Department of Mathematics and Statistics, Georgetown University
Tuesday, March 10, 2020, 3:00–4:00PM
Building 101, Lecture Room A
Tuesday, March 10, 2020, 1:00–2:00PM
Building 1, Room 3304
Host: Gunay Dogan
Abstract: SENSitivity Encoding (SENSE) is an effective mathematical formulation for reconstructing under-sampled MRI data obtained in Parallel Magnetic Resonance Imaging (Parallel MRI). The functional model includes a regularization term and a data fidelity term which needs to be minimized to obtain a high quality MRI result. The proper choice of regularization is essential for image quality. In the first part of the talk, I consider the Total Variation (TV) based model, and I introduce Bregman operator splitting algorithm with variable step size (BOSVS) to solve TV-regularized SENSE. This algorithm is based on operator splitting and alternating direction method of multipliers to tackle the non-smoothness of TV. Moreover, the algorithm employs a novel back-tracking line search and step size strategies to handle the dense structure and ill-conditioning of the operator in the fidelity term. The step size rule starts with a Barzilai-Borwein (BB) stepsize and increases the nominal step until a termination condition is satisfied. Global convergence as well as rate of convergence of BOSVS will be discussed. In the second part, I consider the Euler's elastica regularization, and I solve the Euler's Elastica regularized SENSE problem. Euler's Elastica has an advantage over TV in better connecting the separated parts and enforcing better geometrical features of the image. The Euler’s elastica functional is however complex to minimize as it is nonconvex, nonsmooth, and highly nonlinear. I will propose a new numerical method based on variable splitting approaches, and proper relaxation of the functional to solve it efficiently. Numerical examples will be presented to show the effectiveness and efficiency of the proposed method. In particular, I show that Euler's Elastica based model improves the signal to noise ratio (SNR) and image relative error in comparison to the TV-based model. Since Euler's elastica enforces a better connectivity of separated parts, it also allows the use of much sparser under-sampling pattern hence it improves the imaging speed in addition to image quality.
Bio: Maryam Yashtini is a Tenure-Track Assistant Professor in the Department of Mathematics and Statistics at Georgetown University. Prior to joining the faculty at Georgetown, she was an NSF Postdoctoral Fellow at the Georgia Institute of Technology. She received a PhD Degree in Computational Mathematics and two Master's Degrees one in Applied Mathematics and one in Industrial Systems and Engineering from the University of Florida in 2009. Maryam's areas of expertise are computational optimization and scientific computing. Her research contributes efficient and practical algorithms to solve large-scale problems with applications to imaging. The optimization models of these problems are often non-differentiable, non-convex, ill-conditioned, and/or highly nonlinear. Maryam validates her algorithms by numerical implementation and rigorous theoretical analysis.
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