After [10, 15, 12, 2, 4] minimum cut/maximum ow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial optimization literature provides many min-cut/max-ow algorithms with dierent polynomial time complexity. Their practical eciency, however, has to date been studied mainly outside the scope of computer vision.

Abstract—In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a general-purpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to create the appropriate graph. A software implementation is freely available.

The problem of efficient, interactive foreground/background segmentation in still images is of great practical importance in image editing. Classical image segmentation tools use either texture (colour) information, e.g. Magic Wand, or edge (contrast) information, e.g. Intelligent Scissors. Recently, an approach based on optimization by graph-cut has been developed which successfully combines both types of information. In this paper we extend the graph-cut approach in three respects. First, we have developed a more powerful, iterative version of the optimisation. Secondly, the power of the iterative algorithm is used to simplify substantially the user interaction needed for a given quality of result. Thirdly, a robust algorithm for “border matting ” has been developed to estimate simultaneously the alpha-matte around an object boundary and the colours of foreground pixels. We show that for moderately difficult examples the proposed method outperforms competitive tools.

This paper presents a volumetric formulation for the multi-view stereo problem which is amenable to a computationally tractable global optimisation using Graph-cuts. Our approach is to seek the optimal partitioning of 3D space into two regions labelled as ‘object’ and ‘empty’ under a cost functional consisting of the following two terms: (1) A term that forces the boundary between the two regions to pass through photo-consistent locations and (2) a ballooning term that inflates the ‘object ’ region. To take account of the effect of occlusion on the first term we use an occlusion robust photo-consistency metric based on Normalised Cross Correlation, which does not assume any geometric knowledge about the reconstructed object. The globally optimal 3D partitioning can be obtained as the minimum cut solution of a weighted graph.

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.

Extracting layers from video is very important for video representation, analysis, compression, and synthesis. Assuming that a scene can be approximately described by multiple planar regions, this paper describes a robust and novel approach to automatically extract a set of affine or projective transformations induced by these regions, detect the occlusion pixels over multiple consecutive frames, and segment the scene into several motion layers. First, after determining a number of seed regions using correspondences in two frames, we expand the seed regions and reject the outliers employing the graph cuts method integrated with level set representation. Next, these initial regions are merged into several initial layers according to the motion similarity. Third, an occlusion order constraint on multiple frames is explored, which enforces that the occlusion area increases with the temporal order in a short period and effectively maintains segmentation consistency over multiple consecutive frames. Then the correct layer segmentation is obtained by using a graph cuts algorithm, and the occlusions between the overlapping layers are explicitly determined. Several experimental results are demonstrated to show that our approach is effective and robust. Index Terms Layer-based motion segmentation, video analysis, graph cuts, level set representation, occlusion order constraint. I.

Abstract. We present a new variational method for multi-view stereovision and non-rigid three-dimensional motion estimation from multiple video sequences. Our method minimizes the prediction error of the shape and motion estimates. Both problems then translate into a generic image registration task. The latter is entrusted to a global measure of image similarity, chosen depending on imaging conditions and scene properties. Contrarily to existing deformable surfaces methods, which integrate a matching measure computed independently at each surface point, our approach computes a global image-based matching score between the input images and the predicted images. The matching process fully handles projective distortion and partial occlusions. Neighborhood as well as global intensity information can be exploited to improve the robustness to appearance changes due to non-Lambertian materials and illumination changes, without any approximation of shape, motion or visibility. Moreover, our approach results in a simpler, more flexible, and more efficient implementation than in existing methods. The computation time on large datasets does not exceed thirty minutes on a standard workstation. Finally, our method is compliant with a hardware implementation with graphics processor units. Our stereovision algorithm yields very good results on a variety of datasets including specularities and translucency. We have successfully tested our motion estimation algorithm on a very challenging multi-view video sequence of a non-rigid scene.

Surface reconstruction from multiple calibrated images mainly has been approached using local methods, either as a continuous optimization problem driven by level sets, or by discrete volumetric methods such as space carving. We propose a direct surface reconstruction approach which starts from a continuous geometric functional that is minimized up to a discretization by a global graph-cut algorithm operating on a 3D embedded graph. The method is related to the stereo disparity computation based on graph-cut formulation, but fundamentally different in two aspects. First, existing stereo disparity methods are only interested in obtaining layers of constant disparity, while we focus on high resolution surface geometry. Second, most of the existing graph-cut algorithms only reach approximate solutions, while we guarantee a global minimum. The whole procedure is consistently incorporated into a voxel representation that handles both occlusions and discontinuities. We demonstrate our algorithm on real sequences, yielding remarkably detailed surface geometry up to 1/10th of a pixel.

In [3] we showed that graph cuts can find hypersurfaces of globally minimal length (or area) under any Riemannian metric. Here we show that graph cuts on directed regular grids can approximate a significantly more general class of continuous non-symmetric metrics. Using submodularity condition [1, 11], we obtain a tight characterization of graph-representable metrics. Such “submodular” metrics have an elegant geometric interpretation via hypersurface functionals combining length/area and flux. Practically speaking, we extend “geo-cuts ” algorithm [3] to a wider class of geometrically motivated hypersurface functionals and show how to globally optimize any combination of length/area and flux of a given vector field. The concept of flux was recently introduced into computer vision by [13] but it was mainly studied within variational framework so far. We are first to show that flux can be integrated into graph cuts as well. Combining geometric concepts of flux and length/area within the global optimization framework of graph cuts allows principled discrete segmentation models and advances the state of the art for the graph cuts methods in vision. In particular, we address the “shrinking ” problem of graph cuts, improve segmentation of long thin objects, and introduce useful shape constraints. 1.

This article presents a novel method for acquiring high-quality solid models of complex 3D shapes from multiple calibrated photographs. After the purely geometric constraints associated with the silhouettes found in each image have been used to construct a coarse surface approximation in the form of a visual hull, photoconsistency constraints are enforced in three consecutive steps: (1) the rims where the surface grazes the visual hull are first identified through dynamic programming; (2) with the rims now fixed, the visual hull is carved using graph cuts to globally optimize the photoconsistency of the surface and recover its main features; (3) an iterative (local) refinement step is finally used to recover fine surface details. The proposed approach has been implemented, and experiments with six real data sets are presented, along with qualitative comparisons with several state-of-the-art image-based-modeling algorithms.