Ph.D. dissertation, University of Toronto, March 2012

Modern Foundations of Light Transport Simulation

Christian Lessig

Light transport, or radiative transfer, describes the propagation of visible light energy in macroscopic environments. While applications range from medical imaging over computer graphics to astrophysics, to this date its foundations remain phenomenological. Utilizing recent results, we develop the physical and mathematical structure of light transport from a lifted representation of Maxwell’s equations on the cotangent bundle. At the short wavelength limit, this yields a Hamiltonian description, with the classical formulation over the space of "positions and directions" resulting from a reduction to the cosphere bundle. We establish the connection between light transport and geometrical optics with a Legendre transform and we derive classical concepts such as radiance by considering measurements. We also show that light transport is a Lie-Poisson system for the group of symplectic diffeomorphisms, unveiling a tantalizing similarity to ideal fluid dynamics. Using Stone's theorem, we derive a functional analytic description of light transport that enables us to address one of the central challenges for simulations of everyday environments: how are efficient computations possible when the light energy density can only be evaluated pointwise? Using biorthogonal and possibly overcomplete bases formed by reproducing kernel functions, we develop a comprehensive theory for techniques that are restricted to pointwise information, subsuming for example sampling theorems, interpolation formulas, quadrature rules, density estimation schemes, and Monte Carlo integration. Overcomplete representations makes us thereby robust to imperfect information, and numerical optimization of the sampling locations leads to close to optimal techniques, providing performance that considerably improves over the state of the art in the literature.

The complete dissertation is available here.

Related  Talks:

A talk that tries to explain the central ideas of my disseration to a computer graphics audience is available here. I presented the talk inter alia at Cornell, MIT, and Columbia in May 2012).

A more mathematical version that was presented at the The Fields Institute in Toronto.

lessig (at) caltech (dot) edu May 2012