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108
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 333 (41 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.
The Union of Balls and its Dual Shape
, 1993
"... Efficient algorithms are described for compuiing topological, combinatorial, and metric properties of ihe union of finitely many balls in R^d. These algorithms are based on a simplicial complex dual to a certain decomposition of the union of balls, and on short inclusionexclusion formulas derived f ..."
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Cited by 172 (12 self)
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Efficient algorithms are described for compuiing topological, combinatorial, and metric properties of ihe union of finitely many balls in R^d. These algorithms are based on a simplicial complex dual to a certain decomposition of the union of balls, and on short inclusionexclusion formulas derived from this complex. The algorithms are most relevant in R’3 where unions of finitely many balls are commonly used as models of molecules.
Hierarchical Morse Complexes for Piecewise Linear 2Manifolds
, 2001
"... We present algorithms for constructing a hierarchy of increasingly coarse Morse complexes that decompose a piecewise linear 2manifold. While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simu ..."
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Cited by 66 (2 self)
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We present algorithms for constructing a hierarchy of increasingly coarse Morse complexes that decompose a piecewise linear 2manifold. While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability. We then simplify Morse complexes by cancelling pairs of critical points in order of increasing persistence. Keywords Computational topology, PL manifolds, Morse theory, topological persistence, hierarchy, algorithms, implementation, terrains 1. INTRODUCTION In this paper, we define the Morse complex decomposing a piecewise linear 2manifold and present algorithms for constructing and simplifying this complex. 1.1 Motivation Physical simulation problems often start with a space and measurements over this space. If the measurements are scalar values, we talk about a height function of that space. We use this name throughout the paper, although the functions can ...
Deformable Smooth Surface Design
, 1999
"... A new paradigm for designing smooth surfaces is described. A finite set of points with weights specifies a closed surface in space referred to as skin. It consists of one or more components, each tangent continuous and free of selfintersections and intersections with other components. The skin var ..."
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Cited by 54 (11 self)
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A new paradigm for designing smooth surfaces is described. A finite set of points with weights specifies a closed surface in space referred to as skin. It consists of one or more components, each tangent continuous and free of selfintersections and intersections with other components. The skin varies continuously with the weights and locations of the points, and the variation includes the possibility of a topology change facilitated by the violation of tangent continuity at a single point in space and time. Applications of the skin to molecular modeling and to geometric deformation are discussed.
Dynamic Skin Triangulation
"... This paper describes an algorithm for maintaining an approximating triangulation of a deforming surface in R³. The surface is the envelope of an infinite family of spheres defined and controlled by a finite collection of weighted points. The triangulation adapts dynamically to changing shape, curvat ..."
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Cited by 48 (14 self)
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This paper describes an algorithm for maintaining an approximating triangulation of a deforming surface in R³. The surface is the envelope of an infinite family of spheres defined and controlled by a finite collection of weighted points. The triangulation adapts dynamically to changing shape, curvature, and topology of the surface.
Computing Betti Numbers via Combinatorial Laplacians
 ALGORITHMICA
, 1998
"... We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ”, ..."
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Cited by 48 (1 self)
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We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ”, of eigenvalues which we have yet to understand fully. We numerically verify a conjecture of Björner, Lovász, Vrećica, and ˘Zivaljevic ́ on the chessboard complexes C.4; 6/, C.5; 7/, and C.5; 8/. Our verification suffers a technical weakness, which can be overcome in various ways; we do so for C.4; 6 / and C.5; 8/, giving a completely rigorous (computer) proof of the conjecture in these two cases. This brings up an interesting question in recovering an integral basis from a real basis of vectors.
Measuring proteins and voids in proteins
 In Proc. 28th Ann. Hawaii Int’l Conf. System Sciences
, 1995
"... Common geometric models for proteins and other molecules are the space filling diagram, the solvent accessible surface, and the molecular surface. We descnbe software that compules metric properties of these models, including volume and surface area. It also measures voids or empty space enclosed ..."
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Cited by 46 (12 self)
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Common geometric models for proteins and other molecules are the space filling diagram, the solvent accessible surface, and the molecular surface. We descnbe software that compules metric properties of these models, including volume and surface area. It also measures voids or empty space enclosed by the protein, and it keeps track of surface area contributions of individual atoms. The software is based on Jdimensional alpha complexes and on inclusionexclusion formulas with terms derived from the simplices in this complex. The so,ftware is available via anonymous ftp at ftp.ncsa.uiuc.edu. 1 Introduct ion The space filling dzagram, SF, introduced by Lee and Richards [ll], models a protein as the union of possibly overlapping spherical balls in R3, see figure 1. Each ball represents an atom and its size is determined by the van der Waals radius of the atom. A void is a piece *This work is supported by the National Science Foundation, under grant ASC9200301, the CISE postdoctoral fellow