Structured Nonlinear Least Squares Problems in Biomedical and Biomolecular Imaging

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This project has significant mathematical and computational challenges, and at the same time is focused on specific applications in biomedical (tomosynthesis) and biomolecular (microscopy) imaging.The mathematical models addressed in this project are difficult ill-posed inverse problems. Computed solutions of these problems are very sensitive to errors in the data, and implementation for large scale 3-dimensional images is nontrivial. New image post processing algorithms developed in this project will be based on computing solutions of large scale structured nonlinear least squares problems.Efficiency will be obtained by exploiting algorithmic structure of the nonlinear least squares problem, as well as structure that arises in the applications. Collaborations with researchers in the School of Medicine and Winship Cancer Research Center at Emory University will be used to test and verify developed methods, and will facilitate efforts to transition new software to clinical use.Improved image reconstruction and post processing algorithms for tomosynthesis can have a profound impact on breast cancer screening. In addition to providing better screening capabilities than mammography, tomosynthesis requires less compression of the breast (reducing physical pain to the patient), and it requires a smaller radiation dose than computed tomography (CT). In addition to its application to breast cancer screening, tomosynthesis can be used for many other medical imaging applications where standard x-ray and CT are used. Thus, advances in computational methods for this application can have a very broad impact in the medical field. In the case of biomolecular imaging, improved computational approaches can help provide inexpensive, yet accurate, point-of-care diagnostic imaging systems. This can significantly impact the monitoring of infections that have serious consequences to society; for example, management of HIV infected patients in poor regions of the world. Moreover, the application considered in this project (deconvolution microscopy), can be used to examine many other microscopic quantities, and thus development of new algorithms that provide better and faster reconstructions, can have a broad impact in many scientific fields, including biology, chemistry, neuroscience, and physics.