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18
Power watersheds: A new image segmentation framework extending graph cuts, random walker and optimal spanning forest
"... In this work, we extend a common framework for seeded image segmentation that includes the graph cuts, random walker, and shortest path optimization algorithms. Viewing an image as a weighted graph, these algorithms can be expressed by means of a common energy function with differing choices of a pa ..."
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Cited by 40 (12 self)
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In this work, we extend a common framework for seeded image segmentation that includes the graph cuts, random walker, and shortest path optimization algorithms. Viewing an image as a weighted graph, these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes. Introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watersheds in this same framework. We then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm, which we term power watersheds. Placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watersheds to optimize more general models of use in application beyond image segmentation. 1.
Homology flows, cohomology cuts
 ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2009
"... We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time fo ..."
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Cited by 30 (10 self)
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We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. Except for the special case of planar graphs, for which an O(n log n)time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surfaceembedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimumcost cycle or circulation in a given (real or integer) homology class.
Minimal surfaces extend shortest path segmentation methods to 3D
 IEEE Transactions on PAMI
, 2010
"... Abstract—Shortest paths have been used to segment object boundaries with both continuous and discrete image models. Although these techniques are well defined in 2D, the character of the path as an object boundary is not preserved in 3D. An object boundary in three dimensions is a 2D surface. Howeve ..."
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Cited by 25 (2 self)
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Abstract—Shortest paths have been used to segment object boundaries with both continuous and discrete image models. Although these techniques are well defined in 2D, the character of the path as an object boundary is not preserved in 3D. An object boundary in three dimensions is a 2D surface. However, many different extensions of the shortest path techniques to 3D have been previously proposed in which the 3D object is segmented via a collection of shortest paths rather than a minimal surface, leading to a solution which bears an uncertain relationship to the true minimal surface. Specifically, there is no guarantee that a minimal path between points on two closed contours will lie on the minimal surface joining these contours. We observe that an elegant solution to the computation of a minimal surface on a cellular complex (e.g., a 3D lattice) was given by Sullivan [47]. Sullivan showed that the discrete minimal surface connecting one or more closed contours may be found efficiently by solving a Minimumcost Circulation Network Flow (MCNF) problem. In this work, we detail why a minimal surface properly extends a shortest path (in the context of a boundary) to three dimensions, present Sullivan’s solution to this minimal surface problem via an MCNF calculation, and demonstrate the use of these minimal surfaces on the segmentation of image data. Index Terms—3D image segmentation, minimal surfaces, shortest paths, Dijkstra’s algorithm, boundary operator, total unimodularity, linear programming, minimumcost circulation network flow. Ç 1
Mathematical Morphology and Graphs: Application to Interactive Medical Image Segmentation
, 2008
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Simultaneous searching of globally optimal interacting surfaces with shape priors
 In CVPR
, 2010
"... Multiple surface searching with only image intensity information is a difficult job in the presence of high noise and weak edges. We present in this paper a novel method for globally optimal multisurface searching with a shape prior represented by convex pairwise energies. A 3D graphtheoretic fram ..."
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Cited by 5 (2 self)
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Multiple surface searching with only image intensity information is a difficult job in the presence of high noise and weak edges. We present in this paper a novel method for globally optimal multisurface searching with a shape prior represented by convex pairwise energies. A 3D graphtheoretic framework is employed. An arcweighted graph is constructed based on a shape model built from training datasets. A wide spectrum of constraints is then incorporated. The shape prior term penalizes the local topological change from the original shape model. The globally optimal solution for multiple surfaces can be obtained by computing a maximum flow in loworder polynomial time. Compared with other graphbased methods, our approach provides more local and flexible control of the shape. We also prove that our algorithm can handle the detection of multiple crossing surfaces with no shared voxels. Our method was applied to several application problems, including medical image segmentation, scenic image segmentation, and image resizing. Compared with results without using shape prior information, our improvement was quite impressive, demonstrating the promise of our method. 1.
