Parsimonious Labeling

We propose a new family of discrete energy minimization problems, which we call parsimonious labeling. Specifically, our energy functional consists of unary potentials and high-order clique potentials. While the unary potentials are arbitrary, the clique potentials are proportional to the {\em diversity} of set of the unique labels assigned to the clique. Intuitively, our energy functional encourages the labeling to be parsimonious, that is, use as few labels as possible. This in turn allows us to capture useful cues for important computer vision applications such as stereo correspondence and image denoising. Furthermore, we propose an efficient graph-cuts based algorithm for the parsimonious labeling problem that provides strong theoretical guarantees on the quality of the solution. Our algorithm consists of three steps. First, we approximate a given diversity using a mixture of a novel hierarchical $P^n$ Potts model. Second, we use a divide-and-conquer approach for each mixture component, where each subproblem is solved using an effficient $\alpha$-expansion algorithm. This provides us with a small number of putative labelings, one for each mixture component. Third, we choose the best putative labeling in terms of the energy value. Using both sythetic and standard real datasets, we show that our algorithm significantly outperforms other graph-cuts based approaches

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

GRMA: Generalized Range Move Algorithms for the efficient optimization of MRFs

Markov Random Fields (MRF) have become an important tool for many vision applications, and the optimization of MRFs is a problem of fundamental importance. Recently, Veksler and Kumar et al. proposed the range move algorithms, which are some of the most successful optimizers. Instead of considering only two labels as in previous move-making algorithms, they explore a large search space over a range of labels in each iteration, and significantly outperform previous move-making algorithms. However, two problems have greatly limited the applicability of range move algorithms: 1) They are limited in the energy functions they can handle (i.e., only truncated convex functions); 2) They tend to be very slow compared to other move-making algorithms (e.g., �-expansion and ��-swap). In this paper, we propose two generalized range move algorithms (GRMA) for the efficient optimization of MRFs. To address the first problem, we extend the GRMAs to more general energy functions by restricting the chosen labels in each move so that the energy function is submodular on the chosen subset. Furthermore, we provide a feasible sufficient condition for choosing these subsets of labels. To address the second problem, we dynamically obtain the iterative moves by solving set cover problems. This greatly reduces the number of moves during the optimization.We also propose a fast graph construction method for the GRMAs. Experiments show that the GRMAs offer a great speedup over previous range move algorithms, while yielding competitive solutions

Rounding-based Moves for Semi-Metric Labeling

International audienceSemi-metric labeling is a special case of energy minimization for pairwise Markov random fields. The energy function consists of arbitrary unary potentials, and pairwise potentials that are proportional to a given semi-metric distance function over the label set. Popular methods for solving semi-metric labeling include (i) move-making algorithms, which iteratively solve a minimum st-cut problem; and (ii) the linear programming (LP) relaxation based approach. In order to convert the fractional solution of the LP relaxation to an integer solution, several randomized rounding procedures have been developed in the literature. We consider a large class of parallel rounding procedures, and design move-making algorithms that closely mimic them. We prove that the multiplicative bound of a move-making algorithm exactly matches the approximation factor of the corresponding rounding procedure for any arbitrary distance function. Our analysis includes all known results for move-making algorithms as special cases

