Compressing Hexahedral Volume Meshes

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Compressing Hexahedral Volume Meshes. Martin Isenburg UNC Chapel Hill. Pierre Alliez INRIA Sophia-Antipolis. Overview. Volume Meshes Related Work Compressing Connectivity Coding with Edge Degrees Boundary Propagation Adaptive Traversal Compressing Geometry Parallelogram Prediction - PowerPoint PPT Presentation

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  • CompressingHexahedral Volume Meshes

    Martin IsenburgUNC Chapel HillPierre AlliezINRIA Sophia-Antipolis

  • OverviewVolume MeshesRelated WorkCompressing ConnectivityCoding with Edge DegreesBoundary PropagationAdaptive TraversalCompressing GeometryParallelogram PredictionDemo

  • 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 & simulationunstructured / irregular ( not on a grid )tetrahedral, hexahedral, polyhedral

  • Hexahedral Volume Mesheshave numerical advantages in finite element computationschallenging to generatetheir internal structure looks nice compared to tetrahedral meshes

  • Ingredientsgeometry : positions of verticesconnectivity : which vertices form a hexahedron properties: attached to vertices density, pressure, heat, ...

  • Standard Representationconnectivity geometryvtx1 ( x, y, z )vtx2 ( x, y, z )vtx3 ( x, y, z )

    vtxvhex113647892hex245829167hex3 75

    hexh 71572 hexahedra78618 vertices size: 3.23 MBless than 84 KB

  • Related Work

  • Surface Mesh CompressionGeometry 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]Polygonal Parallelogram Prediction, [Isenburg & Alliez, 02]Schrder, 02]

  • Volume Mesh CompressionGrow & Fold, [Szymczak & Rossignac, 99]Cut-Border Machine, [Gumhold, Guthe & Strasser, 99]!! only for tetrahedral meshes !!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]

  • Surface / Volume Connectivitya mesh with v vertices has maximal

    surfaces:2v-2triangles ~ 6v indicesv-1quadrilaterals ~ 4v indices

    connectivity dominates geometry even more for volume meshes

    volumes:O(v2)tetrahedra ~ 12v indicesO(v2)hexahedra ~ 8v indices

  • Degree Coding for ConnectivityTriangle Mesh Compression, [Touma & Gotsman, 98]Valence-driven Connectivity Coding, [Alliez, Desbrun, 01]Degree Duality Coder, [Isenburg, 02]Near-Optimal Connectivity Coding, [Khodakovsky, Alliez, Desbrun, Schrder, 02] compressed with arithmetic coder converges to entropy

  • Entropyfor a symbol sequence of t types

    # of type tpi =i = 1tEntropy = pi log2( ) bitspi1# total

  • Average Distribution [over a set of 11 polygonal meshes]

  • Worst-case Distribution 3456789vertex degrees[Alliez & Desbrun, 01] 3.241 bpv

    [Tutte, 62]

    Chart2

    0

    0

    0

    1482

    1117

    841

    634

    477

    360

    271

    204

    153

    116

    87

    65

    49

    37

    27

    20

    15

    11

    8

    6

    4

    Sheet1

    0000015360

    000148211178416344773602712041531168765493727201511864321

    Sheet1

    Sheet2

    Sheet3

    Sheet4

    Sheet5

    Sheet6

    Sheet7

    Sheet8

    Sheet9

    Sheet10

    Sheet11

    Sheet12

    Sheet13

    Sheet14

    Sheet15

  • Adaptation to Regularity

  • Degree Coding for Volumes ?

    tri tet vertex degrees edge degrees

  • Regular Volume Meshes?elements for regular 2D tiling regular triangle regular quadrilateral regular hexagon

  • Compressing Connectivity

  • Space Growingsimilar in spirit to region growing:algorithm maintains hull enclosing processed hexahedrainitialize hull with a border faceiterate until done

  • Coding with Edge Degrees

  • Coding with Edge Degreesfocus face

  • Coding with Edge Degreesfocus faceslots

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Coding with Edge Degrees

  • Resulting Symbols border edge degrees interior edge degrees. . .. . .. . .. . .

