以切面為基礎的三維形體分析與幾何處理

Abstract

中間階層的物體表面函數可以有效的表現物體部位的形體特徵與結構，然而 要定義與導出中間階層的表面函數是非常困難的。在本論文中，我們提出了一 個新的中間階層物體表面函數，稱為最小切面周長函數(MSP)，來表示物體的區 域體積資訊並探討其在幾何處理上的應用。對於一個物體表面上的點，其最小 切面周長為是定義在通過該點的切面上，目標是用來表示該點周圍的區域體 積。相對於其他中間階層函數如形體對角函數(SDF)[73]，最小切面周長函數可 以更正確的反應出物體的區域體積資訊。我們也將此一以切面為基礎的函數應 用於網格切割與骨架產生上。所提出之網格切割方法完整的利用到物體區域體 積資訊並能夠產生階層式的部位切割將重要部位首先切出並確保每一階層中所 切出的部位具有類似的重要度。而所提出之網格骨架產生方式是將三維表面透 過一連串的邊交換(edge-swap)操作直接擷取出曲線骨架。所產生之曲線骨架即 使在物體的核心區域或骨架接合處都可以保有高密度的節點，此一特性可以幫 助我們建立網格表面與骨架之間的對應。此外，所提出之方法只仰賴一個重要 度參數來操作，透過調整重要度參數可以得到不同精細度的骨架。最後，我們 也探討目前模型簡化機制只參考幾何資訊而未考量觀察者主觀定義所導致主觀 認定之重要區域會被過度簡化的問題。為了克服此問題，我們提出了一套使用 者可控制之網格簡化機制來讓使用者對網格上的主觀認定重要區域給定權重， 並確保該區域在簡化後的模型上具有與給定權重值相似的解析度提升效果。

Intermediate-level surface functions of 3D objects are useful for representing the object’s part-level shape information and structure. In this thesis, we propose an intermediatelevel surface function and explore its applications to geometry processing. The proposed surface function, called minimum slice perimeter function (MSP), is defined in terms of the slices that pass through the surface point and aims to represent the local volume around the surface point. This slice-based MSP represents more accurate local volume information than previous intermediate-level surface functions, such as Shape Diameter Function (SDF) [73] and ia immediately beneficial to applications such as mesh segmentation and skeletonization. Our proposed mesh segmentation algorithm, which takes advantage of local volume information around the surface point, is able to generate hierarchical segmentation where parts on the same level of the hierarchy share similar salience significance, while parts on a level are less significant than parts on their parental level. The proposed mesh skeletonization scheme employs a greedy edge-swap process that extracts the curve skeleton directly from the 3D surface. The resulting skeleton inherently possesses a dense node distribution at the core part and around the junctions which helps to derive a dense skeleton-surface mapping. Moreover, the single salience parameter for branch removal works well and provides a flexible control for deriving skeleton of varying detail. Finally,existing level-of-detail modeling techniques consider only geometric other than semantic information, and hence areas of semantic importance are often oversimplified. To ameliorate the problem, we propose a user-controllable mesh simplification framework that allows users to assign weights on selected regions and obtain a predictable improvement of the resolution over the regions. ii