Weighted Fourier series representation and its application to quantifying the amount of gray matter

We present a novel weighted Fourier series (WFS) representation for cortical surfaces. The WFS representation is a data smoothing technique that provides the explicit smooth functional estimation of unknown cortical boundary as a linear combination of basis functions. The basic properties of the representation are investigated in connection with a self-adjoint partial differential equation and the traditional spherical harmonic (SPHARM) representation. To reduce steep computational requirements, a new iterative residual fitting (IRF) algorithm is developed. Its computational and numerical implementation issues are discussed in detail. The computer codes are also available at http://www.stat.wisc.edu/~mchung/softwares/weighted-SPHARM /weighted-SPHARM.html. As an illustration, the WFS is applied in quantifying the amount of gray matter in a group of high functioning autistic subjects. Within the WFS framework, cortical thickness and gray matter density are computed and compared. Index Terms--Cortical thickness, diffusion smoothing , gray matter density, iterative residual fitting, SPHARM, spherical harmonics.