Compressing Hexahedral Volume Meshes Martin Isenburg Pierre Alliez

Compressing Hexahedral Volume Meshes Martin Isenburg Pierre Alliez UNC Chapel Hill INRIA Sophia-Antipolis

Overview • Volume Meshes • Related Work • Compressing Connectivity – Coding with Edge Degrees – Boundary Propagation – Adaptive Traversal • Compressing Geometry – Parallelogram Prediction • Demo

Take this home: “The connectivity of a hexahedral mesh can be coded through a sequence of its edge degrees. ” “This encoding naturally exploits the regularity commonly found in such data sets. ”

Volume Meshes

Volume Meshes • scientific & industrial applications – thermodynamics – structural mechanics –… • visualization & simulation • unstructured / irregular ( not on a grid ) • tetrahedral, hexahedral, polyhedral

Hexahedral Volume Meshes • have “numerical advantages in finite element computations” • challenging to generate • their internal structure looks “nice” compared to tetrahedral meshes

Ingredients • geometry : positions of vertices • connectivity : which vertices form a hexahedron • properties: attached to vertices density, pressure, heat, . . .

Standard Representation • connectivity log 2(v) hex 1 1 3 6 4 7 8 9 2 hex 2 4 5 8 2 9 1 6 7 8 h * 32 bits hex 3 7 5 … hexh • geometry vtx 1 ( x, y, z ) vtx 2 ( x, y, z ) 3 v * 32 bits vtx 3 ( x, y, z ) 16 vtxv less than 84 KB 1. 69 size: 3. 23 MB 71572 hexahedra 78618 vertices

Related Work

Surface Mesh Compression – Geometry Compression, [Deering, 95] – Topological Surgery, [Taubin & Rossignac, 98] – Cut-Border Machine, [Gumhold & Strasser, 98] – Triangle Mesh Compression, [Touma & Gotsman, 98] – Edgebreaker, [Rossignac, 99] – Spectral Compression of Geometry, [Karni & Gotsman, 00] – Face Fixer, [Isenburg & Snoeyink, 00] – Valence-driven Connectivity Coding, [Alliez & Desbrun, 01] – Near-Optimal Coding, [Khodakovsky, Alliez, Desbrun & – Degree Duality Coder, [Isenburg, 02] Schröder, 02] – Polygonal Parallelogram Prediction, [Isenburg & Alliez, 02]

Volume Mesh Compression – Grow & Fold, [Szymczak & Rossignac, 99] – Cut-Border Machine, [Gumhold, Guthe & Strasser, 99] – Rendering of compressed volume data, [Yang et al. , 01] Simplification: – Simplification of tetrahedral meshes, [Trotts et al. , 98] – Progressive Tetrahedralizations, [Staadt & Gross, 98] Progressive Compression: – Implant Sprays, [Pajarola, Rossignac & Szymczak, 99] !! only for tetrahedral meshes !!

Surface / Volume Connectivity a mesh with v vertices has maximal surfaces: 2 v-2 triangles ~ 6 v indices v-1 quadrilaterals ~ 4 v indices volumes: O(v 2) tetrahedra ~ 12 v indices O(v 2) hexahedra ~ 8 v indices connectivity dominates geometry even more for volume meshes

Degree Coding for Connectivity • Triangle Mesh Compression, [Touma & Gotsman, 98] • Valence-driven Connectivity Coding, [Alliez, Desbrun, 01] • Degree Duality Coder, [Isenburg, 02] • Near-Optimal Connectivity Coding, [Khodakovsky, Alliez, Desbrun, Schröder, 02] . . . 7. . . 4 3 3 6 4 4 4 7 5 4 6 6 4 4 4 5 6 6 4 4 6. . . 4. . . compressed 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 Distribution [over a set of 11 polygonal meshes] 4 4 3 3 2 5 6 7 8 9+ vertex degrees 5 6 7 8 9+ face degrees

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

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

Degree Coding for Volumes ? tri tet quad hex vertex degrees edge degrees

Regular Volume Meshes? • elements for regular 2 D tiling – regular triangle – regular quadrilateral – regular hexagon • elements for regular 3 D tiling – regular tetrahedron – regular hexahedron

Compressing Connectivity

Space Growing similar in spirit to “region growing”: algorithm maintains hull enclosing processed hexahedra 1. initialize hull with a border face 2. iterate until done • pick incomplete face on hull • process adjacent hexahedra • record degrees of its free edges

Coding with Edge Degrees focus face

Coding with Edge Degrees slots focus face

Coding with Edge Degrees

Coding with Edge Degrees incomplete faces border faces

Coding with Edge Degrees

Coding with Edge Degrees edges on hull maintain slot count

Coding with Edge Degrees edges with a slot count of zero are “zero slots” zero slots

Coding with Edge Degrees

Resulting Symbols – border edge degrees. . . 2 3 3 3 4 3 . . . – interior edge degrees. . . 4 5 4 4 3 4 . . . – border ? . . . N N N Y N N N N N. . . – join ? . . . N N N N Y N N N. . .

