網格參數化在圖學領域中是一項非常重要的核心技術，舉凡高品質顯像、貼圖合成、網格變形...等等，都需要利用到高品質的網格參數化。而在目前圖學研究的領域中，許多研究者已經提出了相當多傑出的研究成果在如何減少參數化的失真度在單一網格上。在本篇論文探討的課題將著重於如何在已知的參數化空間中產生高品質的規律網格，以及如何在多個網格中產生一致性的參數化。
因此我們使用適應性的取樣去得到一個高品質的網格，而且我們一樣希望能夠改善參數化的品質，因此我們提出一個尋找適當參數化空間的演算法。在本論文中，我們提供使用者去描繪其拓撲骨架來建構最好的參數化空間。這樣的演算法，不但幫助我們取得一個最佳的參數化空間，更有效的控制了重建網格上的奇異點，並且可以幫助使用者在短時間內，對多個網格建構出一個一致性的參數化空間。有了一致性的參數化空間，便可使我們原本只是一對一的對應變型動畫變的更多樣化。在論文的最後，我們利用這些重構的規律網格，發展出許多有趣的應用。包括網格錯覺影像，以及可察覺式奇異點線場設計。

Mesh parameterization plays an important role in Computer Graphics, because many applications must make use of mesh parameterization with low distortion, such as realistic rendering, texture synthesis on mesh, and mesh deformation, among other applications.In the current research involving Computer Graphics, many researchers have proposed a lot of outstanding concepts for methods designed to reduce the distortion that is characteristic of single mesh parameterization. For this reason, this thesis focuses on the how to generate high quality mesh in arbitrary non-foldover parameterization, and on the generation of compatible parameterization among multiple meshes.

Therefore, we use adaptive sampling to improve the remesh quality, and we also want to improve the distortion of mesh parameterization. For this reason, we propose an algorithm that demonstrate how to nd a suitable domain for parameterization. In my thesis, we propose an user interface to sketch the skeleton of mesh and construct a suitable parameterization domain via skeleton. This method not only gives us a suitable prameterization domain, but also helps user to construct a consistent domain among multiple meshes and to e ectively control the singularity of the remesh. When
we get a compatible parameterization domain among multiple meshes, we can produce more varied morphing animations. Lastly, we make good use of our remeshing to develop many interesting application for Computer Graphics, such as "Illusory Motions on Surfaces", "Singularity-aware Line Field Design Using Harmonic Functions", among others.

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