Fast And Optimal Assembling Of Partial 3-D Object Models By Procrustes Analysis Techniques

The paper describes an original method for the geometrical size and shape (form) 3-D object reconstruction by way of a least-squares assembling procedure of its partial models. The model components can be independently defined into heterogeneous reference frames as a product of the possible different survey methods employed. The procedure makes use of the Procrustes analysis techniques and is based on the authors' experiences recently acquired solving the photogrammetric block adjustment by independent models with a new formulation offered by these algorithms. The Procrustes analysis furnishes a set of mathematical least squares tools to perform similarity transformations among point-coordinates matrices up to their optimal agreement. The transformation parameters and the tie points coordinates are computed in a fast and efficient way, without any prior information of their approximated values; furthermore the geometric space is not restricted to 2-D or 3-D but could be a generic K-dimensional one. The paper focuses on the principles and the statistical theory behind the Generalised Procrustes Analysis and on the concept of "Centroid configuration" deriving from it. As outlined in this work, when the Centroid vertexes can be assimilated to the tie and control points linking the different partial models, it is then possible to reconstruct the size and the global shape of the object by way of an iterative and fast converging sequence of direct transformations. A numerical example is provided in the article to illustrate the different steps of the procedure and to prove the correctness and efficiency of the algorithms. Among the other suitable applications, the current method has been developed for the industrial manufacturing, especially for the dimensional control of prototypes with respect to their construction plans, for the on-line product quality checks and for reverse engineering applications.