Recovery of global nonrigid motion: a model-based approach without point correspondences

Abstract

We present a novel technique for the estimation of global nonrigid motion of an object's boundary without using point correspondences. A complete description of the motion of an object's boundary involves specifying a displacement vector at each point of the boundary. Such a description provides a large amount of information, which needs to be processed further for study of the global characteristics of the deformation. Nonrigid motion can be studied hierarchically in terms of global nonrigid motion and point-by-point local nonrigid motion. The technique presented gives a method for estimating a global affine or polynomial transformation between two object boundaries. The novelty of the technique lies in the fact that it does not use any point correspondences. Our method uses hyperquadric models to model the data and estimate the global deformation. We show that affine or polynomial transformation between two datasets can be recovered from the hyperquadric parameters. The usefulness of the technique is twofold. First, it paves the way for viewing nonrigid motion hierarchically in terms of global and local motion. Second, it can be used as a front end to other motion analysis techniques that assume small motion. For instance, most nonrigid motion analysis algorithms make some assumptions on the type of nonrigid motion (conformal motion, small motion, etc.) that are not always satisfied in practice. When the motion between two datasets is large, our algorithm can be used to estimate the affine transformation (which includes scale and shear) or a polynomial transformation between the two datasets, which can then be used to warp the first dataset closer to the second so as to satisfy the small-motion assumption. We present experimental results with real and synthetic two- and three-dimensional data.