We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal non-relaxed integral solution. Our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the MAP-inference problem and iteratively prunes those, which do not satisfy our criterion for partial optimality. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous approaches in terms of the number of persistently labelled variables. The method is very general, as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem. Our method's runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem.

Correlation clustering, or multicut partitioning, is widely used in image segmentation for partitioning an undirected graph or image with positive and negative edge weights such that the sum of cut edge weights is minimized. Due to its NP-hardness, exact solvers do not scale and approximative solvers often give unsatisfactory results. We investigate scalable methods for correlation clustering. To this end we define fusion moves for the correlation clustering problem. Our algorithm iteratively fuses the current and a proposed partitioning which monotonously improves the partitioning and maintains a valid partitioning at all times. Furthermore, it scales to larger datasets, gives near optimal solutions, and at the same time shows a good anytime performance.

Recently, unsupervised image segmentation has become increasingly popular. Starting from a superpixel segmentation, an edge-weighted region adjacency graph is constructed. Amongst all segmentations of the graph, the one which best conforms to the given image evidence, as measured by the sum of cut edge weights, is chosen. Since this problem is NP-hard, we propose a new approximate solver based on the move-making paradigm: first, the graph is recursively partitioned into small regions (cut phase). Then, for any two adjacent regions, we consider alternative cuts of these two regions defining possible moves (glue & cut phase). For planar problems, the optimal move can be found, whereas for non-planar problems, efficient approximations exist. We evaluate our algorithm on published and new benchmark datasets, which we make available here. The proposed algorithm finds segmentations that, as measured by a loss function, are as close to the ground-truth as the global optimum found by exact solvers. It does so significantly faster then existing approximate methods, which is important for large-scale problems.

We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random Fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal nonrelaxed integral solution. Our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the MAP-inference problem and iteratively prunes those, which do not satisfy our criterion for partial optimality. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous ap proaches in terms of the number of persistently labelled variables. The method is very general, as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem. Our method's runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem.

Even years ago, Szeliski et al. published an influential study on energy minimization methods for Markov random fields (MRF). This study provided valuable insights in choosing the best optimization technique for certain classes of problems. While these insights remain generally useful today, the phenominal success of random field models means that the kinds of inference problems we solve have changed significantly. Specifically, the models today often include higher order interactions, flexible connectivity structures, large label-spaces of different cardinalities, or learned energy tables. To reflect these changes, we provide a modernized and enlarged study. We present an empirical comparison of 24 state-of-art techniques on a corpus of 2,300 energy minimization instances from 20 diverse computer vision applications. To ensure reproducibility, we evaluate all methods in the OpenGM2 framework and report extensive results regarding runtime and solution quality. Key insights from our study agree with the results of Szeliski et al. for the types of models they studied. However, on new and challenging types of models our findings disagree and suggest that polyhedral methods and integer programming solvers are competitive in terms of runtime and solution quality over a large range of model types.

Discrete graphical models (also known as discrete Markov random fields) are a major conceptual tool to model the structure of optimization problems in computer vision. While in the last decade research has focused on fast approximative methods, algorithms that provide globally optimal solutions have come more into the research focus in the last years. However, large scale computer vision problems seemed to be out of reach for such methods. In this paper we introduce a promising way to bridge this gap based on partial optimality and structural properties of the underlying problem factorization. Combining these preprocessing steps, we are able to solve grids of size 2048×2048 in less than 90 seconds. On the hitherto unsolvable Chinese character dataset of Nowozin et. al we obtain provably optimal results in 56% of the instances and achieve competitive runtimes on other recent benchmark problems. While in the present work only generalized Potts models are considered, an extension to general graphical models seems to be feasible.

We consider energy minimization for undirected graphical models, also known as MAP-inference problem for Markov random fields. Although combinatorial methods, which return a provably optimal integral solution of the problem, made a big progress in the past decade, they are still typically unable to cope with large-scale datasets. On the other hand, large scale datasets are typically defined on sparse graphs, and convex relaxation methods, such as linear programming relaxations often provide good approximations to integral solutions. We propose a novel method of combining combinatorial and convex programming techniques to obtain a global solution of the initial combinatorial problem. Based on the information obtained from the solution of the convex relaxation, our method confines application of the combinatorial solver to a small fraction of the initial graphical model, which allows to optimally solve big problems. We demonstrate the power of our approach on a computer vision energy minimization benchmark.

Approximate inference by decomposition of discrete graphical models and Lagrangian relaxation has become a key technique in computer vision. The resulting dual objective function is convenient from the optimization point-of-view, in principle. Due to its inherent non-smoothness, however, it is not directly amenable to efficient convex optimization. Related work either weakens the relaxation by smoothing or applies variations of the inefficient projected subgradient methods. In either case, heuristic choices of tuning parameters influence the performance and significantly depend on the specific problem at hand. In this paper, we introduce a novel approach based on bundle methods from the field of combinatorial optimization. It is directly based on the non-smooth dual objective function, requires no tuning parameters and showed a markedly improved efficiency uniformly over a large variety of problem instances including benchmark experiments. Our code will be publicly available after publication of this paper.

We study the MAP-labeling problem for graphical models by optimizing a dual problem obtained by Lagrangian decomposition. In this paper, we focus specifically on Nes-terov's optimal first-order optimization scheme for non-smooth convex programs, that has been studied for a range of other problems in computer vision and machine learning in recent years. We show that in order to obtain an efficiently convergent iteration, this approach should be augmented with a dynamic estimation of a corresponding Lip-schitz constant, leading to a runtime complexity of O(1/ϵ) in terms of the desired precision ϵ. Additionally, we devise a stopping criterion based on a duality gap as a sound basis for competitive comparison and show how to compute it efficiently. We evaluate our results using the publicly available Middlebury database and a set of computer generated graphical models that highlight specific aspects, along with other state-of-the-art methods for MAP-inference.

We propose a novel graphical model for probabilistic image segmentation that contributes both to aspects of perceptual grouping in connection with image segmentation, and to globally optimal inference with higher-order graphical models. We represent image partitions in terms of cellular complexes in order to make the duality between connected regions and their contours explicit. This allows us to formulate a graphical model with higher-order factors that represent the requirement that all contours must be closed. The model induces a probability measure on the space of all partitions, concentrated on perceptually meaningful segmentations. We give a complete polyhedral characterization of the resulting global inference problem in terms of the multicut polytope and efficiently compute global optima by a cutting plane method. Competitive results for the Berkeley segmentation benchmark confirm the consistency of our approach.

We present an approach to jointly estimating camera motion and dense scene structure in terms of depth maps from monocular image sequences in driver-assistance scenarios. For two consecutive frames of a sequence taken with a single fast moving camera, the approach combines numerical estimation of egomotion on the Euclidean manifold of motion parameters with variational regularization of dense depth map estimation. Embedding this online joint estimator into a recursive framework achieves a pronounced spatio-temporal filtering effect and robustness. We report the evaluation of thousands of images taken from a car moving at speed up to 100 km/h. The results compare favorably with two alternative settings that require more input data: stereo based scene reconstruction and camera motion estimation in batch mode using multiple frames. The employed benchmark dataset is publicly available.