Results 1  10
of
12
Generalized Roof Duality for PseudoBoolean Optimization
"... The number of applications in computer vision that model higherorder interactions has exploded over the last few years. The standard technique for solving such problems is to reduce the higherorder objective function to a quadratic pseudoboolean function, and then use roof duality for obtaining a ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
(Show Context)
The number of applications in computer vision that model higherorder interactions has exploded over the last few years. The standard technique for solving such problems is to reduce the higherorder objective function to a quadratic pseudoboolean function, and then use roof duality for obtaining a lower bound. Roof duality works by constructing the tightest possible lowerbounding submodular function, and instead of optimizing the original objective function, the relaxation is minimized. We generalize this idea to polynomials of higher degree, where quadratic roof duality appears as a special case. Optimal relaxations are defined to be the ones that give the maximum lower bound. We demonstrate that important properties such as persistency still hold and how the relaxations can be efficiently constructed for general cubic and quartic pseudoboolean functions. From a practical point of view, we show that our relaxations perform better than stateoftheart for a wide range of problems, both in terms of lower bounds and in the number of assigned variables. 1.
Potts model, parametric maxflow and ksubmodular functions
 CoRR
"... The problem of minimizing the Potts energy function frequently occurs in computer vision applications. One way to tackle this NPhard problem was proposed by Kovtun [20, 21]. It identifies a part of an optimal solution by running k maxflow computations, where k is the number of labels. The number o ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
The problem of minimizing the Potts energy function frequently occurs in computer vision applications. One way to tackle this NPhard problem was proposed by Kovtun [20, 21]. It identifies a part of an optimal solution by running k maxflow computations, where k is the number of labels. The number of “labeled ” pixels can be significant in some applications, e.g. 5093 % in our tests for stereo. We show how to reduce the runtime to O(log k) maxflow computations (or one parametric maxflow computation). Furthermore, the output of our algorithm allows to speedup the subsequent alpha expansion for the unlabeled part, or can be used as it is for timecritical applications. To derive our technique, we generalize the algorithm of Felzenszwalb et al. [7] for Tree Metrics. We also show a connection to ksubmodular functions from combinatorial optimization, and discuss ksubmodular relaxations for general energy functions. 1.
Generalized Roof Duality for MultiLabel Optimization: Optimal
"... Abstract. We extend the concept of generalized roof duality from pseudoboolean functions to realvalued functions over multilabel variables. In particular, we prove that an analogue of the persistency property holds for energies of any order with any number of linearly ordered labels. Moreover, we ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We extend the concept of generalized roof duality from pseudoboolean functions to realvalued functions over multilabel variables. In particular, we prove that an analogue of the persistency property holds for energies of any order with any number of linearly ordered labels. Moreover, we show how the optimal submodular relaxation can be constructed in the firstorder case.
Playing with duality: An overview of recent primaldual approaches for . . .
, 2014
"... Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies jointly bringing i ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies jointly bringing into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and nonsmooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primaldual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primaldual algorithms both for solving largescale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness.
Maximum Persistency in Energy Minimization
"... We consider discrete pairwise energy minimization problem (weighted constraint satisfaction, maxsum labeling) and methods that identify a globally optimal partial assignment of variables. When finding a complete optimal assignment is intractable, determining optimal values for a part of variable ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
We consider discrete pairwise energy minimization problem (weighted constraint satisfaction, maxsum labeling) and methods that identify a globally optimal partial assignment of variables. When finding a complete optimal assignment is intractable, determining optimal values for a part of variables is an interesting possibility. Existing methods are based on different sufficient conditions. We propose a new sufficient condition for partial optimality which is: (1) verifiable in polynomial time (2) invariant to reparametrization of the problem and permutation of labels and (3) includes many existing sufficient conditions as special cases. We pose the problem of finding the maximum optimal partial assignment identifiable by the new sufficient condition. A polynomial method is proposed which is guaranteed to
Tighter Relaxations for HigherOrder Models based on Generalized Roof Duality
"... Many problems in computer vision can be turned into a largescale boolean optimization problem, which is in general NPhard. In this paper, we further develop one of the most successful approaches, namely roof duality, for approximately solving such problems for higherorder models. Two new methods ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Many problems in computer vision can be turned into a largescale boolean optimization problem, which is in general NPhard. In this paper, we further develop one of the most successful approaches, namely roof duality, for approximately solving such problems for higherorder models. Two new methods that can be applied independently or in combination are investigated. The first one is based on constructing relaxations using generators of the submodular function cone. In the second method, it is shown that the roof dual bound can be applied in an iterated way in order to obtain a tighter relaxation. We also provide experimental results that demonstrate better performance with respect to the stateoftheart, both in terms of improved bounds and the number of optimally assigned variables.
Lextendable functions and a proximity scaling algorithm for minimum cost multiflow problem
, 2014
"... ..."
Halfintegrality, LPbranching and FPT Algorithms
, 2014
"... A recent trend in parameterized algorithms is the application of polytope tools (specifically, LPbranching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedup ..."
Abstract
 Add to MetaCart
A recent trend in parameterized algorithms is the application of polytope tools (specifically, LPbranching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedups for a range of important problems. However, the existing approaches require the underlying polytope to have very restrictive properties, including halfintegrality and NemhauserTrotterstyle persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994). Taking a slightly different approach, we view halfintegrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from {0, 1}V to {0, 1/2, 1}V such that the new problem admits a polynomialtime exact solution. Using tools from CSP (in particular Thapper and Živný, 2012) to study the existence of such relaxations, we are able to provide a much broader class of halfintegral polytopes with the required properties. Our results unify and significantly extend the previously known cases. In addition to the new insight into problems with halfintegral relaxations, our results yield a range of new and improved FPT algo
PseudoBoolean Optimization: Theory and Applications in Vision
"... Many problems in computer vision, such as stereo, segmentation and denoising can be formulated as pseudoboolean optimization problems. Over the last decade, graphs cuts have become a standard tool for solving such problems. The last couple of years have seen a great advancement in the methods used ..."
Abstract
 Add to MetaCart
Many problems in computer vision, such as stereo, segmentation and denoising can be formulated as pseudoboolean optimization problems. Over the last decade, graphs cuts have become a standard tool for solving such problems. The last couple of years have seen a great advancement in the methods used to minimize pseudoboolean functions of higher order than quadratic. In this paper, we give an overview of how one can optimize higherorder functions via generalized roof duality and how it can be applied to problems in image analysis and vision.