Fractional Fourier Transform Imaging Based on Untrained Neural Networks

Abstract

Fractional Fourier transform is an important branch of optical research, which has been widely used in optical encryption, optical filtering, image watermarking and other fields. The phase retrieval in the case of fractional Fourier transform is widely studied. Also, deep learning has been an intriguing method for optical computational imaging. However, in optical computational imaging, traditional deep learning methods possess some intrinsic disadvantages. In optical imaging experiments, it is often difficult to obtain sufficient quality and quantity of labeled data for training, thus leading to poor robustness of the trained neural network. Even with sufficient datasets, the training time can be particularly long. Recent years have witnessed a growing interest in physic-driven untrained neural networks for computational imaging. Here we apply such a method to study the fractional Fourier transform imaging, which combines neural networks with optical models to achieve phase retrieval of fractional Fourier transform. Different from the traditional neural network training with the original image as the target, in our network framework only a single intensity image is used for the phase retrieval of fractional Fourier transform images. The output image of the neural network will serve as an optical model through fractional Fourier transform, and then the output image of the optical model will be used as a loss function to drive the neural network training with the output image of the neural network. We have studied the fractional Fourier transform reconstruction for the cases with fractional order less than 1 and fractional order greater than 1. The simulations and experiments have shown that the network framework can implement the fractional Fourier transform reconstructions of the intensity objects and phase objects for different fraction orders, in which only 2000 iterations are needed. The experimental results show that the similarity between the reconstructed image and the original image, i.e. the number of normalized correlation coefficient, can reach 99.7%. Therefore, our work offers an efficient scheme for functional Fourier transform reconstruction with physicenhanced deep neutral network.