Date of Award

5-2010

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Advisor

Khan, Taufiquar R

Committee Member

Shier , Douglas R

Committee Member

Brannan , James R

Committee Member

Gao , Bruce Z

Abstract

The focus of this thesis is to formulate an optimal source problem for the medical imaging technique of optical tomography by maximizing certain distinguishability criteria. We extend the concept of distinguishability in electrical impedance tomography to the frequency-domain diffusion approximation model used in optical tomography.
We consider the dependence of the optimal source on the choice of appropriate function spaces, which can be chosen from certain Sobolev or Lp spaces. All of the spaces we consider are Hilbert spaces; we therefore exploit the inner product in several ways. First, we define and use throughout an inner product on the Sobolev space H1(&Omega) that relates to the model we use for optical tomography. Because of the complex term in our model, the natural sesquilinear form related to the model is not an inner product, and we therefore define and use an operator that relates the H1(&Omega) inner product to the sesquilinear form and therefore the variational formulation. We prove the well-posedness of the variational formulation forward problem of the model using the H1(&Omega) inner product we define.
We also describe in detail an orthogonal decomposition of the space H1(&Omega) in terms of our model, and derive the related inner products and Riesz maps for the subspaces involved in the decomposition as well as their dual spaces, which allows us to define an inner product for the dual of H1(&Omega) as well. We complete this process for two sets of inner products, where the more complicated of the two is included in the Appendices.
To accomplish our goal of maximizing distinguishability, it is necessary to compute the adjoints of the operators described by the model. These computations are not trivial, and we therefore include them for all eight pairs of function spaces. Numerically, we employ the Power Method to compute the optimal sources, which turn out to be the dominant eigenfunctions of the associated operators. We present numerical experiments based on the Power Method to demonstrate the effects of different criteria. A localization measure is also used to determine the optimal source that best discriminates inhomogeneities from a known background. We present in the Appendices an analytical determination of the distinguishability criterion for the special case of an inhomogeneity in the center of a unit disk.
We describe steps toward using these ideas in reconstruction of the optical parameters. We also present some preliminary reconstructions of one of the optical parameters in a certain special case. Numerous questions and problems for further investigation result from the work presented here, and we outline several of these in the final chapter of this thesis.

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