The objective of this research is to detect ocean eddies automatically on Synthetic Aperture Radar (SAR) images. We develop a new approach using Mask Region-based Convolutional Neural Networks (Mask R-CNN) and edge enhancement. First, we use Canny edge detector to extract a wide range of edges in SAR images. Then we put both the edge detection results and the corresponding original images into a Mask R-CNN based model for learning, thereby strengthening edge information. The proposed framework has been trained on a sample dataset of Sentinel-1A SAR-C imagery of the Western Mediterranean Sea. Experimental results revealed that the proposed method improved the performance by 2.3% on the MS COCO metrics compared to the method without edge enhancement.