Geometry Processing Approaches for Defective Discrete Surfaces

计算机图形领域的艺术家和软件开发人员不断地将3D对象建模为离散曲面（三角形和四边形网格）。曲面建模涉及多个常规使用的程序，如变形、平滑和平坦。这些程序依赖于几何处理算子。通常情况下，表面不是很好表现，并表现出缺陷，如孔和断开的组件。这些“缺陷曲面”是有问题的，因为它们不适合基本运算，比如导数的计算，而微分运算是几何运算的基础。因此，当前的几何算子无法处理大多数缺陷表面。这代表了一个实质性的实际挑战，因为缺陷表面在计算机图形学中无处不在，尤其是在工业和3D对象存储库中。我们的长期愿景是设计一种新的几何处理范例，可以处理有缺陷的表面。为了推导出适用于所有操作符的一般方法，我们将依次检查操作符对和缺陷类型。我们的第一个目标是调查操作员在处理特定缺陷时的有意义的结果。用户研究将揭示参与者认为有意义的结果。有了这些见解，我们的第二个目标集中在导出缺陷表面的替代表面表示。相对于操作者和缺陷类型的有意义的结果，替代表示法将等同于缺陷表面。通过设计，这种替代表示将允许对几何运算符进行鲁棒计算。第三个目标是推导出一种方法，将操作员的结果转移回原始缺陷表面，再次以保留用户研究中确定的意义的方式。长期目标是调查所有的操作员和缺陷类型，并设计一个统一的替代表示。该研究对学术研究人员和工业终端用户具有重要意义，因为它将扩大几何处理算子的适用范围。它不仅对计算机图形学领域具有重要意义，而且对科学和工程也具有更广泛的意义。研究结果将提供三个核心好处：(1)洞察最终用户认为应用于缺陷表面的几何算子的有意义的结果，(2)缺陷表面的替代表示，以及(3)通过替代表示来回转换将几何算子应用于缺陷表面的过程。加拿大拥有庞大的计算机图形公司生态系统。这些公司依赖于在3D物体表面操作的理论方法的不断进步。此外，通过该研究项目培养的高素质人才将有机会在加拿大获得高质量的工作，同时也增加了加拿大公司的专业知识。

Artists and software developers in the computer graphics field are continually modeling 3D objects as discrete surfaces (triangle and quadrilateral meshes). Modeling surfaces involves multiple routinely used procedures such as deforming, smoothing, and flattening. Such procedures rely on geometry processing operators. Often, the surfaces are not so well-behaved, and exhibit defects such as holes and disconnected components. These "defective surfaces" are problematic as they do not lend themselves to elementary operations, such as computations of derivatives, which form the basis of geometry operators. As a result, current geometry operators cannot handle most of the defective surfaces. This represents a substantial practical challenge, as defective surfaces are all over the place in computer graphics, most notably in industry and in 3D object repositories. Our long-term vision is to design a new geometry processing paradigm that can handle defective surfaces. To derive a general approach that works for all operators, we will successively examine pairs of operators and defect types. Our first objective investigates meaningful results of an operator when dealing with a specific defect. User studies will reveal what participants perceive as meaningful results. With these insights, our second objective focuses on deriving an alternative surface representation for defective surfaces. The alternative representation will be equivalent to the defective surface with respect to the meaningful results for the operator and type of defect. By design, this alternative representation will allow the robust computation of the geometry operator. The third objective is to derive an approach transferring the results of the operator back to the original defective surface, again in a way that preserves the meaningfulness identified in the user study. The long-term goal is to investigate all operators and types of defects, and to design a unified alternative representation. This research is important for academic researchers and end users in industry, as it will broaden the range of applicability of geometry processing operators. It is significant for the computer graphics field, but also more generally, for science and engineering. The outcomes of the research will provide three core benefits: (1) insights into what end users see as meaningful results of geometry operators applied to defective surfaces, (2) an alternative representation for defective surfaces, and (3) a process to apply geometry operators on defective surfaces by back-and-forth conversion through the alternative representation. Canada has a vast ecosystem of companies working in computer graphics. These companies rely on the constant advancement of theoretical approaches operating on the surfaces of 3D objects. Furthermore, the highly qualified personnel trained through this research program will have access to quality jobs here in Canada, while also increasing the expertise of Canadian companies.