在現今科學的急速發展下，三維重建能用在許多領域。一般而言，三維影像所儲存成的電子資訊多為點雲資訊，意即只有掃描到的點座標，因此三維重建旨在處理這些資訊。本篇論文主要以Kazhdan [1] 提出的架構去延伸，從初始的點雲建構出法向量，進一步由此法向量透過解一個帕松方程式求得此點雲所代表的指示函數，最後將此函數透過移動立方體法或是移動四面體法求得此物品的三角網格，然而網格通常的品質(quality) 並不好，透過對此三角網格的後處理點融合(point merge)、邊交換(edge swap) 與球面上的共形映射(spherical conformal map)，進而增進此三角網格的品質。

With the rapid development of science today, 3D reconstruction
can be used in many fields. In general, the
electronic information stored in 3D images is mostly point
cloud information, it means only the scanned point cloud.
This thesis is mainly extended by the architecture proposed by Kazhdan [1]. We construct normal vectors from the initial point cloud. Further we use these normal vectors to solve a poisson equation then we obtain the indicator function representing this point cloud. Finally, the indicator function is used to obtain the triangle mesh of the object by the marching cube method or the marching tetrahedra method. However, the quality of the mesh is generally poor, we improve the quality of the mesh by post-processing the triangular mesh, such as point merge, edge swap, and spherical conformal map.

誌謝
摘要i
Abstract i
List of parameters iii
1 Introduction 1
1.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Our thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Prior knowledge 7
2.0.1 Unit vector . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.0.2 Normal vector . . . . . . . . . . . . . . . . . . . . . . . . 7
2.0.3 Singular value decomposition . . . . . . . . . . . . . . . . 8
2.0.4 Wellposedness
of Poisson equation . . . . . . . . . . . . . 8
2.0.5 Delaunay triangulation . . . . . . . . . . . . . . . . . . . . 9
2.0.6 Voronoi diagram . . . . . . . . . . . . . . . . . . . . . . . 11
2.0.7 Diffuse element method . . . . . . . . . . . . . . . . . . . 12
2.0.8 Moving least squares method . . . . . . . . . . . . . . . . . 12
3 Poisson Surface Reconstruction 15
3.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Normal vector and orientation consistency . . . . . . . . . . . . . . 17
3.2.1 Principal component analysis . . . . . . . . . . . . . . . . 18
3.2.2 Consistent normal vector orientation . . . . . . . . . . . . . 19
3.3 Vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.1 Octree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Smoothing filter Box
filter . . . . . . . . . . . . . . . . . 25
3.3.3 Create vector field . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Methods for solving Poisson equation . . . . . . . . . . . . . . . . 28
3.4.1 Fast Fourier Transform (FFT) . . . . . . . . . . . . . . . . 28
3.4.2 Finite element method (FEM) . . . . . . . . . . . . . . . . 29
3.5 Isosurface extraction . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.1 Marching cubes . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5.2 Marching tetrahedra . . . . . . . . . . . . . . . . . . . . . 41
4 Triangular mesh postprocessing
43
4.1 Point merge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Edge swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Conformal map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Experimental Results and Discussions 51
5.1 Experimental results of surface reconstruction . . . . . . . . . . . . 51
5.1.1 Normal vector orientation . . . . . . . . . . . . . . . . . . 51
5.1.2 Boundary condition . . . . . . . . . . . . . . . . . . . . . . 52
5.1.3 Surface reconstruction with several methods . . . . . . . . 54
5.2 Experimental results of triangular mesh postprocessing
. . . . . . . 57
5.2.1 Point merge and edge swap . . . . . . . . . . . . . . . . . . 57
5.2.2 Conformal map . . . . . . . . . . . . . . . . . . . . . . . . 60
6 Conclusions and future work 65
References 67

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