Surface parameterization using Riemann surface structure
Surface parameterization using Riemann surface structure
复制标题
使用黎曼表面结构进行表面参数化
DOI：
10.1109/iccv.2005.233
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
S. Yau
中科院分区：
文献类型：
--
作者：
Yalin Wang;X. Gu;Kiralee M. Hayashi;T. Chan;P. Thompson;S. Yau
我们提出了一种通用方法，使用黎曼曲面结构对具有复杂（可能分支）拓扑的一般曲面进行参数化。我们没有将表面几何演化为平面或球体，而是利用所有可定向表面都是黎曼表面并允许共形结构的事实，这在表面上引入了特殊的曲线坐标系。然后，我们可以使用连接表面上全局共形结构中的零点的临界图来自动划分表面。等参数曲线的轨迹将表面规范地划分为面片。这些面片中的每一个都是拓扑圆盘或圆柱体，并且可以通过积分表面上定义的全纯 I 形来共形映射到平行四边形。由此产生的表面细分和组件的参数化是固有且稳定的。对于具有相似拓扑和几何形状的表面，我们表明参数化结果是一致的，并且细分表面可以使用约束调和图相互匹配。表面相似度可以通过直接计算两个表面上每对对应点之间的距离来测量。为了说明该技术，我们计算了大脑和人脸表面 MRI 扫描中解剖表面的共形结构。我们发现，所得到的参数化在不同受试者之间是一致的，即使对于心室等分支结构也是如此，否则这些结构很难参数化。我们的方法提供了一个基于表面的框架，用于表面的统计比较以及在表面上生成网格以进行基于偏微分方程的信号处理
We propose a general method that parameterizes general surfaces with complex (possible branching) topology using Riemann surface structure. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces. We can then automatically partition the surface using a critical graph that connects zero points in the global conformal structure on the surface. The trajectories of iso-parametric curves canonically partition a surface into patches. Each of these patches is either a topological disk or a cylinder and can be conformally mapped to a parallelogram by integrating a holomorphic I-form defined on the surface. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. For surfaces with similar topology and geometry, we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other using constrained harmonic maps. The surface similarity can be measured by direct computation of distance between each pair of corresponding points on two surfaces. To illustrate the technique, we computed conformal structures for anatomical surfaces in MRI scans of the brain and human face surfaces. We found that the resulting parameterizations were consistent across subjects, even for branching structures such as the ventricles, which are otherwise difficult to parameterize. Our method provides a surface-based framework for statistical comparison of surfaces and for generating grids on surfaces for PDE-based signal processing