Fig. 2.

Cubature rules for the approximation in Eq. (8) for the 2D integral in Eq. (3). The top graphs show the placement of the quadrature points in the unit disk. The bottom graphs show the weights along s_y=0. (a) 188 points on an equally-spaced Cartesian grid of (s_x, s_y) positions inside the illumination cone. (b) Separation of the 2D integral on the (s_x, s_y) plane into two 1D integrals over the radial coordinates (s, ϕ). The s integral is evaluated using Gauss-Legendre quadrature, and the ϕ integral is evaluated using the midpoint rule. A total of 20×8=160 quadrature points are used. (c) A custom 127-point quadrature rule for the unit disk [16], exact for polynomials $s_x^i{s}_y^j$ where i + j < 25.