| Ph.D. dissertation, University of Toronto, March 2012
Modern Foundations of Light Transport Simulation
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.
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.