Nonlinear parametric inverse problems appear in many applications in science and engineering. We focus on diffuse optical tomography (DOT) in medical imaging. DOT aims to recover an unknown image of interest, such as the absorption coefficient in tissue to locate tumors in the body. Using a mathematical (forward) model to predict measurements given a parametrization of the tissue, we minimize the misfit between predicted and actual measurements up to a given noise level. The main computational bottleneck in such inverse problems is the repeated evaluation of this large-scale forward model, which corresponds to solving large linear systems for each source and frequency at each optimization step. Moreover, to efficiently compute derivative information, we need to solve, repeatedly, linear systems with the adjoint for each detector and frequency. As rapid advances in technology allow for large numbers of sources and detectors, these problems become computationally prohibitive. In this thesis, we introduce two methods to drastically reduce this cost.
To efficiently implement Newton methods, we extend the use of simultaneous random sources to reduce the number of linear system solves to include simultaneous random detectors. Moreover, we combine simultaneous random sources and detectors with optimized ones that lead to faster convergence and more accurate solutions.
We can use reduced order models (ROM) to drastically reduce the size of the linear systems to be solved in each optimization step while still solving the inverse problem accurately. However, the construction of the ROM bases still incurs a substantial cost. We propose to use randomization to drastically reduce the number of large linear solves needed for constructing the global ROM bases without degrading the accuracy of the solution to the inversion problem.
We demonstrate the efficiency of these approaches with 2-dimensional and 3-dimensional examples from DOT; however, our methods have the potential to be useful for other applications as well. / Ph. D. / Medical image reconstruction presents huge computational challenges due to the quantity of data generated by modern equipment. Each stage of processing requires the solution of more than a thousand large, three-dimensional problems. Moreover, as rapid advances in technology allow for ever larger numbers of sources and detectors and using multiple frequencies, these problems become computationally prohibitive. In this thesis, we develop two computational methods to drastically reduce this cost and produce good images from measurements.
First, we focus on efficiently estimating the absorption image while we reduce the cost of each optimization step by solving only for a few linear combinations of sources and of detectors.
Second, we can replace the full mathematical model by a reduced mathematical model to drastically reduce the size of the linear systems in each optimization step while still producing good image reconstructions. However, the computation of this reduced model still poses a formidable cost. Hence, we propose to reduce the cost of building the reduced model by sampling the sources and detectors. Using this reduced model for image reconstruction does not degrade the accuracy of the solutions and the quality of the image reconstruction.
We demonstrate the efficiency of these approaches with 2-dimensional and 3-dimensional examples from medical imaging. However, our methods have the potential to be useful for other applications as well.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/83451 |
| Date | 04 June 2018 |
| Creators | Sariaydin, Selin |
| Contributors | Mathematics, de Sturler, Eric, Kilmer, Misha E., Chung, Matthias, Beattie, Christopher A., Gugercin, Serkan |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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