SGER: Discrete Volumetric Curvature Flow for Graphics Applications

工程应用中的离散体积曲率流给定三维欧氏空间中的一个曲面，该曲面可以变形为三种正则形状之一：单位球面、平面和双曲圆盘。此外，变形是保角的。庞加莱猜想和瑟顿几何化猜想将这一事实推广到三维流形。基本上，三个流形可以以规范的方式分解成多个部分，每个部分包含八个几何中的一个。正则黎曼度量在六面体网格划分、体积参数化及体积样条构造等工程应用中具有重要的应用价值。Poincare猜想的证明为计算此类度量提供了一个强大的工具--Ricci流。Ricci流是使黎曼度规与曲率成比例变形的过程，这样曲率就会随着热扩散而变化。最终，曲率在任何地方都是恒定的，并且达到了正则度量。该方案旨在设计和实现体积的离散曲率流，并将其应用于图形学、几何建模、医学成像等众多工程领域。三维流形被表示为四面体网格。边长和二面角编码了网格的黎曼度量和曲率。离散的曲率流根据曲率使边的长度变形，这样在稳态时，曲率在任何地方都是恒定的。离散曲率流可以表示为特殊能量形式的梯度流。能量可以用牛顿方法来优化；临界点给出了所需的度规。此外，该方案还使用体积调和微分形式来计算基于Hodge理论的几何结构。离散曲率流法是为了解决几个重要的工程应用而提出的。六面体啮合多年来一直是啮合研究领域的圣杯。通过将体积变形为更简单的形状并对变形的体积进行细分，可以直接获得六角网格重新划分。体参数化是纹理映射、形状匹配、配准和比较的基础。曲率流将一般体积映射到标准形状，这导致了自然的参数化。构造与边界面样条一致的体样条是几何造型领域一个长期未解决的问题。曲率流法可以为解决这一问题提供新的见解和工具。

SGER: Discrete Volumetric Curvature Flow for Engineering ApplicationsGiven a surface in the three dimensional Euclidean space, the surface can be deformed to one of the three canonical shapes, the unit sphere, the plane and the hyperbolic disk. Furthermore, the deformation is angle preserving. The Poincare's conjecture and Thurton's geometrization conjecture generalize the fact to three dimensional manifolds. Basically, a three manifold can be decomposed to pieces in a canonical way, with each piece admitting one of eight geometries. The canonical Riemannian metric plays important roles in many engineering applications, such as hexahedral meshing, volumetric parameterization and volumetric spline construction. The proof of Poincare's conjecture offers a powerful tool to compute such metrics, Ricci flow. The Ricci flow is the process to deform the Riemannian metric proportional to the curvature, such that the curvature evolves according to heat diffusion. Eventually, the curvature is constant everywhere, and the canonical metric is achieved. The proposal aims at designing and implementing discrete curvature flow for volumes, and apply it in graphics, geometric modeling, medical imaging and many other engineering fields. The 3-manifolds are represented as tetrahedral meshes. The edge lengths and dihedral angles encode the Riemannian metric and curvatures of the mesh. Discrete curvature flow deforms the edge lengths according to the curvatures, such that at the steady state, the curvature is constant everywhere. Discrete curvature flow can be formulated as the gradient flow of special energy forms. The energies can be optimized using Newton's method; the critical point gives the desired metric. Furthermore, the proposal also uses volumetric harmonic differential forms to compute the geometric structures based on Hodge theory. The discrete curvature flow method is proposed to tackle several important engineering applications. Hexahedral meshing has been the Holy Grail in meshing research field for years. By deforming the volume to simpler shapes and tessellating the deformed volume, hex-remeshing can be obtained straightforwardly. Volumetric parameterization is the foundation for texture mapping, shape matching, registration and comparison. Curvature flow maps general volumes to canonical shapes, which induces natural parameterizations. Constructing volumetric splines, which is consistent with the boundary surface spline, is a long lasting open problem in geometric modeling field. The curvature flow method can offer new insights and tools to tackle the problem.