Approaching phase retrieval with deep learning

Abstract

Phase retrieval is the process of reconstructing images from only magnitude measurements. The problem is particularly challenging as most of the information about the image is contained in the missing phase. An important phase retrieval problem is Fourier phase retrieval, where the magnitudes of the Fourier transform are given. This problem is relevant in many areas of science, e.g., in X-ray crystallography, astronomy, microscopy, array imaging, and optics. In addition to Fourier phase retrieval, we also take a closer look at two additional phase retrieval problems: Fourier phase retrieval with a reference image and compressive Gaussian phase retrieval. Most methods for phase retrieval, e.g., the error-reduction algorithm or Fienup's hybrid-input output algorithms are optimization-based algorithms which solely minimize an error-function to reconstruct the image. These methods usually make strong assumptions about the measured magnitudes which do not always hold in practice. Thus, they only work reliably for easy instances of the phase retrieval problems but fail drastically for difficult instances. With the recent advances in the development of graphics processing units (GPUs), deep neural networks (DNNs) have become fashionable again and have led to breakthroughs in many research areas. In this thesis, we show how DNNs can be applied to solve the more difficult instances of phase retrieval problems when training data is available. On the one hand, we show how supervised learning can be used to greatly improve the reconstruction quality when training images and their corresponding measurements are available. We analyze the ability of these methods to generalize to out-of-distribution data. On the other hand, we take a closer look at an existing unsupervised method that relies on generative models. Unsupervised methods are agnostic toward the measurement process which is particularly useful for Gaussian phase retrieval. We apply this method to the Fourier phase retrieval problem and demonstrate how the reconstruction performance can be further improved with different initialization schemes. Furthermore, we demonstrate how optimizing intermediate representations of the underlying generative model can help overcoming the limited range of the model and, thus, can help to reach better solutions. Finally, we show how backpropagation can be used to learn reference images using a modification of the well-established error-reduction algorithm and discuss whether learning a reference image is always efficient. As it is common in machine learning research, we evaluate all methods on benchmark image datasets as it allows for easy reproducibility of the experiments and comparability to related methods. To better understand how the methods work, we perform extensive ablation experiments, and also analyze the influence of measurement noise and missing measurements.