Feature-based methods for non-rigid registration frequently encounter the correspondence problem. Regardless of whether points, lines, curves or surface parameterizations are used, feature-based non-rigid matching requires us to automatically solve for correspondences between two sets of features. In addition, there could be many features in either set that have no counterparts in the other. This outlier rejection problem further complicates an already di#cult correspondence problem. We formulate feature-based non-rigid registration as a non-rigid point matching problem. After a careful review of the problem and an in-depth examination of two types of methods previously designed for rigid robust point matching (RPM), we propose a new general framework for non-rigid point matching. We consider it a general framework because it does not depend on any particular form of spatial mapping. We have also developed an algorithm---the TPS-RPM algorithm---with the thin-plate spline (TPS) as the parameterization of the non-rigid spatial mapping and the softassign for the correspondence. The performance of the TPS-RPM algorithm is demonstrated and validated in a series of carefully designed synthetic experiments. In each of these experiments, an empirical comparison with the popular iterated closest point (ICP) algorithm is also provided. Finally, we apply the algorithm to the problem of non-rigid registration of cortical anatomical structures which is required in brain mapping. While these results are somewhat preliminary, they clearly demonstrate the applicability of our approach to real world tasks involving feature-based non-rigid registration.

Abstract — We show that the Fisher-Rao Riemannian metric is a natural, intrinsic tool for computing shape geodesics. When a parameterized probability density function is used to represent a landmark-based shape, the modes of deformation are automatically established through the Fisher information of the density. Consequently, given two shapes parameterized by the same density model, the geodesic distance between them under the action of the Fisher-Rao metric is a convenient shape distance measure. It has the advantage of being an intrinsic distance measure and invariant to reparameterization. We first model shape landmarks using a Gaussian mixture model and then compute geodesic distances between two shapes using the Fisher-Rao metric corresponding to the mixture model. We illustrate our approach by computing Fisher geodesics between 2D corpus callosum shapes. Shape representation via the mixture model and shape deformation via the Fisher geodesic are hereby unified in this approach. I.

Abstract. Shape matching plays a prominent role in the analysis of medical and biological structures. Recently, a unifying framework was introduced for shape matching that uses mixture-models to couple both the shape representation and deformation. Essentially, shape distances were defined as geodesics induced by the Fisher-Rao metric on the manifold of mixture-model represented shapes. A fundamental drawback of the Fisher-Rao metric is that it is NOT available in closed-form for the mixture model. Consequently, shape comparisons are computationally very expensive. Here, we propose a new Riemannian metric based on generalized φ- entropy measures. In sharp contrast to the Fisher-Rao metric, our new metric is available in closed-form. Geodesic computations using the new metric are considerably more efficient. Discriminative capabilities of this new metric are studied by pairwise matching of corpus callosum shapes. Comparisons are conducted with the Fisher-Rao metric and the thin-plate spline bending energy. 1

for monotonizing an unconstrained estimator in nonparametric regression