三維點雲的網格重建演算法

Abstract

在本篇論文中，我們介紹一些三維網格生成的演算法，像是 Delaunay triangulation的演算法、曲面擬合(Surface fitting) 的演算法，和隱函數擬合(Implicit function fitting)的演算 法。其中，我們詳細介紹由 kazhdan[11] 提出的隱函數擬合演 算法的細節，先計算三維點雲上的法向量，然後透過解一個帕松 方程(Poisson equation) ， 算出三維指示函數(indicator function) ， 我們透過指示函數， 插值出空間中的等值面 (isosurface)，進而求得三角網格，並且提出一套方法在局部座 標系去計算三角網格點上的微分及曲率，利用Hsieh-Clough- Tocher triangles 建構出𝐶1曲面取代平面表達三角網格。

In this thesis, we introduce some algorithms of mesh reconstruction for point clouds, such as Delaunay triangulation, surface fitting method and implicit function fitting method. The key steps in implicit function fitting method proposed by kazhdan[11] include: estimation of the normal vectors for the point cloud and solving a Poisson equation to obtain the 3D indicator function. The triangular mesh of the surface can be obtained by interpolating the isosurface of the 3D indicator function. Furthermore, the surface can also be represented as a 𝐶1 function in local coordinate by using HCT finite element. The curvature of the surface can then be calculated from this 𝐶1 representation.