The authors are developing an automated reverse engineering system for reconstructing the shape of simple mechanical parts. B-rep models are created by fitting surfaces to point clouds obtained by scanning an object using a 3D laser scanner. The resulting models, although valid, are often not suitable for purposes such as redesign because expected regularities and constraints are not present. This information is lost because each face of the model is determined independently. A global approach is required, in particular one that is capable of finding symmetries originally present. This paper describes a practical algorithm for finding global symmetries in suitable B-rep models built from planes, spheres, cylinders, cones and tori. It has been implemented and used to determine approximate symmetries of models with up to about 200 vertices in reasonable time. The time performance of the algorithm in the worst case is bounded by O(n^3.5 log^4 n), and a justification is given that on common engineering objects it takes about O(n^2 log^4 n), making it a practical tool for use in a reverse engineering package. Details of the algorithm are given, along with some results from a number of illustrative test runs.

This paper considers the task of asymmetry rectification. We start by giving various reasons why the possession of symmetry may be beneficial for designed shapes, and mention how various construction methods may produce shapes which are less symmetric than desired.

Abstract 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 vertices and they are noise resistant. Our method allows the detection of partial as well as approximate equivalences. i