Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


School of Computing

Committee Member

Dr. Jerry Tessendorf, Committee Chair

Committee Member

Dr. Joshua Levine

Committee Member

Dr. Brian Dean

Committee Member

Dr. Amy Apon


The theory of light transport forms the basis by which many computer graphic renderers are implemented. The more general theory of radiative transfer has applications in the wider scientific community, including ocean and atmospheric science, medicine, and even geophysics. Accurately capturing multiple scattering physics of light transport is an issue of great concern. Multiple scattering is responsible for indirect lighting, which is desired for images where high realism is the goal. Additionally, multiple scattering is quite important for scientific applications as it is a routine phenomenon. Computationally, it is a difficult process to model. Many have developed solutions for hard surface scenes where it is assumed that light travels in straight paths, for example, scenes without participating media. However, multiple scattering for participating media is still an open question, especially in developing robust and general techniques for particularly difficult scenes.

Radiative transfer can be expressed mathematically as a Feynman path integral (FPI), and we give background on how the transport kernel of the volume rendering equation can be written in terms of a FPI. To move this model into a numerical setting, we need numerical methods to solve the model. We start by focusing on the spatial and angular integrals of the volume rendering equation, and show a way to generate seed paths without regard as to if they are cast from the emitter or the sensor. Seed paths are converted into a discretized form, and we use an existing numerical method to tackle the FPI. A modified version of this technique shows how to reduce the running time from a quadratic to a linear expression. We then perform experimental analysis of the path integral calculation. The entire numerical method is put to full scale test on a distributed computing platform to calculate beam spread functions and compare the results to experimental data.

The dissertation is laid out as follows. In Chapter 1, we introduce the basic concepts of light propagation for computer graphics, multiple scattering, and volume rendering. Chapter 2 offers background on the subject of FPIs and some mathematical techniques used in their numerical integration for this work. Chapter 3 is a survey of radiative transfer and multiple scattering as it is studied in computer graphics and elsewhere. Chapter 4 is a full description of the current methodology. In Section 4.1 we describe sensor and emitter geometries used for our experiments. We propose a new algorithm for creating seed paths to use in the numerical integration of the FPI in Section 4.2. Section 4.3 introduces past work in the numerical integration, formalizes it, and improves upon its running time. Section 4.4 presents some analysis of the path weighting. In Chapters 5 and 6 we run experiments using the numerical methods. The first characterizes the calculation of the path integral itself using arbitrary spatial parameters, and shows repeatability and unbiased calculation given enough samples. In the second, we calculate beam spread functions, a basic property of scattering media, and compare the calculations to experimentally acquired data. Chapter 7 presents a summary of contributions, a summary of conclusions, and future directions for the research.