Fitting Smooth Paths on Riemannian Manifolds: Endometrial Surface Reconstruction and Preoperative MRI-Based Navigation

We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using C 1 piecewise-B ́ezier func- tions. A property of the method is that, when the manifold reduces to a Euclidean space, the control points minimize the mean square accelera- tion of the path. As an application, we focus on data observations that evolve on certain nonlinear manifolds of importance in medical imag- ing: the shape manifold for endometrial surface reconstruction; the spe- cial 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 modal- ities to improve the localization of the endometrial lesions