Results 1  10
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11
Optimal topological simplification of discrete functions on surfaces
, 2009
"... We present an efficient algorithm for computing a function that minimizes the number of critical points among all functions within a prescribed distance δ from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our metho ..."
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Cited by 11 (2 self)
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We present an efficient algorithm for computing a function that minimizes the number of critical points among all functions within a prescribed distance δ from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2δ, constructively proving that the lower bound on the number of critical points given by the stability theorem of persistent homology is tight in dimension two for any input function. 1
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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Cited by 4 (1 self)
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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
Towards optimality in discrete Morse theory
 Experimental Mathematics
, 2003
"... Morse theory is a fundamental tool for investigating the topology of smooth manifolds. This tool has been extended to discrete structures by Forman, which allows combinatorial analysis and direct computation. This theory relies on discrete gradient vector fields, whose critical elements describe the ..."
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Cited by 4 (1 self)
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Morse theory is a fundamental tool for investigating the topology of smooth manifolds. This tool has been extended to discrete structures by Forman, which allows combinatorial analysis and direct computation. This theory relies on discrete gradient vector fields, whose critical elements describe the topology of the structure. The purpose of this work is to construct optimal discrete gradient vector fields, where optimality means having the minimum number of critical elements. The problem is equivalently stated in terms of maximal hyperforests of hypergraphs. Deduced from this theoretical result, a algorithm constructing almost optimal discrete gradient fields is provided. The optimal parts of the algorithm are proved, and the part of exponential complexity is replaced by heuristics. Although reaching optimality is MAXSNP hard, the experiments on odd topological models are almost always optimal.
Schnyder Woods for Higher Genus Triangulated Surfaces
 SCG'08
, 2008
"... Schnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertexspanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere ..."
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Cited by 4 (2 self)
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Schnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertexspanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. First, we define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and coloration of the edges that extends the one proposed by Schnyder for the planar case. As a byproduct we show how some recent schemes for compression and compact encoding of graphs can be extended to higher genus. All the algorithms presented here have linear time complexity.
Efficient computation of a hierarchy of discrete 3d gradient vector fields
 in Proc. TopoInVis
, 2011
"... Abstract This paper introduces a novel combinatorial algorithm to compute a hierarchy of discrete gradient vector fields for threedimensional scalar fields. The hierarchy is defined by an importance measure and represents the combinatorial gradient flow at different levels of detail. The presented ..."
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Cited by 3 (3 self)
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Abstract This paper introduces a novel combinatorial algorithm to compute a hierarchy of discrete gradient vector fields for threedimensional scalar fields. The hierarchy is defined by an importance measure and represents the combinatorial gradient flow at different levels of detail. The presented algorithm is based on Forman’s discrete Morse theory, which guarantees topological consistency and algorithmic robustness. In contrast to previous work, our algorithm combines memory and runtime efficiency. It thereby lends itself to the analysis of large data sets. A discrete gradient vector field is also a compact representation of the underlying extremal structures – the critical points, separation lines and surfaces. Given a certain level of detail, an explicit geometric representation of these structures can be extracted using simple and fast graph algorithms. 1
Extraction Of Feature Lines On Surface Meshes Based On Discrete Morse Theory
"... We present an approach for extracting extremal feature lines of scalar indicators on surface meshes, based on discrete Morse Theory. By computing initial MorseSmale complexes of the scalar indicators of the mesh, we obtain a candidate set of extremal feature lines of the surface. A hierarchy of Mor ..."
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We present an approach for extracting extremal feature lines of scalar indicators on surface meshes, based on discrete Morse Theory. By computing initial MorseSmale complexes of the scalar indicators of the mesh, we obtain a candidate set of extremal feature lines of the surface. A hierarchy of MorseSmale complexes is computed by prioritizing feature lines according to a novel criterion and applying a cancellation procedure that allows us to select the most significant lines. Given the scalar indicators on the vertices of the mesh, the presented feature line extraction scheme is interpolation free and needs no derivative estimates. The technique is insensitive to noise and depends only on one parameter: the feature significance. We use the technique to extract surface features yielding impressive, non photorealistic images. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Line and Curve Generation I.4.6 [Computer Graphics]: Feature Detection I.4.7 [Computer Graphics]: Feature Measurement
Optimal decision trees on simplicial complexes
 Electron. J. Combin. 12 (2005), Research Paper
"... We consider topological aspects of decision trees on simplicial complexes, concentrating on how to use decision trees as a tool in topological combinatorics. By Robin Forman’s discrete Morse theory, the number of evasive faces of a given dimension i with respect to a decision tree on a simplicial co ..."
