Optimal Cross Parameterization of Surfaces: Computational Methods and Scientific Applications

Mesh-based computations, such as finite-element or finite-volume analysis, are important for a wide range of scientific and engineering applications, ranging from the design of aircrafts and rockets to the understanding of cardiovascular disease. Many modern computational problems involve two or more surfaces or multiple discretizations (such as meshes) of a common surface and require correlating and cross-parameterizing these surfaces to analyze, match, manipulate, or map data between them. Examples include non-conforming domain decomposition, multiphysics simulations, analysis of biomedical images, shape matching, and shape retrieval. The problem of robust cross-parameterization of surfaces is challenging, because in many practical situations the domains do not match well and some major discrepancies (such as singularities and topological changes) may occur in numerical simulations as well as in the physical world. Most existing methods used in practice often assume the distances between corresponding points are at a local minimum. They do not offer theoretical guarantees, may produce inaccurate and even unstable results in the presence of singularities or large discrepancies, and do not offer the accuracy and physical meaningfulness required by scientific and engineering applications. Addressing these problems in a consistent and robust manner is critical for high-fidelity numerical simulations of complex systems and other related applications. In this project, we will devise and analyze a general and unified framework for optimal cross-parameterizations of surfaces and deliver this enabling technology to those numerical applications. Our framework will be based on efficient solutions of variational problems motivated by careful analysis of the accuracy and stability requirements of the underlying physical and numerical applications. Unlike most existing methods, our framework will construct cross-parameterizations directly on curved surfaces without intermediate parameterizations on a template surface, and thus will avoid those spurious sources of numerical errors. We emphasize theoretical foundations for optimal cross-parameterizations, efficient and robust algorithms for static and dynamic surfaces, and applications such as medical imaging, mesh optimization, and manifold learning.