This paper presents the framework for accelerating MR Imaging (MRI) by adopting Compressive Sampling/Sensing (CS) strategy. The major drawback is the extensive time involved in recovering MR Images from the limited k -space data which is often affected by various noises. Specifically speckle noise is seen to affect medical imaging modalities like ultrasound and MR scanning. This work presents the mathematical analysis for the distribution that arises due to the Speckle scattering. Further, a method has been devised to eliminate the Speckle noise to reconstruct MR images from the sub-sampled k -space (spatial frequency) data, while preserving the visual quality. The proposed sparse reconstruction method constructs on a Deep Convolutional Neural Network (DCNN) to reduce the reconstruction time. The CNN training involves the pairs of images which are generated from sub-sampled noisy spatial frequency points and the corresponding fully sampled k -space data. The performance of the proposed method has been evaluated with various sub-sampling schemes. The results show a remarkable reduction in the computation time along with high image quality for various undersampling strategies.