Date of Award

12-2018

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematical Sciences

Committee Member

Dr. Taufiquar R. Khan, Committee Chair

Committee Member

Dr. Hyesuk Lee

Committee Member

Dr. Shitao Liu

Abstract

Diffuse Optical Tomography (DOT) is an emerging modality for soft tissue imaging with medical applications including breast cancer detection. DOT has many benefits, including its use of non ionizing radiation and its ability to produce high contrast images. However, it is well known that DOT image reconstruction is unstable and has low resolution. DOT uses near infra-red light waves to probe inside a body; for example, DOT can be used to measure the changes in the amount of oxygen in tissues, which can detect early stages of cancer in soft tissues such as the breast and brain. In this thesis, we perform dimensional analysis to obtain a dimensionless form of the ODE for the 1-d DOT model and the PDE for the 2-d DOT model. We later solve the 1-d cases using the finite element method (FEM) in MATLAB. We investigate whether the inverse problem using the dimensionless scaled forward DOT model will improve the ill-posedness of the image reconstruction problem in the 1-d case. We solve the inverse problem for DOT image reconstruction by reformulating the inverse problem as a variationally constrained non-linear optimization problem and compare solving the optimization problem for specific cases of the 1-d DOT model with Newton's iteration versus the traditional Gauss-Newton method. We observe the effects of different regularization parameters and step lengths on the reconstructions for Newton's iteration. We also observe the effect of moving the inclusion away from the boundary during image reconstruction. Using the optimally derived regularization parameter from the noise-free data, we reconstructed the parameter space by adding different levels of noise to the synthetic data. Based on our simulations in 1-d, we conclude that the scaled inverse problem is still ill-posed but that the variational approach provides a better reconstruction than the Gauss-Newton method.

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