ABSTRACT: In synthetic aperture radar (SAR) and magnetic resonance imaging (MRI), the acquired data can be modeled as noisy, incomplete and often non-uniformly sampled Fourier measurements of an underlying image. Reconstruction methods typically produce a single estimate that depends on hand-tuned regularization and gives little indication of its reliability. This tutorial presents a Bayesian framework for Fourier imaging that learns regularization from the data and quantifies uncertainty in the reconstruction.
We show how sampling theory and numerical analysis inform the forward model, from non-uniform sampling and data reduction to operator conditioning and fast non-uniform FFT computation for large-scale problems. We then show how approximation theory, which explains the Gibbs phenomenon at edges, motivates sparsity-promoting hierarchical priors, including recent residual prior transforms that go beyond standard differencing. We cover sparse Bayesian learning and computationally tractable uncertainty quantification, and their use in downstream tasks such as coherent change detection. SAR and MRI case studies illustrate the methods throughout. Familiarity with the Fourier transform and basic probability is assumed; no background in Bayesian methods is required.
presented by Anne gelb (Dartmouth College)
biography
Anne Gelb is the John G. Kemeny Parents Professor of Mathematics. She obtained her Ph.D. from the Division of Applied Mathematics at Brown University in 1996. Following her postdoctoral fellowship at the Center for Parallel Computing at the California Institute of Technology, she was a professor in the School of Mathematical and Statistical Sciences at Arizona State University until 2016 when she joined the Department of Mathematics at Dartmouth College. Her research involves developing highly accurate and efficient data-driven numerical methods for extracting important information in applications such as medical imaging, synthetic aperture radar imaging, climatology, signal processing, and fluid dynamics.
Presented by aditya viswanathan (university of michigan - dearborn)
biography
Aditya Viswanathan is an Assistant Professor in the Department of Mathematics and Statistics at the University of Michigan - Dearborn. Between September 2013 and August 2017, he was a Visiting Assistant Professor in the Department of Mathematics at Michigan State University. Prior to that, between September 2010 and August 2013, he was a Postdoctoral Scholar in the Computing and Mathematical Sciences department at Caltech. He received his Ph.D. and M.S. degrees in Electrical Engineering from Arizona State University and a B.E. in Electronics and Communication Engineering from R.V. College of Engineering, Bangalore, India.