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436
DTAM: Dense Tracking and Mapping in RealTime
"... DTAM is a system for realtime camera tracking and reconstruction which relies not on feature extraction but dense, every pixel methods. As a single handheld RGB camera flies over a static scene, we estimate detailed textured depth maps at selected keyframes to produce a surface patchwork with mill ..."
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Cited by 132 (5 self)
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DTAM is a system for realtime camera tracking and reconstruction which relies not on feature extraction but dense, every pixel methods. As a single handheld RGB camera flies over a static scene, we estimate detailed textured depth maps at selected keyframes to produce a surface patchwork with millions of vertices. We use the hundreds of images available in a video stream to improve the quality of a simple photometric data term, and minimise a global spatially regularised energy functional in a novel nonconvex optimisation framework. Interleaved, we track the camera’s 6DOF motion precisely by framerate whole image alignment against the entire dense model. Our algorithms are highly parallelisable throughout and DTAM achieves realtime performance using current commodity GPU hardware. We demonstrate that a dense model permits superior tracking performance under rapid motion compared to a state of the art method using features; and also show the additional usefulness of the dense model for realtime scene interaction in a physicsenhanced augmented reality application. 1.
A primaldual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms
, 2013
"... We propose a new firstorder splitting algorithm for solving jointly the primal and dual formulations of largescale convex minimization problems involving the sum of a smooth function with Lipschitzian gradient, a nonsmooth proximable function, and linear composite functions. This is a full splitti ..."
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Cited by 56 (9 self)
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We propose a new firstorder splitting algorithm for solving jointly the primal and dual formulations of largescale convex minimization problems involving the sum of a smooth function with Lipschitzian gradient, a nonsmooth proximable function, and linear composite functions. This is a full splitting approach, in the sense that the gradient and the linear operators involved are applied explicitly without any inversion, while the nonsmooth functions are processed individually via their proximity operators. This work brings together and notably extends several classical splitting schemes, like the forward–backward and Douglas–Rachford methods, as well as the recent primal–dual method of Chambolle and Pock designed for problems with linear composite terms.
Generalized forwardbackward splitting
, 2011
"... This paper introduces the generalized forwardbackward splitting algorithm for minimizing convex functions of the form F + ∑ n i=1 Gi, where F has a Lipschitzcontinuous gradient and the Gi’s are simple in the sense that their Moreau proximity operators are easy to compute. While the forwardbackwar ..."
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Cited by 48 (9 self)
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This paper introduces the generalized forwardbackward splitting algorithm for minimizing convex functions of the form F + ∑ n i=1 Gi, where F has a Lipschitzcontinuous gradient and the Gi’s are simple in the sense that their Moreau proximity operators are easy to compute. While the forwardbackward algorithm cannot deal with more than n = 1 nonsmooth function, our method generalizes it to the case of arbitrary n. Our method makes an explicit use of the regularity of F in the forward step, and the proximity operators of the Gi’s are applied in parallel in the backward step. This allows the generalized forwardbackward to efficiently address an important class of convex problems. We prove its convergence in infinite dimension, and its robustness to errors on the computation of the proximity operators and of the gradient of F. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.
Realtime minimization of the piecewise smooth MumfordShah functional
 in Proceedings of the European Conference on Computer Vision (ECCV), 2014
"... Abstract. We propose an algorithm for eciently minimizing the piecewise smooth MumfordShah functional. The algorithm is based on an extension of a recent primaldual algorithm from convex to nonconvex optimization problems. The key idea is to rewrite the proximal operator in the primaldual algor ..."
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Cited by 36 (26 self)
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Abstract. We propose an algorithm for eciently minimizing the piecewise smooth MumfordShah functional. The algorithm is based on an extension of a recent primaldual algorithm from convex to nonconvex optimization problems. The key idea is to rewrite the proximal operator in the primaldual algorithm using Moreau’s identity. The resulting algorithm computes piecewise smooth approximations of color images at 1520 frames per second at VGA resolution using GPU acceleration. Compared to convex relaxation approaches [18], it is orders of magnitude faster and does not require a discretization of color values. In contrast to the popular AmbrosioTortorelli approach [2], it naturally combines piecewise smooth and piecewise constant approximations, it does not require an epsilonapproximation and it is not based on an alternation scheme. The achieved energies are in practice at most 5% o ↵ the optimal value for onedimensional problems. Numerous experiments demonstrate that the proposed algorithm is wellsuited to perform discontinuitypreserving smoothing and realtime video cartooning.
Iterationcomplexity of blockdecomposition algorithms and the alternating minimization augmented Lagrangian method
, 2010
"... In this paper, we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a pointtoset maximal monotone operator with a separable twoblock structure and introduce a framework of blockdecomposition proxtype algorithms for solving it which allows for each on ..."
