Higher order differential structure of images

Abstract

This paper is meant as a tutorial on the basic concepts for vision in the 'Koenderink' school. The concept of scale-space is a necessity, if the extraction of structure from measured physical signals (i.e. images) is at stage. The Gaussian derivative kernels then are for physical signals the natural analogs of the mathematical differential operators. This paper discusses first some interesting properties of the Gaussian derivative kernels, like their orthogonality and behaviour with noisy input data. Geometrical structure to extract is expressed as differential invariants, in this paper limited to invariants under orthogonal transformations. Three representations are summarized: Cartesian, gauge and manifest invariant notation. Many explicit examples are given. A section is included about computer implementation of the calculation of higher order invariant structure.