Compressing Polygon Mesh Connectivity with Degree Duality Prediction

Compressing Polygon Mesh Connectivity with Degree Duality Prediction Martin Isenburg University of North Carolina at Chapel Hill

Overview • • Background Connectivity Compression Coding with Degrees Duality Prediction Adaptive Traversal Example Run Conclusion

Background

Polygon Meshes • connectivity face 1 1 2 3 4 face 2 3 4 3 k v log 2 (v) face 3 5 2 1 3 facef : k ~ 4 : k ~ 6 • geometry vertex 1 ( x, y, z ) vertex 2 ( x, y, z ) 24 ~ 96 v vertex 3 ( x, y, z ) vertexv 4 5

Mesh Compression • Geometry Compression [Deering, 95] – Fast Rendering – Progressive Transmission – Maximum. Compression • Geometry • Connectivity – Triangle Meshes – Polygon. Meshes

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

Results model triceratops galleon cessna … tommygun cow teapot bits per vertex (bpv) Face Fixer Degree Duality 1. 189 2. 093 2. 543 2. 115 2. 595 2. 841 … … 2. 611 2. 213 1. 669 2. 258 1. 781 1. 127 gain 44 % 19 % 11 %. . . 14 % 20 % 33 % min / max / average gain [%] = 11 / 55 / 26

Connectivity Compression

Connectivity Compression assumption • order of vertices does not matter advantage • no need to “preserve” indices approach • code only the “connectivity graph” • re-order vertices appropriately

Connectivity Graphs • connectivity of simple meshes is homeomorphic to planar graph enumeration asymptotic bounds [William Tutte 62 / 63] number of planar triangulations with v vertices 3. 24 bpv << 6 log (v) bpv 2

Spanning Tree • Succinct Representations of Graphs [Turan, 84] • Short encodings of planar graphs and maps [Keeler & Westbrook, 95] • Geometric Compression through Topological Surgery [Taubin & Rossignac, 98] extends to meshes of non-zero genus

Region Growing • Triangle Mesh Compression [Touma & Gotsman, 98] • Cut-Border Machine [Gumhold & Strasser, 98] • Edgebreaker [Rossignac, 99] • Simple Sequential Encoding [de Floriani et al. , 99] • Dual Graph Approach [Lee & Kuo, 99] • Face Fixer [Isenburg & Snoeyink, 00]

Classification • code symbols are associated with edges, faces, or vertices: boundary processed region boundary unprocessed region focus edge-based face-based focus vertex-based

Edge-Based Compression Schemes

Edge-Based • Dual Graph Approach, [Lee & Kuo, 99] • Face Fixer, [Isenburg & Snoeyink, 00] processed region unprocessed region focus . . . F F R

Edge-Based • Dual Graph Approach, [Lee & Kuo, 99] • Face Fixer, [Isenburg & Snoeyink, 00] processed region unprocessed region . . . F F R F F

Edge-Based • Dual Graph Approach, [Lee & Kuo, 99] • Face Fixer, [Isenburg & Snoeyink, 00] processed region unprocessed region . . . F F R F F

Edge-Based • Dual Graph Approach, [Lee & Kuo, 99] • Face Fixer, [Isenburg & Snoeyink, 00] processed region unprocessed region . . . F F R

Edge-Based • Dual Graph Approach, [Lee & Kuo, 99] • Face Fixer, [Isenburg & Snoeyink, 00] processed region F unprocessed region . . . F F R F F F R

Edge-Based • Dual Graph Approach, [Lee & Kuo, 99] • Face Fixer, [Isenburg & Snoeyink, 00] processed region R F unprocessed region . . . F F R F R F F R

Edge-Based ? ? • Dual Graph Approach, [Lee & Kuo, 99] • Face Fixer, [Isenburg & Snoeyink, 00] processed region R F F unprocessed region F R F . . . F F R F R F. . . 5 4 3 6 4. . . ?

Face-Based Compression Schemes

Face-Based • Cut-Border Machine, [Gumhold & Strasser, 98] • Edgebreaker, [Rossignac, 99] processed region unprocessed region focus . . . C R

Face-Based • Cut-Border Machine, [Gumhold & Strasser, 98] • Edgebreaker, [Rossignac, 99] processed region unprocessed region . . . C R C C

Face-Based • Cut-Border Machine, [Gumhold & Strasser, 98] • Edgebreaker, [Rossignac, 99] processed region R unprocessed region . . . C R C

Face-Based • Cut-Border Machine, [Gumhold & Strasser, 98] • Edgebreaker, [Rossignac, 99] processed region R R unprocessed region . . . C R R C

Face-Based ? ? ? • Cut-Border Machine, [Gumhold & Strasser, 98] • Edgebreaker, [Rossignac, 99] processed region C unprocessed region . . . C R R C. . . 5 4 3 6 4. . . R R C

Vertex-Based Compression Schemes

Vertex-based • Triangle Mesh Compression, [Touma & Gotsman, 98] processed region unprocessed region focus . . . 6

Vertex-based • Triangle Mesh Compression, [Touma & Gotsman, 98] processed region 5 unprocessed region . . . 6 5

Vertex-based • Triangle Mesh Compression, [Touma & Gotsman, 98] processed region 5 unprocessed region . . . 6 5

Vertex-based • Triangle Mesh Compression, [Touma & Gotsman, 98] processed region 5 unprocessed region . . . 6 5

