On invariants of Lie group actions and their application to some equivalence problems

This thesis studies the existence of joint invariants, their computation, their relation to differential invariants and their application to object recognition and symmetry detection. Our main tool is the moving frame method as developed by M. Fels and P. J. Olver.; We start by studying prolonged Lie group actions on the Cartesian product of n copies of a manifold. We show that the orbit dimensions of such actions do not pseudo-stabilize when n increases and obtain a bound on the stabilization order. We also obtain a discrete analogue to a famous theorem by Ovsiannikov and Olver. These facts are important relative to the existence and computation of joint invariants. Interesting corollaries are presented.; Based on these theoretical results, we show how joint invariants can be used to solve two equivalence problems, namely curve and polygon recognition (and symmetry detection). Our approach to curve recognition and symmetry detection is based on a paper by Calabi et al. and relies on the concept of differential invariant signature. The main idea consists in obtaining numerically invariant approximations of the differential invariants parameterizing the signature. This signature uniquely characterizes the equivalence class of a given curve under the action of a Lie group. We correct the numerically invariant approximations initially proposed by Calabi et al. for the special Euclidean and equi-affine differential invariants of a planar curve and solve the problem of spatial curve recognition modulo the action of the special Euclidean group.; Our approach to polygon recognition and symmetry detection is based on moving frames and contains a general method. The cases of the special Euclidean, Euclidean, equi-affine, skewed-affine and similarity Lie groups are discussed in detail. The time complexity of our algorithms is linear in the number of 300 vertices and they are noise resistant. Our method allows the detection of partial as well as approximate equivalences.