Lognormal Distributions and Geometric Averages of Positive Definite Matrices

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (PD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. PD matrices: the log-Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of PD matrices. The limiting distributions of these averages are used to provide large-sample confidence regions for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, permitting a comparison between the various average types from the point of view of their sampling variability.Comment: 28 pages, 8 figure

Proceedings of the Fourth International Workshop on Mathematical Foundations of Computational Anatomy - Geometrical and Statistical Methods for Biological Shape Variability Modeling (MFCA 2013), Nagoya, Japan

International audienceComputational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information. The Mathematical Foundations of Computational Anatomy (MFCA) workshop aims at fostering the interactions between the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum for the exchange of the theoretical ideas and aims at being a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations. Following the first edition of this workshop in 2006, second edition in New-York in 2008, the third edition in Toronto in 2011, the forth edition was held in Nagoya Japan on September 22 2013

Statistical analysis for longitudinal MR imaging of dementia

Serial Magnetic Resonance (MR) Imaging can reveal structural atrophy in the brains of subjects with neurodegenerative diseases such as Alzheimer’s Disease (AD). Methods of computational neuroanatomy allow the detection of statistically significant patterns of brain change over time and/or over multiple subjects. The focus of this thesis is the development and application of statistical and supporting methodology for the analysis of three-dimensional brain imaging data. There is a particular emphasis on longitudinal data, though much of the statistical methodology is more general. New methods of voxel-based morphometry (VBM) are developed for serial MR data, employing combinations of tissue segmentation and longitudinal non-rigid registration. The methods are evaluated using novel quantitative metrics based on simulated data. Contributions to general aspects of VBM are also made, and include a publication concerning guidelines for reporting VBM studies, and another examining an issue in the selection of which voxels to include in the statistical analysis mask for VBM of atrophic conditions. Research is carried out into the statistical theory of permutation testing for application to multivariate general linear models, and is then used to build software for the analysis of multivariate deformation- and tensor-based morphometry data, efficiently correcting for the multiple comparison problem inherent in voxel-wise analysis of images. Monte Carlo simulation studies extend results available in the literature regarding the different strategies available for permutation testing in the presence of confounds. Theoretical aspects of longitudinal deformation- and tensor-based morphometry are explored, such as the options for combining within- and between-subject deformation fields. Practical investigation of several different methods and variants is performed for a longitudinal AD study

Geometry of logarithmic strain measures in solid mechanics

We consider the two logarithmic strain measures$$\omega_{\rm iso}=\|\mathrm{dev}_n\log U\|=\|\mathrm{dev}_n\log \sqrt{F^TF}\|\quad\text{ and }\quad \omega_{\rm vol}=|\mathrm{tr}(\log U)|=|\mathrm{tr}(\log\sqrt{F^TF})|\,,$$which are isotropic invariants of the Hencky strain tensor $\log U$, and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group $\mathrm{GL}(n)$. Here, $F$ is the deformation gradient, $U=\sqrt{F^TF}$ is the right Biot-stretch tensor, $\log$ denotes the principal matrix logarithm, $\|.\|$ is the Frobenius matrix norm, $\mathrm{tr}$ is the trace operator and $\mathrm{dev}_n X$ is the $n$-dimensional deviator of $X\in\mathbb{R}^{n\times n}$. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor $\varepsilon=\mathrm{sym}\nabla u$, which is the symmetric part of the displacement gradient $\nabla u$, and reveals a close geometric relation between the classical quadratic isotropic energy potential $$\mu\,\|\mathrm{dev}_n\mathrm{sym}\nabla u\|^2+\frac{\kappa}{2}\,[\mathrm{tr}(\mathrm{sym}\nabla u)]^2=\mu\,\|\mathrm{dev}_n\varepsilon\|^2+\frac{\kappa}{2}\,[\mathrm{tr}(\varepsilon)]^2$$in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy$$\mu\,\|\mathrm{dev}_n\log U\|^2+\frac{\kappa}{2}\,[\mathrm{tr}(\log U)]^2=\mu\,\omega_{\rm iso}^2+\frac\kappa2\,\omega_{\rm vol}^2\,,$$where $\mu$ is the shear modulus and $\kappa$ denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor $R$, where $F=R\,U$ is the polar decomposition of $F$. We also contrast our approach with prior attempts to establish the logarithmic Hencky strain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity