Meshes: Catmull-Clark Subdivision and Simplification

Transcription

1 Meshes: Catmull-Clark Subdivision and Simplification Part 1: What I did CS 838, Project 1 Eric Aderhold My main goal with this project was to learn about and better understand three-dimensional mesh surfaces. To that end, I implemented Catmull-Clark subdivision and a rudimentary mesh simplification method using the OpenMesh C++ halfedge mesh data structure library. Catmull-Clark Subdivision My Catmull-Clark program subdivides a mesh surface using the Catmull-Clark algorithm. It runs on the command line, and requires three arguments: the number of subdivision iterations to perform, the input mesh file, and the output mesh file. The input and output files can be in any of the file types supported by OpenMesh's import functions: OBJ, OFF, STL, and OM formats. I made two simple meshes to demonstrate the algorithm: a cube and a "star." Pictures of both of these after various stages of subdivision appear below. Cube, subdivided 0, 1, 2, 3, and 5 times (respectively)

2 Star, 0 iterations Star, 1 iteration Star, 2 iterations Star, 3 iterations Star, 4 iterations Star, 5 iterations

3 Mesh Simplification I read a number of papers about mesh simplification techniques, particularly Michael Garland's papers about using quadric error matrices to find a good approximation of a detailed mesh using fewer polygons, and some of the earlier work that he referenced. Unfortunately, I was not able to get to this phase of the project until less than a week was remaining before the due date, so there was not time to implement any of these more complicated simplification algorithms. I instead implemented a greedy algorithm that progressively collapses edges in the mesh. My algorithm uses a simple heuristic to estimate the relative local curvature in the area of each edge. For each vertex, my code sums the measure of the angles around that vertex, since a flat surface (with no curvature) will have angles that sum to exactly 360º. A highly curved surface, on the other hand, will often have angles around a point that sum to less than 360º, getting very small indeed in the case of a sharp point. For each vertex, the difference between the angle sum and 360º is squared. Then for each edge, this result is averaged across the two vertices, with the smallest sum being likely to be among the flattest regions on the surface. This edge is then collapsed, removing two triangles from the overall mesh. This process is repeated until the desired number of polygons is reached. I tested my simplification algorithm on a model of Darth Vader s helmet that contains over 60,000 polygons. Results of my simplification algorithm at varying numbers of result polygons are shown below: Darth Vader s helmet (original, 50k polygons, 30k polygons, 10k polygons, and 5k polygons)

4 Part 2: Evaluation Catmull-Clark Subdivision I feel that my implementation of Catmull-Clark subdivision works quite well. I was originally concerned that I had implemented the algorithm incorrectly because my subdivided cube did not match images I had seen on Wikipedia and other places of what a Catmull-Clark subdivided cube should look like after a given number of iteration. However, the problem turned out to be in the OpenMesh provided mesh viewer, and how it renders non-flat quadrilaterals (i.e. quadrilaterals whose vertices are not all on the same plane). The example renderings I had seen used some method to ensure that the displayed faces were all flat, while the OpenMesh viewer simply decomposes non-flat quadrilaterals into two triangles. I confirmed this by testing my output against a known implementation. I was able to find another implementation of Catmull-Clark subdivision (by Hennig Biermann and Denis Zorin at NYU, available at The points calculated by my implementation matched theirs, and so I concluded that my implementation was likely correct. In addition to correctness, I believe my implementation performs relatively well. It is able to subdivide a cube through five iterations (from six faces to 12,288) in 0.2 seconds on a two-year-old Apple MacBook Pro. A more complete version of this algorithm would use exact evaluation techniques to compute the location on a limit surface for each point in the subdivided mesh. I did not have time to complete this step yet, but I do plan to do so over the upcoming break. Mesh Simplification I am less pleased with my mesh simplification algorithm. The algorithm successfully keeps the general shape of the original figure, but the output starts to look quite bad at a higher polygon count than I would like. Also, there is an issue where certain areas appear as holes in the simplified figure. I originally thought that my algorithm might not be processing normal vectors properly, but I dismissed this possibility because all that is stored in the mesh file is a set of vertices and faces; the viewer computes all normals from there. My next best guess is that the models I used contained certain discontinuities that are not immediately apparent by looking at the original version, and this caused holes to form as edges collapsed and spread the discontinuous pieces apart. I have unfortunately not had time to fully test this theory. In terms of performance, my algorithm is quite poor. Its process of recalculating every edge s relative curvature heuristic for every edge removed is incredibly inefficient, and could be improved significantly with a bit more work. The simplifications of Darth Vader s helmet as shown on the previous page took a couple of hours to complete. Part 3: Description of Process The main stumbling block I encountered during this project is that I severely underestimated the amount of time required to get up and running with OpenMesh. This is true both in terms of the amount of time needed to simply compile the library and associated components, and also in terms of the time required to fully understand how to make use of OpenMesh s data structure and functionality in my own programs. I did not realize during the planning stages how arduous the installation process would be, or how poorly the documentation explains certain key concepts, and this held my progress back significantly. There are a couple of major extra things I would have tried to accomplish if this were a longer project. First and foremost, I would have implemented a much better mesh simplification algorithm. Due to the long time I needed to get Catmull-Clark working properly, I did not have time to attempt a more sophisticated simplification algorithm such as those demonstrated by Michael Garland in his several

