A Spline-based Volumetric Data Modeling Framework and Its Applications

Abstract

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 volumetric modeling framework to represent 3D solids. The need to explore more efficient and robust 3D modeling framework has gained the prominence. The traditional surface representation (e.g., triangle mesh) is incapable of expressing the interior space and materials. Such a serious drawback overshadows many potential modeling and analysis applications. Consequently, it is desirable to explore one efficient 3D volumetric data modeling framework to suffice above great potential of paradigm shift from surface to volume data. Contrary to surface modeling techniques, two fundamental changes seriously impede this shifting to volumetric data: dramatic explosion on total quantity of data size and unprecedent strong demand for faster and more accurate scientific computations. Volumetric data modeling thus has an extraordinarily intense need for a regular, continuous and compact representation. This dissertation presents the challenging research issue of developing a spline-based modeling framework to bridge this gap. This methodology adopts regular cube parametric domain and provides compact and precise mathematical representation, to sufficiently comply with the requirements in volumetric data modeling. Moreover, the regular tensor-product structure enables our new developed methods to be embedded into the industry standard seamlessly. These properties make spline-base framework highly preferable in many physically-based applications including mechanical analysis, shape deformation and editing, virtual surgery training, etc. Nevertheless, using this new framework to represent general volumetric models involves many theoretically fundamental obstacles. This dissertation focuses on the most important problems, and seeks accurate and efficient solutions. 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. Using GPCs can effectively reduce the number of domain and improve the domain quality. We then further investigate the technique to allow trivariate splines supporting arbitrary topology. Through 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 volumetric datasets (e.g., volumetric image) into trivariate splines. To capture the fine features in the data, we construct an as-smooth-as-possible frame field based on 3D principal curvatures to align with a sparse set of directional features. The frame field naturally conducts a volumetric parameterization and thus a spline representation. Next, we focus on promoting broader applications of our powerful modeling techniques. We present a novel methodology based on geometric deformation metrics to simulate magnification lens that can be utilized for the Focus+Context (F+C) visualization. We apply this methodology to both 2D image and 3D volume visualization. Through our extensive experiments, we demonstrate that our framework is an effective and powerful tool for comprehensive existing models. The great potential of our modeling framework will be highlighted through many valuable applications. We also envision further research directions and broader application scopes including many potential theoretical problems and useful applications.