Abstract.A main focus of statistical shape analysis is the description of variability of a population of geometric objects. In this paper, we present work in progress towards modeling the shape and pose variability of sets of multiple objects. Principal geodesic analysis (PGA) is the extension of the standard technique of principal component analysis (PCA) into the nonlinear Riemannian symmetric space of pose and our medial m-rep shape description, a space in which use of PCA would be incorrect. In this paper, we discuss the decoupling of pose and shape in multi-object sets using different normalization settings. Further, we introduce new methods of describing the statistics of object pose using a novel extension of PGA, which previously has been used for global shape statistics. These new pose statistics are then combined with shape statistics to form a more complete description of multi-object complexes. We demonstrate our methods in an application to a longitudinal pediatric autism study with object sets of 10 subcortical structures in a population of 20 subjects. The results show that global scale accounts for most of the major mode of variation across time. Furthermore, the PGA components and the corresponding distribution of different subject groups vary significantly depending on the choice of normalization, which illustrates the importance of global and local pose alignment in multi-object shape analysis.

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In the context of Alzheimer's disease (AD), state-of-the-art methods separating normal control (NC) from AD patients or CN from progressive MCI (mild cognitive impairment patients converting to AD) achieve decent classification rates. However, they all perform poorly at separating stable MCI (MCI patients not converting to AD) and progressive MCI. Instead of using features extracted from a single temporal point, we address this problem using descriptors of the hippocampus evolutions between two time points. To encode the transformation, we use the framework of large deformations by diffeomorphisms that provides geodesic evolutions. To perform statistics on those local features in a common coordinate system, we introduce an extension of the Kärcher mean algorithm that defines the template modulo rigid registrations, and an initialization criterion that provides a final template leading to better matching with the patients. Finally, as local descriptors transported to this template do not directly perform as well as global descriptors (e.g. volume difference), we propose a novel strategy combining the use of initial momentum from geodesic shooting, extended Kärcher algorithm, density transport and integration on a hippocampus subregion, which is able to outperform global descriptors.

This chapter introduces the basic concepts of differential geometry: Manifolds, charts, curves, their derivatives, and tangent spaces. The addition of a Riemannian metric enables length and angle measurements on tangent spaces giving rise to the notions of curve length, geodesics, and thereby the basic constructs for statistical analysis of manifold valued data. Lie groups appear when the manifold in addition has smooth group structure, and homogeneous spaces arise as quotients of Lie groups. We discuss invariant metrics on Lie groups and their geodesics. The goal is to establish the mathematical bases that will allow in the sequel to build a simple but consistent statistical computing framework on manifolds. In the later part of the chapter, we describe computational tools, the Exp and Log maps, derived from the Riemannian metric. The implementation of these atomic tools will then constitute the basis to build more complex generic algorithms in the following chapters.