Magnetic resonance imaging (MRI), which assists doctors in determining clinical staging and expected surgical range, has high medical value. A large number of MRI images require a large amount of storage space and the transmission bandwidth of the PACS system in offline storage and remote diagnosis. Therefore, high-quality compression of MRI images is very research-oriented. Current compression methods for MRI images with high compression ratio cause loss of information on lesions, leading to misdiagnosis; compression methods for MRI images with low compression ratio does not achieve the desired effect. Therefore, a fast fractal-based compression algorithm for MRI images is proposed in this paper. First, three-dimensional (3D) MRI images are converted into a two-dimensional (2D) image sequence, which facilitates the image sequence based on the fractal compression method. Then, range and domain blocks are classified according to the inherent spatiotemporal similarity of 3D objects. By using self-similarity, the number of blocks in the matching pool is reduced to improve the matching speed of the proposed method. Finally, a residual compensation mechanism is introduced to achieve compression of MRI images with high decompression quality. The experimental results show that compression speed is improved by 2-3 times, and the PSNR is improved by nearly 10. It indicates the proposed algorithm is effective and solves the contradiction between high compression ratio and high quality of MRI medical images.