A Variational Framework of Multivariate Splines and Its Applications
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Multivariate spline technique has proved to be a powerful mathematicaltool for solving variational problems in a great number of researchand engineering tasks, such as computer vision, scientific computing,engineering design, etc. As present, tensor-product B-splines andNURBS are the prevailing industrial standards and have been widelyused in different disciplines. There have been a few new multivariatespline techniques developed recently bestowed with unique andfavorable features, e.g., triangular B-splines and manifold splines.However, their potentials in facilitating practical scientific andindustrial applications have not yet been fully explored.In this dissertation, we presented a variational framework built upona range of newly proposed multivariate splines, and then applied it tosolve a few research problems in medical imaging, scientific computingand geometric design. More specifically, we introduced a novel imageregistration method empowered by triangular B-splines, which iscapable of modeling local rigidities inside a global non-rigidtransformation. We also developed triangular B-spline finite elementmethod (TBFEM) and solved an elastic problem on a pseudo breast modelfor temporal mammogram registration. Combining B-spline with featuredetection and matching techniques, we proposed a registrationalgorithm that specifically registers mammogram images with littlehuman interventions. In addition, we simulated elastic deformations onthin-shell objects with complicated geometries and arbitrarytopologies, which are rigorously represented by manifold splines.Moreover, we proposed the new RTP-spline, a trivariate spline withrestricted boundaries and defined over polycubic parametric domain. Itis virtually a sub-class of trivariate T-splines, but constructed in adifferent top-down fashion such that semi-standardness can bepreserved via knot insertion and blending function refinement.RTP-splines are featured with the ability of local refinement,restricted boundaries, domain flexibility and efficient evaluation ofbasis functions, all of which would greatly benefit a variety ofapplications working on solid objects and/or volumetric data.Through extensive experiments, we demonstrated that while the uniqueand advantageous properties of those new multivariate splines areexploited and applied to appropriate applications, our proposedframework would turn into an effective and powerful tool for solvingvariational problems in many science and engineering areas.