Manifold Homotopy via the Flow Complex
It is known that the critical points of the distance function induced by a dense sample P of a submanifold Σ of ℝn are distributed into two groups, one lying close to Σ itself, called the shallow, and the other close to medial axis of Σ, called deep critical points. We prove that under (uniform) sam...
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| Published in: | Computer graphics forum Vol. 28; no. 5; pp. 1361 - 1370 |
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| Format: | Journal Article |
| Language: | English |
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Oxford, UK
Blackwell Publishing Ltd
01.07.2009
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | It is known that the critical points of the distance function induced by a dense sample P of a submanifold Σ of ℝn are distributed into two groups, one lying close to Σ itself, called the shallow, and the other close to medial axis of Σ, called deep critical points. We prove that under (uniform) sampling assumption, the union of stable manifolds of the shallow critical points have the same homotopy type as Σ itself and the union of the stable manifolds of the deep critical points have the homotopy type of the complement of Σ. The separation of critical points under uniform sampling entails a separation in terms of distance of critical points to the sample. This means that if a given sample is dense enough with respect to two or more submanifolds of ℝn, the homotopy types of all such submanifolds together with those of their complements are captured as unions of stable manifolds of shallow versus those of deep critical points, in a filtration of the flow complex based on the distance of critical points to the sample. This results in an algorithm for homotopic manifold reconstruction when the target dimension is unknown. |
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| AbstractList | It is known that the critical points of the distance function induced by a dense sample P of a submanifold S of Rn are distributed into two groups, one lying close to S itself, called the shallow, and the other close to medial axis of S, called deep critical points. We prove that under (uniform) sampling assumption, the union of stable manifolds of the shallow critical points have the same homotopy type as S itself and the union of the stable manifolds of the deep critical points have the homotopy type of the complement of S. The separation of critical points under uniform sampling entails a separation in terms of distance of critical points to the sample. This means that if a given sample is dense enough with respect to two or more submanifolds of Rn, the homotopy types of all such submanifolds together with those of their complements are captured as unions of stable manifolds of shallow versus those of deep critical points, in a filtration of the flow complex based on the distance of critical points to the sample. This results in an algorithm for homotopic manifold reconstruction when the target dimension is unknown. It is known that the critical points of the distance function induced by a dense sample P of a submanifold Σ of ℝn are distributed into two groups, one lying close to Σ itself, called the shallow, and the other close to medial axis of Σ, called deep critical points. We prove that under (uniform) sampling assumption, the union of stable manifolds of the shallow critical points have the same homotopy type as Σ itself and the union of the stable manifolds of the deep critical points have the homotopy type of the complement of Σ. The separation of critical points under uniform sampling entails a separation in terms of distance of critical points to the sample. This means that if a given sample is dense enough with respect to two or more submanifolds of ℝn, the homotopy types of all such submanifolds together with those of their complements are captured as unions of stable manifolds of shallow versus those of deep critical points, in a filtration of the flow complex based on the distance of critical points to the sample. This results in an algorithm for homotopic manifold reconstruction when the target dimension is unknown. It is known that the critical points of the distance function induced by a dense sample P of a submanifold Σ of ℝ n are distributed into two groups, one lying close to Σ itself, called the shallow , and the other close to medial axis of Σ, called deep critical points. We prove that under (uniform) sampling assumption, the union of stable manifolds of the shallow critical points have the same homotopy type as Σ itself and the union of the stable manifolds of the deep critical points have the homotopy type of the complement of Σ. The separation of critical points under uniform sampling entails a separation in terms of distance of critical points to the sample. This means that if a given sample is dense enough with respect to two or more submanifolds of ℝ n , the homotopy types of all such submanifolds together with those of their complements are captured as unions of stable manifolds of shallow versus those of deep critical points, in a filtration of the flow complex based on the distance of critical points to the sample. This results in an algorithm for homotopic manifold reconstruction when the target dimension is unknown. It is known that the critical points of the distance function induced by a dense sample P of a submanifold ? of ?n are distributed into two groups, one lying close to ? itself, called the shallow, and the other close to medial axis of ?, called deep critical points. We prove that under (uniform) sampling assumption, the union of stable manifolds of the shallow critical points have the same homotopy type as ? itself and the union of the stable manifolds of the deep critical points have the homotopy type of the complement of ?. The separation of critical points under uniform sampling entails a separation in terms of distance of critical points to the sample. This means that if a given sample is dense enough with respect to two or more submanifolds of ?n, the homotopy types of all such submanifolds together with those of their complements are captured as unions of stable manifolds of shallow versus those of deep critical points, in a filtration of the flow complex based on the distance of critical points to the sample. This results in an algorithm for homotopic manifold reconstruction when the target dimension is unknown. [PUBLICATION ABSTRACT] |
| Author | Sadri, Bardia |
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| References_xml | – reference: Herbert Edelsbrunner. Surface reconstruction by wrapping finite point sets in space. Discrete & Computational Geometry, 32:231-244, 2004. – reference: Karl Grove. Critical point theory for distance functions. Symposia in Pure Mathematics, 54(3):357-385, 1993. – reference: Nina Amenta, Thomas J. Peters, and Alexander Russell. Computational topology: Ambient isotopic approximation of 2-manifolds. Theo. Comp. Sci., 305(1-3):3-15, 2003. – reference: Kevin Buchin, Tamal K. Dey, Joachim Giesen, and Matthias John Recursive geometry of the flow complex and topology of the flow complex filtration. Comput. Geom. Theory Appl., 40:115-157, 2008. – reference: H. Edelsbrunner. The union of balls and its dual shape. Discrete Comput. Geom., 13:415-440, 1995. – reference: Nina Amenta, Marshall W. Bern, and David Eppstein. The crust and the β-skeleton: Combinatorial curve reconstruction. Graphical Models and Processing, 60(2):125-135, 1998. – reference: Allen Hatcher. 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| SubjectTerms | Algorithms Computer graphics F.2.2 [Theory of Computation]: Nonnumerical Algorithms and Prolems-Geometric Problems and Computation Geometry Sampling Studies |
| Title | Manifold Homotopy via the Flow Complex |
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