Mathematical methods for image synthesis

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. First, my previous work on “Perturbation Methods for Image Synthesis” is extended here in several ways: (1) I propose a coherent framework using closed-form path Jacobians and path Hessians to perturb a general ray-traced path involving both specular reflections and refractions, and an algorithm similar to that used for interactive specular reflections is employed to simulate lens effects. (2) The original path Jacobian formula is simplified by means of matrix manipulations. (3) Path Jacobians and Hessians are extended to parametric surfaces which may not have an implicit definition. (4) Theoretical comparisons and connections are made with related work including pencil tracing and ray differentials. (5) Identify potential applications of perturbation methods of this nature in rendering and computer vision. Next, a closed-form solution is derived for the irradiance at a point on a surface due to an arbitrary polygonal Lambertian luminaire with linearly-varying radiant exitance. The solution consists of elementary functions and a single well-behaved special function known as the Clausen integral. The expression is derived from the Taylor expansion and a recurrence formula derived for an extension of double-axis moments, and then verified by Stokes' theorem and Monte Carlo simulation. The study of linearly-varying luminaires introduces much of the machinery needed to derive closed-form solutions for the general case of luminaires with radiance distributions that vary polynomially in both position and direction. Finally, the concept of irradiance tensors is generalized to account for inhomogeneous radiant exitance distributions from luminaires. These tensors are comprised of scalar elements that consist of constrained rational polynomials integrated over regions of the sphere, which arise frequently in simulating some rendering effects due to polynomially-varying luminaires. Several recurrence relations are derived for generalized irradiance tensors and their scalar elements, which reduce the surface integrals associated with spatially-varying luminaires to one-dimensional boundary integrals, leading to closed-form solutions in polyhedral environments. These formulas extend the range of illumination and non-Lambertian effects that can be computed deterministically, which includes illumination from polynomially-varying luminaires, reflections from and transmissions through glossy surfaces due to these emitters. Particularly, we have derived a general tensor formula for the irradiance due to a luminaire whose radiant exitance varies according to a monomial of any order, which subsumes Lambert's formula and expresses the solution for higher order monomials in terms of those for lower-order cases.