LS-CS-Residual (LS-CS): Compressive Sensing on Least Squares Residual

We consider the problem of recursively and causally reconstructing time sequences of sparse signals (with unknown and time-varying sparsity patterns) from a limited number of noisy linear measurements. The sparsity pattern is assumed to change slowly with time. The key idea of our proposed solution, LS-CS-residual (LS-CS), is to replace compressed sensing (CS) on the observation by CS on the least squares (LS) residual computed using the previous estimate of the support. We bound CS-residual error and show that when the number of available measurements is small, the bound is much smaller than that on CS error if the sparsity pattern changes slowly enough. Most importantly, under fairly mild assumptions, we show “stability” of LS-CS over time for a signal model that allows support additions and removals, and that allows coefficients to gradually increase (decrease) until they reach a constant value (become zero). By “stability,” we mean that the number of misses and extras in the support estimate remain bounded by time-invariant values (in turn implying a time-invariant bound on LS-CS error). Numerical experiments, and a dynamic MRI example, backing our claims are shown.