# 臺灣博碩士論文加值系統

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 曲面參數化是將曲面映射至一個平面：給定任兩個拓樸等價的曲面，我們可以在這兩個曲面之間，找到一對一且映成的對應關係。如果將曲面以三角網格離散化，則計算此兩曲面間對應關係的問題稱為網格參數化 (mesh parameterization)，曲面所對應的二維平面被稱作參數值域 (parameter domain)。參數化的方法可分為固定邊界 (fixed boundary) 及自由邊界 (free boundary) 兩種，而此兩種方法都可再細分為線性與非線性方法。一般來說，固定邊界源自於物理中的彈簧模型：給定一個二維平面上的區域，將三維曲面的邊界固定在指定區域的邊界上，接著將曲面內部的邊視為彈簧，透過不同彈力係數的選取，可得到不同參數化結果。此篇論文計算了以下六種不同的 weight：Uniform weight, Chord Length weight, Wachspress weight, Harmonic weight, Mean Value weight, Semi-Cotangent weight。而非線性的部分，我們實作了理論上保角的 Riemann Mapping。自由邊界主要採用最小化目標函數 (objective function) 的能量，目前常見的方法有：Least Square Confomral Maps (LSCM) 以及 Linear Angle-Based Flattening (LABF)。在這篇論文中，我們對單一邊界的網格作不同的參數化，並比較各種不同方法的保角性。由於好的參數化是希望最小化角度形變量 (angle distortion)，也就是做到保角的參數化 (conformal parameterization )。在我們的實驗數據中，Riemann Mapping 的保角性及效能是相當好的，對面數在30至90K的模型大都能在1分鐘內作完，除驗證保角性與理論相符，也應證了保角性與網格間的關係。
 Surface parameterization is the process of mapping a surface to a planar region. Given any two surfaces with similar topology, it is possible to compute a one-to-one and onto mapping between them. If one of these surfaces is represented by a triangular mesh, the problem of computing such a mapping is referred to as mesh parameterization. The surface that the mesh is mapped to is typically referred to as the parameter domain. The parameterization methods can be roughly classified into fixed boundary methods and free boundary methods. In general, fixed boundary methods are based on the spring model. According to the different choice of weight, results may vary from method to method. In this paper, we parameterize the single boundary meshes with six linear methods by choosing different weight and the non-linear Riemann mapping method. Then we compare the conformality of different methods. Theoretically, the Riemann mapping is an angle-preserving method. On the other hand, the free boundary methods is always consid- ered to minimize energy of the objective function. In our numerical results, Riemann mapping method reaches high efficiency and good conformality. For larger models with 90,000 number of areas, parameterization can be finished within one minute. When several methods address the same parameterization problem, this paper strives to provide an objective comparison between them based on criteria such as parameterization quality, efficiency and robustness.
 Tables ixFigures xi1 Introduction 11.1 Background 11.2 Overview 32 Fixed Boundary Methods 42.1 The Barycentric Mappings 52.2 Riemann mapping 63 Free Boundary Methods 73.1 Least Square Conformal Maps Method 73.2 Linear Angle-based Flattening Method 94 Measurements of Conformality 114.1 Conformality 124.2 L2 shear 144.3 Squared Sum of Angle Differences 155 Numerical Results 155.1 Mesh Preparation 155.2 Discussion 386 Conclusions 39
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