Max-Plank Institut für Informatik systematic error parallelogram rule polygonal rules exact prediction Geometry Prediction for High Degree Polygons Martin Isenburg Stefan Gumhold Ioannis Ivrissimtzis Hans-Peter Seidel INFORMATIK

Compression real stuff – sleeping bags – compressed air polygon meshes – faster downloads / less storage – collaborative CAD – distribution of simulation results – archival of spare parts / history

Movies “Rustboy” animated short by Brian Taylor

Engineering “Audi A8” created by Roland Wolf

Architectural Visualization “Atrium” created by Karol Myszkowski and Frederic Drago

Product Catalogues “Bedroom set-model Assisi” created by Stolid

Historical Study scanning of “Michelangelo’s David” courtesy of Marc Levoy

Computer Games screen shot of “The village of Gnisis”, The Elder Scrolls III

– Efficient Rendering – Progressive Transmission – Maximum Compression Connectivity Geometry Properties Mesh Compression Geometry Compression [ Deering, 95 ] storage / network main memory Maximum Compression

Most Popular Coder Triangle Mesh Compression [ Touma & Gotsman, 98 ] M 54 S 66 6 connectivity with vertex degrees ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) geometry with corrective vectors

Coding Connectivity

Predicting Geometry

Not Triangles … Polygons! Face Fixer [ Isenburg & Snoeyink, 00 ]

Coding Polygon Connectivity Compressing Polygon Mesh Connectivity with Degree Duality … [ Isenburg, 02 ]  same compression in primal and dual !!

Predicting Polygon Geometry Compressing Polygon Mesh Geometry with Parallelogram … [ Isenburg & Alliez, 02 ]  but … does not work well in the dual !!

High Degree Polygons v2v2 v1v1 v0v0 v4v4 v2v2 v1v1 v0v0 v3v3 c 0 = c 1 = c 2 = c 3 = c 0 = 0.8 c 1 =-0.6 c 2 =-0.4 c 3 =1.2 v3v3 v4v4 c 0 = c 1 = c 2 = c 3 = c 4 = c 0 = v2v2 v1v1 v0v0 v3v3 v4v4 v5v5 v6v6 v2v2 v1v1 v3v3 v0v0 c 0 = 1.0 c 1 =-1.0 c 2 =1.0 polygonal rules: v p = c 0 v 0 + c 1 v 1 + … + c p-1 v p-1 v2v2 v1v1 v2v2 v1v1 v0v0 v3v3 v3v3 v0v0 v2v2 v1v1 v0v0 v3v3 v2v2 v1v1 v0v0 v3v3 parallelogram rule: v 3 = v 0 – v 1 + v 2

Overview Motivation Geometry Compression Frequency of Polygons Polygonal Prediction Rules Results Discussion INFORMATIK

Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Geometry Compression [ Deering, 95 ] Geometric Compression through topological surgery [ Taubin & Rossignac, 98 ] Triangle Mesh Compression [ Touma & Gotsman, 98 ] Java3DMPEG - 4Virtue3D

Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates

Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates Spectral Compression of Mesh Geometry [ Karni & Gotsman, 00 ] expensive numerical computations

Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates Progressive Geometry Compression [ Khodakovsky et al., 00 ] modifies mesh prior to compression

Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates Geometric Compression for interactive transmission [ Devillers & Gandoin, 00 ] poly-soups; complex geometric algorithms

Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates Vertex data compression for triangle meshes [ Lee & Ko, 00 ] local coord-system + vector-quantization

Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates Angle-Analyzer: A triangle- quad mesh codec [ Lee, Alliez & Desbrun, 02 ] dihedral + internal = heavy trigonometry

Geometry Compression Classic approaches [ 95 – 98 ]: – linear prediction Recent approaches [ 00 – 03 ]: – spectral – re-meshing – space-dividing – vector-quantization – angle-based – delta coordinates High-Pass Quantization for Mesh Encoding [ Sorkine et al., 03 ] basis transformation with Laplacian matrix

Linear Prediction Schemes 1. quantize positions with b bits 2. traverse positions 3. linear prediction from neighbors 4. store corrective vector ( , , ) ( 1008, 68, 718 ) floating point integer

Linear Prediction Schemes 1. quantize positions with b bits 2. traverse positions 3. linear prediction from neighbors 4. store corrective vector use traversal order implied by the connectivity coder

Linear Prediction Schemes 1. quantize positions with b bits 2. traverse positions 3. linear prediction from neighbors 4. store corrective vector ( 1004, 71, 723 ) apply prediction rule prediction

