This research focuses on recursive algorithms for causally reconstructing a time sequence of (approximately) sparse signals from a small number of ``incoherent" linear projections. The algorithms will be useful for real-time dynamic magnetic resonance imaging (MRI) in interventional radiology applications such as image-guided surgery or in functional-MRI. MRI is currently not usable for such real-time applications due to its "relatively slow image acquisition" (large data acquisition times and/or slow image reconstruction algorithms). Other potential applications include dynamic tomography for solar imaging or real-time single-pixel video imaging.
Since the recent introduction of compressive sensing (CS), the static version of the above problem has been thoroughly studied. But most existing algorithms for the dynamic problem just use CS to jointly reconstruct the entire time sequence in one go. This is a batch solution and has very high complexity. The alternative - CS at each time (simple CS) - requires many more measurements. This research is the first to develop and analyze recursive algorithms for signal sequence reconstruction, which have the same complexity as simple CS, but which (a) achieve exact reconstruction using much fewer noise-free measurements than those needed by simple CS; (b) achieve provably smaller reconstruction error than simple CS, when using noisy measurements, especially when the number of measurements is small; and (c) are provably stable over time (reconstruction error remains bounded). Fewer measurements means reduced scan times for MRI, while recursive reconstruction means real-time imaging is possible. By exploiting the fact that sparsity patterns change slowly over time, the problem is formulated as one of compressive sensing with partially known support.
CS and sequential CS are incorporated into the graduate/undergraduate curriculum and into senior-design at appropriate levels.
During the fourth and last year of this project, the only funding left was the REU supplement. This money was used to support a summer research project of an excellent sophomore year undergraduate student, Christopher Sheafe. He also collaborated with a group of Mathematics REU site students and some of his summer research funding also came from that grant. The entire group was mentored by the PI and two mathematics graduate students whom she co-advises and by Professor Leslie Hogben from Mathematics. Christopher worked on studying the performance of weighted $ell_1$ cite{hassibi} for sparse recovery with partial support knowledge. In earlier work cite{modcsjournal}, the PI and her graduate student had first studied the above problem and introduced a solution approach called modified-CS. Denote the partial support knowledge by $T$. Weighted $ell_1$ can be interpreted as a generalization of modified-CS and it solves $$min_x ||x_{T^c}||_1 + lambda ||x_T||_1 s.t. y=Ax$$ (modified-CS solves the above problem with $lambda=0$). Using extensive simulations, Christopher compared its reconstruction performance (number of measurements required for exact recovery) with that of modified-CS and simple $ell_1$. It was observed that this outperforms modified-CS in situations where the number of extras in the set $T$ is large. Along with the rest of the Mathematics REU group, he also derived exact recovery conditions for weighted $ell_1$.
