A Fast Spectral Method for Active 3D Shape Reconstruction

Variational energy minimization techniques for surface reconstruction are implemented by evolving an active surface according to the solutions of a sequence of elliptic partial differential equations (PDE's). For these techniques, most current approaches to solving the elliptic PDE are iterative involving the implementation of costly finite element methods (FEM) or finite difference methods (FDM). The heavy computational cost of these methods makes practical application to 3D surface reconstruction burdensome. In this paper, we develop a fast spectral method which is applied to 3D active surface reconstruction of star-shaped surfaces parameterized in polar coordinates. For this parameterization the Euler-Lagrange equation is a Helmholtz-type PDE governing a diffusion on the unit sphere. After linearization, we implement a spectral non-iterative solution of the Helmholtz equation by representing the active surface as a double Fourier series over angles in spherical coordinates. We show how this approach can be extended to include region-based penalization. A number of 3D examples and simulation results are presented to illustrate the performance of our fast spectral active surface algorithms.