Linear Prediction Schemes 1. quantize positions with b bits 2. traverse positions 3. linear prediction from neighbors 4. store corrective vector position distribution corrector distribution ( 1004, 71, 723 )( 1008, 68, 718 ) position ( 4, -3, -5 ) correctorprediction

Deering, 95 Prediction: Delta-Coding A processed region unprocessed region P P = A

Taubin & Rossignac, 98 Prediction: Spanning Tree A B C D E processed region unprocessed region P P = α A + βB + γC + δD + εE + …

Touma & Gotsman, 98 Prediction: Parallelogram Rule processed region unprocessed region P P = A – B + C A B C

 “within”- predictions often find existing parallelograms ( i.e. quadrilateral faces ) “within” versus “across”  “within”- predictions avoid creases within-prediction across-prediction

Overview Motivation Geometry Compression Frequency of Polygons Polygonal Prediction Rules Results Discussion INFORMATIK

Discrete Fourier Transform ( 1 ) where. The Discrete Fourier Transform ( DFT ) of a complex vector is a basis transform that is described by the matrix:

Discrete Fourier Transform ( 2 ) Here is the Fourier Transform of.

Discrete Fourier Transform ( 3 ) Rewriting the equation makes the change of basis more obvious. This basis is called the Fourier Basis. basis vectors

Geometric Interpretation v2v2 v1v1 v0v0 v3v3 v4v4

The parallelogram rule predicts the highest frequency to be zero: Predict with Low Frequencies v2v2 v1v1 v0v0 v3v3 v3v3 v2v2 v1v1 v0v0 v3v3 v3v3

Overview Motivation Geometry Compression Frequency of Polygons Polygonal Prediction Rules Results Discussion INFORMATIK

Eliminate High Frequencies ( 1 ) v3v3 v2v2 v1v1 v4v4 v0v0 v3v3 v5v5 v3v3 v2v2 v1v1 v4v4 v0v0 v3v3 v5v5

Eliminate High Frequencies ( 2 ) v2v2 v3v3 v1v1 v0v0 v4v4 v1v1 v0v0 v4v4 v2v2 v3v3

Eliminate High Frequencies ( 3 ) v1v1 v0v0 v4v4 v3v3 v2v2 v1v1 v0v0 v4v4 v3v3 v2v2

Computing the Coefficients given k of n points are known: 1. write polygon in Fourier basis 2. put n-k highest frequencies to zero 3. invert known sub-matrix 4. calculate prediction coefficients known points missing points

Example: n = 5, k = 3 v0v0 v1v1 v4v4 v2v2 v3v3 missing points v0v0 v1v1 v4v4 v2v2 v3v3 known points

Example: n = 5, k = 3 v0v0 v1v1 v4v4 v2v2 v3v3 missing points known points

Polygonal Predictors v2v2 v1v1 v0v0 v3v3 v2v2 v1v1 v0v0 v3v3 c 0 = 1.0 c 1 = c 2 = c 0 = 1.0 c 1 =-2.0 c 2 =2.0 v2v2 v1v1 v0v0 v3v3 c 0 = 1.0 c 1 = c 2 = v2v2 v1v1 v0v0 v3v3 c 0 = 1.0 c 1 = c 2 = three vertices are known v2v2 v1v1 v0v0 v4v4 v3v3 c 0 = c 1 = c 2 = c 3 = v3v3 v2v2 v0v0 v3v3 c 0 = 1.0 c 1 =-1.0 c 2 =1.0 c 3 =-1.0 c 4 =1.0 v4v4 v5v5 c 0 = c 1 = c 2 = c 3 = c 4 = c 5 = v2v2 v1v1 v0v0 v3v3 v4v4 v5v5 v6v6 v2v2 v1v1 v0v0 v3v3 v4v4 c 0 =1.0 c 1 =-1.0 c 2 =1.0 c 3 =-1.0 c 4 =1.0 c 5 =-1.0 c 6 = 1.0 v7v7 v5v5 v6v6 v1v1 one vertex is missing

Overview Motivation Geometry Compression Frequency of Polygons Polygonal Prediction Rules Results Discussion INFORMATIK

Test Set of Dual Meshes

Parallelogram vs. Polygonal

Prediction Rule Histogram hexagon heptagon pentagon octagon

Dual vs. Primal Compression ( coordinates quantized at 14 bits of precision )

Average Prediction Error

Overview Motivation Geometry Compression Frequency of Polygons Polygonal Prediction Rules Results Discussion INFORMATIK

Validation of Predictors eliminating the highest frequency in a mesh element – + + parallelogram predictor [ Touma & Gotsman, 98 ] + – + – + – + Lorenzo predictor [ Ibarria et al, 03 ]

Exact barycentric prediction after dualization polygons of even order have a highest frequency of zero

Thank You INFORMATIK