Vertex-based ? ? ? • Triangle Mesh Compression, [Touma & Gotsman, 98] processed region 6 unprocessed region . . . 6. . . 5 6 5 4 3 6 . . . 4. . . 5

Coding with Vertex and Face Degrees

Coding with Degrees while ( unprocessed faces ) move focus to a face degree for ( free vertices ) case switch ( case ) “add”: vertex degree “split”: offset “merge”: index,

Example Traversal

“add” free vertex boundary focus processed region exit focus end slotregion unprocessed 3 free vertices 5 focus (widened) 4 boundary slots start slot . . . 4. . . 5 5 3 4 4 4 3 5 4 3

free vertex “splits” boundary unprocessed region S split offset processed region . . . 4. . . 5 5 3 4 4 4 3 focus

free vertex “merges” boundary in stack unprocessed region stack focus processed region M merge offset . . . 4. . . 5 processed region 5 3 4 4 4 3

Resulting Code • two symbol sequences – vertex degrees (+ “split” / “merge”). . . 4 3 S 4 4 M 6 4 5 4 4. . . – face degrees. . . 4 3 4 4 5 4 4 4 6 4 4 4. . . • compress with arithmetic coder converges to entropy

Entropy for a symbol sequence of t types Entropy = t i =1 1 pi • log 2( ) bits pi pi = 0. 2 bits 1. 3 bits # of type t # total 2. 0 bits

Average Distributions add 4 4 3 split merge case 3 2 5 6 7 8 9+ vertex degrees 5 6 7 8 9+ face degrees

Adaptation to Regularity 6 . . . 3 . . . vertex degrees . . . 3 . . . face degrees . . . 6 . . . vertex degrees . . . 4 . . . face degrees . . . 4 . . . vertex degrees . . . face degrees

“Worst-case” Distribution 3 3 pi = 1 i =3 4 5 i • pi = 6 6 [Alliez & Desbrun, 01] 3. 241… bpv [Tutte, 62] i =3 7 8 9 …… … vertex degrees face degrees

Compressing with Duality Prediction

Degree Correlation • high-degree faces are “likely” to be surrounded by low-degree vertices • and vice-versa mutual degree prediction

Face Degree Prediction fdc 3. 3 3 focus (widened) 4 3 average degree of focus vertices 3+4+3 fdc = = 3. 333 3 “face degree context” 3. 3 fdc 4. 9 fdc

Vertex Degree Prediction 6 vdc = 3 vdc = 44 degree of focus face vdc = 55 vdc = 6 “vertex degree context” vdc 66

Compression Gain model triceratops galleon cessna … tommygun cow teapot without bits per vertex with bits per vertex 1. 192 2. 371 2. 811 1. 189 2. 093 2. 543 … … 2. 917 1. 781 1. 632 2. 258 1. 781 1. 127 min / max / average gain [%] = 0 / 31 / 17

Reducing the Number of Splits

Occurance of “splits”

Occurance of “splits”

Occurance of “splits”

Occurance of “splits”

Occurance of “splits” unprocessed region

Occurance of “splits” unprocessed region

Occurance of “splits” unprocessed region

Occurance of “splits” unprocessed region

Occurance of “splits” unprocessed region split processed region

Adaptive Traversal • Valence-driven connectivity encoding for 3 D meshes [Alliez & Desbrun, 01] avoid creation of cavities exit focus

Compression Gain model triceratops galleon cessna … tommygun cow teapot without splits bpv 53 78 172 … 131 154 10 1. 311 2. 309 2. 882 … 2. 449 2. 313 1. 167 with splits bpv 25 18 28 … 32 13 3 1. 189 2. 093 2. 543 … 2. 258 1. 781 1. 127 min / max / average gain [%] = 4 / 23 / 10

Example Decoding Run

Example Decoding Run 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 4 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run focus 6 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 3 free vertex 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run exit focus 6 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run free vertices 5 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 3 4 6 6 3 3 5 4 4 3 4 5 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 3 5 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run exit focus 4 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 4 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 5 4 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 4 exit focus 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 5 end slot focus (widened) 3 3 start slot 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 4 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run exit focus 4 5 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run focus (widened) 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 4 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run exit focus 4 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 3 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 4 4 6 6 3 3 5 4 4 3 5 4 6 exit focus 5 4 4 5. . . 2 4 4. . .

Example Decoding Run focus (widened) 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 6 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 2 6 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run 4 exit focus 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run focus (widened) 4 6 6 3 3 5 4 4 3 5 4 6 5 5. . . 4 4 2 4 4. . .

Example Decoding Run . . . 5 4 6 6 3 3 5 4 4 3 5 4 6 5 5 . . . 4 4 2 4 4. . .

Conclusion

Summary • degree coding for polygonal connectivity • duality prediction • adaptive traversal • proof-of-concept implementation using Shout 3 D http: //www. cs. unc. edu/~isenburg/degreedualitycoder/

Similar Result • Near-Optimal Connectivity Coding of 2 -manifold polygon meshes [Khodakovsky, Alliez, Desbrun, Schröder] analysis of worst-case face degree and vertex degree distribution entropy Tutte’s bounds Martin France

Current Work (w. Pierre Alliez) • use polygons for better predictive geometry coding “fairly planar & convex” • extend degree coding to volume mesh connectivity “edge degrees”

Thank You!