Globally optimal surface segmentation using regional properties of segmented objects
 In: Proc. of SPIE. Lake Buena
, 2009
"... Efficient segmentation of globally optimal surfaces in volumetric images is a central problem in many medical image analysis applications. Intraclass variance has been successfully utilized, for instance, in the ChanVese model especially for images without prominent edges. In this paper, we study ..."
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Cited by 4 (1 self)
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Efficient segmentation of globally optimal surfaces in volumetric images is a central problem in many medical image analysis applications. Intraclass variance has been successfully utilized, for instance, in the ChanVese model especially for images without prominent edges. In this paper, we study the optimization problem of detecting a region (volume) between two coupled smooth surfaces by minimizing the intraclass variance using an efficient polynomialtime algorithm. Our algorithm is based on the shape probing technique in computational geometry and computes a sequence of minimumcost closed sets in a derived parametric graph. The method has been validated on computersynthetic volumetric images and in Xray CTscanned datasets of plexiglas tubes of known sizes. Its applicability to clinical data sets was demonstrated in human CT image data. The achieved results were highly accurate with mean signed surface positioning errors of the inner and outer walls of the tubes of +0.013mm and 0.012mm, respectively, given a voxel size of 0.39 × 0.39 × 0.6mm 3. Comparing with the original ChanVese method [8], our algorithm expressed higher robustness. With its polynomialtime efficiency, our algorithm is ready to be extended to higherdimensional image segmentation. In addition, the developed technique is of its own interest. We expect that it can shed some light on solving other important optimization problems arising in computer vision. To the best of our knowledge, the shape probing technique is for the first time introduced into the field of computer vision. 1.
Combinatorial Optimization of Cycles and Bases
 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
"... We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups. ..."
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Cited by 4 (0 self)
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We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups.
Combining Radiometric and Spatial Structural Information in a New Metric for Minimal Surface Segmentation
 in Information Processing in Medical Imaging, IPMI, volume LNCS 4584
, 2007
"... Abstract. Segmentation of anatomical structures via minimal surface extraction using gradientbased metrics is a popular approach, but exhibits some limits in the case of weak or missing contour information. We propose a new framework to define metrics, robust to missing image information. Given an ..."
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Cited by 3 (3 self)
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Abstract. Segmentation of anatomical structures via minimal surface extraction using gradientbased metrics is a popular approach, but exhibits some limits in the case of weak or missing contour information. We propose a new framework to define metrics, robust to missing image information. Given an object of interest we combine graylevel information and knowledge about the spatial organization of cerebral structures, into a fuzzy set which is guaranteed to include the object’s boundaries. From this set we derive a metric which is used in a minimal surface segmentation framework. We show how this metric leads to improved segmentation of subcortical gray matter structures. Quantitative results on the segmentation of the caudate nucleus in T1 MRI are reported on 18 normal subjects and 6 pathological cases. Index terms: minimal surface segmentation, level sets, spatial relations, fuzzy knowledge representation. 1
Discrete minimum ratio curves and surfaces
"... Graph cuts have proven useful for image segmentation and for volumetric reconstruction in multiple view stereo. However, solutions are biased: the cost function tends to favour either a short boundary (in 2D) or a boundary with a small area (in 3D). This bias can be avoided by instead minimising the ..."
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Cited by 2 (0 self)
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Graph cuts have proven useful for image segmentation and for volumetric reconstruction in multiple view stereo. However, solutions are biased: the cost function tends to favour either a short boundary (in 2D) or a boundary with a small area (in 3D). This bias can be avoided by instead minimising the cut ratio, which normalises the cost by a measure of the boundary size. This paper uses ideas from discrete differential geometry to develop a linear programming formulation for finding a minimum ratio cut in arbitrary dimension, which allows constraints on the solution to be specified in a natural manner, and which admits an efficient and globally optimal solution. Results are shown for 2D segmentation and for 3D volumetric reconstruction. 1.