Methods for Structural Pattern Recognition: Complexity and Applications

Katedra kybernetik

High-Order Inference, Ranking, and Regularization Path for Structured SVM

This thesis develops novel methods to enable the use of structured prediction in computer vision and medical imaging. Specifically, our contributions are four fold. First, we propose a new family of high-order potentials that encourage parsimony in the labeling, and enable its use by designing an accurate graph cuts based algorithm to minimize the corresponding energy function. Second, we show how the average precision SVM formulation can be extended to incorporate high-order information for ranking. Third, we propose a novel regularization path algorithm for structured SVM. Fourth, we show how the weakly supervised framework of latent SVM can be employed to learn the parameters for the challenging deformable registration problem.In more detail, the first part of the thesis investigates the high-order inference problem. Specifically, we present a novel family of discrete energy minimization problems, which we call parsimonious labeling. It is a natural generalization of the well known metric labeling problems for high-order potentials. In addition to this, we propose a generalization of the Pn-Potts model, which we call Hierarchical Pn-Potts model. In the end, we propose parallelizable move making algorithms with very strong multiplicative bounds for the optimization of the hierarchical Pn-Potts model and the parsimonious labeling.Second part of the thesis investigates the ranking problem while using high-order information. Specifically, we introduce two alternate frameworks to incorporate high-order information for the ranking tasks. The first framework, which we call high-order binary SVM (HOB-SVM), optimizes a convex upperbound on weighted 0-1 loss while incorporating high-order information using joint feature map. The rank list for the HOB-SVM is obtained by sorting samples using max-marginals based scores. The second framework, which we call high-order AP-SVM (HOAP-SVM), takes its inspiration from AP-SVM and HOB-SVM (our first framework). Similar to AP-SVM, it optimizes upper bound on average precision. However, unlike AP-SVM and similar to HOB-SVM, it can also encode high-order information. The main disadvantage of HOAP-SVM is that estimating its parameters requires solving a difference-of-convex program. We show how a local optimum of the HOAP-SVM learning problem can be computed efficiently by the concave-convex procedure. Using standard datasets, we empirically demonstrate that HOAP-SVM outperforms the baselines by effectively utilizing high-order information while optimizing the correct loss function.In the third part of the thesis, we propose a new algorithm SSVM-RP to obtain epsilon-optimal regularization path of structured SVM. We also propose intuitive variants of the Block-Coordinate Frank-Wolfe algorithm (BCFW) for the faster optimization of the SSVM-RP algorithm. In addition to this, we propose a principled approach to optimize the SSVM with additional box constraints using BCFW and its variants. In the end, we propose regularization path algorithm for SSVM with additional positivity/negativity constraints.In the fourth and the last part of the thesis (Appendix), we propose a novel weakly supervised discriminative algorithm for learning context specific registration metrics as a linear combination of conventional metrics. Conventional metrics can cope partially - depending on the clinical context - with tissue anatomical properties. In this work we seek to determine anatomy/tissue specific metrics as a context-specific aggregation/linear combination of known metrics. We propose a weakly supervised learning algorithm for estimating these parameters conditionally to the data semantic classes, using a weak training dataset. We show the efficacy of our approach on three highly challenging datasets in the field of medical imaging, which vary in terms of anatomical structures and image modalities.Cette thèse présente de nouvelles méthodes pour l'application de la prédiction structurée en vision numérique et en imagerie médicale.Nos nouvelles contributions suivent quatre axes majeurs.La première partie de cette thèse étudie le problème d'inférence d'ordre supérieur.Nous présentons une nouvelle famille de problèmes de minimisation d'énergie discrète, l'étiquetage parcimonieux, encourageant la parcimonie des étiquettes.C'est une extension naturelle des problèmes connus d'étiquetage de métriques aux potentiels d'ordre élevé.Nous proposons par ailleurs une généralisation du modèle Pn-Potts, le modèle Pn-Potts hiérarchique.Enfin, nous proposons un algorithme parallélisable à proposition de mouvements avec de fortes bornes multiplicatives pour l'optimisation du modèle Pn-Potts hiérarchique et l'étiquetage parcimonieux.La seconde partie de cette thèse explore le problème de classement en utilisant de l'information d'ordre élevé.Nous introduisons deux cadres différents pour l'incorporation d'information d'ordre élevé dans le problème de classement.Le premier modèle, que nous nommons SVM binaire d'ordre supérieur (HOB-SVM), optimise une borne supérieure convexe sur l'erreur 0-1 pondérée tout en incorporant de l'information d'ordre supérieur en utilisant un vecteur de charactéristiques jointes.Le classement renvoyé par HOB-SVM est obtenu en ordonnant les exemples selon la différence entre la max-marginales de l'affectation d'un exemple à la classe associée et la max-marginale de son affectation à la classe complémentaire.Le second modèle, appelé AP-SVM d'ordre supérieur (HOAP-SVM), s'inspire d'AP-SVM et de notre premier modèle, HOB-SVM.Le modèle correspond à une optimisation d'une borne supérieure sur la précision moyenne, à l'instar d'AP-SVM, qu'il généralise en permettant également l'incorporation d'information d'ordre supérieur.Nous montrons comment un optimum local du problème d'apprentissage de HOAP-SVM peut être déterminé efficacement grâce à la procédure concave-convexe.En utilisant des jeux de données standards, nous montrons empiriquement que HOAP-SVM surpasse les modèles de référence en utilisant efficacement l'information d'ordre supérieur tout en optimisant directement la fonction d'erreur appropriée.Dans la troisième partie, nous proposons un nouvel algorithme, SSVM-RP, pour obtenir un chemin de régularisation epsilon-optimal pour les SVM structurés.Nous présentons également des variantes intuitives de l'algorithme Frank-Wolfe pour l'optimisation accélérée de SSVM-RP.De surcroît, nous proposons une approche systématique d'optimisation des SSVM avec des contraintes additionnelles de boîte en utilisant BCFW et ses variantes.Enfin, nous proposons un algorithme de chemin de régularisation pour SSVM avec des contraintes additionnelles de positivité/negativité.Dans la quatrième et dernière partie de la thèse, en appendice, nous montrons comment le cadre de l'apprentissage semi-supervisé des SVM à variables latentes peut être employé pour apprendre les paramètres d'un problème complexe de recalage déformable.Nous proposons un nouvel algorithme discriminatif semi-supervisé pour apprendre des métriques de recalage spécifiques au contexte comme une combinaison linéaire des métriques conventionnelles.Selon l'application, les métriques traditionnelles sont seulement partiellement sensibles aux propriétés anatomiques des tissus.Dans ce travail, nous cherchons à déterminer des métriques spécifiques à l'anatomie et aux tissus, par agrégation linéaire de métriques connues.Nous proposons un algorithme d'apprentissage semi-supervisé pour estimer ces paramètres conditionnellement aux classes sémantiques des données, en utilisant un jeu de données faiblement annoté.Nous démontrons l'efficacité de notre approche sur trois jeux de données particulièrement difficiles dans le domaine de l'imagerie médicale, variables en terme de structures anatomiques et de modalités d'imagerie

Rounding-based Moves for Metric Labeling

International audienceMetric labeling is a special case of energy minimization for pairwise Markov random fields. The energy function consists of arbitrary unary potentials, and pairwise potentials that are proportional to a given metric distance function over the label set. Popular methods for solving metric labeling include (i) move-making algorithms, which iteratively solve a minimum st-cut problem; and (ii) the linear programming (LP) relaxation based approach. In order to convert the fractional solution of the LP relaxation to an integer solution, several randomized rounding procedures have been developed in the literature. We consider a large class of parallel rounding procedures, and design move-making algorithms that closely mimic them. We prove that the multiplicative bound of a move-making algorithm exactly matches the approximation factor of the corresponding rounding procedure for any arbitrary distance function. Our analysis includes all known results for move-making algorithms as special cases