ABSTRACT
The parallelogram rule is a simple, yet effective scheme to predict the position of a vertex from a neighboring triangle. It was introduced by Touma and Gotsman [1998] to compress the vertex positions of triangular meshes. Later, Isenburg and Alliez [2002] showed that this rule is especially efficient for quad-dominant polygon meshes when applied "within" rather than across polygons. However, for hexagon-dominant meshes the parallelogram rule systematically performs miss-predictions.In this paper we present a generalization of the parallelogram rule to higher degree polygons. We compute a Fourier decomposition for polygons of different degrees and assume the highest frequencies to be zero for predicting missing points around the polygon. In retrospect, this theory also validates the parallelogram rule for quadrilateral surface mesh elements, as well as the Lorenzo predictor [Ibarria et al. 2003] for hexahedral volume mesh elements.
- Berlekamp, E., Gilbert, E., and Sinden, F. 1965. A polygon problem. Amer. Math. Monthly 72, 233--241.]]Google ScholarCross Ref
- Chazelle, B. 2000. The Discrepancy Method --- Randomness and Complexity. Cambridge University Press.]] Google ScholarDigital Library
- Cohen-Or, D., Cohen, R., and Irony, R. 2002. Multi-way geometry encoding. Tech. rep., Computer Science, Tel Aviv University.]]Google Scholar
- Coors, V., and Rossignac, J. 2004. Delphi: Geometry-based connectivity prediction in triangle mesh compression. The Visual Computer 20, 8-9, 507--520.]] Google ScholarDigital Library
- Davis, P. 1979. Circulant matrices. Wiley-Interscience.]]Google Scholar
- Devillers, O., and Gandoin, P.-M. 2002. Progressive and lossless compression of arbitrary simplicial complexes. In SIGGRAPH'02 Conference Proceedings, 372--379.]] Google ScholarDigital Library
- Ibarria, L., Lindstrom, P., Rossignac, J., and Szymczak, A. 2003. Out-of-core compression and decompression of large n-dimensional scalar fields. In Eurographics'03 Proceedings, 343--348.]]Google Scholar
- Isenburg, M., and Alliez, P. 2002. Compressing polygon mesh geometry with parallelogram prediction. In Visualization'02 Conference Proceedings, 141--146.]] Google ScholarDigital Library
- Isenburg, M., and Gumhold, S. 2003. Out-of-core compression for gigantic polygon meshes. In SIGGRAPH'03 Proceedings, 935--942.]] Google ScholarDigital Library
- Isenburg, M., and Snoeyink, J. 2000. Face Fixer: Compressing polygon meshes with properties. In SIGGRAPH'00 Proceedings, 263--270.]] Google ScholarDigital Library
- Isenburg, M., and Snoeyink, J. 2005. Early-split coding of triangle mesh connectivity. manuscript.]]Google Scholar
- Isenburg, M. 2002. Compressing polygon mesh connectivity with degree duality prediction. In Graphics Interface'02 Proceedings, 161--170.]]Google Scholar
- Kälberer, F., Polthier, K., Reitebuch, U., and Wardetzky, M. 2005. Freelence: Compressing triangle meshes using geometric information. to appear in Eurographics'05 Proceedings, September 2005.]]Google Scholar
- Karni, Z., and Gotsman, C. 2000. Spectral compression of mesh geometry. In SIGGRAPH'00 Conference Proceedings, 279--286.]] Google ScholarDigital Library
- Khodakovsky, A., Schroeder, P., and Sweldens, W. 2000. Progressive geometry compression. In SIGGRAPH'00 Proceedings, 271--278.]] Google ScholarDigital Library
- Khodakovsky, A., Alliez, P., Desbrun, M., and Schroeder, P. 2002. Near-optimal connectivity encoding of 2-manifold polygon meshes. Graphical Models 64, 3-4, 147--168.]] Google ScholarDigital Library
- Kronrod, B., and Gotsman, C. 2002. Optimized compression of triangle mesh geometry using prediction trees. In International Symposium on 3D Data Processing Visualization and Transmission, 602--608.]]Google Scholar
- Lee, H., Alliez, P., and Desbrun, M. 2002. Angle-analyzer: A triangle-quad mesh codec. In Eurographics'02 Proceedings, 198--205.]]Google Scholar
- Shikhare, D., Bhakar, S., and Mudur, S. 2001. Compression of 3D engineering models using automatic discovery of repeating geometric features. In Proc. of Vision Modeling and Visualization'01, 233--240.]] Google ScholarDigital Library
- Sorkine, O., Cohen-Or, D., and Toledo, S. 2003. High-pass quantization for mesh encoding. In Proceedings of Symposium on Geometry Processing'03, 42--51.]] Google ScholarDigital Library
- Surazhsky, V., Alliez, P., and Gotsman, C. 2003. Isotropic remeshing of surfaces: a local parameterization approach. In Proceedings of 12th International Meshing Roundtable, 215--224.]]Google Scholar
- Touma, C., and Gotsman, C. 1998. Triangle mesh compression. In Graphics Interface'98 Conference Proceedings, 26--34.]]Google Scholar
- Witten, I. H., Neal, R. M., and Cleary, J. G. 1987. Arithmetic coding for data compression. Communications of the ACM 30, 6, 520--540.]] Google ScholarDigital Library
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