Understanding how to compare and interpolate complex data, such as images, shapes, and network structures, is a fundamental challenge across science and engineering, especially in the context of artificial intelligence. This project develops new mathematical and computational tools that extend the theory of optimal transport, a well-established framework for measuring distances between probability distributions. The proposed methods are tailored to settings which more closely reflect specialized real world data structures than those considered in classical optimal transport. These advances will enable more adequate quantitative analysis methods for medical images, dynamic crowd movements, and biological network structures. An integral outcome of the project will consist in the production of robust and open-source software packages, which will make these generalized optimal transport methods accessible to researchers and practitioners in biomedical imaging, machine learning, and network analysis. Importantly, these algorithms will be firmly grounded in mathematical theory. The project will also train graduate and postdoctoral researchers through cross disciplinary collaborations, foster community engagement via a workshop, and engage with the broader community via a coding-focused course and K-12 outreach activities. The project pursues three interlocking aims. First, it formulates a new Constrained Unbalanced Optimal Transport model for comparing general positive measures