  • Average Distributions 23456+border degrees34567+interior degreesborder?2join?noyesnoyes

  • Possible Configurations

  • Possible Configurations

  • Configuration: huthut

  • Configuration: stepstep0

  • Configuration: bridgebridge00

  • hut or roof?roof join operation

  • other join operationsfree vertex already on hullfree edge already on hullhutstep

  • Adaptive traversalto avoid joinoperations

  • Reason for join operationshull

  • Reason for join operationshull

  • Reason for join operationshull

  • Reason for join operationshull

  • Reason for join operationsprocessed regionunprocessed regionhull

  • Reason for join operationsprocessed regionunprocessed regionhull

  • Reason for join operationsprocessed regionunprocessed regionhull

  • Reason for join operationsprocessed regionunprocessed regionhull

  • Reason for join operationsprocessed regionunprocessed regionhull

  • Adaptive TraversalValence-driven connectivity encoding for 3D meshes [Alliez & Desbrun, 01]

    avoid creation of cavities

  • Propagating theborder information

  • Explaining Example

  • Explaining Example

  • Explaining Example

  • Explaining Example

  • Results (Connectivity)hangerrabumpwarpedhutchc1 5.32.92.1 0.20.30.6 rawbits per hexahedron (bph)model compressed72.0 80.0 88.0 112.0 112.0 136.0 1 : 141 : 281 : 42 ... 1 : 6211 : 3611 : 226 ratio average compression ratio = 1 : 163

  • Results (Connectivity)hangerrabumpwarpedhutchc1 5.32.92.1 0.20.30.6 rawbits per hexahedron (bph)model compressed72.0 80.0 88.0 112.0 112.0 136.0 1 : 141 : 281 : 42 ... 1 : 6211 : 3611 : 226 ratio average compression ratio = 1 : 163

  • Compressing Geometry

  • Predictive Compressionquantize positions with b bits

  • Predictive Compressionquantize positions with b bits traverse positions

  • Predictive Compressionquantize positions with b bits traverse positionspredict position from neighbors(1004, 71, 723)prediction

  • Predictive Compressionquantize positions with b bits traverse positionspredict position from neighborsstore corrective vector(4, -3, -5)corrector(1004, 71, 723)prediction

  • Parallelogram Rule

    acrossnon-convextrianglesacrossnon-planartrianglesTriangle Mesh Compression, [Touma & Gotsman, 98]

  • Position Predictionsinithut0123v0 0v1 v0v2 v1v3 v0 - v1 + v2v4 2v0 v8 (or v0 )v5 v1 v0 + v4v6 v2 v1 + v5v7 v3 v2 + v6vertex prediction rule

  • Results (Geometry)hangerrabumpwarpedhutchc1 23.230.824.4 10.519.95.9 rawbits per vertex (bpv)model compressed48.0 48.0 48.0 48.0 48.0 48.0 1 : 2.11 : 1.61 : 2.0 ... 1 : 4.61 : 2.41 : 8.1 ratio average compression ratio = 1 : 3.7

  • Demo

  • Summarydegree coding for volume mesh connectivity edge degrees boundary propagation adaptive traversalparallelogram prediction for volume mesh geometry within predictions

  • Current / Future WorkMixed Volume Meshes hex + tet + prism + pyramid cellsUniversal 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

  • Acknowledgementsdata setsAlla ShefferSteven OwenScott MitchellClaudio Silva financial supportARC TlGo grant from INRIA

  • Thank You!

  • Old Beijing & DuckWhoever had expressed interest in going to eat duck in the ancient-style hutong area:meet me 10-15 minutes afterend of PG in front of hotelbring :mapaddress card of your hotel~100 yuan (smaller bills for subway / bus)

  • Configurations: roofroofzero-slots:0adjacent faces:1

    free faces4 free edges4 free vertices--

  • Configurations: tunneltunnelzero-slots:2adjacent faces:4

    free faces2 free edges-- free vertices--0000

  • Configuration: cornercornerzero-slots:2adjacent faces:3

    free faces3 free edges3 free vertices1000

  • Configuration: gapgapzero-slots:3adjacent faces:4

    free faces2 free edges1 free vertices--00000

  • Configurations: pit0000zero-sl

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