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25
Power Watershed: A Unifying GraphBased Optimization Framework
, 2011
"... In this work, we extend a common framework for graphbased 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 ..."
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Cited by 40 (8 self)
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In this work, we extend a common framework for graphbased 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 watershed 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 watershed. In particular when q = 2, the power watershed leads to a multilabel, scale and contrast invariant, unique global optimum obtained in practice in quasilinear time. Placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watershed to optimize more general models of use in applications beyond image segmentation.
Curvature Regularity for Regionbased Image Segmentation and Inpainting: A Linear Programming Relaxation
"... We consider a class of regionbased energies for image segmentation and inpainting which combine region integrals with curvature regularity of the region boundary. To minimize such energies, we formulate an integer linear program which jointly estimates regions and their boundaries. Curvature regula ..."
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Cited by 31 (8 self)
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We consider a class of regionbased energies for image segmentation and inpainting which combine region integrals with curvature regularity of the region boundary. To minimize such energies, we formulate an integer linear program which jointly estimates regions and their boundaries. Curvature regularity is imposed by respective costs on pairs of adjacent boundary segments. By solving the associated linear programming relaxation and thresholding the solution one obtains an approximate solution to the original integer problem. To our knowledge this is the first approach to impose curvature regularity in regionbased formulations in a manner that is independent of initialization and allows to compute a bound on the optimal energy. In a variety of experiments on segmentation and inpainting, we demonstrate the advantages of higherorder regularity. Moreover, we demonstrate that for most experiments the optimality gap is smaller than 2 % of the global optimum. For many instances we are even able to compute the global optimum. 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.
The least spanning area of a knot and the optimal bounding chain problem
 In Proceedings of the 27th Annual ACM Symposium on Computational Geometry (SoCG’11). ACM
, 2011
"... Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3dimensional manifold. When the knot is embedded in a general 3manifold, the problems of finding these surfaces were shown to be NPcomplete and NPhard respectively. However, th ..."
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Cited by 15 (2 self)
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Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3dimensional manifold. When the knot is embedded in a general 3manifold, the problems of finding these surfaces were shown to be NPcomplete and NPhard respectively. However, there is evidence that the special case when the ambient manifold is R3, or more generally when the second homology is trivial, should be considerably more tractable. Indeed, we show here that a natural discrete version of the least area surface can be found in polynomial time. The precise setting is that the knot is a 1dimensional subcomplex of a triangulation of the ambient 3manifold. The main tool we use is a linear programming formulation of the Optimal Bounding Chain Problem (OBCP), where one is required to find the smallest norm chain with a given boundary. While the decision variant of OBCP is NPcomplete in general, we give conditions under which it can be solved in polynomial time. We then show that the least area surface can be constructed from the optimal bounding chain using a standard desingularization argument from 3dimensional topology. We also prove that the related Optimal Homologous Chain Problem is NPcomplete for homology with integer coefficients, complementing the corresponding result of Chen and Freedman for mod 2 homology.
D.: Geometrically consistent elastic matching of 3d shapes: A linear programming solution
 In Proc. International Conference on Computer Vision
, 2011
"... We propose a novel method for computing a geometrically consistent and spatially dense matching between two 3D shapes. Rather than mapping points to points we match infinitesimal surface patches while preserving the geometric structures. In this spirit we consider matchings as diffeomorphisms betwee ..."
