Recursive Projected Modified Compressed Sensing for Undersampled Measurements

We study the problem of recursively reconstructing a time sequence of sparse vectors St from measurements of the form Mt = ASt+BLt where A and B are known measurement matrices, and Lt lies in a slowly changing low dimensional subspace. We assume that the signal of interest (St) is sparse, and has support which is correlated over time. We introduce a solution which we call Recursive Projected Modified Compressed Sensing (ReProMoCS), which exploits the correlated support change of St. We show that, under weaker assumptions than previous work, with high probability, ReProMoCS will exactly recover the support set of St and the reconstruction error of St is upper bounded by a small time-invariant value. A motivating application where the above problem occurs is in functional MRI imaging of the brain to detect regions that are “activated” in response to stimuli. In this case both measurement matrices are the same (i.e. A = B). The active region image constitutes the sparse vector St and this region changes slowly over time. The background brain image changes are global but the amount of change is very little and hence it can be well modeled as lying in a slowly changing low dimensional subspace, i.e. this constitutes Lt.