Relative illumination in an optical system is a function of many variables including distortion, vignetting and pupil aberration. It can be obtained by determining the size of the exit pupil in direction cosine space, using the appropriate coordinate system. In general, the calculation is done by tracing bundles of rays through the system. In many cases of interest, it is possible to calculate relative illumination accurately by tracing one on-axis ray and three off-axis rays.
Matthew P Rimmer, Matthew P Rimmer,
"Relative Illumination Calculations", Proc. SPIE 0655, Optical System Design, Analysis, Production for Advanced Technology Systems, (31 October 1986); doi: 10.1117/12.938414; https://doi.org/10.1117/12.938414