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.

In this paper we describe a new technique for general purpose interactive segmentation of N-dimensional images. The user marks certain pixels as “object” or “background” to provide hard constraints for segmentation. Additional soft constraints incorporate both boundary and region information. Graph cuts are used to find the globally optimal segmentation of the N-dimensional image. The obtained solution gives the best balance of boundary and region properties among all segmentations satisfying the constraints. The topology of our segmentation is unrestricted and both “object” and “background” segments may consist of sev-eral isolatedparts. Some experimental results are presented in the context ofphotohideo editing and medical image seg-mentation. We also demonstrate an interesting Gestalt example. A fast implementation of our segmentation method is possible via a new mar-$ow algorithm in [2].

Many application domains suffer from not having enough labeled training data for learning. However, large amounts of unlabeled examples can often be gathered cheaply. As a result, there has been a great deal of work in recent years on how unlabeled data can be used to aid classification. We consider an algorithm based on finding minimum cuts in graphs, that uses pairwise relationships among the examples in order to learn from both labeled and unlabeled data. Our algorithm

We address the problem of computing the 3-dimensional shape of an arbitrary scene from a set of images taken at known viewpoints.

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

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.

This paper describes a novel approach for reconstructing a closed continuous surface of an object from multiple calibrated color images and silhouettes. Any accurate reconstruction must satisfy (1) photo-consistency and (2) silhouette consistency constraints. Most existing techniques treat these cues identically in optimization frameworks where silhouette constraints are traded off against photo-consistency and smoothness priors. Our approach strictly enforces silhouette constraints, while optimizing photo-consistency and smoothness in a global graph-cut framework. We transform the reconstruction problem into computing max-flow / mincut in a geometric graph, where any cut corresponds to a surface satisfying exact silhouette constraints (its silhouettes should exactly coincide with those of the visual hull); a minimum cut is the most photo-consistent surface amongst them. Our graph-cut formulation is based on the rim mesh, (the combinatorial arrangement of rims or contour generators from many views) which can be computed directly from the silhouettes. Unlike other methods, our approach enforces silhouette constraints without introducing a bias near the visual hull boundary and also recovers the rim curves. Results are presented for synthetic and real datasets.

In this paper, we investigate what can be inferred from several silhouette probability maps, in multi-camera environments. To this aim, we propose a new framework for multi-view silhouette cue fusion. This framework uses a space occupancy grid as a probabilistic 3D representation of scene contents. Such a representation is of great interest for various computer vision applications in perception, or localization for instance. Our main contribution is to introduce the occupancy grid concept, popular in the robotics community, for multi-camera environments. The idea is to consider each camera pixel as a statistical occupancy sensor. All pixel observations are then used jointly to infer where, and how likely, matter is present in the scene. As our results illustrate, this simple model has various advantages. Most sources of uncertainty are explicitly modeled, and no premature decisions about pixel labeling occur, thus preserving pixel knowledge. Consequently, optimal scene object localization, and robust volume reconstruction, can be achieved, with no constraint on camera placement and object visibility. In addition, this representation allows to improve silhouette extraction in images.

This paper presents a new volumetric stereo algorithm to reconstruct the 3D shape of an arbitrary object. Our method is based on finding the minimum cut in an octahedral graph structure embedded into the volumetric grid, which establishes a well defined relationship between the integrated photo-consistency function of a region in space and the corresponding edge weights of the embedded graph. This new graph structure allows for a highly efficient hierarchical implementation supporting high volumetric resolutions and large numbers of input images. Furthermore we will show how the resulting cut surface can be directly converted into a consistent, closed and manifold mesh. Hence this work provides a complete multi-view stereo reconstruction pipeline. We demonstrate the robustness and efficiency of our technique by a number of high quality reconstructions of real objects.