Fitting smooth paths on riemannian manifolds: Endometrial surface reconstruction and preoperative MRI-based navigation

Abstract

We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using C1 piecewise-Bézier functions. A property of the method is that, when the manifold reduces to a Euclidean space, the control points minimize the mean square acceleration of the path. As an application, we focus on data observations that evolve on certain nonlinear manifolds of importance in medical imaging: the shape manifold for endometrial surface reconstruction; the special orthogonal group SO(3) and the special Euclidean group SE(3) for preoperative MRI-based navigation. Results on real data show that our method succeeds in meeting the clinical goal: combining different modalities to improve the localization of the endometrial lesions.