| The human cerebral cortex is complicated in terms of the depth and irregularity of its convolutions as well as its variability from one subject to another. These complications are big challenges for generating quantitative measurements in diagnosing mental illness or understanding brain development. In the last decade, neuroimaging techniques (e.g. MRI, PET/CT) have rapidly emerged, allowing to non-invasively record brain anatomy and function. They have been accompanied by the development of mathematical methods for the representation and comparison of the cortex in the emerging field of Computational Anatomy (CA).; The aim of this thesis is to develop intrinsic and extrinsic methods in CA for studying anatomical coordinates M to permit statistical inference on random physiological signals F representing morphometric measurements and functional activations of the cortex in clinical populations. Such signals F(·) on M can be described by infinite number of random variables F i = Fx yxdM , where the psii are termed as generalized partition functions on M . In the intrinsic case, the psii are defined intrinsically for each of the individual anatomical coordinate system M . In contrast, the extrinsic analysis only needs one set of partition functions for the template coordinates, and then the partition functions are applied to other anatomical coordinate systems via diffeomorphic transformations &phis;. Hence, the challenges are to find partition functions psi and diffeomorphic mappings &phis; between anatomies.; We choose a surface (the boundary between white matter and gray matter) as representation of the cortex. The thesis starts with deriving a large deformation diffeomorphic curve mapping algorithm to find diffeomorphic maps &phis; between anatomies. Subsequently, we define partition functions psi based on Courant's theorem on nodal domain analysis via self-adjoint differential operators. Then for modelling F(x), we generalize the Fourier basis representation of signals on the regular grid to orthonormal basis representation of signals on the cortical manifold. In particular, we choose the eigenfunctions of the Laplace-Beltrami operator as orthonormal basis functions. To test group difference in F(x ), we introduce a Gaussian random field model indexed over the cortical surface and a statistical model within nodal domains in both intrinsic and extrinsic frameworks. At the end of the thesis, two clinical studies are presented and the statistical power of the intrinsic and extrinsic analyses is compared. |
