<title>Point-based registration under a similarity transform</title>

Abstract

This paper investigates the problem of point-based registration under a similarity transformation. This is a transformation that consists of rotation, translation, and isotropic scaling. There are many applications for registration under a similarity transform. First, the medical applications that usually use rigid-body registration may in some cases be improved by using a scale factor to account for particular types of distortion (for example, drift in gradient strength in MR image volumes). Second, similarity transforms are often used in biometrics to analyze and compare different sets of data. It was shown by Gower in 1971 that the choice of scale factor is independent from the choice of rotation and translation. We use a well-known solution for the rotation and translation parts of the transformation, and concentrate on the problem of choosing the scale factor. We examine three different methods of scaling, one of which is a novel maximum likelihood approach. We derive the target registration error and show the bias for each method. We introduce two different models of fiducial localization error, and we show that for one error model, Gower's method of scaling to minimize the sum of squared distances between corresponding points is also the maximum likelihood solution. Under the other error model, however, maximum likelihood leads to a new method of scaling.