Recursive Reconstruction of Sparse Signal Sequences

Namrata Vaswani
Iowa State University

Tuesday, May 25, 2010 at 11:00am
Engr IV Maxwell Room 57-124

Abstract
Consider the problem of recursively and causally reconstructing a time sequence of sparse signals from a greatly reduced number of linear projection measurements at each time. The signals are sparse in some transform domain referred to as the sparsity basis and their sparsity patterns can change with time. Some key applications where this problem occurs include dynamic MR imaging for real-time applications such as MR image-guided surgery or real-time single-pixel video imaging. Since the recent introduction of compressive sensing (CS), the static sparse reconstruction problem has been thoroughly studied. But most existing algorithms for the dynamic problem are batch solutions with very high complexity. Using the empirically observed fact that sparsity patterns change slowly over time, the recursive reconstruction problem can be formulated as one of sparse reconstruction with partially known support. We develop two classes of approaches to solve this problem - CS-Residual and Modified-CS, both of which have the same complexity as CS at a single time instant (simple CS), but achieve exact/accurate reconstruction using much fewer measurements.

Under the practically valid assumption of slowly changing support, Modified-CS achieves provably exact reconstruction using much fewer noise-free measurements than those needed to provide the same guarantee for simple CS. When using noisy measurements, under fairly mild assumptions, and again using fewer measurements, it can be shown that the error bounds for both Modified-CS and CS-Residual are much smaller; and most importantly, their errors are "stable" (remain bounded by small time-invariant values) over time. The proof of stability is critical for any recursive algorithm since it ensures that the error does not blow up over time. Experiments for the dynamic MRI application back up these claims for real data. Important extensions that also use the slow change of signal values over time are developed.

Biography
Namrata Vaswani received a B.Tech. from the Indian Institute of Technology (IIT), Delhi, in August 1999 and a Ph.D. from the University of Maryland, College Park, in August 2004, both in electrical engineering. From 2004 to 2005, she was a research scientist at Georgia Tech. Since Fall 2005, she has been an Assistant Professor in the ECE department at Iowa State University. She is currently serving as an Associate Editor for the IEEE Transactions on Signal Processing (2009-present). Her research interests are in estimation and detection problems in sequential signal processing and in biomedical imaging with current focus being on recursive sparse reconstruction problems, sequential compressive sensing and large dimensional tracking problems.