Discrete spherical means of directional derivatives and Veronese maps

Alexander Belyaev, Boris Khesin, Serge Tabachnikov

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension, we present a general construction for obtaining discrete spherical means of directional derivatives. The construction is based on using Minkowski's existence theorem and Veronese maps. Approximating the directional derivatives by appropriate finite differences allows one to obtain finite difference operators with good rotation invariance properties. In particular, we use discrete circular and spherical means to derive discrete approximations of various linear and nonlinear first- and second-order differential operators, including discrete Laplacians. A practical potential of our approach is demonstrated by considering applications to nonlinear filtering of digital images and surface curvature estimation.

Original languageEnglish (US)
Pages (from-to)124-136
Number of pages13
JournalJournal of Geometry and Physics
Issue number1
StatePublished - Jan 2012

All Science Journal Classification (ASJC) codes

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology


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