Computational anatomy is an emerging discipline that aims at analyzing and modeling the individual anatomy of organs and their biological variability across a population. The goal is not only to model the normal variations among a population, but also discover morphological diferences between normal and pathological populations, and possibly to detect, model and classify the pathologies from structural abnormalities. Applications are very important both in neuroscience, to minimize the inuence of the anatomical variability in functional group analysis, and in medical imaging, to better drive the adaptation of generic models of the anatomy (atlas) into patient-specic data.

However, understanding and modeling the shape of organs is made di- cult by the absence of physical models for comparing diferent subjects, the complexity of shapes, and the high number of degrees of freedom implied. Moreover, the geometric nature of the anatomical features usually extracted raises the need for statistics and computational methods on objects like curves, surfaces and deformations that do not belong to standard Euclidean spaces. We investigate in this chapter the Riemannian metric as a basis for developing generic algorithms to compute on manifolds.

We show that few computational tools derive this structure can be used in practice as the basic atoms to build more complex generic algorithms such as mean computation, Mahalanobis distance, interpolation, ltering and anisotropic difusion on elds of geometric features. This computational framework is illustrated with the analysis of the shape of the scoliotic spine and the modeling of the brain variability from sulcal lines where the results suggest new anatomical findings.