TY - JOUR
T1 - Discrete spherical means of directional derivatives and Veronese maps
AU - Belyaev, Alexander
AU - Khesin, Boris
AU - Tabachnikov, Serge
PY - 2012/1
Y1 - 2012/1
N2 - 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.
AB - 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.
U2 - 10.1016/j.geomphys.2011.09.005
DO - 10.1016/j.geomphys.2011.09.005
M3 - Article
VL - 62
SP - 124
EP - 136
JO - Journal of Geometry and Physics
JF - Journal of Geometry and Physics
SN - 0393-0440
IS - 1
ER -