A spectral method for solving elliptic equations for surface reconstruction and 3D active contours

The solution of elliptic partial differential equations arises in 3D surface reconstruction and active contours. Most current approaches are iterative including finite element methods (FEM) and finite difference methods (FDM). We describe a fast spectral method for solving elliptic equations over the unit sphere. A double Fourier series expansion is applied to model convex or star-shaped 3D surfaces. The Helmholtz equation governing a diffusion on the unit sphere is solved by spectral methods using the double Fourier series as orthogonal basis functions. The optimization of the regularization parameter, which controls the tradeoff between denoising and matching high spatial frequencies, is studied for different 3D shapes and noise models. We show how the resultant solution can be combined with active contour methods to speed up 3D medical image segmentation. A number of examples and simulation results are presented to illustrate the algorithm.