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Computational Topology for Shape Modeling
, 1999
"... This paper expands the role of the new field of computational topology by surveying methods for incorporating connectedness in shape modeling. Two geometric representations in particular, recurrent models and implicit surfaces, can (often unpredictably) become connected or disconnected based on typi ..."
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This paper expands the role of the new field of computational topology by surveying methods for incorporating connectedness in shape modeling. Two geometric representations in particular, recurrent models and implicit surfaces, can (often unpredictably) become connected or disconnected based on typical changes in modeling parameters. Two methodologies for controlling connectedness are identified: connectedness loci and Morse theory. The survey concludes by identifying several open problems remaining in shape modeling for computational topology to solve. 1 Introduction One might ask "what is topology?" and receive the proper definition: "the study of open sets." While this answer may enlighten some, the novice puzzles "what is an open set?" at about the same time the definition continues on to add that open sets are whatever a topology defines them to be (so long as they pass a few conditions regarding closure under union and intersection, etc.). This might be the main reason students ...
Cubical Approximation and Computation of Homology
, 1998
"... The purpose of this article is to introduce a method for computing the homology groups of cellular complexes composed of cubes. We will pay attention to issues of storage and efficiency in performing computations on large complexes which will be required in applications to the computation of the Con ..."
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Cited by 12 (1 self)
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The purpose of this article is to introduce a method for computing the homology groups of cellular complexes composed of cubes. We will pay attention to issues of storage and efficiency in performing computations on large complexes which will be required in applications to the computation of the Conley index. The algorithm used in the homology computations is based on a local reduction procedure, and we give a subquadratic estimate of its computational complexity. This estimate is rigorous in two dimensions, and we conjecture its validity in higher dimensions. 1 Introduction The computability of homology groups is wellknown and is found in most standard textbooks, e.g. [18], and the classical algorithm is based on performing row and column operations on the boundary matrices as a whole and reducing them to the Smith Normal Form (SNF), which is known to exist for any integer matrix, [20]. The homology groups can then be immediately determined from this canonical form. However, explici...
Isomorphism free lexicographic enumeration of triangulated surfaces and 3manifolds
, 2006
"... We present a fast enumeration algorithm for combinatorial 2 and 3manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3manifolds with 11 vertices. We further determine all equivelar maps on the nonorientable surface of genus 4 as well as a ..."
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We present a fast enumeration algorithm for combinatorial 2 and 3manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3manifolds with 11 vertices. We further determine all equivelar maps on the nonorientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the nonorientable surfaces of genus 5 and 6. 1
Topology Verification for Isosurface Extraction
, 2010
"... The importance of properly implemented isosurface extraction for verifiable visualization led to a previously published paper on the general Method of Manufactured Solutions (MMS), inclusive of a supportive software infrastructure. This work builds upon that foundation, while significantly extending ..."
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Cited by 11 (5 self)
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The importance of properly implemented isosurface extraction for verifiable visualization led to a previously published paper on the general Method of Manufactured Solutions (MMS), inclusive of a supportive software infrastructure. This work builds upon that foundation, while significantly extending it. Specifically, we extend previous work on verification of geometrical properties to ensuring correctness of considerably more subtle topological characteristics that are crucial for the extracted surfaces. We first show a new theoretical synthesis of results from stratified Morse theory and digital topology for algorithms created to verify topological invariants and then we demonstrate how the MMS approach can be extended to embrace topology, consistent with the design intent for MMS. The transition to topological verification motivated these considerable theoretical advances and algorithmic development, consistent with general MMS principles. The methodology reported reveals unexpected behavior and even coding mistakes in publicly available popular isosurface codes, as presented in a case study for visualization tools that documents the
SLIDING WINDOWS AND PERSISTENCE: AN APPLICATION OF TOPOLOGICAL METHODS TO SIGNAL ANALYSIS
"... Abstract. We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistenc ..."
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Abstract. We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the pointcloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets, and compare the results with those obtained using stateoftheart methods in gene expression analysis. We call this new method SW1PerS which stands for Sliding Windows and 1dimensional Persistence Scoring. 1.
Analysis of Blood Vessel Topology by Cubical Homology
 Proceedings of International Conference on Imagine Processing
"... In this note, we segment and topologically classify brain vessel data obtained from magnetic resonance angiography (MRA). The segmentation is done adaptively and the classification by means of cubical homology, i.e. the computation of homology groups. In this way the number of connected components ( ..."
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In this note, we segment and topologically classify brain vessel data obtained from magnetic resonance angiography (MRA). The segmentation is done adaptively and the classification by means of cubical homology, i.e. the computation of homology groups. In this way the number of connected components (measured byH0), the tunnels (given byH1) and the voids (given byH2) are determined, resulting in a topological characterization of the blood vessels. 1.
Computational topology for reconstruction of surfaces with boundary: integrating experiments and theory
 Proceedings of the IEEE International Conference on Shape Modeling and Applications, June 15 17, 2005
, 2005
"... Abstract. This paper presents new mathematical foundations for topologically correct surface reconstruction techniques that are applicable to 2manifolds with boundary, where provable techniques previously had been limited to surfaces without boundary. This is done by an intermediate construction of ..."
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Abstract. This paper presents new mathematical foundations for topologically correct surface reconstruction techniques that are applicable to 2manifolds with boundary, where provable techniques previously had been limited to surfaces without boundary. This is done by an intermediate construction of the envelope (as defined herein) of the original surface. For any compact C 2 manifold M it is shown that its envelope is C 1,1 and this envelope can be reconstructed with topological guarantees. Then it is shown that there exists a piecewise linear (PL) subset of the reconstruction of the envelope that is ambient isotopic to M, whenever M is orientable. The emphasis of this paper is upon the mathematical proofs needed for these extensions, where more practical applications and examples are presented in a companion paper.
Constructing Discrete Morse Functions
, 2002
"... Morse theory has been considered a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It o ..."
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Morse theory has been considered a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several categories of interesting objects (particularly meshes) to applications of Morse theory. Once a