Riviere, BeatriceFuentes, David T2021-04-142021-11-012021-052021-04-13May 2021Actor, Jonas Albert. "Mathematical Approaches to Liver and Tumor Medical Image Segmentation." (2021) Diss., Rice University. <a href="https://hdl.handle.net/1911/110280">https://hdl.handle.net/1911/110280</a>.https://hdl.handle.net/1911/110280Hepatocellular carcinoma (HCC), more commonly known as liver cancer, is the most common cause of liver-related deaths in the United States, and a leading cause of cancer deaths worldwide. Diagnosis and treatment methods for HCC are often obtained by CT imaging; necessitating image segmentation to provide pixel-wise labels of what is liver, tumor, or healthy tissue. Such segmentations are costly, in time, effort, and money, to obtain manually. In contrast, automated segmentation methods, such as PDE-based methods or deep learning algorithms, are more efficient, but they too suffer from their own flaws; specifically, deep learning models can achieve state-of-the-art segmentation accuracies, but are viewed as “black boxes” that cannot cope with noise. This thesis describes mathematical approaches to overcome these issues, namely, to provide a mathematical foundation for deep learning segmentation methods that build upon classical applied mathematics techniques. First, we propose various improvements on existing deep learning architectures to perform liver and tumor segmentation. Second, we build upon classical techniques from applied mathematics, such as the discretization of partial differential equations and such as tensor factorization, to make sense of the underlying structure of the operators in convolutional neural networks for image segmentation. Third, we analyze the stability and uncertainty in deep learning segmentation models, and we derive a new bound for lower the Lipschitz constants of deep convolutional neural networks for image segmentation, improving the resilience of these networks to imaging noise.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.image segmentationscientific machine learningdeep learningcomputed tomographyapplied mathematicsThesis2021-04-14