Computational approaches for diffusive light transport: finite-elements, grid adaption, and error estimation

Subsurface light transport in highly scattering media is a problem of great interest to both computer graphics and biomedical optics researchers. Specifically, computer graphics researchers strive to develop more accurate simulations of light physics to in turn generate more realistic synthetic images. Likewise, biomedical optics researchers are concerned with accurately simulating light propagation to aid in design of equipment for diagnostic medical use. The mathematical formulation for diffusive light transport is presented along with a derivation for both a finite difference and finite element numerical solution for two and three dimensions. Efficient implementations are proposed which use Cholesky factorization to efficiently update the light scatter calculations if the source changes. Furthermore, the use data structures are proposed to accelerate (by an order of magnitude) parts of the source calculation and finite element matrix construction proposed by current biomedical optics literature. Furthermore, a novel technique for grid refinement for the finite element formulation using hanging nodes is presented. This technique allows for simple mesh refinement while maintaining the flux continuity on the finite element formulation. In conjunction with this technique, a per-element error estimator derived from a Green's function is presented along with a discussion on why traditional Galerkin style a posteriori estimation techniques fail. These techniques can then be combined to drive an adaptive finite element grid refinement method. Finally, to demonstrate the practicality of these techniques are demonstrated on simulations of numerical phantoms whose geometry and scattering properties were selected using segmented MRI data of human tissues. Furthermore, the results are visually comparable to images derived from a real world device which visualizes subsurface vasculature in human tissue by transilluminating tissue with near infrared light.

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