Separating sparse and low-dimensional signal sequences from time-varying undersampled projections of their sums

The goal of this work is to recover a sequence of sparse vectors, s_t; and a sequence of dense vectors, ℓ_t, that lie in a “slowly changing” low dimensional subspace, from time-varying undersampled linear projections of their sum. This type of problem typically occurs when the quantity being imaged can be split into a sum of two layers, one of which is sparse and the other is low-dimensional. A key application where this problem occurs is in undersampled functional magnetic resonance imaging (fMRI) to detect brain activation patterns in response to a stimulus. The brain image at time t can be modeled as being a sum of the active region image, st, (equal to the activation in the active region and zero everywhere else) and the background brain image, ℓ_t, which can be accurately modeled as lying in a slowly changing low dimensional subspace. We introduce a novel solution approach called matrix completion projected compressive sensing or MatComProCS. Significantly improved performance of MatComProCS over existing work is shown for the undersampled fMRI based brain active region detection problem.