Deep (convolutional) neural networks (DCNN) have recently gained popularity, and shown improved performance in the field of image enhancement (de-noising and super-resolution, for instance). However, the central issue of recovering finer texture details in images still remains unsolved. State-of-the-art objective functions used in DCNN mostly focus on minimizing the mean squared reconstruction error. The resulting image estimates have high peak signal-to-noise ratios, but they are often lacking high-frequency details, and are therefore error-prone with respect to fine-scale, possibly clinically relevant details.

In this article, we present GANReDL, a generative adversarial network (GAN) for image enhancement equipped with a real-order derivative induced loss functions (ReDL) which we will show gives improved images, in particular wrt to the reconstruction of fine-scale details. To the best of our knowledge, this is the first framework that incorporates non-integer order derivatives in loss functions. To this aim, we propose a discriminator network that is trained to differentiate between the enhanced images and ground-truth images, and propose a new loss function motivated by real-order derivatives which is capable of also capturing global image features rather than pixel-wise features only. We show, with several numerical experiments, that GANReDL is better in reconstructing the high-frequency image details, and therefore show improved performance for image enhancement over other state-of-the-art methods.