Regularized Newton Methods for X-ray Phase Contrast and General Imaging Problems

Like many other advanced imaging methods, x-ray phase contrast imaging and tomography require mathematical inversion of the observed data to obtain real-space information. While an accurate forward model describing the generally nonlinear image formation from a given object to the observations is often available, explicit inversion formulas are typically not known. Moreover, the measured data might be insufficient for stable image reconstruction, in which case it has to be complemented by suitable a priori information. In this work, regularized Newton methods are presented as a general framework for the solution of such ill-posed nonlinear imaging problems. For a proof of principle, the approach is applied to x-ray phase contrast imaging in the near-field propagation regime. Simultaneous recovery of the phase- and amplitude from a single near-field diffraction pattern without homogeneity constraints is demonstrated for the first time. The presented methods further permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval and tomographic inversion. We demonstrate the potential of this approach by three-dimensional imaging of a colloidal crystal at 95 nm isotropic resolution.Comment: (C)2016 Optical Society of America. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibite

Phase Retrieval for Sparse Signals: Uniqueness Conditions

In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts the recovery of the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. A fundamental question then is: "Under which conditions can we uniquely recover the signal of interest from its measured magnitudes?" In this paper, we assume the measured signal to be sparse. This is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. In this work, we derive a sufficient condition for the uniqueness of the solution of the phase retrieval (PR) problem for both discrete and continuous domains, and for one and multi-dimensional domains. More precisely, we show that there is a strong connection between PR and the turnpike problem, a classic combinatorial problem. We also prove that the existence of collisions in the autocorrelation function of the signal may preclude the uniqueness of the solution of PR. Then, assuming the absence of collisions, we prove that the solution is almost surely unique on 1-dimensional domains. Finally, we extend this result to multi-dimensional signals by solving a set of 1-dimensional problems. We show that the solution of the multi-dimensional problem is unique when the autocorrelation function has no collisions, significantly improving upon a previously known result.Comment: submitted to IEEE TI

Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data

In this paper we study a Tikhonov-type method for ill-posed nonlinear operator equations \gdag = F( ag) where \gdag is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density t\gdag where $t>0$ may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations, and phase retrieval problems in optics. Our approach uses a Kullback-Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as $t\to\infty$ both for an a priori and for a Lepski{\u\i}-type parameter choice rule

Solving 3D Radar Imaging Inverse Problems with a Multi-cognition Task-oriented Framework

This work focuses on 3D Radar imaging inverse problems. Current methods obtain undifferentiated results that suffer task-depended information retrieval loss and thus don't meet the task's specific demands well. For example, biased scattering energy may be acceptable for screen imaging but not for scattering diagnosis. To address this issue, we propose a new task-oriented imaging framework. The imaging principle is task-oriented through an analysis phase to obtain task's demands. The imaging model is multi-cognition regularized to embed and fulfill demands. The imaging method is designed to be general-ized, where couplings between cognitions are decoupled and solved individually with approximation and variable-splitting techniques. Tasks include scattering diagnosis, person screen imaging, and parcel screening imaging are given as examples. Experiments on data from two systems indicate that the pro-posed framework outperforms the current ones in task-depended information retrieval

Approaching phase retrieval with deep learning

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

Algorithmic Guarantees for Inverse Imaging with Untrained Neural Network Priors

Deep neural networks as image priors have been recently introduced for problems such as denoising, super-resolution and inpainting with promising performance gains over hand-crafted image priors such as sparsity and low-rank. Unlike learned generative priors they do not require any training over large datasets. However, few theoretical guarantees exist in the scope of using untrained network priors for inverse imaging problems. We explore new applications and theory for untrained neural network priors. Specifically, we consider the problem of solving linear inverse problems, such as compressive sensing, as well as non-linear problems, such as compressive phase retrieval. We model images to lie in the range of an untrained deep generative network with a fixed seed. We further present a projected gradient descent scheme that can be used for both compressive sensing and phase retrieval and provide rigorous theoretical guarantees for its convergence. We also show both theoretically as well as empirically that with deep network priors, one can achieve better compression rates for the same image quality as compared to when hand crafted priors are used