Page 69 - Textos de Matemática Vol. 43

A LOCALLY QUADRATIC CONVERGENT METHOD TO COMPUTE THE KARCHER MEAN OF SPD MATRICES
L. MACHADO AND R. SEPULCHRE
Dedicated to Professor F´atima Leite
Abstract. In the past few years, it has been unquestionable the importance of the Karcher mean of symmetric and positive definite matrices in several applications of many disciplines.
However, in general, there is no a closed form for the Karcher mean and there- fore numerical methods are being required to yield satisfactory approximations for the mean.
In this work, we will use the Newton’s method on Riemannian manifolds to compute the Karcher mean in comparison with the traditional steepest descent method.
1. Introduction
The Karcher mean of a finite set of points lying on a Riemannian manifold has been defined in [12] as the minimizer of the sum of its square geodesic distances to each of the given points. Contrary to what happens in the Euclidean spaces there is no guarantee, in general, about its uniqueness. Such is not the case for the open cone of the symmetric and positive definite matrices (SPD). Since it has non-positive sectional curvature, the Karcher mean of a finite set of SPD matrices is a singleton [12, 13].
In the past few years there has been an astounding interest in studying the geometric mean for a finite set of SPD matrices. This growing interest is related with several important applications from di↵erent areas of research. We shortly refer to some of them for purposes of illustration.
In the medical imaging area, the geometric mean can be used to intrinsically in- terpolate scattered di↵usion tensor fields from di↵usion tensor magnetic resonance imaging (DT-MRI), [19], [8]. The DT-MRI is a recent MRI modality used for relat- ing image intensities to the relative mobility of endogenous tissue water molecules. We refer to [26] and [15] for a detailed description and several clinical applications of this technique.
2000 Mathematics Subject Classification. 53A35, 53B21, 57R35, 58C35, 65K10.
Key words and phrases. Karcher mean, Riemannian Hessian, steepest descent method, New- ton’s method.
The work was partly supported by the Institute of Systems and Robotics - Coimbra, while the first author visited the Department of Electrical Engineering and Computer Science at the University of Li`ege.
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