自由邊界的保角參數化在Matlab上實現

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姓名

黃淑如(Shu-ju Huang) 查詢紙本館藏

畢業系所

數學系

論文名稱

自由邊界的保角參數化在Matlab上實現
(Matlab Implementation for Conformal Parameterizations with Free Boundary)

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摘要(中)

三角網格在幾何學中是經典的離散化表示方式。近年來，網格參數化研究已經有許多研究成果被提出。它廣泛的被應用在人臉貼圖、工程及醫學上，不同的應用方法也需要不同的參數化方式，而目的都是需要一個容易交互處理的參數域。在本篇論文我們回顧四個自由邊界參數化方法，包含最小平方共形映射、基於角度攤平法、線性基於角度攤平法以及保相似參數化方法，並透過Matlab的實現進行比較。而為了確認參數化方法的優劣，我們利用三個已知的度量方式進行比較：角度失真、L_2 stretch、L_2 shear。
在實驗結果我們得到ABF++所得到的參數化結果最佳，然而所花費的時間遠大於其他的方法。

摘要(英)

Triangular mesh is the classical representation in geometry. Mesh parameterization have been proposed for many of studies in recent years. It is used in a wide range of texture mapping , engineering and medicine. Different applications require different methods of parameterization methods. The goal need an easy interactive processing of the parameter domain. In this thesis, we review four free boundary parameterization methods, (1) least squares conformal maps (LSCM), (2) angle based flattening (ABF++), (3) linear angle based flattening (LABF), (4) As-Similar-As-Possible Planar Parameterization (ASAP).
Through implementation of Matlab comparison parameterization. We use three metrics to measure the parametric distortion: angle distortion, L_2 stretch, and L_2 shear.
In the experimental results we have obtained ABF + + parameterization the best results, but the time it takes is much larger than other method.

關鍵字(中)

★ 自由邊界
★ 保角參數化
★ 參數域
★ 最小失真度

關鍵字(英)

★ free boundary
★ Conformal Parameterizations
★ parameter domain
★ minimization distortion

論文目次

中文摘要....................................i
英文摘要....................................ii
致謝........................................iii
圖目錄......................................iv
表目錄......................................v
1. 緒論.......................................... 1
1.1 前言...................................... 1
1.2 論文結構...............................1
1.3 符號表..................................... 2
2. 網格參數化.................................... 3
2.1 前言....................................... 3
2.2 最小平方共形映射........................... 3
(Least Squares Conformal Maps, LSCM)
2.3 基於角度攤平............................... 6
(Fast and Robust Angle Based Flattening, ABF++)
2.4 線性基於角度攤平.......................... 11
(Linear Angle Based Parameterization, LABF)
2.5 保相似的參數化............................ 13
(As-Similar-As-Possible Planar Parameterization, ASAP)
3. 參數化測量....................................15
3.1 前言...................................... 15
3.2 angle distortion..............................15
3.3 L2 stretch.................................. 15
3.4 L2 shear................................... 16
4. 網格資料實例應用.............................17
4.1 前言.................................. 17
4.2 實例測試.................................. 18
4.3 實驗分析及結論............................ 27
5. 參考文獻......................................28

參考文獻

[1] A. Bu, Conformality of Planar Parameterization for Single Boundary Triangulated Surface Meshes. 國立中央大學數學所碩士論文,2012.
[2] X. David and S.T. Yau, Computational Conformal Geometry. International Press of Boston, 2008.
[3] K. Hormann, B. Levy, and A. Sheffer, Mesh Parameterization: Theory and Practice. August. HAL - CCSD URL: {http://hal.inria.fr/inria-00186795/en/}, 2007.
[4] L. Kharevych, B. Springborn, and P. Schröder. Discrete conformal mappings via circle patterns. ACM Transactions on Graphics, 25(2):412–438, 2006.
[5] B. Lévy, S. Petitjean, N. Ray, and J. Maillot. Least squares conformal maps for automatic texture atlas generation. ACM Transactions on Graphics, 21(3):362–371, 2002.
[6] L. Liu, L. Zhang, Y. Xu, C. Gostsman, and S.J. Gortler. A local/global approach to mesh parameterization. Eurographics Symposium on Geometry Processing(SGP 2008) Volume 27, Number 5.
[7] P. V. Sander, J. Snyder, S. J. Gortler, and H. Hoppe. Texture mapping progressive meshes. In Proceedings of SIGGRAPH 2001, pages 409–416. ACM Press, 2001.
[8] A. Sheffer and E. de Sturler. Parameterization of faceted surfaces for meshing using angle-based flattening. Engineering with Computers, 17(3):326–337, 2001
[9] A. Sheffer, B. Lévy, M. Mogilnitsky, and A. Bogomyakov. ABF++: fast and robust angle based flattening. ACM Transactions on Graphics, 24(2):311–330, 2005.
[10] R. Zayer, B. Levy, and H.P. Seidel. Linear angle based parameterization. Geometry Processing, (SGP’07):349-360, 2007.
[11] AIM@SHAPE Shape Repository. http://shapes.aimatshape.net/

指導教授

單維彰(Wei-chang Shann)

審核日期

2013-7-30

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