Smoothing the Silhouettes of Polyhedral Meshes by Boundary Curve Interpolation

This dissertation, "Smoothing the Silhouettes of Polyhedral Meshes by Boundary Curve Interpolation" by Sing-on, Wu, 胡成安, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: 1. Introduction Efficient and accurate rendering of smooth curved surfaces1 is a fundamental problem in computer graphics. It is because much of computer graphics is concerned with modeling the real world, and many real-world objects are inherently smooth on a certain scale [FOLE90]. For interactive computer graphics applications nowadays such as multimedia, computer animation, virtual reality, computer games, and so on, we not only want to render smooth surfaces such that they look like their real counterparts as much as possible, but we also want to do it as fast as possible. To render a smooth surface for the purpose of interactive display, it is now almost a standard practice to first obtain a polyhedral mesh, which is a set of polygons, typically triangles, as an approximate representation of the curved surface, and then pass the vertex data including the vertex coordinates, normal vectors, color values, etc., to the graphics sub-system, which is then responsible for rendering the mesh as a 2D image. The reason for using polyhedral approximations for curved surfaces is that commercial graphics hardware, which is dedicated to rendering smoothly shaded polygons efficiently, is now commonly available. This kind of hardware is virtually a hardware implementation of the standard scan-line algorithm2 integrated with Gouraud shading [GOUR71], together with texture mapping and more recently, even bump mapping. Workstations or personal computers equipped with such graphics hardware can render smoothly shaded polygons at a very high speed. This is arguably the primary reason for the computer graphics community to represent every 3D objects they want to render by polygons, or a polyhedral mesh. Even though people do not have access to graphics hardware, they would still like to use polygons and the scan-line algorithm implemented in software since the algorithm itself is fast and simple. Though techniques for rendering curves surfaces other than using polyhedral approximations and the scan-line algorithm do exist, no one seems to bother to use them for interactive computer graphics applications, probably due to their inherent complexity and inefficiency. It is almost certain that polyhedral meshes will continue to be used for representing smooth curved surfaces in computer graphics. 1 We will use the terms smooth curved surface, smooth surface and curved surface interchangeably.However, using polyhedral meshes to represent curves surfaces for rendering is not without its problems. A polyhedral mesh rendered may not look as faithful to the curved surface it represents as one might expect. One important criterion for the mesh to look 'realistic' is the smoothness of its shading. A curved surface should have a continuous, smooth shading. Simply shading the polygons individually as they are obviously does not work, since it gives a faceted appearance instead of a smooth one. Fortunately, much effort was devoted to smoothly shading a polyhedral mesh to simulate a smooth surface, while still using the fast and simple scan-line algorithm. This is the well-known Gouraud shading [GOUR71] and Phong [BUIT75] shading methods. These methods give a smoothly shaded appearance to a polyhedral mesh so that it looks like a curved surface in terms of shading, though they both suffer more or less from the well-known Mach band effect3, especially Gouraud shading. Phong shading in general is superior in quality to Go