconstructions at higher resolutions. 1
Object Segmentation in Video: A Hierarchical Variational Approach for Turning Point Trajectories into Dense Regions
"... Point trajectories have emerged as a powerful means to obtain high quality and fully unsupervised segmentation of objects in video shots. They can exploit the long term motion difference between objects, but they tend to be sparse due to computational reasons and the difficulty in estimating motion ..."
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Cited by 35 (5 self)
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Point trajectories have emerged as a powerful means to obtain high quality and fully unsupervised segmentation of objects in video shots. They can exploit the long term motion difference between objects, but they tend to be sparse due to computational reasons and the difficulty in estimating motion in homogeneous areas. In this paper we introduce a variational method to obtain dense segmentations from such sparse trajectory clusters. Information is propagated with a hierarchical, nonlinear diffusion process that runs in the continuous domain but takes superpixels into account. We show that this process raises the density from 3% to 100 % and even increases the average precision of labels. 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