A Spline-Based Data Modeling Framework Over Regular Domains

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All are welcome to attend Hongyu Wang's thesis proposal (prelim) defense. Time: Wednesday 1/14, 1:30pm, Place: Rm.2311 (Wireless Seminar Room)

With the rapid advancement of modern 3D scanning technologies, CAD-based digital prototypes are routinely acquired in forms of raw points and/or triangular meshes. In order to enable geometric design and downstream product development processes (e.g., accurate shape analysis, finite element simulation, and e-manufacturing) in CAE environments, discrete data inputs must be converted into continuous, compact representations for scientific computing and engineering applications. In this thesis proposal, we present a novel spline-based data modeling framework to directly define tensor-product splines over any manifolds (serving as parametric domains). Since tensor-product B-splines and NURBS are current standards in CAD software industry, our entire mesh-to-spline data transformation pipeline enables and expedites the manifold surface design over existing CAD software platform industry (without any trimming), and thus, has great potential in shape modeling and reverse engineering applications of complicated real-world objects.

Tensor-product spline schemes require the parametric domains have the regular (rectangular) structures, and constructing the domain manifold with regular structures in an efficient way still remains a challenge. In this proposal, we study and present efficient regular domain construction methods, and demonstrate their applications in modeling 3D objects of arbitrary topology.

First, we propose the novel concept of polycube splines by defining splines directly upon the polycube map, serving as its parametric domain. We present a systematic way to construct polycube maps for surfaces of arbitrary topology based on global conformal parameterization, and demonstrate the modeling efficacy of the proposed polycube splines in solid modeling and shape computing. We also articulate our strategy for hole-filling.

We then further improve our polycube map construction by introducing the user-controllable polycube maps, which allows users to directly select the corner points of the polycubes on the original 3D surfaces, then construct the polycube maps by using the discrete Euclidean Ricci flow. The location of singularities can be interactively placed where no important geometric features exist, which makes the entire hole-filling process much easier to accomplish.

We also develop an effective method to construct polycube maps in an automatic fashion. The proposed algorithm can both construct a similar polycube of high geometric fidelity and compute a high-quality polycube map. In addition, it is theoretically guaranteed to output a one-to-one map.

Finally, we propose a geometry-aware domain decomposition algorithm for T-spline-based manifold modeling by which objects with arbitrary topology (especially objects with long branches) can be modeled elegantly. The segmentation process simultaneously respects local geometric features and global topological structures.

Through our experiments, we demonstrate that the proposed framework is very flexible and can potentially serve as a geometric standard for product data representation and model conversion in shape design and geometric processing. We also apply our proposed polycube maps to quite a few other valuable applications. We describe the applications in detail and outline a few new directions including skeleton-assisted corner point selection for user-controllable polycube map construction, the extension of our current bivariate polycube splines to trivariate volumetric splines to facilitate applications in volume modeling and scientific visualization, GPU-accelerated surface/volume rendering, texture mapping and synthesis, and other potential applications, which lead towards our future plans and will comprise a part of the PhD dissertation topics.