Higher order differential structure of images

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 for physical signals are then the natural analogues of the mathematical differential operators. This paper discusses some interesting properties of the Gaussian derivative kernels, like their orthogonality and behaviour with noisy input data. Geometrical structure to extract is expressed in terms of 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 the computer implementation of the calculation of higher order invariant structure