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32
An introduction to total variation for image analysis
 in Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyter
, 2010
"... These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize th ..."
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Cited by 44 (3 self)
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These notes address various theoretical and practical topics related to Total Variationbased image reconstruction. They focuse first on some theoretical results on functions which minimize the total variation, and in a second part, describe a few standard and less standard algorithms to minimize the total variation in a finitedifferences setting, with a series of applications from simple denoising to stereo, or deconvolution issues, and even more exotic uses like the minimization of minimal partition problems.
Globally Consistent Depth Labeling of 4D Lightfields
 In Proc. CVPR, 41
, 2012
"... We present a novel paradigm to deal with depth reconstruction from 4D light fields in a variational framework. Taking into account the special structure of light field data, we reformulate the problem of stereo matching to a constrained labeling problem on epipolar plane images, which can be thoug ..."
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Cited by 34 (5 self)
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We present a novel paradigm to deal with depth reconstruction from 4D light fields in a variational framework. Taking into account the special structure of light field data, we reformulate the problem of stereo matching to a constrained labeling problem on epipolar plane images, which can be thought of as vertical and horizontal 2D cuts through the field. This alternative formulation allows to estimate accurate depth values even for specular surfaces, while simultaneously taking into account global visibility constraints in order to obtain consistent depth maps for all views. The resulting optimization problems are solved with stateoftheart convex relaxation techniques. We test our algorithm on a number of synthetic and realworld examples captured with a light field gantry and a plenoptic camera, and compare to ground truth where available. All data sets as well as source code are provided online for additional evaluation. 1.
Continuous Multiclass Labeling Approaches and Algorithms
 SIAM J. Imag. Sci
, 2011
"... We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the originally combinatorial problem. We focus on two specific r ..."
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Cited by 27 (4 self)
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We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the originally combinatorial problem. We focus on two specific relaxations that differ in flexibility and simplicity – one can be used to tightly relax any metric interaction potential, while the other one only covers Euclidean metrics but requires less computational effort. For solving the nonsmooth discretized problem, we propose a globally convergent DouglasRachford scheme, and show that a sequence of dual iterates can be recovered in order to provide a posteriori optimality bounds. In a quantitative comparison to two other firstorder methods, the approach shows competitive performance on synthetical and realworld images. By combining the method with an improved binarization technique for nonstandard potentials, we were able to routinely recover discrete solutions within 1%–5 % of the global optimum for the combinatorial image labeling problem. 1 Problem Formulation The multiclass image labeling problem consists in finding, for each pixel x in the image domain Ω ⊆ Rd, a label `(x) ∈ {1,..., l} which assigns one of l class labels to x so that the labeling function ` adheres to some local data fidelity as well as nonlocal spatial coherency constraints. This problem class occurs in many applications, such as segmentation, multiview reconstruction, stitching, and inpainting [PCF06]. We consider the variational formulation inf `:Ω→{1,...,l} f(`), f(`):= Ω s(x, `(x))dx ︸ ︷ ︷ ︸ data term + J(`). ︸ ︷ ︷ ︸ regularizer
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
Agapito. Robust trajectoryspace tvl1 optical flow for nonrigid sequences
 In EMMCVPR
, 2011
"... Abstract. This paper deals with the problem of computing optical flow between each of the images in a sequence and a reference frame when the camera is viewing a nonrigid object. We exploit the high correlation between 2D trajectories of different points on the same nonrigid surface by assuming th ..."
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Cited by 12 (3 self)
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Abstract. This paper deals with the problem of computing optical flow between each of the images in a sequence and a reference frame when the camera is viewing a nonrigid object. We exploit the high correlation between 2D trajectories of different points on the same nonrigid surface by assuming that the displacement sequence of any point can be expressed in a compact way as a linear combination of a lowrank motion basis. This subspace constraint effectively acts as a long term regularization leading to temporally consistent optical flow. We formulate it as a robust soft constraint within a variational framework by penalizing flow fields that lie outside the lowrank manifold. The resulting energy functional includes a quadratic relaxation term that allows to decouple the optimization of the brightness constancy and spatial regularization terms, leading to an efficient optimization scheme. We provide a new benchmark dataset, based on motion capture data of a flag waving in the wind, with dense ground truth optical flow for evaluation of multiview optical flow of nonrigid surfaces. Our experiments, show that our proposed approach provides comparable or superior results to state of the art optical flow and dense nonrigid registration algorithms. 1
A Variational Approach to Video Registration with Subspace Constraints
"... Abstract This paper addresses the problem of nonrigid video registration, or the computation of optical flow from a reference frame to each of the subsequent images in a sequence, when the camera views deformable objects. We exploit the high correlation between 2D trajectories of different points o ..."
