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757
Topological Persistence and Simplification
, 2000
"... We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast ..."
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Cited by 338 (44 self)
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We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast algorithms for computing persistence and experimental evidence for their speed and utility.
On the existence of harmonic maps
, 1977
"... (1.1) A map between Riemannian manifolds is harmonic if the divergence of its differential vanishes. (Those terms will be defined in §3.) Such maps are the extrema ( = critical points) of the energy functional. More precisely, if <$>: M* • N is a map between Riemannian manifolds, we define ..."
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Cited by 283 (5 self)
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(1.1) A map between Riemannian manifolds is harmonic if the divergence of its differential vanishes. (Those terms will be defined in §3.) Such maps are the extrema ( = critical points) of the energy functional. More precisely, if <$>: M* • N is a map between Riemannian manifolds, we define
Computing Contour Trees in All Dimensions
, 1999
"... We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al. ..."
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Cited by 161 (11 self)
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We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al.
MorseSmale Complexes for Piecewise Linear 3Manifolds
, 2003
"... We define the MorseSmale complex of a Morse function over a 3manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatori ..."
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Cited by 133 (28 self)
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We define the MorseSmale complex of a Morse function over a 3manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatorial algorithm for constructing such complexes for piecewise linear data.
A topology preserving level set method for geometric deformable models
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2003
"... Active contour and surface models, also known as deformable models, are powerful image segmentation techniques. Geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior, parameterization independence, and ease of implement ..."
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Cited by 121 (7 self)
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Active contour and surface models, also known as deformable models, are powerful image segmentation techniques. Geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior, parameterization independence, and ease of implementation. However, a long claimed advantage of geometric deformable models—the ability to automatically handle topology changes—turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved. In this paper, we present a new class of geometric deformable models designed using a novel topologypreserving level set method, which achieves topology preservation by applying the simple point concept from digital topology. These new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces. Moreover, since the topologypreserving constraint is enforced efficiently through local computations, the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models. Several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm.
Diffusion Kernels on Statistical Manifolds
, 2004
"... A family of kernels for statistical learning is introduced that exploits the geometric structure of statistical models. The kernels are based on the heat equation on the Riemannian manifold defined by the Fisher information metric associated with a statistical family, and generalize the Gaussian ker ..."
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Cited by 116 (8 self)
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A family of kernels for statistical learning is introduced that exploits the geometric structure of statistical models. The kernels are based on the heat equation on the Riemannian manifold defined by the Fisher information metric associated with a statistical family, and generalize the Gaussian kernel of Euclidean space. As an important special case, kernels based on the geometry of multinomial families are derived, leading to kernelbased learning algorithms that apply naturally to discrete data. Bounds on covering numbers and Rademacher averages for the kernels are proved using bounds on the eigenvalues of the Laplacian on Riemannian manifolds. Experimental results are presented for document classification, for which the use of multinomial geometry is natural and well motivated, and improvements are obtained over the standard use of Gaussian or linear kernels, which have been the standard for text classification.
Balancing For Nonlinear Systems
 Systems & Control Letters
, 1993
"... We present a method of balancing for nonlinear systems which is an extension of balancing for linear systems in the sense that it is based on the input and output energy of a system. We deal with the input and output energy function of a stable nonlinear systems and propose a method to use these fun ..."
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Cited by 94 (15 self)
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We present a method of balancing for nonlinear systems which is an extension of balancing for linear systems in the sense that it is based on the input and output energy of a system. We deal with the input and output energy function of a stable nonlinear systems and propose a method to use these functions to get a balanced form for a stable nonlinear system. It is a local result, but gives `broader' results then we obtain by just linearizing the system. Keywords: balancing, nonlinear systems, energy functions, HamiltonJacobi equations, Hankel singular values. 1 Introduction Balancing for linear systems is a well known subject on which there has been a lot of research in the last decade. It started with a paper of Moore [6] in 1981, where balancing is introduced with the aim of using it as a tool for model reduction. If a linear system is in balanced form the Hankel singular values are a measure for the importance of state components. This means that the influence of the correspondin...
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 90 (20 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Featurebased surface parameterization and texture mapping
 ACM Transactions on Graphics
, 2005
"... and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this article, we present an automatic parameterization method for segmenting a surface into patches that are then flattened with little stretch. Many objects consist o ..."
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Cited by 90 (5 self)
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and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this article, we present an automatic parameterization method for segmenting a surface into patches that are then flattened with little stretch. Many objects consist of regions of relatively simple shapes, each of which has a natural parameterization. Based on this observation, we describe a threestage featurebased patch creation method for manifold surfaces. The first two stages, genus reduction and feature identification, are performed with the help of distancebased surface functions. In the last stage, we create one or two patches for each feature region based on a covariance matrix of the feature’s surface points. To reduce stretch during patch unfolding, we notice that stretch is a 2 × 2 tensor, which in ideal situations is the identity. Therefore, we use the GreenLagrange tensor to measure and to guide the optimization process. Furthermore, we allow the boundary vertices of a patch to be optimized by adding scaffold triangles. We demonstrate our featurebased patch creation and patch unfolding methods for several textured models. Finally, to evaluate the quality of a given parameterization, we describe an imagebased error measure that takes into account stretch, seams, smoothness, packing efficiency, and surface visibility.
Removing excess topology from isosurfaces
 ACM Trans. Graph
, 2004
"... Many highresolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studie ..."
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Cited by 85 (1 self)
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Many highresolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studied. However, isosurfaces often contain many topological errors in the form of tiny handles. These nearly invisible artifacts hinder subsequent operations like mesh simplification, remeshing, and parametrization. In this paper we present an efficient method for removing handles in an isosurface. Our algorithm makes an axisaligned sweep through the volume to locate handles, compute their sizes, and selectively remove them. The algorithm is designed for outofcore execution. It finds the handles by incrementally constructing and analyzing a surface Reeb graph. The size of a handle is measured by a short surface loop that breaks it. Handles are removed robustly by modifying the volume rather than attempting “mesh surgery. ” Finally, the volumetric modifications are spatially localized to preserve geometrical detail. We demonstrate topology simplification on several complex models, and show its benefit for subsequent surface processing.