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Geometry Prediction for High Degree Polygons

Martin Isenburg Ioannis Ivrissimtzis Stefan Gumhold Hans-Peter Seidel

__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 to compress
the vertex positions of triangular meshes. Later,
we 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 for hexahedral volume mesh elements.

__main contributions:__

extending the parallelogram rule to higher degree polygons
using polygonal Fourier decomposition for designing the predictor
giving existing prediction rules a retroactive theoretic blessing

__publications:__

[iigs-gphdp-05.pdf slides] Martin Isenburg, Ioannis Ivrissimtzis, Stefan Gumhold, Hans-Peter Seidel, *Geometry Prediction for High Degree Polygons*, Proceedings of SCCG'05, pages 147-152, May 2005.
__related publications:__

[ia-cpmgpp-02.pdf slides] Martin Isenburg, Pierre Alliez, *Compressing Polygon Mesh Geometry with Parallelogram Prediction*, in Proceedings of Visualization 2002, pages 141-146, October 2002.
[ils-lcpfpg-05.pdf slides] Martin Isenburg, Peter Lindstrom, Jack Snoeyink, *Lossless Compression of Predicted Floating-Point Geometry*, in Computer-Aided Design, Volume 37, Issue 8, pages 869-877, July 2005.

__download:__

slides: gphdp.ppt (6 MB)
__funding:__

Max Planck Center for Visual Computing and Communication, Saarbruecken, Germany.