Riemannian Laplace Distribution on the Space of Symmetric Positive Definite Matrices

Hatem Hajri (1) , Ioana Ilea (2, 1) , Salem Said (1) , Lionel Bombrun (1) , Yannick Berthoumieu (1)

1 IMS - Laboratoire de l'intégration, du matériau au système
2 Technical University of Cluj-Napoca

The Riemannian geometry of the space P m , 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 P m. 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.

Mots clés: symmetric positive definite matrices; Laplace distribution; expectation-maximization; Bayesian information criterion; texture classification

Domaines: Traitement du signal et de l'image [eess.SP]

Hatem Hajri, Ioana Ilea, Salem Said, Lionel Bombrun, Yannick Berthoumieu. Riemannian Laplace Distribution on the Space of Symmetric Positive Definite Matrices. Entropy, 2016, 18 (3), pp.98. ⟨10.3390/e18030098⟩. ⟨hal-01379726⟩