曲面的保角與保面積參數化及其應用

Abstract

曲面的參數化已被廣泛應用於三維影像的處理。在本論文中，作者分別提出了單連通的開曲面之保角與保面積參數化的高效能演算法。更進一步，作者推廣了所提出的保角參數化算法至零虧格的多重連接曲面的保角參數化之計算。此外，作者基於曲面的參數化，提出了兩曲面之間平滑配準函數的快速演算法。最後，作者展示了幾個曲面配準的應用，包含：曲面形變、三維影片壓縮、三維動畫系統，以及材質與幾何的重定向。拜所提出的參數化方法所賜，兩曲面之間平滑配準函數的計算是快速而穩健的。

Surface parameterizations have been widely applied to digital geometry processing. In this thesis, the author proposes efficient numerical algorithms for the computation of conformal and equiareal parameterizations of simply-connected open surfaces with very small distortions and highly improved computational efficiencies. In addition, the author generalized the proposed conformal parameterization algorithm to computing conformal parameterizations of genus-zero multiply-connected surfaces. Furthermore, the author proposed an efficient algorithm for the construction of smooth registration mappings between surfaces based on the proposed parameterization algorithms. Applications of the surface registration on the surface morphing, 3D video compression, 3D animation system, texture retargeting, and geometry retargeting are demonstrated thereafter. Thanks to the proposed algorithms, the computation for the registration mappings between surfaces can be performed efficiently and robustly.