Document Type

Dissertation

Date of Degree

Summer 2015

Degree Name

PhD (Doctor of Philosophy)

Degree In

Applied Mathematical and Computational Sciences

First Advisor

Weimin Han

Second Advisor

Jianfeng Cai

Abstract

As one of the emerging optical molecular imaging modalities, bioluminescence tomography is for reconstruction of the light source distribution inside a small animal body from the measured photon data on its surface. Such a light source distribution can be either induced through the excitation of an external light source, or generated by an internal bioluminescent source. The propagation of light within a biological medium is accurately described by the radiative transfer equation.

In this thesis, a bioluminescent source recovery problem is considered for the steady state radiative transfer equation. The recovery model is based on the minimization of combined effects of equation residual, boundary condition residual, and boundary measurement residual. The total variation of the light source is taken as one of the regularization terms so that the minimization of surface area of a light source is achieved. An alternating direction multiplier method is applied to decouple the system governing the light distribution and source data. Three different formulations and algorithms are introduced for the total variation regularization term. Convergence analysis is provided for each algorithm and numerical experiments are presented to show the performance of the algorithms.

Public Abstract

Bioluminescence is the process of light emission in living organisms. Bioluminescence tomography is developed to reveal molecular and cellular signatures inside a small animal body by the photon data measured by bioluminescent probes placed on its surface. The data is a result of a light source distribution inside the animal body, which can be either induced through an external light source, or generated by an internal bioluminescent source. Such process can be accurately described by the radiative transfer equation.

We consider a bioluminescent source recovery problem for the time independent radiative transfer equation. The recovery model is based on the minimization of combined effects of equation residual, inflow boundary condition residual, and outflow boundary measurement residual. The ill-posedness of the recovery problem triggers the technique of regularization applied to the source term. Along with the standard L2 regularization term, the total variation of the light source is considered so that the minimization of surface area of a light source is achieved.

We apply an alternating direction multiplier method to decouple the system governing the light distribution and source data. Due to the non-smoothness of the total variation term, three different formulations and corresponding algorithms are introduced and discussed. Convergence analysis is provided for each algorithm and numerical experiments are presented to show the performance of the algorithms.

Keywords

publicabstract

Pages

vii, 89 pages

Bibliography

Includes bibliographical references (pages 86-89).

Copyright

Copyright 2015 Tianyi Zhang

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