%0 Journal Article
%T Approximating Sets of Symmetric and Positive-Definite Matrices by Geodesics
%A L. Machado
%A F. Silva Leite
%J Conference Papers in Science
%D 2013
%R 10.1155/2013/425608
%X We formulate a generalized version of the classical linear regression problem on Riemannian manifolds and derive the counterpart to the normal equations for the manifold of symmetric and positive definite matrices, equipped with the only metric that is invariant under the natural action of the general linear group. 1. Introduction The geometry of the set of symmetric and positive definite (SPD) matrices is in the focus of intensive research activity involving tensor analysis. The importance of SPD matrices lies in the fact that they encode image information. As a consequence, they appear in several contexts of computer vision, such as, for instance, in [1, 2], in medical image analysis to interpolate scattered diffusion tensor magnetic resonance imaging, [3, 4], but also in the area of continuum physics related to averaging methods for the case of the elasticity tensor of the generalized Hooke¡¯s law [5]. Since the set of SPD matrices has a natural structure of Riemannian manifold, the rich theory of differential geometry can be used to solve real problems that may be formulated on this manifold. One particular problem of interest, that as far as we know has not been studied before, is that of approximating a set of data points in the SPD manifold by a geodesic. In the present paper, we first formulate the problem of finding the geodesic that best fits a given set of time-labelled points on a general Riemannian manifold. This corresponds to the natural generalization of the classical linear regression problem in . Solving this problem on a Riemannian manifold requires knowing explicit formulas for geodesics and for the geodesic distance between two points. Such is the case of connected and compact Lie groups, where geodesics are one-parameter subgroups or their translations, or of Euclidean spheres, where geodesics are the great circles. The geodesic regression problem has been studied for these two cases in [6], and the numerical implementation of the spherical case will appear soon in [7]. Our main objective is to derive the counterpart of the normal equations when the given data lies in the SPD manifold, equipped with a particular Riemannian metric that is affine-invariant. The paper is organized as follows. In Section 2, we start with the formulation of the geodesic fitting problem for the general case of a geodesically complete Riemannian manifold. In Section 3, we specialize to the case of the SPD manifold endowed with its natural affine-invariant Riemannian metric and gather all the necessary background to achieve our goal. The main result appears
%U http://www.hindawi.com/journals/cpis/2013/425608/