The deep unfolding network (DUN) provides an efficient framework for image restoration. It consists of a regularization module and a data fitting module. In existing DUN models, it is common to directly use a deep convolution neural network (DCNN) as the regularization module, and perform data fitting before regularization in each iteration/stage. In this work, we present a DUN by incorporating a new regularization module, and putting the regularization module before the data fitting module. The proposed regularization model is deducted by using the regularization by denoing (RED) and plugging in it a newly designed DCNN. For the data fitting module, we use the closed-form solution with Faster Fourier Transform (FFT). The resulted DRED-DUN model has some major advantages. First, the regularization model inherits the flexibility of learned image-adaptive and interpretability of RED. Second, the DRED-DUN model is an end-to-end trainable DUN, which learns the regularization network and other parameters jointly, thus leads to better restoration performance than the plug-and-play framework. Third, extensive experiments show that, our proposed model significantly outperforms the-state-of-the-art model-based methods and learning based methods in terms of PSNR indexes as well as the visual effects. In particular, our method has much better capability in recovering salient image components such as edges and small scale textures.