A study of the anisotropically weighted procrustes problem [optical image-guided surgery application]

The authors studied the problem of finding an optimal rigid-body transformation that maps one configuration of points to a corresponding configuration when the alignment errors are anisotropically weighted. Such a problem arises in point-to-point registration, in particular when an optical device (e.g. microscope or video camera) is involved, as in image-guided surgery. The authors review the existing literature and algorithms and study the mathematical difficulties. After stating a symmetry condition for critical points, they use it to establish some results. In particular, they show that the starting configuration and weight matrices can have a special form (orthogonal columns), they show that the rotational part can be computed from configurations translated so that their centroids lie at the origin, and they give an estimate for the total number of critical points. The authors then compare the 2 existing algorithms, studying carefully their dependence on the different parameters, and propose an independent technique in order to validate them. This technique reduces the problem to the problem of finding the roots of a system of 3 polynomials in 3 variables