摘要
三維圖學的發展由來已久，而目前的物件資料大多是由掃描系統掃描所得，因此這類的原始資料不但十分龐大，亦常有許多冗餘的部份，致使繪圖時耗費大量時間及資源。基於此，如何將物件以更有效實際的資料作表示一直是一個重要的議題。
本篇論文以隱函數作為主要的物件描述方式。相較於多邊形資料表示方式，以隱函數表示物件具有很大的彈性，因為以數學形式表示物件的方式可以讓使用者用不同的取樣資料來描述這個物件。
本篇論文是以放射基礎函數來求取隱函數，並採用二元搜尋樹分割物件以求取切割部份的隱函數的方式，減少其運算量。同時以漸進式物件壓縮的方式同時減少運算量及儲存空間，最後在個別的隱函數上取樣畫出整個物件。

Abstract
The representation of 3D Computer Graphics has been studied for a long time. Most 3D object models are obtained by 3D scan systems. These kinds of data are not only very huge, but also have a lot of redundancy. It consumes a large mount of time and resources. For this reason, how to represent the object efficiently is always an important issue. The purpose of this study is to present the objects by implicit functions. Different with the presentation of polygon mesh, implicit function is very compact to present objects because that the mathematical form of object can be obtained in different data forms. The implicit function used is Radial Basis Function, and then the BSP tree is used to partition the object to reduce the amount of computing. We also use the compression of progressive mesh to decrease the storage and the computing time. In addition, the object can be rendered according to the sampling points on each implicit surface.

目 錄
1. 簡介 1
1.1 背景 3
1.2 問題 5
1.3 目的 6
1.4 貢獻 7
2. 相關研究及背景理論 9
2.1 相關研究 9
2.1.1 隱函數及放射基礎函數 9
2.1.2 曲面方程式取樣 13
2.1.3 壓縮 14
2.2 背景理論 15
2.2.1 隱含曲面 15
2.2.2 放射基礎函數(Radial Basic Function RBF) 16
3. 演算法 19
3.1 問題陳述 19
3.2 演算法陳述 20
3.2.1 漸進式物件壓縮 20
3.2.2 物件空間分割及階層樹的建立 24
3.2.3 隱含曲面的建立 25
3.2.4 取樣並畫出物件 28
4. 實作結果 28
5. 結論 34
參考文獻 37


圖 目 錄
圖 1 演算法流程圖 21
圖 2 輸入物件 30
圖 3 beethoven運算後的取樣物件 30
圖 4 經漸近式壓縮後再運算的取樣物件 31
圖 5 venus運算後的取樣物件 31
圖 6 經漸近式壓縮後再運算的取樣物件 32


表 目 錄
表 1 物件特性表 29
表 2 前處理時間及節點數比較表 32

參考文獻
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