Robust Sparse Signal Recovery for Compressed Sensing with Sampling and Dictionary Uncertainties

Compressed sensing (CS) shows that a signal having a sparse or compressible representation can be recovered from a small set of linear measurements. In classical CS theory, the sampling matrix and dictionary are assumed be known exactly in advance. However, uncertainties exist due to sampling distortion, finite grids of the parameter space of dictionary, etc. In this paper, we take a generalized sparse signal model, which simultaneously considers the sampling and dictionary uncertainties. Based on the new signal model, a new optimization model for robust sparse signal recovery is proposed. This optimization model can be deduced with stochastic robust approximation analysis. Both convex relaxation and greedy algorithms are used to solve the optimization problem. For the convex relaxation method, a sufficient condition for recovery by convex relaxation method is given; For the greedy algorithm, it is realized by the introduction of a pre-processing of the sensing matrix and the measurements. In numerical experiments, both simulated data and real-life ECG data based results show that the proposed method has a better performance than the current methods. Index Terms compressed sensing, robust sparse signal recovery, sampling uncertainty, dictionary uncertainty. Yipeng Liu and Sabine Van Huffel are with ESAT-STADIUS and iMinds Future Health Department, Dept. of Electrical Engineering, KU Leuven, Kasteelpark Arenberg 10, box 2446, 3001 Leuven, Belgium. (email:[email protected]; [email protected]) Maarten De Vos is with Methods in Neurocognitive Psychology Lab, Department of Psychology, Cluster of excellence Hearing4all, European Medical School, Carl von Ossietzky University, Oldenburg, Germany; and he is also with Research Center Neurosensory Science, Carl von Ossietzky University, Oldenburg, Germany. (email:[email protected]) Manuscript received Month Day, 2014; revised Month Day, Year. ACHA, VOL. X, NO. X, MONTH YEAR 2