Geometry prediction for high degree polygons

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.

[1]  M. Isenburg.  Compressing Polygon Mesh Connectivity with Degree Duality Prediction , 2002, Graphics Interface.

[2]  Ian H. Witten,et al.  Arithmetic coding for data compression , 1987, CACM.

[3]  Martin Isenburg,et al.  Out-of-core compression for gigantic polygon meshes , 2003, ACM Trans. Graph..

[4]  Martin Isenburg,et al.  Compressing polygon mesh geometry with parallelogram prediction , 2002, IEEE Visualization, 2002. VIS 2002..

[5]  Bernard Chazelle,et al.  The discrepancy method - randomness and complexity , 2000 .

[6]  Olivier Devillers,et al.  Progressive lossless compression of arbitrary simplicial complexes , 2002, SIGGRAPH.

[7]  Andrei Khodakovsky,et al.  Progressive geometry compression , 2000, SIGGRAPH.

[8]  Martin Isenburg,et al.  Early-split coding of triangle mesh connectivity , 2006, Graphics Interface.

[9]  D. Cohen-Or,et al.  Multi-way Geometry Encoding , 2002 .

[10]  Craig Gotsman,et al.  Optimized compression of triangle mesh geometry using prediction trees , 2002, Proceedings. First International Symposium on 3D Data Processing Visualization and Transmission.

[11]  Elwyn R. Berlekamp,et al.  A Polygon Problem , 1965 .

[12]  Volker Coors,et al.  Delphi: geometry-based connectivity prediction in triangle mesh compression , 2004, The Visual Computer.

[13]  Craig Gotsman,et al.  Triangle Mesh Compression , 1998, Graphics Interface.

[14]  Jarek Rossignac,et al.  Out‐of‐core compression and decompression of large n‐dimensional scalar fields , 2003, Comput. Graph. Forum.

[15]  Pierre Alliez,et al.  Near-Optimal Connectivity Encoding of 2-Manifold Polygon Meshes , 2002, Graph. Model..

[16]  Pierre Alliez,et al.  Isotropic Remeshing of Surfaces: A Local Parameterization Approach , 2003, IMR.

[17]  Konrad Polthier,et al.  FreeLence : Compressing Triangle Meshes using Geometric Information , 2022 .

[18]  Pierre Alliez,et al.  Angle‐Analyzer: A Triangle‐Quad Mesh Codec , 2002, Comput. Graph. Forum.

[19]  Sudhir P. Mudur,et al.  Compression of Large 3D Engineering Models using Automatic Discovery of Repeating Geometric Features , 2001, VMV.

[20]  Martin Isenburg,et al.  Face fixer: compressing polygon meshes with properties , 2000, SIGGRAPH.

[21]  Sivan Toledo,et al.  High-Pass Quantization for Mesh Encoding , 2003, Symposium on Geometry Processing.

[22]  Craig Gotsman,et al.  Spectral compression of mesh geometry , 2000, EuroCG.

[23]  Geoffrey C. Shephard,et al.  A Polygon Problem , 1996 .