5 papers on the subject. If I had an extra two weeks to work with from the beginning, I would have attempted to do that instead of the simpler algorithm I actually implemented. Also, as already stated, I would have added some exact evaluation features to the Catmull-Clark algorithm. Reading List Sharp, Brian. Subdivision Surface Theory. Gamasutra magazine, April 10, Sharp, Brian. Implementing Subdivision Surface Theory. Gamasutra magazine, April 25, Akenine-Moller, T. et al. Curves and Curved Surfaces. Real-Time Rendering, Chapter 13. Catmull Clark subdivision surface. Wikipedia. OpenMesh documentation, Garland, M. and Shaffer, E. A Multiphase Approach to Efficient Surface Simplification. IEEE Visualization Garland, M. and Heckbert, P. Surface Simplification Using Quadric Error Metrics. SIGGRAPH Roufard, R. and Rossignac, J. Full-range approximation of triangulated polyhedra. EUROGRAPHICS Hoppe, Hugues. Progressive Meshes. SIGGRAPH Hoppe, Hugues, et al. Mesh optimization. SIGGRAPH For learning about surfaces in general and subdivision algorithms, I found the chapter of the Real-Time Rendering book to be useful. When it came time to actually implement the Catmull-Clark algorithm, I found the Wikipedia article to be the most concise, easy-to-follow source. I found Michael Garland s 1996 paper to be a useful explanation of his general methods, and his later paper was an interesting look at some of his follow-up work. Part 4: Self-Evaluation Overall, I give myself a medium-level rating on this project. I put a good deal of time and effort into the project, but I wish I had accomplished a bit more. I think my overall topic choice was appropriate, and I do not regret deciding to explore this topic. However, the selection of OpenMesh as a vehicle for exploring meshes, subdivision, and simplification was a mistake. Too much time was spent on simply getting the framework to function and then after that determining how to best use it, and this left too little time for the meat of the project. I am unsure whether there exists a better data structure library to use for a project such as this, but OpenMesh is much less than ideal. In terms of graphics topics, I learned quite a bit about surfaces and meshes above and beyond what I took away from CS 559. That was my general goal coming into the project, and I feel that I succeeded at this goal despite the subpar deliverables. I also gained experience with coming up to speed on the use of poorly documented library code, a skill that is sadly all-too-necessary in the software engineering field these days. I also learned more about using Apple s Xcode development environment, both for graphics use and in general. If I had to do the project again, I think I would have chosen the same topic. In the planning stages, I would do a more thorough job looking for suitable library code, and I would have allotted more time to actually learning how to use this code. It seems like the coding went fairly quickly when I actually understood what I was doing with OpenMesh, and so with a better library I believe I could have produced much better results.

6 One thing I really do wish I had been told earlier in the project was how it s pretty much impossible to get Qt built and running correctly on Windows. Trying to do this and other things on Windows to get OpenMesh running cost me a couple of full days of work and convinced me to try it on the Mac, where it got working in a fraction of the time.