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We consider topological aspects of decision trees on simplicial complexes, concentrating on how to use decision trees as a tool in topological combinatorics. By Robin Forman’s discrete Morse theory, the number of evasive faces of a given dimension i with respect to a decision tree on a simplicial complex is greater than or equal to the ith reduced Betti number (over any field) of the complex. Under certain favorable circumstances, a simplicial complex admits an “optimal ” decision tree such that equality holds for each i; we may hence read off the homology directly from the tree. We provide a recursive definition of the class of seminonevasive simplicial complexes with this property. A certain generalization turns out to yield the class of semicollapsible simplicial complexes that admit an optimal discrete Morse function in the analogous sense. In addition, we develop some elementary theory about seminonevasive and semicollapsible complexes. Finally, we provide explicit optimal decision trees for several wellknown simplicial complexes.
MemoryEfficient Computation of Persistent Homology for 3D Images using Discrete Morse Theory
"... Abstract—We propose a memoryefficient method that computes persistent homology for 3D grayscale images. The basic idea is to compute the persistence of the induced MorseSmale complex. Since in practice this complex is much smaller than the input data, significantly less memory is required for the ..."
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Abstract—We propose a memoryefficient method that computes persistent homology for 3D grayscale images. The basic idea is to compute the persistence of the induced MorseSmale complex. Since in practice this complex is much smaller than the input data, significantly less memory is required for the subsequent computations. We propose a novel algorithm that efficiently extracts the MorseSmale complex based on algorithms from discrete Morse theory. The proposed algorithm is thereby optimal with a computational complexity of O(n 2). The persistence is then computed using the MorseSmale complex by applying an existing algorithm with a good practical running time. We demonstrate that our method allows for the computation of persistent homology for large data on commodity hardware. Keywordspersistent homology, MorseSmale complex, discrete Morse theory, large data I.
COMPUTER GRAPHICS forum Extraction of Dominant Extremal Structures in Volumetric Data using Separatrix Persistence
"... Extremal lines and surfaces are features of a 3D scalar field where the scalar function becomes minimal or maximal with respect to a local neighborhood. These features are important in many applications, e.g., computer tomography, fluid dynamics, cell biology. We present a novel topological method t ..."
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Extremal lines and surfaces are features of a 3D scalar field where the scalar function becomes minimal or maximal with respect to a local neighborhood. These features are important in many applications, e.g., computer tomography, fluid dynamics, cell biology. We present a novel topological method to extract these features using discrete Morse theory. In particular, we extend the notion of Separatrix Persistence from 2D to 3D, which gives us a robust estimation of the feature strength for extremal lines and surfaces. Not only does it allow us to determine the most important (parts of) extremal lines and surfaces, it also serves as a robust filtering measure of noiseinduced structures. Our purely combinatorial method does not require derivatives or any other numerical computations.
Noname manuscript No. (will be inserted by the editor) Efficient Computation of 3D MorseSmale Complexes and Persistent Homology using Discrete Morse Theory
"... Abstract We propose an efficient algorithm that computes the MorseSmale complex for 3D grayscale images. This complex allows for a efficient computation of persistent homology since it is, in general, much smaller than the input data but still contains all necessary information. Our method improve ..."
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Abstract We propose an efficient algorithm that computes the MorseSmale complex for 3D grayscale images. This complex allows for a efficient computation of persistent homology since it is, in general, much smaller than the input data but still contains all necessary information. Our method improves a recently proposed algorithm to extract the MorseSmale complex in terms of memory consumption and running time. It also allows for a parallel computation of the complex. The computational complexity of the MorseSmale complex extraction solely depends on the topological complexity of the input data. The persistence is then computed using the MorseSmale complex by applying an existing algorithm with a good practical running time. We demonstrate that our method allows for the computation of persistent homology for large data on commodity hardware.