Results 1 
9 of
9
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 ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
(Show Context)
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
What is optimized in tight convex relaxations for multilabel problems
 in Proc. CVPR
, 2012
"... In this work we present a unified view onMarkov random fields and recently proposed continuous tight convex relaxations for multilabel assignment in the image plane. These relaxations are far less biased towards the grid geometry than Markov random fields. It turns out that the continuous methods ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
In this work we present a unified view onMarkov random fields and recently proposed continuous tight convex relaxations for multilabel assignment in the image plane. These relaxations are far less biased towards the grid geometry than Markov random fields. It turns out that the continuous methods are nonlinear extensions of the local polytope MRF relaxation. In view of this result a better understanding of these tight convex relaxations in the discrete setting is obtained. Further, a wider range of optimization methods is now applicable to find a minimizer of the tight formulation. We propose two methods to improve the efficiency of minimization. One uses a weaker, but more efficient continuously inspired approach as initialization and gradually refines the energy where it is necessary. The other one reformulates the dual energy enabling smooth approximations to be used for efficient optimization. We demonstrate the utility of our proposed minimization schemes in numerical experiments. 1.
Nonmetric Priors for Continuous Multilabel Optimization
"... Abstract. We propose a novel convex prior for multilabel optimization which allows to impose arbitrary distances between labels. Only symmetry, d(i, j) ≥ 0 and d(i, i) = 0 are required. In contrast to previous grid based approaches for the nonmetric case, the proposed prior is formulated in the co ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We propose a novel convex prior for multilabel optimization which allows to impose arbitrary distances between labels. Only symmetry, d(i, j) ≥ 0 and d(i, i) = 0 are required. In contrast to previous grid based approaches for the nonmetric case, the proposed prior is formulated in the continuous setting avoiding grid artifacts. In particular, the model is easy to implement, provides a convex relaxation for the MumfordShah functional and yields comparable or superior results on the MSRC segmentation database comparing to metric or grid based approaches.
A FAST CONTINUOUS MAXFLOW APPROACH TO NONCONVEX MULTILABELING PROBLEMS
"... This work addresses a class of multilabeling problems over a spatially continuous image domain, where the data fidelity term can be any bounded function, not necessarily convex. Two total variation based regularization terms are considered, the first favoring a linear relationship between the labe ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
This work addresses a class of multilabeling problems over a spatially continuous image domain, where the data fidelity term can be any bounded function, not necessarily convex. Two total variation based regularization terms are considered, the first favoring a linear relationship between the labels and the second independent of the label values (Pott’s model). In the spatially discrete setting, Ishikawa [33] showed that the first of these labeling problems can be solved exactly by standard maxflow and mincut algorithms over specially designed graphs. We will propose a continuous analogue of Ishikawa’s graph construction [33] by formulating continuous maxflow and mincut models over a specially designed domain. These maxflow and mincut models are equivalent under a primaldual perspective. They can be seen as exact convex relaxations of the original problem and can be used to compute global solutions. Fast continuous maxflow based algorithms are proposed based on the maxflow models whose efficiency and reliability can be validated by both standard optimization theories and experiments. In comparison to previous work [53, 52] on continuous generalization of Ishikawa’s construction, our approach differs in the maxflow dual treatment which leads to the following main advantages: A new theoretical framework which embeds the label order constraints implicitly and naturally results in optimal labeling functions taking values in any predefined finite label set; A more general thresholding theorem which, under some conditions, allows to produce a larger set of nonunique solutions to the original problem; Numerical experiments show the new maxflow algorithms converge faster than the fast primaldual algorithm of [53, 52]. The speedup factor is especially significant at high precisions. In the end, our dual formulation and algorithms are extended to a recently proposed convex relaxation of Pott’s model [50], thereby avoiding expensive iterative computations of projections without closed form solution.
