Total Least Squares Fitting of Two Point Sets in m-D

The problem of estimating the motion and orientation parameters of a rigid object from two m-D point set patterns is of significant importance in medical imaging, computer assisted surgery, mobile robot navigation, computer vision, and fingerprint matching. Several least squares algorithms which make use of the singular value decomposition (SVD) have appeared in the literature. These algorithms consider the noise as coming from one image only, when in fact both images are corrupted by noise. In addition, these algorithms may suffer from roundoff accumulation errors due to a SVD of a matrix product between two noise corrupted matrices. This motivates the use of total least squares, where both data sets are treated as noisy. The formulation also avoids computing a SVD of a product of two noise corrupted matrices. This formulation is also convenient for an online implementation. In this paper we treat the image registration problem from a mixed least squares-total least squares point of view. The advantages of such an approach are various: (1) there is no matrix product in the singular value decomposition formulation, (2) the algorithm is given as a nonsingular matrix transformation, T, and (3) it can be easily implemented recursively. Contrary to previous least squares algorithms, the new algorithm takes advantage of the noise structure in the data. That is, we assume there is noise present in both data patterns, as well as a noise-free column for the translation vector. Also, the proposed algorithm computes all the parameters at once, without averaging the data.