Constrained Optimal Fitting of Three-dimensional Vector Patterns

Abstract — This paper addresses the problem of finding whether a given set of three-dimensional (3-D) vectors (the object) can be brought to match a second set of vectors (the template) by means of an affine motion, minimizing a measure of the mismatch error and satisfying an assigned set of geometrical constraints. This problem is encountered in many applications of computer vision, robotics, and manufacturing processes, and has been tackled by several authors in the unconstrained case. Spherical, ellipsoidal and polyhedral constraints are here introduced in the problem, and a solution scheme based on an efficient convex optimization algorithm is proposed. An example of application of the proposed methodology to a manufacturing tolerancing problem is also provided. Index Terms—Absolute orientation, automated tolerancing, computer vision, convex optimization, least squares, visual servo control