Geometric Statistics
Mathematical foundations and applications in computational anatomy

Ecole d'ete du Gretsi a Peyresq, 1er au 5 juillet 2019

Xavier Pennec
Equipe Asclepios
2004, route des Lucioles B.P. 93
06902 Sophia Antipolis Cedex
http://www-sop.inria.fr/epione
04 92 38 76 64
Xavier.Pennec@inria.fr

The material provided below is copyrighted. Author versions are provided as a personnal communication to each of the participant of the summer school for scholar purposes only and shall not be redistributed.

• 1/ Intrinsic Statistics on Riemannian Manifolds
• Introduction to differential and Riemannian geometry Chapter 1 of Riemannian Geometric Statistics in Medical Image Analysis. X. Pennec, S. Sommer and T. Fletcher Ed. Elsevier, 2019.
• Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. J. Math Imaging and Vision 25(1):127-154, July 2006.
• 2/ SPD matrices and manifold-valued image processing
  • Manifold-valued image processing with SPD matrices. Chapter 3 of Riemannian Geometric Statistics in Medical Image Analysis. X. Pennec, S. Sommer and T. Fletcher Ed. Elsevier, 2019.
  • Historical reference: A Riemannian Framework for Tensor Computing. International Journal of Computer Vision, 66(1):41-66, Jan. 2006.

• 3/ Metric and affine geonetric settings for Lie groups
  • Beyond Riemannian Geometry The affine connection setting for transformation groups Chapter 5 of Riemannian Geometric Statistics in Medical Image Analysis. X. Pennec, S. Sommer and T. Fletcher Ed. Elsevier, 2019.
  • Historical reference: Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups. In Matrix Information Geometry, pages 123-166. Springer, May 2012.

• 4/ Parallel transport to analyze longitudinal deformations
  • Geodesics, Parallel Transport and One-parameter Subgroups for Diffeomorphic Image Registration. Int. Journal of Computer Vision, 105(2), November 2013.
  • Parallel Transport with Pole Ladder: a Third Order Scheme in Affine Connection Spaces which is Exact in Affine Symmetric Spaces. [arXiv:1805.11436]

• 5/ Advanced statistics: central limit theorem and extension of PCA
  • Curvature effects on the empirical mean in Riemannian and affine Manifolds: a non-asymptotic high concentration expansion in the small-sample regime. 2019. [arXiv:1906.07418]
  • Barycentric Subspace Analysis on Manifolds. Annals of Statistics. 46(6A):2711-2746, 2018. [arXiv:1607.02833]
  • Additional reference for PGA: Statistics on manifolds Chapter 2 of Riemannian Geometric Statistics in Medical Image Analysis. X. Pennec, S. Sommer and T. Fletcher Ed. Elsevier, 2019.

Upcoming Workshop on this very subject (Free, you just have to register)

• Workshop on Geometric Statistics, Toulouse, August 30 - September 5 2019. Part of the Labex CIMI trimester on Statistics with Geometry and Topology, Toulouse August-November 2019.