Results 1  10
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10
Random walks for image segmentation
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2006
"... Abstract—A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with userdefined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach on ..."
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Cited by 218 (18 self)
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Abstract—A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with userdefined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels. By assigning each pixel to the label for which the greatest probability is calculated, a highquality image segmentation may be obtained. Theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits. This algorithm is formulated in discrete space (i.e., on a graph) using combinatorial analogues of standard operators and principles from continuous potential theory, allowing it to be applied in arbitrary dimension on arbitrary graphs. Index Terms—Image segmentation, interactive segmentation, graph theory, random walks, combinatorial Dirichlet problem, harmonic functions, Laplace equation, graph cuts, boundary completion. Ç 1
Parametric Tilings and Scattered Data Approximation
 International Journal of Shape Modeling
, 1998
"... Abstract: This paper is concerned with methods for mapping meshes in IR 3 to meshes in IR 2 in such a way that the local geometry of the mesh is as far as possible preserved. Two linear methods are analyzed and compared. The first, which is more general and more stable, is based on convex combinatio ..."
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Cited by 21 (8 self)
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Abstract: This paper is concerned with methods for mapping meshes in IR 3 to meshes in IR 2 in such a way that the local geometry of the mesh is as far as possible preserved. Two linear methods are analyzed and compared. The first, which is more general and more stable, is based on convex combinations while the second is based on minimizing weighted squared lengths of edges. These methods can be used to approximate scattered data points in IR 3 with smooth parametric spline surfaces. 1.
Simple and Optimal OutputSensitive Construction of Contour Trees Using Monotone Paths
, 2004
"... Contour trees are used when highdimensional data are preprocessed for efficient extraction of isocontours for the purpose of visualization. So far, efficient algorithms for contour trees are based on processing the data in sorted order. We present a new algorithm that avoids sorting of the whole da ..."
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Cited by 19 (1 self)
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Contour trees are used when highdimensional data are preprocessed for efficient extraction of isocontours for the purpose of visualization. So far, efficient algorithms for contour trees are based on processing the data in sorted order. We present a new algorithm that avoids sorting of the whole dataset, but sorts only a subset of socalled componentcritical points. They form only a small fraction of the vertices in the dataset, for typical data that arise in practice. The algorithm is simple, achieves the optimal outputsensitive bound in running time, and works in any dimension. Our experiments show that the algorithm compares favorably with the previous best algorithm.
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
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Cited by 12 (1 self)
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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...
F.: Mesh parameterization by minimizing the synthesized distortion metric with the coefficient optimizing algorithm
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract—The parameterization of a 3D mesh into a planar domain requires a distortion metric and a minimizing process. Most previous work has sought to minimize the average area distortion, the average angle distortion, or a combination of these. Typical distortion metrics can reflect the overall pe ..."
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Cited by 1 (0 self)
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Abstract—The parameterization of a 3D mesh into a planar domain requires a distortion metric and a minimizing process. Most previous work has sought to minimize the average area distortion, the average angle distortion, or a combination of these. Typical distortion metrics can reflect the overall performance of parameterizations but discount high local deformations. This affects the performance of postprocessing operations such as uniform remeshing and texture mapping. This paper introduces a new metric that synthesizes the average distortions and the variances of both the area deformations and the angle deformations over an entire mesh. Experiments show that, when compared with previous work, the use of synthesized distortion metric performs satisfactorily in terms of both the average area deformation and the average angle deformation; furthermore, the area and angle deformations are distributed more uniformly. This paper also develops a new iterative process for minimizing the synthesized distortion, the coefficientoptimizing algorithm. At each iteration, rather than updating the positions immediately after the local optimization, the coefficientoptimizing algorithm first update the coefficients for the linear convex combination and then globally updates the positions by solving the Laplace system. The high performance of the coefficientoptimizing algorithm has been demonstrated in many experiments. Index Terms—Mesh parameterization, texture mapping, barycentric mapping, conformal mapping, harmonic mapping. æ 1
Metric Approaches to Invariant Shape Similarity
, 2009
"... Nonrigid shapes are ubiquitous in Nature and are encountered at all levels of life, from macro to nano. The need to model such shapes and understand their behavior arises in many applications in imaging sciences, pattern recognition, computer vision, and computer graphics. Of particular importance ..."
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Nonrigid shapes are ubiquitous in Nature and are encountered at all levels of life, from macro to nano. The need to model such shapes and understand their behavior arises in many applications in imaging sciences, pattern recognition, computer vision, and computer graphics. Of particular importance is understanding which properties of the shape are attributed to deformations and which are invariant, i.e., remain unchanged. This chapter presents an approach to nonrigid shapes from the point of view of metric geometry. Modeling shapes as metric spaces, one can pose the problem of shape similarity as the similarity of metric spaces and harness tools from theoretical metric geometry for the computation of such a similarity. 1
The Cycle Space of a 3Connected Graph is Generated by Its Finite and Infinite Locally Peripheral Circuits
, 2003
"... We extend Tutte's result that in a finite 3connected graph the cycle space is generated by the peripheral circuits to locally finite graphs. Such a generalization becomes possible by the admission of infinite circuits in the graph compactified by its ends. ..."
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We extend Tutte's result that in a finite 3connected graph the cycle space is generated by the peripheral circuits to locally finite graphs. Such a generalization becomes possible by the admission of infinite circuits in the graph compactified by its ends.
Combinatorial Optimization: Some . . .
, 1992
"... In this working paper I discuss some issues, with the aim of suggesting research topics for the DIMACS special year "Combinatorial Optimization". I plan to focus on the unorthodox, hoping that at least one of the areas I describe below will gain new momentum as a result of this special year. A compa ..."
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In this working paper I discuss some issues, with the aim of suggesting research topics for the DIMACS special year "Combinatorial Optimization". I plan to focus on the unorthodox, hoping that at least one of the areas I describe below will gain new momentum as a result of this special year. A companion preprint, written by Martin Grötschel and myself as a chapter of the Handbook of Combinatorics, describes the state of the art, concentrating on the wellestablished methods and models like polyhedral combinatorics, and should be available soon.