Heat kernel smoothing on unit sphere

In brain imaging, cortical data such as the cortical thickness, cortical surface curvatures and surface coordinates have been mapped to a unit sphere for the purpose of visualization, surface registration and statistical analysis. Since the unit sphere provides a readily available parametrization and basis functions, cortical data can be easily quantified with respect to the spherical parametrization. For the cortical data on the unit sphere, it is necessary to smooth them to increase the signal-to-noise ratio and the smoothness for the subsequent statistical analysis. We present a mathematical framework for smoothing data on a unit sphere using the heat kernel. The heat kernel is analytically constructed using the spherical harmonics and O(n) heat kernel smoothing algorithm is presented