Approximation and convergence of the intrinsic volume

Herbert Edelsbrunner; Florian Pausinger

doi:10.1016/j.aim.2015.10.004

Approximation and convergence of the intrinsic volume

Herbert Edelsbrunner, Florian Pausinger^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We introduce a modification of the classic notion of intrinsic volume using persistence moments of height functions. Evaluating the modified first intrinsic volume on digital approximations of a compact body with smoothly embedded boundary in Rⁿ, we prove convergence to the first intrinsic volume of the body as the resolution of the approximation improves. We have weaker results for the other modified intrinsic volumes, proving they converge to the corresponding intrinsic volumes of the n-dimensional unit ball.

Original language	English
Pages (from-to)	674-703
Journal	Advances in Mathematics
Volume	287
Early online date	20 Nov 2015
DOIs	https://doi.org/10.1016/j.aim.2015.10.004
Publication status	Early online date - 20 Nov 2015
Externally published	Yes

Keywords

Crofton formula
Digital image processing
Distorted normals
Intrinsic volume
Persistent homology

ASJC Scopus subject areas

Mathematics(all)

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10.1016/j.aim.2015.10.004Licence: Unspecified

Cite this

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title = "Approximation and convergence of the intrinsic volume",

abstract = "We introduce a modification of the classic notion of intrinsic volume using persistence moments of height functions. Evaluating the modified first intrinsic volume on digital approximations of a compact body with smoothly embedded boundary in Rn, we prove convergence to the first intrinsic volume of the body as the resolution of the approximation improves. We have weaker results for the other modified intrinsic volumes, proving they converge to the corresponding intrinsic volumes of the n-dimensional unit ball.",

keywords = "Crofton formula, Digital image processing, Distorted normals, Intrinsic volume, Persistent homology",

author = "Herbert Edelsbrunner and Florian Pausinger",

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AU - Pausinger, Florian

PY - 2015/11/20

Y1 - 2015/11/20

N2 - We introduce a modification of the classic notion of intrinsic volume using persistence moments of height functions. Evaluating the modified first intrinsic volume on digital approximations of a compact body with smoothly embedded boundary in Rn, we prove convergence to the first intrinsic volume of the body as the resolution of the approximation improves. We have weaker results for the other modified intrinsic volumes, proving they converge to the corresponding intrinsic volumes of the n-dimensional unit ball.

AB - We introduce a modification of the classic notion of intrinsic volume using persistence moments of height functions. Evaluating the modified first intrinsic volume on digital approximations of a compact body with smoothly embedded boundary in Rn, we prove convergence to the first intrinsic volume of the body as the resolution of the approximation improves. We have weaker results for the other modified intrinsic volumes, proving they converge to the corresponding intrinsic volumes of the n-dimensional unit ball.

KW - Crofton formula

KW - Digital image processing

KW - Distorted normals

KW - Intrinsic volume

KW - Persistent homology

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DO - 10.1016/j.aim.2015.10.004

M3 - Article

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VL - 287

SP - 674

EP - 703

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -

Approximation and convergence of the intrinsic volume

Abstract

Keywords

ASJC Scopus subject areas

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Florian Pausinger

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Approximation and convergence of the intrinsic volume

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

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Fingerprint

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Florian Pausinger

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