以有號距離場實現對三角形網格模型挖空之可靠方法

Abstract

三角形網格模型(Triangle Mesh)是現今普遍使用於表現三維模型的格式，藉由拼接一定數量的三角形，可以近似物體的表面。理論上三角形總數愈多即可產生愈精細的模型。近年流行的3D列印，預設輸入格式也是使用此格式。 Triangle Mesh只會描述物體的表面，在3D列印時，會將其視為實心的物體。但在許多情況下，如製作燈飾、樂器或本文討論的撲滿時，我們會需要一個中空的模型，為此我們必須編輯原本的Triangle Mesh。 在模型較單純時，可以使用CAD軟體手動將內部挖空，但對較複雜的模型進行這樣的處理費時又費力，於是我們設計一個系統，可以自動且可靠地將Triangle Mesh的內部挖空。 我們的系統會先對輸入之Triangle Mesh計算有號距離場(Signed Distance Field，以下簡稱為SDF)，可以記錄空間中的點到模型表面的距離，距離的正或負分別代表該點位於此模型的外部或內部。利用SDF與著名的Marching Cubes演算法可以產生一個內部的模型來表示內部空洞的表面，最後將此內部模型與輸入模型結合後輸出，即可產生內部挖空的模型。 經過實驗，只要輸入模型符合SDF的基本假設，我們的系統就可以正確產生目標的挖空模型，外殼的厚度均勻，且可以參數控制此厚度。

Triangle mesh is widely used for representing 3D models. By stitching an amount of triangles, it can approximate the surface of an object. Theoretically, the more the triangles, the more accurate it would be. It is also the default input format for 3D printing. Triangle mesh only describes the surface of an object. For 3D printing, it is viewed as a solid object. However, in many circumstance, e.g. making lanterns, instruments or, as we would discuss later, piggy banks, an empty model is required. So we have to edit the original mesh. When the mesh is simple, one could manually hollow it out with CAD software. But for those more complicated models, it could be quite challenging and time-consuming. As a result, we developed a system that can automatically and robustly hollow out triangle meshes. Our program will first calculate Signed Distance Field (abbreviated as SDF) for the triangle mesh. SDF records the distance from points in 3D space to the surface of the model. The sign of the distance being positive or negative indicates the point is outside or inside the model, respectively. Then, we will apply a well-known algorithm, Marching Cubes, to regenerate the surface of the inner hole. And finally combine this inner model with the input mesh to get an empty model. In our experiments, as long as the input model follows the basic assumptions of SDF, our system can robustly generate corresponding hollow model.