This thesis presents the application of some advanced mathematical methods to image synthesis. The mainstream of our work is to formulate and analyze some rendering problems in terms of mathematical concepts, and develop some new mathematical machineries to pursue analytical solutions or nearly analytical approximations to them. An enhanced Taylor expansion formula is derived for the perturbation of a general ray-traced path and new theoretical results are presented for spatially-varying luminaires. On top of them, new deterministic algorithms are presented for simulating direct lighting and other scattering effects involving a wide range of non-diffuse surfaces and spatially-varying luminaires. Our work greatly extends the repertoire of non-Lambertian effects that can be handled in a deterministic fashion.