Existed inherent sparsity of natural images in some domains helps to reconstruct the signal with a lower number of measurements. To benefit from the sparsity, one should solve the reweighted $\ell_{1}$-norm minimization algorithms. Although, the existed reweighted $\ell_{1}$-norm minimization approaches work well for k-sparse signals, but, the performance of these methods for compressible signals are not competitive with unweighted one. Motivated by this challenge, in this paper, we propose a new reweighted $\ell_{1}$-minimization algorithm based on singular value decomposition (SVD) of compressible signals like images. Moreover, we develop our proposed algorithm on the block-based compressed sensing (BCS) to make it applicable to large-size images. Simulation results also demonstrated the superiority of our proposed method over current state-of-the-art reweighted CS reconstruction algorithms for natural images.