Invariant geometric evolutions of surfaces and volumetric smoothing

Peter J. Olver, Guillermo Sapiro, Allen Tannenbaum

Research output: Contribution to journalArticlepeer-review

52 Scopus citations

Abstract

The study of geometric flows for smoothing, multiscale representation, and analysis of two- and three-dimensional objects has received much attention in the past few years. In this paper, we first survey the geometric smoothing of curves and surfaces via geometric heat-type flows, which are invariant under the groups of Euclidean and affine motions. Second, using the general theory of differential invariants, we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group. As an application, we present the simplest affine invariant flow for (convex) surfaces in three-dimensional space, which, like the affine-invariant curve shortening flow, will be of fundamental importance in the processing of three-dimensional images.

Original languageEnglish (US)
Pages (from-to)176-194
Number of pages19
JournalSIAM Journal on Applied Mathematics
Volume57
Issue number1
DOIs
StatePublished - Feb 1997

Keywords

  • Geometric smoothing
  • Invariant surface evolutions
  • Partial differential equations
  • Symmetry groups

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