Abstract
In the field of Compressed Sensing, estimation of sparsity level is very essential as the sparsity level determines the minimum number of (i) measurements to be obtained of a sparse signal during acquisition and (ii) iterations to be performed for many of the greedy techniques for the perfect recovery of the sparse signal from the obtained measurements. In this paper, we propose a Maximum Likelihood (ML) estimator to estimate the instantaneous sparsity level during acquisition and an ML sequence (MLS) estimator of sparsity levels during recovery. As the sparsity level varies in time due to the continuous birth of newer supporting components and death of existing supporting components, this paper models the sparsity level variation as a stochastic birth-death process. The real-world applications of the proposed estimators are presented on the compression of aircraft vibration signals and the estimation of wireless channels. The simulation results on real-world and model-generated data demonstrate the performance merits of the proposed estimators compared to other existing methods.
Highlights
I N recent years, Compressed Sensing (CS) [1]–[5] has emerged as a powerful technique for acquiring high-dimensional sparse signals with fewer measurements than what is dictated by classical Shannon-Nyquist theory
In the CS recovery process, the N −dimensional sparse signal is reconstructed from those m CS measurements using either convex relaxation based algorithms [6]–[8] or greedy techniques [9], [10] with the prior knowledge of the sensing basis involved during acquisition
To determine the minimum number of Binary Sensing sub-Matrix (BSM) measurements required for obtaining an accurate estimate of sparsity level of the underlying sparse signal, one can minimize the variance of τs
Summary
I N recent years, Compressed Sensing (CS) [1]–[5] has emerged as a powerful technique for acquiring high-dimensional sparse signals with fewer measurements than what is dictated by classical Shannon-Nyquist theory. The collection of indices of those active components forms the support S and the cardinality of the set S is the sparsity level k. In the CS acquisition process, m samples or measurements are obtained using m × N −dimensional random or deterministic sensing basis, where m < N , and m ≥ ck log(N/k) for some constant c [4], [5]. In the CS recovery process, the N −dimensional sparse signal is reconstructed from those m CS measurements using either convex relaxation based algorithms [6]–[8] or greedy techniques [9], [10] with the prior knowledge of the sensing basis involved during acquisition. The basic theoretical concepts underlying CS can be found in [11]
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