A generic proximal algorithm for convex optimization  Application to totalvariation
 IEEE Signal Processing Letters
"... Abstract—We propose new optimization algorithms to minimize a sum of convex functions, which may be smooth or not and composed or not with linear operators. This generic formulation encompasses various forms of regularized inverse problems in imaging. The proposed algorithms proceed by splitting: t ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract—We propose new optimization algorithms to minimize a sum of convex functions, which may be smooth or not and composed or not with linear operators. This generic formulation encompasses various forms of regularized inverse problems in imaging. The proposed algorithms proceed by splitting: the gradient or proximal operators of the functions are called individually, without inner loop or linear system to solve at each iteration. The algorithms are easy to implement and have proven convergence to an exact solution. The classical Douglas–Rachford and forward–backward splitting methods, as well as the recent and efficient algorithm of Chambolle–Pock, are recovered as particular cases. The application to inverse imaging problems regularized by the total variation is detailed. Index Terms—Convex nonsmooth optimization, proximal splitting algorithm, regularized inverse problem, total variation
Simultaneous SuperResolution of Depth and Images using a Single Camera
"... In this paper, we propose a convex optimization framework for simultaneous estimation of superresolved depth map and images from a single moving camera. The pixel measurement error in 3D reconstruction is directly related to the resolution of the images at hand. In turn, even a small measurement er ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
In this paper, we propose a convex optimization framework for simultaneous estimation of superresolved depth map and images from a single moving camera. The pixel measurement error in 3D reconstruction is directly related to the resolution of the images at hand. In turn, even a small measurement error can cause significant errors in reconstructing 3D scene structure or camera pose. Therefore, enhancing image resolution can be an effective solution for securing the accuracy as well as the resolution of 3D reconstruction. In the proposed method, depth map estimation and image superresolution are formulated in a single energy minimization framework with a convex function and solved efficiently by a firstorder primaldual algorithm. Explicit interframe pixel correspondences are not required for our superresolution procedure, thus we can avoid a huge computation time and obtain improved depth map in the accuracy and resolution as well as highresolution images with reasonable time. The superiority of our algorithm is demonstrated by presenting the improved depth map accuracy, image superresolution results, and camera pose estimation. 1.
Mathematics of Information
, 2013
"... Our world has been shaped by the outcome of the scientific revolution and its offspring, the industrial revolution. The essence of this scientific revolution, shaped by its pioneers – Newton, Gallileo, Descartes, Kepler, Huygens, Boyle, Leibnitz – was in the rigorous understanding of the physical wo ..."
Abstract
 Add to MetaCart
(Show Context)
Our world has been shaped by the outcome of the scientific revolution and its offspring, the industrial revolution. The essence of this scientific revolution, shaped by its pioneers – Newton, Gallileo, Descartes, Kepler, Huygens, Boyle, Leibnitz – was in the rigorous understanding of the physical world, the laws underlying matter, energy and their interaction. The word “rigorous ” is a hint to the fundamental role of mathematics in this endeavour but, one way or the other, the process has been driven by physics. Ultimately, physics led to applied physics (also known as engineering) and to the industrial revolution. Humanity has been changed forever. We are living in interesting times, arguably at a juncture of an equal importance and impact to the original scientific revolution. The world around us is changing and it is clear that the future will be as unrecognisable to us as our world – the Internet, mobile telephony, medical imaging, satellite navigation, social networks, the entire panoply of personal computers, laptops, tablets, smart phones so ubiquitous in our daily life of work and leasure... – would have been
Simultaneous SuperResolution of Depth and Images using a Single Camera
"... In this paper, we propose a convex optimization framework for simultaneous estimation of superresolved depth map and images from a single moving camera. The pixel measurement error in 3D reconstruction is directly related to the resolution of the images at hand. In turn, even a small measurement e ..."
Abstract
 Add to MetaCart
(Show Context)
In this paper, we propose a convex optimization framework for simultaneous estimation of superresolved depth map and images from a single moving camera. The pixel measurement error in 3D reconstruction is directly related to the resolution of the images at hand. In turn, even a small measurement error can cause significant errors in reconstructing 3D scene structure or camera pose. Therefore, enhancing image resolution can be an effective solution for securing the accuracy as well as the resolution of 3D reconstruction. In the proposed method, depth map estimation and image superresolution are formulated in a single energy minimization framework with a convex function and solved efficiently by a firstorder primaldual algorithm. Explicit interframe pixel correspondences are not required for our superresolution procedure, thus we can avoid a huge computation time and obtain improved depth map in the accuracy and resolution as well as highresolution images with reasonable time. The superiority of our algorithm is demonstrated by presenting the improved depth map accuracy, image superresolution results, and camera pose estimation. 1.