Tensor-based Cortical Morphometry via Weighted Spherical Harmonic Representation

We present a new tensor-based morphometric framework that quantifies cortical shape variations using the local area element. The local area element is obtained from the Riemannian metric tensors, which are, in turn, obtained from the smooth functional parametrization of a triangle mesh. For the smooth parametrization, we have developed a novel weighted spherical harmonic (SPHARM) representation. The weighted-SPHARM differs from the classical SPHARM in a regularizing cost function. The classical SPHARM is a special case of the weighted-SPHARM. Further, for a specific choice of weights, the weighted-SPHARM is shown to be the finite least squares approximation to the solution of an isotropic heat diffusion on a unit sphere. The main aims of this paper are to present a theoretical framework for the weighted-SPHARM, and to show how it can be used in the tensor-based morphometry. As an illustration, the methodology has been applied in the problem of detecting abnormal cortical regions in a clinical population.