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• We compared the proposed algorithm MSP with three similar existing algorithms, namely; SGCS [29], PCG [22] and CGD [30]. • Numerical experiments show that MSP outperforms PCG and CGD in signal recovery problems and it restored image with high quality than SGCS and CGD methods. • The proposed algorithm is an extension of the method for solving signal and image restoration problems. • Some nice properties of the algorithm are that it is derivative-free as well as matrix-free. We present a Barzilai-Borwein gradient method using the hyperplane projection technique of Solodov and Svaiter (1998) for solving the non-smooth nonlinear monotone equation arising from the reformulation of the ℓ 1 -norm regularized problem. The proposed method is an extension of the modified method by Liu and Duan (J. Inequal. Appl. 2015(1), 8, 2015) for solving signal and image restoration problems. The method is derivative-free and its search direction satisfies the sufficient descent condition. Numerical experiments presented show that the proposed method can recover sparse signals in fewer iterations and less CPU time and can reconstruct blurred images with higher quality compared to similar methods in the literature. [ABSTRACT FROM AUTHOR]