Greedy Pursuits Based Gradual Weighting Strategy for Weighted $\ell_{1}$-Minimization

In Compressive Sensing (CS) of sparse signals, standard <tex>$\ell_{1}$</tex> minimization can be effectively replaced with Weighted <tex>$\ell_{1}$</tex> -minimization <tex>$(\mathbf{W}\ell_{1})$</tex> if some information about the signal or its sparsity pattern is available. If no such information is available, Re-Weighted <tex>$\ell_{1}$</tex> -minimization <tex>$(\mathbf{ReW}\ell_{1})$</tex> can be deployed. ReW <tex>$\ell _{1}$</tex> solves a series of W <tex>$\ell_{1}$</tex> problems, and therefore, its computational complexity is high. An alternative to ReW <tex>$\ell_{1}$</tex> is the Greedy Pursuits Assisted Basis Pursuit (GPABP) which employs multiple Greedy Pursuits (GPs) to obtain signal information which in turn is used to run W <tex>$\ell_{1}$</tex>. Although GPABP is an effective fusion technique, it adapts a binary weighting strategy for running W <tex>$\ell_{1}$</tex>, which is very restrictive. In this article, we propose a gradual weighting strategy for W <tex>$\ell_{1}$</tex>, which handles the signal estimates resulting from multiple GPs more effectively compared to the binary weighting strategy of GPABP. The resulting algorithm is termed as Greedy Pursuits assisted Weighted <tex>$\ell_{1}$</tex> -minimization <tex>$(\mathbf{GP-W}\ell_{1})$</tex>. For GP-W <tex>$\ell_{1}$</tex>, we derive the theoretical upper bound on its reconstruction error. Through simulation results, we show that the proposed GP-W <tex>$\ell_{1}$</tex> outperforms ReW <tex>$\ell_{1}$</tex> and the state-of-the-art GPABP.

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