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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 Variation-based 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 (4 self)
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These notes address various theoretical and practical topics related to Total Variation-based 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 finite-differences 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 recon-struction 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 con-strained 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 recon-struction 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 con-strained 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 ac-curate depth values even for specular surfaces, while simul-taneously taking into account global visibility constraints in order to obtain consistent depth maps for all views. The re-sulting optimization problems are solved with state-of-the-art convex relaxation techniques. We test our algorithm on a number of synthetic and real-world 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 con-tinuous 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 28 (5 self)
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We study convex relaxations of the image labeling problem on a con-tinuous 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 conver-gent Douglas-Rachford 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 first-order methods, the approach shows competitive performance on synthetical and real-world images. By combining the method with an improved binarization technique for non-standard 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 multi-class 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, mul-tiview reconstruction, stitching, and inpainting [PCF06]. We consider the vari-ational formulation inf `:Ω→{1,...,l} f(`), f(`):= Ω s(x, `(x))dx ︸ ︷ ︷ ︸ data term + J(`). ︸ ︷ ︷ ︸ regularizer
TIGHT CONVEX RELAXATIONS FOR VECTOR-VALUED LABELING
"... Abstract. Multi-label 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 15 (7 self)
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Abstract. Multi-label 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 real-world 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 large-scale 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
A Survey and Comparison of Discrete and Continuous Multi-label 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 multi-label 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 12 (8 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 multi-label 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 multi-label problem is known to be NP hard and thus one can only expect near-optimal 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 short-comings 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 multi-label case.
Agapito. Robust trajectory-space tv-l1 optical flow for non-rigid 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 non-rigid object. We exploit the high correlation between 2D trajectories of different points on the same non-rigid 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 non-rigid object. We exploit the high correlation between 2D trajectories of different points on the same non-rigid surface by assuming that the displacement sequence of any point can be expressed in a compact way as a linear combination of a low-rank 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 low-rank 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 multi-view optical flow of non-rigid surfaces. Our experiments, show that our proposed approach provides comparable or superior results to state of the art optical flow and dense non-rigid registration algorithms. 1
A Variational Approach to Video Registration with Subspace Constraints
"... Abstract This paper addresses the problem of non-rigid 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 12 (2 self)
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Abstract This paper addresses the problem of non-rigid 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 non-rigid 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 low-rank 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 low-rank 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 non-rigid surfaces. Our experiments show that our proposed approach outperforms state of the art optical flow and dense non-rigid registration algorithms.
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 vi-sion. The framework unifies a broad range of data and regularization terms, and is particularly suited for nonsmooth problems such as To-tal Variation-based 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 vi-sion. The framework unifies a broad range of data and regularization terms, and is particularly suited for nonsmooth problems such as To-tal Variation-based approaches to image labeling. However, for many interesting problems the constraint sets involved are difficult to han-dle numerically. State-of-the-art methods rely on using nested iterative projections, which induces both theoretical and practical convergence is-sues. We present a dual multiple-constraint Douglas-Rachford splitting approach that is globally convergent, avoids inner iterative loops, en-forces 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 log-likelihoods 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 log-likelihoods 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 NP-hard functionals (Zhu-Yuille, Chan-Vese, Grab-Cut, 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 NP-hard segmentation functionals unnecessary. Our one cut algorithm effectively replaces approximate iterative optimization techniques based on block coordinate descent. 1.
Optimality bounds for a variational relaxation of the image partitioning problem
- IN: ENERGY
, 2011
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