Centroidal Voronoi tessellations (CVTs) are special Voronoi tessellations having the property that the generators of the Voronoi tessellation are also the centers of mass, with respect to a given density function, of the corresponding Voronoi cells. CVTs have been very usedful in a wide range of research fields, including image and data analysis, vector quantization, computer graphics, resource optimization, cell biology, numerical solution of partial differential equations, optimal control, mobile sensing networks, and so on. This project aims at further analysis on CVTs' properties and algorithms, and the broadening of applications for which CVTs and related concepts can be used as a basis for more efficient and accurate treatments.
In the theoretical aspects, the convergence and acceleration schemes of popular algorithms for computing CVTs and generalization of CVTs in other metric settings and with hierarchical structures will be studied. In the application aspects, the investigator will develop and implement robust CVT-based mesh generation and optimization algorithms, and then incorporate them in adaptive computations of numerical partial differential equations using finite element methods or finite volume methods, and in the solution of some challenging physical problems on the sphere and other surfaces such as geophysical flows, in light of the high-quality CVT-based surface meshing.
Also considered will be the cortical surface-flattening techniques based on the CVT methodology, which are very important to brain-imaging data analysis, including quantitative mapping of functional variability and construction of probabilistic brain-surface atlases. The proposed research will offer new insight into a number of outstanding theoretical issues and lead to renovation of computational algorithms for diverse important applications in science and engineering. The software resulting from this project will be actively disseminated, so that it can be used not only by researchers in the scientific computing area, but also by practitioners in a much broader community for application to problems in interdisciplinary sciences.
