The image edge information plays an important role in positioning in the automatic detection system (ADS) of medical images. Low resolution MR medical image lacking of edge information may easily lead to misdiagnosis and missed diagnosis by the ADS. This paper proposes a deep learning-based super-resolution algorithm called ESR-EASGAN, which can automatically achieves super-resolution restoration of edge information in MR medical images. ESR-EASGAN comprises two parallel dual modules: the edge super-resolution module (ESRM) is responsible for edge information restoration and the edge artifacts suppression module (EASM) is in charge of edge artifacts suppression. This algorithm is capable of automatically restoring the edge information in MR medical images while suppressing the generation of unnecessary artifacts during the recovery process. The experimental results show that the proposed ESR-EASGAN can effectively achieve high-quality restoration of edge information in medical MR images, both in terms of numerical evaluation metrics and human visual perception.