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Cited by 11 (2 self)
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Abstract This paper addresses the problem of nonrigid video registration, or the computation of optical flow from a reference frame to each of the subsequent images in a sequence, when the camera views deformable objects. We exploit the high correlation between 2D trajectories of different points on the same nonrigid surface by assuming that the displacement of any point throughout the sequence can be expressed in a compact way as a linear combination of a lowrank motion basis. This subspace constraint effectively acts as a trajectory regularization term leading to temporally consistent optical flow. We formulate it as a robust soft constraint within a variational framework by penalizing flow fields that lie outside the lowrank manifold. The resulting energy functional can be decoupled into the optimization of the brightness constancy and spatial regularization terms, leading to an efficient optimization scheme. Additionally, we propose a novel optimization scheme for the case of vector valued images, based on the dualization of the data term. This allows us to extend our approach to deal with colour images which results in significant improvements on the registration results. Finally, we provide a new benchmark dataset, based on motion capture data of a flag waving in the wind, with dense ground truth optical flow for evaluation of multiframe optical flow algorithms for nonrigid surfaces. Our experiments show that our proposed approach outperforms state of the art optical flow and dense nonrigid registration algorithms.
A Survey and Comparison of Discrete and Continuous Multilabel Optimization Approaches for the Potts Model
 INT J COMPUT VIS
, 2013
"... We present a survey and a comparison of a variety of algorithms that have been proposed over the years to minimize multilabel optimization problems based on the Potts model. Discrete approaches based on Markov Random Fields as well as continuous optimization approaches based on partial differential ..."
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Cited by 11 (7 self)
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We present a survey and a comparison of a variety of algorithms that have been proposed over the years to minimize multilabel optimization problems based on the Potts model. Discrete approaches based on Markov Random Fields as well as continuous optimization approaches based on partial differential equations can be applied to the task. In contrast to the case of binary labeling, the multilabel problem is known to be NP hard and thus one can only expect nearoptimal solutions. In this paper, we carry out a theoretical comparison and an experimental analysis of existing approaches with respect to accuracy, optimality and runtime, aimed at bringing out the advantages and shortcomings of the respective algorithms. Systematic quantitative comparison is done on the Graz interactive image segmentation benchmark. This paper thereby generalizes a previous experimental comparison (Klodt et al. 2008) from the binary to the multilabel case.
Fast and exact primaldual iterations for variational problems in computer vision
 In Computer Vision–ECCV 2010
, 2010
"... Abstract. The saddle point framework provides a convenient way to formulate many convex variational problems that occur in computer vision. The framework unifies a broad range of data and regularization terms, and is particularly suited for nonsmooth problems such as Total Variationbased approach ..."
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Cited by 9 (1 self)
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Abstract. The saddle point framework provides a convenient way to formulate many convex variational problems that occur in computer vision. The framework unifies a broad range of data and regularization terms, and is particularly suited for nonsmooth problems such as Total Variationbased approaches to image labeling. However, for many interesting problems the constraint sets involved are difficult to handle numerically. Stateoftheart methods rely on using nested iterative projections, which induces both theoretical and practical convergence issues. We present a dual multipleconstraint DouglasRachford splitting approach that is globally convergent, avoids inner iterative loops, enforces the constraints exactly, and requires only basic operations that can be easily parallelized. The method outperforms existing methods by a factor of 4−20 while considerably increasing the numerical robustness. 1
GrabCut in One Cut
"... Among image segmentation algorithms there are two major groups: (a) methods assuming known appearance models and (b) methods estimating appearance models jointly with segmentation. Typically, the first group optimizes appearance loglikelihoods in combination with some spacial regularization. This p ..."
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Cited by 6 (4 self)
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Among image segmentation algorithms there are two major groups: (a) methods assuming known appearance models and (b) methods estimating appearance models jointly with segmentation. Typically, the first group optimizes appearance loglikelihoods in combination with some spacial regularization. This problem is relatively simple and many methods guarantee globally optimal results. The second group treats model parameters as additional variables transforming simple segmentation energies into highorder NPhard functionals (ZhuYuille, ChanVese, GrabCut, etc). It is known that such methods indirectly minimize the appearance overlap between the segments. We propose a new energy term explicitly measuring L1 distance between the object and background appearance models that can be globally maximized in one graph cut. We show that in many applications our simple term makes NPhard segmentation functionals unnecessary. Our one cut algorithm effectively replaces approximate iterative optimization techniques based on block coordinate descent. 1.
C.: Optimality bounds for a variational relaxation of the image partitioning problem
 In: Energy
, 2011
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