對於物體的拓樸結構描述

Abstract

隨著科技的進步，想要得到高解析度及複雜的三維影像資料是不困難的。然而，我們每次都要處理這麼大量的資料其實是滿浪費資源的。因此，拓樸結構在分析物體上是不可或缺的。首先，我們會先介紹Reeb graph的方法，Reeb graph是一種對函數的拓樸結構，如果能找到一個適當的函數來描述物體，那Reeb graph算是對物體的一個拓樸結構的表現。接著我們會介紹skeleton的方法，這是一種我們可以最直觀想像的提取方法。而最後我們會對於數學拓樸上找到一個好的基底來描述一個物體結構。利用基底的剪開來實現Poincaré-Klein-Koebe Uniformization Theorem。

With the recent advances in mesh acquisition device, polygonal mesh with high resolution and complex structure can be easily to get. Topological structure and is crucial for analyzing the shape of complex mesh model. We introduce Reeb graph first. Reeb graph is the topological structure of the function. The Reeb graph can be regarded as a topological structure if there exists a suitable function defined on mesh. Second, we will introduce the skeleton, which is the most intuitively for us to realize the topological structure of a shape. Finally we want to show the basis of the shape in topology, the homotopy basis and the homology basis. In the end, we compare this method and show our experiment.