Computation of circular area and spherical volume invariants via boundary integrals∗

Riley C.W. O’neill; Pedro Angulo-Umaña; Jeff Calder; Bo Hessburg; Peter J. Olver; Chehrzad Shakiban; Katrina Yezzi-Woodley

doi:10.1137/19M1260803

Computation of circular area and spherical volume invariants via boundary integrals^∗

Riley C.W. O’neill, Pedro Angulo-Umaña, Jeff Calder, Bo Hessburg, Peter J. Olver, Chehrzad Shakiban, Katrina Yezzi-Woodley

School of Mathematics

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

2 Scopus citations

Abstract

We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the divergence theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our results also extend to higher-dimensional hypersurfaces. The resulting surface integrals are computable analytically on a triangulated mesh. This gives a simple computational algorithm for computing the spherical volume invariant for triangulated surfaces that does not involve discretizing the ambient space. We discuss potential applications to feature detection on broken bone fragments of interest in anthropology.

Original language	English (US)
Pages (from-to)	53-77
Number of pages	25
Journal	SIAM Journal on Imaging Sciences
Volume	13
Issue number	1
DOIs	https://doi.org/10.1137/19M1260803
State	Published - 2019

Bibliographical note

Funding Information:
∗Received by the editors May 9, 2019; accepted for publication (in revised form) October 17, 2019; published electronically January 7, 2020. https://doi.org/10.1137/19M1260803 Funding: The work of the authors was supported by the National Science Foundation grant DMS-1816917, the University of St. Thomas Center for Applied Mathematics, and a University of Minnesota Grant in Aid award. †Department of Mathematics, University of St. Thomas, St Paul, MN 55105 ([email protected], [email protected]). ‡School of Mathematics, University of Minnesota, Minneapolis, MN 55454 ([email protected], jcalder@umn. edu, [email protected], [email protected]). §Department of Anthropology, University of Minnesota, Minneapolis, MN 55455 ([email protected]).

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics and by SIAM. Unauthorized reproduction of this article is prohibited.

Keywords

Boundary integral methods
Computational geometry
Curvature
Integral invariants
PCA on local neighborhoods
Spherical volume invariant

Publisher link

10.1137/19M1260803

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Cite this

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title = "Computation of circular area and spherical volume invariants via boundary integrals∗",

abstract = "We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the divergence theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our results also extend to higher-dimensional hypersurfaces. The resulting surface integrals are computable analytically on a triangulated mesh. This gives a simple computational algorithm for computing the spherical volume invariant for triangulated surfaces that does not involve discretizing the ambient space. We discuss potential applications to feature detection on broken bone fragments of interest in anthropology.",

keywords = "Boundary integral methods, Computational geometry, Curvature, Integral invariants, PCA on local neighborhoods, Spherical volume invariant",

author = "O{\textquoteright}neill, {Riley C.W.} and Pedro Angulo-Uma{\~n}a and Jeff Calder and Bo Hessburg and Olver, {Peter J.} and Chehrzad Shakiban and Katrina Yezzi-Woodley",

note = "Funding Information: ∗Received by the editors May 9, 2019; accepted for publication (in revised form) October 17, 2019; published electronically January 7, 2020. https://doi.org/10.1137/19M1260803 Funding: The work of the authors was supported by the National Science Foundation grant DMS-1816917, the University of St. Thomas Center for Applied Mathematics, and a University of Minnesota Grant in Aid award. †Department of Mathematics, University of St. Thomas, St Paul, MN 55105 ([email protected], [email protected]). ‡School of Mathematics, University of Minnesota, Minneapolis, MN 55454 ([email protected], jcalder@umn. edu, [email protected], [email protected]). §Department of Anthropology, University of Minnesota, Minneapolis, MN 55455 ([email protected]). Publisher Copyright: {\textcopyright} 2020 Society for Industrial and Applied Mathematics and by SIAM. Unauthorized reproduction of this article is prohibited.",

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AU - Shakiban, Chehrzad

AU - Yezzi-Woodley, Katrina

N1 - Funding Information: ∗Received by the editors May 9, 2019; accepted for publication (in revised form) October 17, 2019; published electronically January 7, 2020. https://doi.org/10.1137/19M1260803 Funding: The work of the authors was supported by the National Science Foundation grant DMS-1816917, the University of St. Thomas Center for Applied Mathematics, and a University of Minnesota Grant in Aid award. †Department of Mathematics, University of St. Thomas, St Paul, MN 55105 ([email protected], [email protected]). ‡School of Mathematics, University of Minnesota, Minneapolis, MN 55454 ([email protected], jcalder@umn. edu, [email protected], [email protected]). §Department of Anthropology, University of Minnesota, Minneapolis, MN 55455 ([email protected]). Publisher Copyright: © 2020 Society for Industrial and Applied Mathematics and by SIAM. Unauthorized reproduction of this article is prohibited.

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N2 - We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the divergence theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our results also extend to higher-dimensional hypersurfaces. The resulting surface integrals are computable analytically on a triangulated mesh. This gives a simple computational algorithm for computing the spherical volume invariant for triangulated surfaces that does not involve discretizing the ambient space. We discuss potential applications to feature detection on broken bone fragments of interest in anthropology.

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