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Cited by 33 (4 self)
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In this paper, we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a pointtoset maximal monotone operator with a separable twoblock structure and introduce a framework of blockdecomposition proxtype algorithms for solving it which allows for each one of the singleblock proximal subproblems to be solved in an approximate sense. Moreover, by showing that any method in this framework is also a special instance of the hybrid proximal extragradient (HPE) method introduced by Solodov and Svaiter, we derive corresponding convergence rate results. We also describe some instances of the framework based on specific and inexpensive schemes for solving the singleblock proximal subproblems. Finally, we consider some applications of our methodology to: i) propose new algorithms for the monotone inclusion problem consisting of the sum of two maximal monotone operators, and; ii) study the complexity of the classical alternating minimization augmented Lagrangian method for a class of linearly constrained convex programming problems with proper closed convex objective functions.
X.: Convergence analysis of primaldual algorithms for total variation image restoration
, 2010
"... Abstract. Recently, some attractive primaldual algorithms have been proposed for solving a saddlepoint problem, with particular applications in the area of total variation (TV) image restoration. This paper focuses on the convergence analysis of existing primaldual algorithms and shows that the i ..."
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Cited by 31 (2 self)
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Abstract. Recently, some attractive primaldual algorithms have been proposed for solving a saddlepoint problem, with particular applications in the area of total variation (TV) image restoration. This paper focuses on the convergence analysis of existing primaldual algorithms and shows that the involved parameters of those primaldual algorithms (including the step sizes) can be significantly enlarged if some simple correction steps are supplemented. As a result, we present some primaldualbased contraction methods for solving the saddlepoint problem. These contraction methods are in the predictioncorrection fashion in the sense that the predictor is generated by a primaldual method and it is corrected by some simple correction step at each iteration. In addition, based on the context of contraction type methods, we provide a novel theoretical framework for analyzing the convergence of primaldual algorithms which simplifies existing convergence analysis substantially.
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 28 (5 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
Distributed basis pursuit
 IEEE Trans. Sig. Proc
, 2012
"... Abstract—We propose a distributed algorithm for solving the optimization problem Basis Pursuit (BP). BP finds the leastnorm solution of the underdetermined linear system and is used, for example, in compressed sensing for reconstruction. Our algorithm solves BP on a distributed platform such as a s ..."
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Cited by 28 (6 self)
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Abstract—We propose a distributed algorithm for solving the optimization problem Basis Pursuit (BP). BP finds the leastnorm solution of the underdetermined linear system and is used, for example, in compressed sensing for reconstruction. Our algorithm solves BP on a distributed platform such as a sensor network, and is designed to minimize the communication between nodes. The algorithm only requires the network to be connected, has no notion of a central processing node, and no node has access to the entire matrix at any time. We consider two scenarios in which either the columns or the rows of are distributed among the compute nodes. Our algorithm, named DADMM, is a decentralized implementation of the alternating direction method of multipliers. We show through numerical simulation that our algorithm requires considerably less communications between the nodes than the stateoftheart algorithms. Index Terms—Augmented Lagrangian, basis pursuit (BP), distributed optimization, sensor networks.
Parallel proximal algorithm for image restoration using hybrid regularization
 IEEE Transactions on Image Processing
, 2011
"... Regularization approaches have demonstrated their effectiveness for solving illposed problems. However, in the context of variational restoration methods, a challenging question remains, namely how to find a good regularizer. While total variation introduces staircase effects, wavelet domain regula ..."
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Cited by 25 (8 self)
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Regularization approaches have demonstrated their effectiveness for solving illposed problems. However, in the context of variational restoration methods, a challenging question remains, namely how to find a good regularizer. While total variation introduces staircase effects, wavelet domain regularization brings other artefacts, e.g. ringing. However, a tradeoff can be made by introducing a hybrid regularization including several terms non necessarily acting in the same domain (e.g. spatial and wavelet transform domains). While this approachwas shown to provide good results for solving deconvolution problems in the presence of additive Gaussian noise, an important issue is to efficiently deal with this hybrid regularization for more general noise models. To solve this problem, we adopt a convex optimization framework where the criterion to be minimized is split in the sum of more than two terms. For spatial domain regularization, isotropic or anisotropic total variation definitions using various gradient filters are considered. An accelerated version of the Parallel Proximal Algorithm is proposed to perform the minimization. Some difficulties in the computation of the proximity operators involved in this algorithm are also addressed in this paper. Numerical experiments performed in the context of Poisson data recovery, show the good behaviour of the algorithm as well as promising results concerning the use of hybrid regularization techniques.
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 (10 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.