A variational framework of multivariate splines and its applications

Multivariate spline technique has proved to be a powerful mathematical tool for solving variational problems in a great number of research and engineering tasks, such as computer vision, scientific computing, engineering design, etc. As present, tensor-product B-splines and NURBS are the prevailing industrial standards and have been widely used in different disciplines. There have been a few new multivariate spline techniques developed recently bestowed with unique and favorable features, e.g., triangular B-splines and manifold splines. However, their potentials in facilitating practical scientific and industrial applications have not yet been fully explored.In this dissertation, we presented a variational framework built upon a range of newly proposed multivariate splines, and then applied it to solve a few research problems in medical imaging, scientific computing and geometric design. More specifically, we introduced a novel image registration method empowered by triangular B-splines, which is capable of modeling local rigidities inside a global non-rigid transformation. We also developed triangular B-spline finite element method (TBFEM) and solved an elastic problem on a pseudo breast model for temporal mammogram registration. Combining B-spline with feature detection and matching techniques, we proposed a registration algorithm that specifically registers mammogram images with little human interventions. In addition, we simulated elastic deformations on thin-shell objects with complicated geometries and arbitrary topologies, which are rigorously represented by manifold splines. Moreover, we proposed the new RTP-spline, a trivariate spline with restricted boundaries and defined over polycubic parametric domain. It is virtually a sub-class of trivariate T-splines, but constructed in a different top-down fashion such that semi-standardness can be preserved via knot insertion and blending function refinement. RTP-splines are featured with the ability of local refinement, restricted boundaries, domain flexibility and efficient evaluation of basis functions, all of which would greatly benefit a variety of applications working on solid objects and/or volumetric data. Through extensive experiments, we demonstrated that while the unique and advantageous properties of those new multivariate splines are exploited and applied to appropriate applications, our proposed framework would turn into an effective and powerful tool for solving variational problems in many science and engineering areas.