@UNPUBLISHED{publication1, title = "$\backslash$(k$\backslash$)-Splines on SPD manifolds", author = "{CAMARINHA, Margarida} and {MACHADO, Lu{\ยด{i}}s} and {SILVA LEITE, F{\'{a}}tima}", year = "2023-05-09", abstract = "

The generalization of Euclidean splines to Riemannian manifolds was initially motivated by trajectory planning problems for rigid body motion. The increased interest in non-Euclidean splines was essentially due to its relevance in many areas of science and technology. Lie groups and symmetric spaces play an important role in this context. The manifold of symmetric positive definite (SPD) matrices is used, in particular, in computer vision, with emphasis in medical imaging. Different Riemannian structures have been considered in the SPD, in part to reduce the computational effort. In this paper, we\  first review the theory of highorder geometric splines for general Riemannian manifolds and its specialization to Lie groups. We then solve the variational problem that gives rise to spline curves on the manifold of symmetric positive definite matrices, equipped with the Log-Cholesky metric and having a Lie group group structure introduced in [4]. This enables considerable simplifications and, as a consequence, closed form expressions for higher-order polynomial splines are obtained.

", url = "http://www.mat.uc.pt/preprints/eng\_2023.html", note = "" }