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Cited by 11 (5 self)
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We propose a novel method for computing a geometrically consistent and spatially dense matching between two 3D shapes. Rather than mapping points to points we match infinitesimal surface patches while preserving the geometric structures. In this spirit we consider matchings as diffeomorphisms between the objects ’ surfaces which are by definition geometrically consistent. Based on the observation that such diffeomorphisms can be represented as closed and continuous surfaces in the product space of the two shapes we are led to a minimal surface problem in this product space. The proposed discrete formulation describes the search space with linear constraints. Computationally, our approach leads to a binary linear program whose relaxed version can be solved efficiently in a globally optimal manner. As cost function for matching, we consider a thin shell energy, measuring the physical energy necessary to deform one shape into the other. Experimental results demonstrate that the proposed LP relaxation allows to compute highquality matchings which reliably put into correspondence articulated 3D shapes. Moreover a quantitative evaluation shows improvements over existing works. Figure 1. We propose to cast the dense elastic matching of surfaces in 3D as a codimensiontwo minimal surface problem which aims at minimizing the distortion when transforming one shape into the other. We show that a consistent discretization of this minimal surface problem gives rise to an integer linear program. By means of LP relaxation we can compute nearoptimal matchings such as the one shown above. These matchings are dense trianglewise matchings. (For visualization we combined triangles to patches and colored them consistently with their corresponding patch.) 1.
Efficient algorithms for image and high dimensional data processing using eikonal equation on graphs
 in Proc. ISVC
, 2010
"... Abstract. In this paper we propose an adaptation of the static eikonal equation over weighted graphs of arbitrary structure using a framework of discrete operators. Based on this formulation, we provide explicit solutions for the L1, L2 and L ∞ norms. Efficient algorithms to compute the explicit sol ..."
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Cited by 3 (2 self)
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Abstract. In this paper we propose an adaptation of the static eikonal equation over weighted graphs of arbitrary structure using a framework of discrete operators. Based on this formulation, we provide explicit solutions for the L1, L2 and L ∞ norms. Efficient algorithms to compute the explicit solution of the eikonal equation on graphs are also described. We then present several applications of our methodology for image processing such as superpixels decomposition, region based segmentation or patchbased segmentation using nonlocal configurations. By working on graphs, our formulation provides an unified approach for the processing of any data that can be represented by a graph such as highdimensional data. 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.
Falcão, “Elucidating the relations among seeded image segmentation methods and their possible extensions
 in Proceedings of Sibgrapi
, 2011
"... Abstract—Many image segmentation algorithms have been proposed, specially for the case of binary segmentation (object/background) in which hard constraints (seeds) are provided interactively. Recently, several theoretical efforts were made in an attempt to unify their presentation and clarify their ..."
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Cited by 1 (0 self)
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Abstract—Many image segmentation algorithms have been proposed, specially for the case of binary segmentation (object/background) in which hard constraints (seeds) are provided interactively. Recently, several theoretical efforts were made in an attempt to unify their presentation and clarify their relations. These relations are usually pointed out textually or depicted in the form of a table of parameters of a general energy formulation. In this work we introduce a more general diagram representation which captures the connections among the methods, by means of conventional relations from set theory. We formally instantiate several methods under this diagram, including graph cuts, power watersheds, fuzzy connectedness, grow cut, distance cuts, and others, which are usually presented as unrelated methods. The proposed diagram representation leads to a more elucidated view of the methods, being less restrictive than the tabular representation. It includes new relations among methods, besides bringing together the connections gathered from different works. It also points out some promising unexplored intermediate regions, which can lead to possible extensions of the existing methods. We also demonstrate one of such possible extensions, which is used to effectively combine the strengths of region and local contrast features. Keywordsgraph search algorithms; image foresting transform; graphcut segmentation; watersheds; fuzzy connectedness; I.
Contiguous minimum singlesourcemultisink cuts in weighted planar graphs
"... Abstract. We present a fast algorithm for uniform sampling of contiguous minimum cuts separating a source vertex from a set of sink vertices in a weighted undirected planar graph with n vertices embedded in the plane. The algorithm takes O(n) time per sample, after an initial O(n 3) preprocessing ti ..."
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Cited by 1 (1 self)
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Abstract. We present a fast algorithm for uniform sampling of contiguous minimum cuts separating a source vertex from a set of sink vertices in a weighted undirected planar graph with n vertices embedded in the plane. The algorithm takes O(n) time per sample, after an initial O(n 3) preprocessing time during which the algorithm computes the number of all such contiguous minimum cuts. Contiguous cuts (that is, cuts where a naturally defined boundary around the cut set forms a simply connected planar region) have applications in computer vision and medical imaging [6, 14]. 1