A Spline-based Volumetric Data Modeling Framework and Its Applications

We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation. 2012 The rapid advances in 3D scanning and acquisition techniques have given rise to the explosive increase of volumetric digital models in recent years. This dissertation systematically trailblazes a novel volu-metric modeling framework to represent 3D solids. The need to explore more efficient and robust 3D modeling framework has gained the prominence. Although the traditional surface representation (e.g., triangle mesh) has many attractive properties, it is incapable of expressing the interior space and materials. Such a serious drawback overshadows many potential modeling and analysis applications. Consequently volumetric modeling techniques become the well-known solution to this problem. Nevertheless, many unsolved research issues remain when developing an efficient modeling paradigm for existing 3D models: complex geometry (fine details and extreme concaveness), arbitrary topology, heterogenous materials, large-scale data storage and processing, etc. In this dissertation, we concentrate on the challenging research issue of developing a spline-based modeling framework, which converts the iv iv iv conventional data (e.g., surface meshes) to tensor-product trivariate splines. This methodology can represent both boundary/volumetric geometry and real volumetric physical attributes in a compact and continuous fashion. The regular tensor-product structure enables our new developed methods to be embedded into the industry standard seamlessly. These properties make our techniques highly preferable in many physically-based applications including mechanical analysis, shape deformation and editing, virtual surgery training, etc. Using tensor-product trivariate splines to reconstruct existing 3D objects is highly challenging, which always involves component-based decomposition, volumetric parameterization and trivariate spline approximation. This dissertation seeks accurate and efficient solutions to these fundamental and important problems, and demonstrates their applications in modeling 3D objects of arbitrary topology. First, in order to achieve a " surface model to trivariate splines " conversion , we define our new splines upon a novel parametric domain called generalized poly-cubes (GPCs), which comprise a set of regular cube domains topologically glued together. We then further improve our trivariate splines to support arbitrary topol-ogy by allowing the divide-and-conquer scheme: The user can decompose the model into components and represent them by trivariate spline patches. Then the key contribution is our powerful merging strategy that can glue tensor-product spline solids together, while preserving many attractive advantages. We also develop an effective method to reconstruct discrete volumet-ric datasets (e.g., volumetric image) into continuous trivariate splines. To capture the fine features in …