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

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)53-77
Number of pages25
JournalSIAM Journal on Imaging Sciences
Volume13
Issue number1
DOIs
StatePublished - 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 (riley.oneill@stthomas.edu, cshakiban@stthomas.edu). ‡School of Mathematics, University of Minnesota, Minneapolis, MN 55454 (angul010@umn.edu, jcalder@umn. edu, hessb017@umn.edu, olver@umn.edu). §Department of Anthropology, University of Minnesota, Minneapolis, MN 55455 (yezz0003@umn.edu).

Keywords

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

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