Author ORCID Identifier

Date of Award


Document Type

Thesis (Ph.D.)

Department or Program

Computer Science

First Advisor

Wojciech Jarosz


In computer graphics, we use geometry representations to model a wide range of virtual scenes—from the fantastical worlds shown in animated movies to intricate mechanical parts.
These representations provide the context for transport problems—light transport is used to produce images of virtual scenes and diffusive transport to simulate distributions of quantities like heat.

There are many types of representations each with their own advantages.
For example, explicit ones make it easy to directly manipulate surfaces, while implicit representations allow for intuitive modeling by non-technical users and straightforward integration into machine learning systems.
Unfortunately, many algorithms that work on these digital scenes make strong assumptions on the information the representation is able to provide.
This causes interoperability issues—geometry has to be lossily converted while being passed from one method to the next.

Our goal is to develop transport methods agnostic of the underlying geometry representation.
We first extend the surface editing capabilities of implicit surfaces—bringing them closer to explicit representations—and recognize that systems used in practice already favor algorithms agnostic of the underlying representation.
We then pose that random walk algorithms for transport use a minimal geometry "API" that can be easily supported by a wide range of representations.
Light transport has long been primarily solved by such random walk methods, but diffusive transport solvers on the other hand have been focused on finite element methods which require an explicit representation of the scene.
We provide a unified view of radiative and diffusive transport and use it to extend the field of random walk methods for diffusive transport.

Beyond simulating transport, stochastic processes are also used to model microscopic irregularities in the geometry of the scene—resulting in models for rough surfaces and participating media. Transport can then be computed as the ensemble average over all realizations of a process, but explicit representation of realizations is intractable. By using geometry agnostic random walk methods we can compute the ensemble averaged transport on the stochastic implicit representation directly, allowing us to use stochastic geometry as a new geometry representation that seamlessly transitions between surface and volume models.