Abstract. One of the most exciting advances in early vision has been the development of efficient energy minimization algorithms. Many early vision tasks require labeling each pixel with some quantity such as depth or texture. While many such problems can be elegantly expressed in the language of Markov Random Fields (MRF’s), the resulting energy minimization problems were widely viewed as intractable. Recently, algorithms such as graph cuts and loopy belief propagation (LBP) have proven to be very powerful: for example, such methods form the basis for almost all the top-performing stereo methods. Unfortunately, most papers define their own energy function, which is minimized with a specific algorithm of their choice. As a result, the tradeoffs among different energy minimization algorithms are not well understood. In this paper we describe a set of energy minimization benchmarks, which we use to compare the solution quality and running time of several common energy minimization algorithms. We investigate three promising recent methods—graph cuts, LBP, and tree-reweighted message passing—as well as the well-known older iterated conditional modes (ICM) algorithm. Our benchmark problems are drawn from published energy functions used for stereo, image stitching and interactive segmentation. We also provide a general-purpose software interface that allows vision researchers to easily switch between optimization methods with minimal overhead. We expect that the availability of our benchmarks and interface will make it significantly easier for vision researchers to adopt the best method for their specific problems. Benchmarks, code, results and images are available at

In this paper, we propose a symmetric stereo model to handle occlusion in dense two-frame stereo. Our occlusion reasoning is directly based on the visibility constraint that is more general than both ordering and uniqueness constraints used in previous work. The visibility constraint requires occlusion in one image and disparity in the other to be consistent. We embed the visibility constraint within an energy minimization framework, resulting in a symmetric stereo model that treats left and right images equally. An iterative optimization algorithm is used to approximate the minimum of the energy using belief propagation. Our stereo model can also incorporate segmentation as a soft constraint. Experimental results on the Middlebury stereo images show that our algorithm is state-of-the-art. 1

In this paper, we formulate a stereo matching algorithm with careful handling of disparity, discontinuity, and occlusion. The algorithm works with a global matching stereo model based on an energy-minimization framework. The global energy contains two terms, the data term and the smoothness term. The data term is first approximated by a color-weighted correlation, then refined in occluded and low-texture areas in a repeated application of a hierarchical loopy belief propagation algorithm. The experimental results are evaluated on the Middlebury data sets, showing that our algorithm is the top performer among all the algorithms listed there.

This tutorial reviews image alignment and image stitching algorithms. Image alignment algorithms can discover the correspondence relationships among images with varying degrees of overlap. They are ideally suited for applications such as video stabilization, summarization, and the creation of panoramic mosaics. Image stitching algorithms take the alignment estimates produced by such registration algorithms and blend the images in a seamless manner, taking care to deal with potential problems such as blurring or ghosting caused by parallax and scene movement as well as varying image exposures. This tutorial reviews the basic motion models underlying alignment and stitching algorithms, describes effective direct (pixel-based) and feature-based alignment algorithms, and describes blending algorithms used to produce seamless mosaics. It ends with a discussion of open research problems in the area. 1

Many vision tasks can be formulated as graph partition problems that minimize energy functions. For such problems, the Gibbs...

Belief propagation (BP) has become widely used for low-level vision problems and various inference techniques have been proposed for loopy graphs. These methods typically rely on ad hoc spatial priors such as the Potts model. In this paper we investigate the use of learned models of image structure, and demonstrate the improvements obtained over previous ad hoc models for the image denoising problem. In particular, we show how both pairwise and higher-order Markov random fields with learned clique potentials capture rich image structures that better represent the properties of natural images. These models are learned using the recently proposed Fields-of-Experts framework. For such models, however, traditional BP is computationally expensive. Consequently we propose some approximation methods that make BP with learned potentials practical. In the case of pairwise models we propose a novel approximation of robust potentials using a finite family of quadratics. In the case of higher order MRFs, with 2 × 2 cliques, we use an adaptive state space to handle the increased complexity. Extensive experiments demonstrate the power of learned models, the benefits of higher-order MRFs and the practicality of BP for these problems with the use of simple principled approximations.

State-of-the-art stereo vision algorithms utilize color changes as important cues for object boundaries. Most methods impose heuristic restrictions or priors on disparities, for example by modulating local smoothness costs with intensity gradients. In this paper we seek to replace such heuristics with explicit probabilistic models of disparities and intensities learned from real images. We have constructed a large number of stereo datasets with ground-truth disparities, and we use a subset of these datasets to learn the parameters of Conditional Random Fields (CRFs). We present experimental results illustrating the potential of our approach for automatically learning the parameters of models with richer structure than standard hand-tuned MRF models. 1. Introduction and

A new framework is presented for both understanding and developing graph-cut based combinatorial algorithms suitable for the approximate optimization of a very wide class of MRFs that are frequently encountered in computer vision. The proposed framework utilizes tools from the duality theory of linear programming in order to provide an alternative and more general view of state-of-the-art techniques like the α-expansion algorithm, which is included merely as a special case. Moreover, contrary to α-expansion, the derived algorithms generate solutions with guaranteed optimality properties for a much wider class of problems, e.g. even for MRFs with non-metric potentials. In addition, they are capable of providing per-instance suboptimality bounds in all occasions, including discrete Markov Random Fields with an arbitrary potential function. These bounds prove to be very tight in practice (i.e. very close to 1), which means that the resulting solutions are almost optimal. Our algorithms ’ effectiveness is demon-strated by presenting experimental results on a variety of low level vision tasks, such as stereo matching, image restoration, image completion and optical flow estimation, as well as on synthetic problems.

Depth estimation in computer vision and robotics is most commonly done via stereo vision (stereopsis), in which images from two cameras are used to triangulate and estimate distances. However, there are also numerous monocular visual cues— such as texture variations and gradients, defocus, color/haze, etc.—that have heretofore been little exploited in such systems. Some of these cues apply even in regions without texture, where stereo would work poorly. In this paper, we apply a Markov Random Field (MRF) learning algorithm to capture some of these monocular cues, and incorporate them into a stereo system. We show that by adding monocular cues to stereo (triangulation) ones, we obtain significantly more accurate depth estimates than is possible using either monocular or stereo cues alone. This holds true for a large variety of environments, including both indoor environments and unstructured outdoor environments containing trees/forests, buildings, etc. Our approach is general, and applies to incorporating monocular cues together with any off-the-shelf stereo system. 1

In this paper we introduce a novel method to address minimization of static and dynamic MRFs. Our approach is based on principles from linear programming and, in particular, on primal dual strategies. It generalizes prior state-of-the-art methods such as α-expansion, while it can also be used for efficiently minimizing NP-hard problems with complex pair-wise potential functions. Furthermore, it offers a substantial speedup- of a magnitude ten- over existing techniques, due to the fact that it exploits information coming not only from the original MRF problem, but also from a dual one. The proposed technique consists of recovering pair of solutions for the primal and the dual such that the gap between them is minimized. Therefore, it can also boost performance of dynamic MRFs, where one should expect that the new new pair of primal-dual solutions is closed to the previous one. Promising results in a number of applications, and theoretical, as well as numerical comparisons with the state of the art demonstrate the extreme potentials of this approach.