Invariance Theory

Abstract

A linear scale space of a 2-dimensional grey-valued image consists of a set of spatial positions and scales (see chapter 6). As the input image is normally modeled with respect to a fixed Cartesian frame a Euclidean movement of the frame within the image plane induces a local transformation of the linear scale space preserving the metrical relations. If the linear scale space is subjected to a similarity transformation, which is a particular scaling of the spatial coordinates, the scale parameter and the input image simultaneously, then certain similarity relations are preserved. The question arises which are the intrinsic invariants of a linear scale space that are invariant under the group of Euclidean movements and under the similarity group. Normally such an equivalence problem is tackled by means of differential and integral geometry (Salden et al., 1995a). Intrinsic invariants of a linear scale space can be read out by means of differential geometric slot-machines such as the torsion and curvature tensor or by means of integral geometric slot-machines such as circuit integrals measuring Burgers and Frank vectors. The integral geometric slot-machines can in turn be used to define topological currents, which are true physical observables. In this chapter, however, after stating the equivalence problem for a linear scale space, the group of Euclidean movements and the similarity group, a complete and irreducible set of intrinsic invariants of a linear scale space is derived by means of invariance theory.