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Since the recent introduction of compressive sensing (CS) and its guarantees, the problem of sparse recovery, and more recently, various other structured signal recovery problems, such as low-rank or low-rank plus sparse matrix recovery, have been extensively studied. Sparse recovery or CS refers to the problem of recovering a sparse signal from a highly subsampled set of its projected measurements. Many medical imaging techniques image cross-sections of human organs non-invasively by acquiring their linear projections one at a time and then reconstructing the image from these projections. For example, in magnetic resonance imaging (MRI), one acquires Fourier projections one at a time. For all these applications, the ability to accurately reconstruct using fewer measurements directly translates into reduced scan times and hence sparse recovery methods have had a huge impact in these areas. In dynamic imaging, e.g. in dynamic MRI, one needs to do the above for a time sequence of images or volumes of the heart or brain or other organs. Most existing works for sparse recovery consider batch methods that try to recover the entire image sequence in one go. However, batch methods cannot be used for real-time applications e.g. MRI-guided surgery; and even for offline applications, they are often very slow and memory-intensive.
Our recent works on Kalman Filtered CS (KF-CS) and later on Modified-CS were among the first works to propose and analyze low-complexity recursive solutions to the above problem. The key idea of Modified-CS is to reformulate the problem as one of sparse recovery with partially known support. Several applications in functional MRI require the identification of sparse outliers (activity regions in the brain) from slow-changing but dense background images. We demonstrate that this can be accomplished by treating the problem as one of sparse recovery in the presence of large but structured (low-dimensional) noise. We propose a recursive robust PCA algorithm called ReProCS for this purpose. Our key theoretical results include exact and stable (over time) recovery conditions for modified-CS based dynamic sparse recovery; and conditions for stable recovery with high probability for ReProCS based dynamic sparse plus low-rank matrix recovery. Based on preliminary experiments with real data, our proposed algorithms have the potential to greatly reduce scan times for dynamic MRI, functional MRI or dynamic optical coherence tomography (OCT) and thus enable real-time imaging of fast changing physiological phenomena.
Namrata Vaswani received a B.Tech. from Indian Institute of Technology (IIT), Delhi, in 1999 and a Ph.D. from University of Maryland, College Park, in 2004, both in Electrical Engineering. During 2004-05, she was a research scientist at Georgia Tech. Since Fall 2005, she has been with the Iowa State University where she is currently an associate professor of electrical and computer engineering. She held the Harpole-Pentair assistant professorship at ISU during 2008-09. From 2009 to 2013, she served as an Associate Editor for IEEE Transactions on Signal Processing. Her research interests are in signal and information processing for problems motivated by bio-imaging applications. Her current work focuses on recursive sparse recovery, robust PCA, matrix completion and applications in medical imaging.