A novel method for sparsity estimation by means of the divide and conquer method is presented. Also, the underestimation and overestimation criteria for signal sparsity is proposed and proven. Then the blind-sparsity subspace pursuit (BSP) algorithm for sparse reconstruction is discussed. Based on the estimation, BSP combines the support set and inherits the backtracking refinement that attaches to compressive sampling matching pursuit (CoSaMP)/subspace pursuit (SP), whereas the pruning process of BSP is improved by introducing the weakly matching backtracking strategy. With the said improvement, there is no need for BSP to require the sparsity as an input parameter. Furthermore, experiments demonstrate that the divide and conquer method is effective for sparsity estimation when the isometry constant is known. In addition, the simulation results also validate the superior performance of the new algorithm and show that BSP is an excellent algorithm for blind sparse reconstruction and is robust when the estimate of sparsity is not perfectly accurate.

A type of back-track algorithm has recently been proposed to handle sparse reconstruction problems arising in compressed sensing. This sort of algorithm is attractive owing to its recovery performance; however, it requires the sparsity level to be known a priori. A blind sparsity algorithm is presented based on subspace pursuit and weak pruning, and its convergence is demonstrated. The experiments validate the proposed algorithm's superior performance to that of several other back-tracking-type and optimisation methods.

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