Hybrid Method for Solving the Radiative Transport Equation ^†

The spherical harmonics method ($P_N$ method) is often used for solving the radiative transport equation in terms of analytical functions. A severe and unsolved problem in this context was the evaluation of the angle-resolved radiance near sources and boundaries, which is a serious limitation of this method in view of concrete applications, e.g., in biomedical optics for investigating the different types of optical microscopy, within NIR spectroscopy, such as for the determination of ingredients in foods or in pharmaceuticals, and within physics-based rendering. In this article, we report on a hybrid method that enables accurate evaluation of the angle-resolved radiance directly at the boundary of an anisotropically scattering medium, avoiding the problems of the traditional $P_N$ methods. The derived integral equation needed for the realization of the hybrid $P_N$ method is formally valid for an arbitrary convex bounded medium. The proposed approach can be evaluated with practically the same computational effort as the traditional $P_N$ method while being far more accurate.