#### Date of Award

Fall 2022

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematical and Statistical Sciences

#### Program

Computational Mathematical and Statistical Sciences

#### First Advisor

Hamilton, Sarah

#### Second Advisor

Rowe, Daniel B.

#### Third Advisor

Clough, Anne V.

#### Abstract

This thesis addresses the electrical impedance tomography (EIT) image reconstruction problem where samples may have irregular discretizations and presents two, new, learned reconstruction algorithms which leverage a graph framework. These new frameworks consider the irregular, non-uniform data as a graph thus allowing graph neural networks to be applied directly to the data defined over irregular meshes. Currently in imaging, convolutional neural networks are used most frequently in learned methods because they are spatially invariant and have the ability to leverage localized information. In addition, many images are represented by rows and columns of uniformly sized pixels which can easily be used as input or output to a convolutional network. When images do not exist in a pixel grid format and are instead defined over an irregular grid or mesh, an embedding, or interpolation, step must precede the convolution operation. Instead, the graph framework presented in this thesis allows the irregular data to be used directly as input to a graph neural network which has many of the same benefits as convolutional neural networks. A model-based, learned image reconstruction method on graphs and a post-processing graph variant of the U-net architecture are developed here. The model-based method integrates graph neural networks directly into each iteration of an iterative algorithm and has a Gauss-Newton-like feel. The post-processing network replaces the convolutional, transpose convolutional, and max pool layers of the traditional U-net with graph variants such that the entire network no longer requires regular pixel grids. Both methods are demonstrated using the application of EIT but may be flexible to other inverse problems utilizing discretized meshes. The graph methods were shown to require about one quarter of the inference time and produce equal or superior reconstructions when compared to baseline, non-learned methods across a variety of simulated and experimental test cases. In addition, the graph methods performed similarly to convolutional neural network alternatives. These results and the demonstrated flexibility of the graph frameworks in general are encouraging for the outlook of their use on other inverse problems where the samples can more easily be represented as graph data than as uniform pixel grids.