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Entropy 2016, 18(3), 98; doi:10.3390/e18030098

Riemannian Laplace Distribution on the Space of Symmetric Positive Definite Matrices

1,†,* , 1,2,†
,
1,†
,
1,†
and
1,†
1
Groupe Signal et Image, CNRS Laboratoire IMS, Institut Polytechnique de Bordeaux, Université de Bordeaux, UMR 5218, Talence 33405, France
2
Communications Department, Technical University of Cluj-Napoca, 71-73 Dorobantilor street, Cluj-Napoca 3400, Romania
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editors: Frédéric Barbaresco and Frank Nielsen
Received: 19 December 2015 / Revised: 7 March 2016 / Accepted: 8 March 2016 / Published: 16 March 2016
(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)
View Full-Text   |   Download PDF [952 KB, uploaded 16 March 2016]   |  

Abstract

The Riemannian geometry of the space Pm, of m × m symmetric positive definite matrices, has provided effective tools to the fields of medical imaging, computer vision and radar signal processing. Still, an open challenge remains, which consists of extending these tools to correctly handle the presence of outliers (or abnormal data), arising from excessive noise or faulty measurements. The present paper tackles this challenge by introducing new probability distributions, called Riemannian Laplace distributions on the space Pm. First, it shows that these distributions provide a statistical foundation for the concept of the Riemannian median, which offers improved robustness in dealing with outliers (in comparison to the more popular concept of the Riemannian center of mass). Second, it describes an original expectation-maximization algorithm, for estimating mixtures of Riemannian Laplace distributions. This algorithm is applied to the problem of texture classification, in computer vision, which is considered in the presence of outliers. It is shown to give significantly better performance with respect to other recently-proposed approaches. View Full-Text
Keywords: symmetric positive definite matrices; Laplace distribution; expectation-maximization; Bayesian information criterion; texture classification symmetric positive definite matrices; Laplace distribution; expectation-maximization; Bayesian information criterion; texture classification
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Hajri, H.; Ilea, I.; Said, S.; Bombrun, L.; Berthoumieu, Y. Riemannian Laplace Distribution on the Space of Symmetric Positive Definite Matrices. Entropy 2016, 18, 98.

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