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].

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.

Abstract. 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. 1

In this note we present an implementation of the fast marching algorithm for solving Eikonal equations that reduces the original run-time from O(N log N) to linear. This lower run-time cost is obtained while keeping an error bound of the same order of magnitude as the original algorithm. This improvement is achieved introducing the straight forward untidy priority queue, obtained via a quantization of the priorities in the marching computation. We present the underlying framework, estimations on the error, and examples showing the usefulness of the proposed approach. Key words: Fast marching, Hamilton-Jacobi and Eikonal equations, distance functions, bucket sort, untidy priority queue.

This work reviews the watershed in the graph framework of a shortest-path forest problem using a lexicographic path cost formulation. This formulation reects the behavior of the ordered queue-based watershed algorithm. This algorithm is compared with our proposed shortest-path forest (IFT{Image Foresting Transform), concluding that the watershed is a special case of that. Recently many dierent watershed approaches are being used. We point out that in some cases the watershed algorithm does not keep the optimality of the shortest-path forest solution unless the IFT algorithm is used. The main dierence between the algorithms is related to permanently labeling a pixel when inserting or removing it from the queue. The watershed based on the pixel dissimilarity using IFT can segment one-pixel width regions while keeping the optimality of the shortest-path forest solution.

In this work, we present a common framework for seeded image segmentation algorithms that yields two of the leading methods as special cases- The Graph Cuts and the Random Walker algorithms. The formulation of this common framework naturally suggests a new, third, algorithm that we develop here. Specifically, the former algorithms may be shown to minimize a certain energy with respect to either an ℓ1 or an ℓ2 norm. Here, we explore the segmentation algorithm defined by an ℓ ∞ norm, provide a method for the optimization and show that the resulting algorithm produces an accurate segmentation that demonstrates greater stability with respect to the number of seeds employed than either the Graph Cuts or Random Walker methods.

Figure 1: These four volume renderings utilize a fully opaque transfer function, but are segmented using the method discussed in this paper. The segmented volumes show: (a) abdominal aortic branch vessels, (b) an aortic aneurysm, (c) an aorta, and (d) peripheral blood vessels in the lung. The yellow arrows indicate the location of the user’s initial seeds that were evolved to form the presented segmentations. Segmentation of structures from measured volume data, such as anatomy in medical imaging, is a challenging data-dependent task. In this paper, we present a segmentation method that leverages the parallel processing capabilities of modern programmable graphics hardware in order to run significantly faster than previous methods. In addition, collocating the algorithm computation with the visualization on the graphics hardware circumvents the need to transfer data across the system bus, allowing for faster visualization and interaction. This algorithm is unique in that it utilizes sophisticated graphics hardware functionality (i.e., floating point precision, render to texture, computational masking, and fragment programs) to enable fast segmentation and interactive visualization.

Abstract. We present a fast and accurate tool for semiautomatic segmentation of volumetric medical images based on the live wire algorithm, shape-based interpolation and a new optimization method. While the user-steered live wire algorithm represents an efficient, precise and reproducible method for interactive segmentation of selected twodimensional images, the shape-based interpolation allows the automatic approximation of contours on slices between user-defined boundaries. The combination of both methods leads to accurate segmentations with significantly reduced user interaction time. Moreover, the subsequent automated optimization of the interpolated object contours results in a better segmentation quality or can be used to extend the distances between user-segmented images and for a further reduction of interaction time. Experiments were carried out on hepatic computer tomographies from three different clinics. The results of the segmentation of liver parenchyma have shown that the user interaction time can be reduced more than 60% by the combination of shape-based interpolation and our optimization method with volume deviations in the magnitude of inter-user differences. 1

Efficient segmentation of globally optimal surfaces representing object boundaries in volumetric data sets is important and challenging in many medical image analysis applications. We have developed an optimal surface detection method capable of simultaneously detecting multiple interacting surfaces, in which the optimality is controlled by the cost functions designed for individual surfaces and by several geometric constraints defining the surface smoothness and interrelations. The method solves the surface segmentation problem by transforming it into computing a minimum s-t cut in a derived arc-weighted directed graph. The proposed algorithm has a low-order polynomial time complexity and is computationally efficient. It has been extensively validated on more than 300 computer-synthetic volumetric images, 72 CT-scanned data sets of different-sized plexiglas tubes, and tens of medical images spanning various imaging modalities. In all cases, the approach yielded highly accurate results. Our approach can be readily extended to higher-dimensional image segmentation.

Abstract. A framework is proposed for the segmentation of brain tumors from MRI. Instead of training on pathology, the proposed method trains exclusively on healthy tissue. The algorithm attempts to recognize deviations from normalcy in order to compute a fitness map over the image associated with the presence of pathology. The resulting fitness map may then be used by conventional image segmentation techniques for honing in on boundary delineation. Such an approach is applicable to structures that are too irregular, in both shape and texture, to permit construction of comprehensive training sets. The technique is an extension of EM segmentation that considers information on five layers: voxel intensities, neighborhood coherence, intra-structure properties, inter-structure relationships, and user input. Information flows between the layers via multi-level Markov random fields and Bayesian classification. A simple instantiation of the framework has been implemented to perform preliminary experiments on synthetic and MRI data. 1