Discrete spherical means of directional derivatives and Veronese maps

TL;DR

This paper introduces a geometric framework for constructing discrete spherical means of directional derivatives using Minkowski's theorem and Veronese maps, enabling rotation-invariant finite difference operators for image processing and surface analysis.

Contribution

It presents a novel general construction for discrete spherical means of directional derivatives applicable in any dimension, enhancing rotation invariance in finite difference approximations.

Findings

Discrete spherical means improve rotation invariance of differential operators.

Application to nonlinear image filtering demonstrates practical effectiveness.

Method enables accurate surface curvature estimation.

Abstract

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 the 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.