We wish to solve for in equation 1. is a direction vector and can be described by spherical polar coordinates i.e. . Hence we can view 1 as a five dimensional integro-differential equation.

We discretize equation 1 spatially on a 3D grid, and angularly in either 6 (axial) or 26 (axial+diagonal) directions, using a forward-differencing to approximate the directional derivative.

To evaluate the integral we initially tried to approximate as a sum of spherical harmonics as defined above. Unfortunately this turned out to be a pathological case for interpolating with due to our fixed set of directions falling directly on the lobes of the basis functions causing ill-conditioning, evidenced by zeros in the singular value decomposition of the of the interpolation matrix.

To obtain a smooth interpolation function (see figure 1) we chose a set of independant basis functions composed of products of sine and cosine functions (see appendix A). We want to interpolate a function defined on a sphere -

where k ranges over 0..5 or 0..25. Given our discrete array of function values

hence

where

and we can find . To integrate the function over the sphere we let

where

**Figure 1:** Intensity distribution.

Note that the vector can be precalculated.

While maintaining suitable boundary conditions equation 2 can be iterated until it converges, i.e. the transport of light reaches a steady state.

Thu Nov 17 10:01:16 EST 1994