Date of Award

12-2012

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Advisor

Yoon, Jeong-Rock

Committee Member

Brannan , Jim

Committee Member

Ervin , Vincent

Committee Member

Khan , Taufiquar

Abstract

or the detection of early stage cancer. MRE utilizes interior data for its inverse problems, which greatly reduces the ill-posedness from which most traditional inverse problems suffer.
In this thesis, we first establish a sensitivity analysis for viscoelastic scalar medium with complex wave number and compare it with the purely elastic case. Also we estimate the smallest detectable inclusion for breast and liver, which is about twice larger than using the purely elastic model. We also found the existence of optimal frequency (50 Hz) that maximizes the detectability when the Voigt model is used.
Second, we propose a local wavelength reconstruction based on the wave direction estimate for purely elastic medium. The main observation is that the wave looks primarily like a plane wave on a small window. On the small window, we first estimate the wave direction by solving a one dimensional optimization problem related to the minimum variance of shifted identical signals. Then along the wave direction, we use a non-periodic Fourier transform to reconstruct the wave number. This algorithm is extremely resilient to the noise and combined with another direct inversion method, this hybrid reconstruction becomes accurate as well. Extensive test reconstruc- tions on simulated and experimental data provided by the Mayo Clinic are included in this thesis. For the viscoelastic medium, this local wavelength reconstruction method will need an additional parameter for a scaling factor which leads to a two dimensional minimization problem. A slight modification in the Fourier transform part will also need to be created which is left for future work.

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