Heat Kernel Smoothing and Statistical Inference on Manifolds

In computational neuroanatomy, there is need for analyzing data collected on the cortical surface of the human brain. Gaussian kernel smoothing has been widely used in this area in conjunction with random field theory for analyzing data residing in Euclidean spaces. The Gaussian kernel is isotropic in Euclidian space so it assigns the same weights to observations equal distance apart. However, when we smooth data residing on a curved surface, it fails to be isotropic. On the curved surface, a straight line between two points is not the shortest distance so one may assign smaller weights to closer observations. For this reason smoothing data residing on manifolds requires constructing a kernel that is isotropic along the geodesic curves. With this motivation in mind, we construct the kernel of a heat equation on manifolds that should be isotropic in the local conformal coordinates and develop a framework for heat kernel smoothing and statistical inference on manifolds. As an illustration, we apply our approach in comparing the cortical thickness of autistic children to that of normal children.