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 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. We also obtain conditions for "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). The concept is meaningful only if the bounds are small compared to the support size. Numerical experiments backing our claims are shown.Comment: Accepted (with mandatory minor revisions) to IEEE Trans. Signal Processing. 12 pages, 5 figure

Modified-CS: Modifying Compressive Sensing for Problems with Partially Known Support

We study the problem of reconstructing a sparse signal from a limited number of its linear projections when a part of its support is known, although the known part may contain some errors. The ``known" part of the support, denoted T, may be available from prior knowledge. Alternatively, in a problem of recursively reconstructing time sequences of sparse spatial signals, one may use the support estimate from the previous time instant as the ``known" part. The idea of our proposed solution (modified-CS) is to solve a convex relaxation of the following problem: find the signal that satisfies the data constraint and is sparsest outside of T. We obtain sufficient conditions for exact reconstruction using modified-CS. These are much weaker than those needed for compressive sensing (CS) when the sizes of the unknown part of the support and of errors in the known part are small compared to the support size. An important extension called Regularized Modified-CS (RegModCS) is developed which also uses prior signal estimate knowledge. Simulation comparisons for both sparse and compressible signals are shown.Comment: To Appear in IEEE Trans. Signal Processing, September 2010, shorter version presented at ISIT 200

Stability (over time) of regularized modified CS (noisy) for recursive causal sparse reconstruction

Accurate signal recovery from under-determined system of equations is a topic of considerable interest. Compressive sensing (CS) gives an approach to find a solution to this system when the unknown signal is sparse. Regularized modied CS (noisy) propose an approach to find the solution to the under-determined system of equations when we are provided with 1- Partial part of signal support denoted by T and 2- A prior estimate of signal value on this support denoted by mu_T . In many applications, e.g sequential MRI reconstruction, the sparse signal support and its nonzero signal values change slowly over time. Inspired by this fact, we propose an algorithm utilizing reg-mod-CSN for sequential signal reconstruction such that the prior estimate of T and mu_T is generated from the previous time instant. Our major focus in this work is to study the stability of the proposed algorithm for recursive reconstruction of sparse signal sequences from noisy measurements. By stability we mean that the number of misses from the current support estimate; the number of extras in it; and the l_2 norm of the reconstruction error remain bounded by a time-invariant value at all times. For achieving this goal, we need a signal model that can represent the sequential signals in real applications. It should satisfy three constraint; 1- The distribution of the signal entries should follow the same distribution as real sequential signals; 2- It follows the same evolutionary pattern as the real sequential signals over time and 3-The signal support changes dynamically over time. In the two proposed signal model, we tried to satisfy these three constraints. Using these signal models, we analyzed the performance of the proposed algorithm and found the condition such that the system remain stable. These conditions are weaker in compare with older methods like CS and mod-CS. At the end, we show empirically that reg-mod-CS achieves a lower reconstruction error in compare with mod-CS and CS