In this paper, a novel image denoising scheme based on the weighted nuclear norm minimization (WNNM) in the discrete periodic Radom transform (DPRT) domain is proposed. While the traditional patch-based low rank minimization approach, such as WNNM, has shown highly competitive image denoising performance, they treat all image patch groups with the same strategy hence cannot be optimum since image patches can have different properties. Particularly for patches with sharp edges, they need to be carefully handled as any error in their denoising can lead to significant degradation to the visual quality of the image. For effective denoising of natural lines/edges with prominent singularities, we apply the WNNM operator in the DPRT domain which allows the edges of different orientations to be effectively represented by different DPRT projections. The proposed algorithm first identifies the image patches with strong edges in the DPRT domain. Then, the new DPRT based WNNM operator is applied for their denoising. For the smooth patches, the conventional WNNM operator is performed in the spatial domain. Simulation results unto the various testing images show that the proposed approach achieves a substantial improvement in terms of both peak signal-to-noise (PSNR) ratio and in visual quality as compared with other state-of-the-art image denoising approaches.