Yuri Boykov

Fast approximate energy minimization via graph cuts

Many tasks in computer vision involve assigning a label (such as disparity) to every pixel. A common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist, e.g., at object boundaries. These tasks are naturally stated in terms of energy minimization. The authors consider a wide class of energies with various smoothness constraints. Global minimization of these energy functions is NP-hard even in the simplest discontinuity-preserving case. Therefore, our focus is on efficient approximation algorithms. We present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves, namely expansion moves and swap moves. These moves can simultaneously change the labels of arbitrarily large sets of pixels. In contrast, many standard algorithms (including simulated annealing) use small moves where only one pixel changes its label at a time. Our expansion algorithm finds a labeling within a known factor of the global minimum, while our swap algorithm handles more general energy functions. Both of these algorithms allow important cases of discontinuity preserving energies. We experimentally demonstrate the effectiveness of our approach for image restoration, stereo and motion. On real data with ground truth, we achieve 98 percent accuracy.

Graph Cuts and Efficient N-D Image Segmentation

Combinatorial graph cut algorithms have been successfully applied to a wide range of problems in vision and graphics. This paper focusses on possibly the simplest application of graph-cuts: segmentation of objects in image data. Despite its simplicity, this application epitomizes the best features of combinatorial graph cuts methods in vision: global optima, practical efficiency, numerical robustness, ability to fuse a wide range of visual cues and constraints, unrestricted topological properties of segments, and applicability to N-D problems. Graph cuts based approaches to object extraction have also been shown to have interesting connections with earlier segmentation methods such as snakes, geodesic active contours, and level-sets. The segmentation energies optimized by graph cuts combine boundary regularization with region-based properties in the same fashion as Mumford-Shah style functionals. We present motivation and detailed technical description of the basic combinatorial optimization framework for image segmentation via s/t graph cuts. After the general concept of using binary graph cut algorithms for object segmentation was first proposed and tested in Boykov and Jolly (2001), this idea was widely studied in computer vision and graphics communities. We provide links to a large number of known extensions based on iterative parameter re-estimation and learning, multi-scale or hierarchical approaches, narrow bands, and other techniques for demanding photo, video, and medical applications.

Markov random fields with efficient approximations

Markov Random Fields (MRFs) can be used for a wide variety of vision problems. In this paper we focus on MRFs with two-valued clique potentials, which form a generalized Potts model. We show that the maximum a posteriori estimate of such an MRF can be obtained by solving a multiway minimum cut problem on a graph. We develop efficient algorithms for computing good approximations to the minimum multiway, cut. The visual correspondence problem can be formulated as an MRF in our framework; this yields quite promising results on real data with ground truth. We also apply our techniques to MRFs with linear clique potentials.

Fast Approximate Energy Minimization with Label Costs

The α-expansion algorithm has had a significant impact in computer vision due to its generality, effectiveness, and speed. It is commonly used to minimize energies that involve unary, pairwise, and specialized higher-order terms. Our main algorithmic contribution is an extension of α-expansion that also optimizes “label costs” with well-characterized optimality bounds. Label costs penalize a solution based on the set of labels that appear in it, for example by simply penalizing the number of labels in the solution.Our energy has a natural interpretation as minimizing description length (MDL) and sheds light on classical algorithms like K-means and expectation-maximization (EM). Label costs are useful for multi-model fitting and we demonstrate several such applications: homography detection, motion segmentation, image segmentation, and compression. Our C++ and MATLAB code is publicly available http://vision.csd.uwo.ca/code/.

Interactive Organ Segmentation Using Graph Cuts

An N-dimensional image is divided into “object” and “background” segments using a graph cut approach. A graph is formed by connecting all pairs of neighboring image pixels (voxels) by weighted edges. Certain pixels (voxels) have to be a priori identified as object or background seeds providing necessary clues about the image content. Our objective is to find the cheapest way to cut the edges in the graph so that the object seeds are completely separated from the background seeds. If the edge cost is a decreasing function of the local intensity gradient then the minimum cost cut should produce an object/background segmentation with compact boundaries along the high intensity gradient values in the image. An efficient, globally optimal solution is possible via standard min-cut/max-flow algorithms for graphs with two terminals. We applied this technique to interactively segment organs in various 2D and 3D medical images.

A variable window approach to early vision

Early vision relies heavily on rectangular windows for tasks such as smoothing and computing correspondence. While rectangular windows are efficient, they yield poor results near object boundaries. We describe an efficient method for choosing an arbitrarily shaped connected window, in a manner that varies at each pixel. Our approach can be applied to several problems, including image restoration and visual correspondence. It runs in linear time, and takes a few seconds on traditional benchmark images. Performance on both synthetic and real imagery appears promising.

Energy-Based Geometric Multi-model Fitting

Geometric model fitting is a typical chicken-&-egg problem: data points should be clustered based on geometric proximity to models whose unknown parameters must be estimated at the same time. Most existing methods, including generalizations of RANSAC, greedily search for models with most inliers (within a threshold) ignoring overall classification of points. We formulate geometric multi-model fitting as an optimal labeling problem with a global energy function balancing geometric errors and regularity of inlier clusters. Regularization based on spatial coherence (on some near-neighbor graph) and/or label costs is NP hard. Standard combinatorial algorithms with guaranteed approximation bounds (e.g. α-expansion) can minimize such regularization energies over a finite set of labels, but they are not directly applicable to a continuum of labels, e.g.

Graph Cuts in Vision and Graphics: Theories and Applications

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Global Optimization for Shape Fitting

in line fitting. Our proposed approach (PEaRL) combines model sampling from data points as in RANSAC with iterative re-estimation of inliers and models’ parameters based on a global regularization functional. This technique efficiently explores the continuum of labels in the context of energy minimization. In practice, PEaRL converges to a good quality local minimum of the energy automatically selecting a small number of models that best explain the whole data set. Our tests demonstrate that our energy-based approach significantly improves the current state of the art in geometric model fitting currently dominated by various greedy generalizations of RANSAC.

Active Graph Cuts

Combinatorial min-cut algorithms on graphs have emerged as an increaseingly useful tool for problems in vision. Typically, the use of graph-cuts is motivated by one of the following two reasons. Firstly, graph-cuts allow geometric interpretation; under certain conditions a cut on a graph can be seen as a hypersurface in N-D space embedding the corresponding graph. Thus, many applications in vision and graphics use min-cut algorithms as a tool for computing optimal hypersurfaces. Secondly, graph-cuts also work as a powerful energy minimization tool for a fairly wide class of binary and nonbinary energies that frequently occur in early vision. In some cases graph cuts produce globally optimal solutions. More generally, there are iterative techniques based on graph-cuts that produce provably good approximations which (were empirically shown to) correspond to high-quality solutions in practice. Thus, another large group of applications use graph-cuts as as an optimization technique for low-level vision problems based on global energy formulations.