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378
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 235 (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.
Shellable nonpure complexes and posets. I
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 1996
"... The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of ..."
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Cited by 125 (7 self)
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The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed fvectors and hvectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their StanleyReisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the kequal partition lattice (the intersection lattice of the kequal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the kequal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.
The Topological Structure of Asynchronous Computability
 JOURNAL OF THE ACM
, 1996
"... We give necessary and sufficient combinatorial conditions characterizing the tasks that can be solved by asynchronous processes, of which all but one can fail, that communicate by reading and writing a shared memory. We introduce a new formalism for tasks, based on notions from classical algebra ..."
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Cited by 117 (11 self)
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We give necessary and sufficient combinatorial conditions characterizing the tasks that can be solved by asynchronous processes, of which all but one can fail, that communicate by reading and writing a shared memory. We introduce a new formalism for tasks, based on notions from classical algebraic and combinatorial topology, in which a task's possible input and output values are each associated with highdimensional geometric structures called simplicial complexes. We characterize computability in terms of the topological properties of these complexes. This characterization has a surprising geometric interpretation: a task is solvable if and only if the complex representing the task's allowable inputs can be mapped to the complex representing the task's allowable outputs by a function satisfying certain simple regularity properties. Our formalism thus replaces the "operational" notion of a waitfree decision task, expressed in terms of interleaved computations unfolding ...
Global Conformal Surface Parameterization
, 2003
"... We solve the problem of computing global conformal parameterizations for surfaces with nontrivial topologies. The parameterization is global in the sense that it preserves the conformality everywhere except for a few points, and has no boundary of discontinuity. We analyze the structure of the space ..."
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Cited by 114 (22 self)
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We solve the problem of computing global conformal parameterizations for surfaces with nontrivial topologies. The parameterization is global in the sense that it preserves the conformality everywhere except for a few points, and has no boundary of discontinuity. We analyze the structure of the space of all global conformal parameterizations of a given surface and find all possible solutions by constructing a basis of the underlying linear solution space. This space has a natural structure solely determined by the surface geometry, so our computing result is independent of connectivity, insensitive to resolution, and independent of the algorithms to discover it. Our algorithm is based on the properties of gradient fields of conformal maps, which are closedness, harmonity, conjugacy, duality and symmetry. These properties can be formulated by sparse linear systems, so the method is easy to implement and the entire process is automatic. We also introduce a novel topological modification method to improve the uniformity of the parameterization. Based on the global conformal parameterization of a surface, we can construct a conformal atlas and use it to build conformal geometry images which have very accurate reconstructed normals.
Finding the homology of submanifolds with high confidence from random samples
, 2004
"... Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn ” the hom ..."
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Cited by 114 (7 self)
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Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn ” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learningtheoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to selfintersection of the submanifold. We are also able to treat the situation where the data is “noisy ” and lies near rather than on the submanifold in question.
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 105 (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.
Computing Persistent Homology
 Discrete Comput. Geom
"... We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enabl ..."
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Cited by 101 (20 self)
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We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This results generalizes and extends the previously known algorithm that was restricted to subcomplexes of S and Z2 coefficients. Finally, our study implies the lack of a simple classification over nonfields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.
An Incremental Algorithm for Betti Numbers of Simplicial Complexes
, 1993
"... A general and direct method for computing the betti numbers of the homology groups of a finite simplicial complex is given. For subcomplexes of a triangulation of S³ this method has implementations that run in time 0(’na(n)) and O(n), where n is the number of simplices in the triangulation. If app!i ..."
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Cited by 93 (16 self)
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A general and direct method for computing the betti numbers of the homology groups of a finite simplicial complex is given. For subcomplexes of a triangulation of S³ this method has implementations that run in time 0(’na(n)) and O(n), where n is the number of simplices in the triangulation. If app!ied to the family of ashapes of a finite point set in R³ ittakes time O(ncz(n)) to compute the betti numbers of all crshapes.
Topology Preserving Edge Contraction
 Publ. Inst. Math. (Beograd) (N.S
, 1998
"... We study edge contractions in simplicial complexes and local conditions under which they preserve the topological type. The conditions are based on a generalized notion of boundary, which lends itself to defining a nested hierarchy of triangulable spaces measuring the distance to being a manifold. ..."
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Cited by 62 (6 self)
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We study edge contractions in simplicial complexes and local conditions under which they preserve the topological type. The conditions are based on a generalized notion of boundary, which lends itself to defining a nested hierarchy of triangulable spaces measuring the distance to being a manifold.
Poseoblivious shape signature
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 13
, 2007
"... A 3D shape signature is a compact representation for some essence of a shape. Shape signatures are commonly utilized as a fast indexing mechanism for shape retrieval. Effective shape signatures capture some global geometric properties which are scale, translation, and rotation invariant. In this pa ..."
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Cited by 60 (3 self)
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A 3D shape signature is a compact representation for some essence of a shape. Shape signatures are commonly utilized as a fast indexing mechanism for shape retrieval. Effective shape signatures capture some global geometric properties which are scale, translation, and rotation invariant. In this paper, we introduce an effective shape signature which is also poseoblivious. This means that the signature is also insensitive to transformations which change the pose of a 3D shape such as skeletal articulations. Although some topologybased matching methods can be considered poseoblivious as well, our new signature retains the simplicity and speed of signature indexing. Moreover, contrary to topologybased methods, the new signature is also insensitive to the topology change of the shape, allowing us to match similar shapes with different genus. Our shape signature is a 2D histogram which is a combination of the distribution of two scalar functions defined on the boundary surface of the 3D shape. The first is a definition of a novel function called the localdiameter function. This function measures the diameter of the 3D shape in the neighborhood of each vertex. The histogram of this function is an informative measure of the shape which is insensitive to pose changes. The second is the centricity function that measures the average geodesic distance from one vertex to all other vertices on the mesh. We evaluate and compare a number of methods for measuring the similarity between two signatures, and demonstrate the effectiveness of our poseoblivious shape signature within a 3D search engine application for different databases containing hundreds of models.