Efficient Algorithms for Solving Nonlinear Inverse Problems in Image Reconstruction

Throughout many fields of science, images are used to display information in a relatable matter about objects which may not be directly visible. Instead, sensors are used–beyond standard cameras–to capture measurements of the object. These measurements are then processed to reconstruct an image of the object which created them. This process of reconstructing the cause of the observed effects is known as an inverse problem. In this thesis, algorithms are proposed for solving various inverse problems in image reconstruction. These algorithms are then analyzed to demonstrate their statistical and computational efficiency. The main through-line tying these problems together is that the proposed solutions leverage inherent structural information.

The thesis begins by demonstrating how to design effective spectral methods for estimating an image from phaseless measurements given approximate knowledge of the structure of the noise effecting the system. Next, an amplitude-based loss function is proposed for solving a generalized matrix phaseless sensing problem and algorithms are derived which reach a critical point of such a loss function. Continuing, stochastic variance-reduction gradient techniques are applied to an algorithmic framework for reconstructing an image known as Plug-and-Play (PnP) to achieve faster computation times while maintaining high accuracy. The thesis closes by analyzing how to reconstruct a large set of images when there exists a global structure relating the images to each other. The methods presented in this thesis are applied to phase retrieval, compressive sensing magnetic resonance imaging, and electron back-scattered diffraction microscopy.