Groupwise and Local Analysis of Image Warps

Most non-rigid image registration algorithms are based on aligning pairs of images, but there has been recent interest in groupwise algorithms, which enable the registration of a set of images into a single common frame of reference. The set of deformation fields obtained from such a registration contains information about the variability of structures across the group, meaning that the quantitative analysis and modelling of the set of deformation fields provides useful information about the images in the set. Such analysis must be based (either implicitly or explicitly) on a particular common mathematical representation of the set of deformation fields. Previous work on the analysis of shape variability has used a range of representations; examples include (Cootes et al., 1995; Bookstein, 1989; Pizer et al., 1999). Recent work on modelling dense 2D and 3D deformation fields has either used the densely-sampled deformation vectors directly (e.g., LeBriquer and Gee (1997)), or has employed a smooth, continuous representation of them (e.g.,B-splines, Rueckert et al. (2001)). In this paper we describe a class of representations for such deformation fields based on interpolating splines. We present both diffeomorphic and non-diffeomorphic representations; the non-diffeomorphic is computationally much simpler, and suitable for use within a non-rigid registration algorithm (Marsland and Twining, 2003), whereas the diffeomorphic representation leads to useful metrics for the analysis of a set of parameterised warps. We demonstrate that these metrics are superior to ad hoc Euclidean metrics for the task of classifying legal and illegal variations of a set of shapes. We also discuss differential-geometric constructions of invariant scalars for the analysis of the spatial variation of a warp.