3D eigenfunction expansion of sparsely sampled 2D cortical data

Various cortical measures such as cortical thickness are routinely computed along the vertices of cortical surface meshes. These metrics are used in surface-based morphometric studies. If one wishes to compare the surface-based morphometric studies to 3D volume-based studies at a voxel level, 3D interpolation of the sparsely sampled 2D cortical data is needed. In this paper, we have developed a new computational framework for explicitly representing sparsely sampled cortical data as a linear combination of eigenfunctions of the 3D Laplacian. The eigenfunctions are expressed as the product of spherical Bessel functions and spherical harmonics. The coefficients of the expansion are estimated in the least squares fashion iteratively by breaking the problem into smaller subproblems to reduce a computational bottleneck.