Average Distributions 3 2 no 4 4 5 6+ 2 3 5 6 7+ border degrees interior degrees yes no yes border? join?

Possible Configurations

Possible Configurations hut step bridge roof corner tunnel gap pit den

Configuration: “hut” free vertex free face free edge hut not a zero slot zero-slots: 0 free faces free edges free vertices 5 8 4

Configuration: “step” step 0 this is a zero slot zero-slots: 1 free faces free edges free vertices 4 5 2

Configuration: “bridge” 0 bridge 0 zero-slots: 2 free faces free edges free vertices 3 2 --

“hut” or “roof” “join” operation hut ? for the free face of “hut” that potentially forms a roof: “join? ” roof Y / N

other “join” operations • free vertex already on hull • free edge already on hull hut for every free vertex / edge: “join? ” step Y / N

Adaptive traversal to avoid “join”operations

Reason for “join” operations hull

Reason for “join” operations unprocessed region hull processed region

Reason for “join” operations unprocessed region join hull processed region

Adaptive Traversal • Valence-driven connectivity encoding for 3 D meshes [Alliez & Desbrun, 01] avoid creation of cavities pick face with largest # of zero slots focus face

Propagating the border information

Explaining Example border face & slot count = 1 border face

Explaining Example border face border edges

Explaining Example interior edges incomplete faces

Explaining Example remaining elements for everything else: “border? ” Y / N

Results (Connectivity) model hanger ra bump … warped hutch c 1 bits per hexahedron (bph) raw compressed 5. 3 2. 9 2. 1 72. 0 80. 0 88. 0 … … 112. 0 136. 0 0. 2 0. 3 0. 6 ratio 1 : 14 1 : 28 1 : 42. . . 1 : 621 1 : 361 1 : 226 average compression ratio = 1 : 163

Compressing Geometry

Predictive Compression 1. quantize positions with b bits floating point (1. 2045, -0. 2045, 0. 7045) integer (1008, 67, 718)

Predictive Compression 1. quantize positions with b bits floating point (1. 2045, -0. 2045, 0. 7045) 2. traverse positions integer (1008, 67, 718)

Predictive Compression 1. quantize positions with b bits floating point (1. 2045, -0. 2045, 0. 7045) integer (1008, 67, 718) 2. traverse positions 3. predict position from neighbors prediction (1004, 71, 723)

Predictive Compression 1. quantize positions with b bits floating point (1. 2045, -0. 2045, 0. 7045) integer (1008, 67, 718) 2. traverse positions 3. predict position from neighbors 4. store corrective vector prediction (1004, 71, 723) corrector (4, -3, -5)

Parallelogram Rule Triangle Mesh Compression, [Touma & Gotsman, 98] across “non-convex” triangles across “non-planar” triangles within “planar” & “convex” quadrilateral

Position Predictions 2 3 vertex v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 1 0 6 init prediction rule 0 v 1 v 0 - v 1 + v 2 2 v 0 – v 8 (or v 0 ) v 1 – v 0 + v 4 v 2 – v 1 + v 5 v 3 – v 2 + v 6 7 5 4 2 3 1 0 8 hut

Results (Geometry) model hanger ra bump … warped hutch c 1 bits per vertex (bpv) raw compressed 23. 2 30. 8 24. 4 48. 0 … … 48. 0 10. 5 19. 9 5. 9 ratio 1 : 2. 1 1 : 1. 6 1 : 2. 0. . . 1 : 4. 6 1 : 2. 4 1 : 8. 1 average compression ratio = 1 : 3. 7

Demo

Summary • degree coding for volume mesh connectivity – edge degrees – boundary propagation – adaptive traversal • parallelogram prediction for volume mesh geometry – “within” predictions

Current / Future Work • Mixed Volume Meshes – hex + tet + prism + pyramid cells • Universal Connectivity Coder – – – face, vertex, and edge degrees tri / quad / poly surfaces tet / hex / poly volumes surface mesh = cell of volume mesh bit-rate like specialized coder

Acknowledgements data sets • • Alla Sheffer Steven Owen Scott Mitchell Claudio Silva financial support • ARC TéléGéo grant from INRIA

Thank You!

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