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Magnetic Resonance Imaging (MRI) is one of the prominent medical imaging techniques. This process is time-consuming and can take several minutes to acquire one image. The aim of this research is to reduce the imaging process time of MRI. This issue is addressed by reducing the number of acquired measurements using theory of Compressive Sensing (CS). Compressive Sensing exploits sparsity in MR images. Randomly under sampled k-space generates incoherent noise which can be handled using a nonlinear image reconstruction method. In this paper, a new framework is presented based on the idea to exploit non-uniform nature of sparsity in MR images, where local sparsity constrains were used instead of traditional global constraint, to further reduce the sample set. Experimental results and comparison with CS using global constraint are demonstrated.