Image reconstructions are a challenging task in MRI images. The performance of the MRI image can be measure by following parameters like mean squared error (MSE), peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM). Compromising the above parameters and reconstructing the MRI image leads to false diagnosing. To avoid the false diagnosis, we have combined sparse based compressive sensing and block-based compressive sensing algorithm, and we introduced the hybrid sparse and block-based compressive sensing algorithm (HSBCS). In compressive stage, however, image reconstruction performance is decreased, hence, in the image reconstruction module, we have introduced convex relaxation algorithm. This proposed algorithm is obtained by relaxing some of the constraints of the original problem and meanwhile extending the objective function to the larger space. The performance is compared with the existing algorithm, block-based compressive sensing algorithm (BCS), BCS based on discrete wavelet transform (DWT), and sparse based compress-sensing algorithm (SCS). The experimentation is carried out using BRATS dataset, and the performance of image compression HSBCS evaluated based on SSIM, and PSNR, which attained 56.19 dB, and 0.9812. © 2024 Elsevier B.V., All rights reserved.