Hepatocellular 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.