Abstract. The exterior surface of the brain is characterized by a juxtaposition of crests and troughs that together form a folding pattern. The majority of the deformations that occur in the normal course of adult human development result in folds changing their length or width. Current statistical shape analysis methods cannot easily discriminate between these two cases. Using discrete exterior calculus and Tikhonov regularization, we develop a method to estimate a dense orientation field in the tangent space of a surface described by a triangulated mesh, in the direction of its folds. We then use this orientation field to distinguish between shape differences in the direction parallel to folds and those in the direction across them. We test the method quantitatively on synthetic data and qualitatively on a database consisting of segmented cortical surfaces of 92 healthy subjects and 97 subjects with Alzheimer’s disease. The method estimates the correct fold directions and also indicates that the healthy and diseased subjects are distinguished by shape differences that are in the direction perpendicular to the underlying hippocampi, a finding which is consistent with the neuroscientific literature. These results demonstrate the importance of direction specific computational methods for shape analysis. 1

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Computational anatomy is an emerging discipline that aims at analyzing and modeling the individual anatomy of organs and their biological variability across a population. However, understanding and modeling the shape of organs is made difficult by the absence of physical models for comparing different 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 on objects like curves, surfaces and deformations that do not belong to standard Euclidean spaces. We explain in this chapter how the Riemannian structure can provide a powerful framework to build generic statistical computing tools. We show that few computational tools derive for each Riemannian metric can be used in practice as the basic atoms to build more complex generic algorithms such as interpolation, filtering and anisotropic diffusion on fields of geometric features. This computational framework is illustrated with

Abstract. The folding pattern of the human cortical surface is organized in a coherent set of troughs and ridges, which mark important anatomical demarcations that are similar across subjects. Cortical surface shape is often analyzed in the literature using isotropic diffusion, a strategy that is questionable because many anatomical regions are known to follow the direction of folds. This paper introduces anisotropic diffusion kernels to follow neighboring fold directions on surfaces, extending recent literature on enhancing curve-like patterns in images. A second contribution is to map deformations that affect sulcal length, i.e., are parallel to neighboring folds, with other deformations that affect sulcal length, within the diffusion process. Using the proposed method, we demonstrate anisotropic shape differences of the cortical surface associated with aging in a database of 95 healthy subjects, such as a contraction of the cingulate sulcus, shorter gyri in the temporal lobe and a contraction in the frontal lobe.