In this project, the PI will investigate Sparsity Constrained Regularization (SCR) for solving the Diffuse Optical Tomography (DOT) inverse problem. Two recent algorithms are available for the implementation of SCR in DOT: the Generalized Conditional Gradient Method with Sparsity Constraint (GCGM-SC) and the Generalized Semi-smooth Newton's Method with Sparsity Constraint (GSNM-SC). The efficacy of GCGM-SC has been demonstrated with both theoretical and numerical results for linear and some non-linear problems. GCGM-SC, which has also been derived for linear problems, minimizes a given functional, not necessarily convex, with sparsity constraint with respect to a given basis. In this project, the computational aspect of the DOT inverse problem will be investigated to (i) develop a reconstruction algorithm for DOT using GCGM-SC, (ii) devise a strategy for computing adaptive basis such as finite elements to take full advantage of the sparsity constraint idea, (iii) extend GSNM-SC to nonlinear problems and compare GSNM-SC to Tikhonov regularization, and (iv) perform convergence analysis of the proposed GSNM-SC and GCGM-SC in DOT. The interdisciplinary research project seeks to integrate education and research and foster an international exchange of students. The educational activities in the project will include: (i) international research experience for students, (ii) student exchange between Clemson University and the University of Bremen, Germany, (iii) the incorporation of much of the research activities into teaching activities to help students bridge course materials with research, (iv) exposure of high school students to applied mathematics research, and (v) attraction of underrepresented groups to pursue applied mathematics.
Diffuse Optical Tomography (DOT) is a method for imaging a highly scattering medium using near infrared and visible light. For example, optical tomography of biological tissue has potential applications for the early detection of breast cancer. One major advantage of DOT is that it is less expensive and non-invasive as compared with x-ray mammography. DOT imaging technology also shows great promise as a tool for initiating discoveries in physics, biology, and medicine. However, despite its great potential, it has yet to be commercially or medically successful as the instability in image reconstruction results in the blurring of any resulting image. To overcome these difficulties, we will investigate novel mathematical techniques for image reconstruction. Research applications will vary from cancer detection in biomedical imaging to land mine detection in remote sensing to imaging objects in the ocean. Research results are also expected to contribute to scientific knowledge in neutron transport, transport in atmospheric science, photothermal spectroscopies and microscopies, laser pump probes, diffuse photon density waves and new tomography technologies, such as optical, electronic and thermal imaging, and biomedical diagnostics.
In this project, we developed a new image reconstruction algorithm using sparse representation of optical parameters that has potential applications to breast cancer screening, neonatal brain imaging etc. We formulated the theoretical framework for analysis of the sparsity algorithm including the optimal laser source for reconstruction. We found that sparsity approach outperforms the least square approach typically used for image reconstruction. We proved the parameter differentiability of the cost functional in optical tomography. We published four research articles in international journals, one report, nine conference and invited presentations based on this project. Our approach has been successfully applied to electrical impedance tomography with applications to medical and geo-physical imaging. Two domestic PhD students were trained and supported through this project. Both students have written PhD dissertations based on this project. Both of the students have graduated and are now employed at the Rochester Institute of Technology as a faculty member and National Geospatial Intelligence Agency as a researcher respectively. The project partially supported a new graduate exchange program between the Center of Industrial Mathematics in Germany and the Department of Mathematical Sciences at Clemson University. More than half a dozen domestic students from Clemson have participated in the exchange program and have visited Bremen to develop master's and PhD projects. The NSF supported work of this project was also disseminated to high school students and our work was featured in local media at Clemson when the collaborators from Germany visited Clemson University.
