INTECH’INTECH’ April, the 28 April, the 28thth 2005 2005

Mesh ParameterizationMesh Parameterization

Bruno Lévy, Bruno Lévy, INRIA, project ALICEINRIA, project ALICE

OverviewOverview

1. Geometry in the ALICE Project:1. Geometry in the ALICE Project:

Vision, Background and Goals,Vision, Background and Goals,

2. Parameterization (1998 - 2001),2. Parameterization (1998 - 2001),

3. Atlas (2002 - 2003),3. Atlas (2002 - 2003),

4. PGP [Nicolas Ray] (2004 - …)4. PGP [Nicolas Ray] (2004 - …)

1.Vision1.Vision

From the 70’s ….From the 70’s ….

To the 2000’s ….To the 2000’s ….

1.1. VisionVisionComputer Graphics and 3D modellingComputer Graphics and 3D modelling

MeshesMeshesdiscretediscrete

SplinesSplinescontinuouscontinuous

ScientificScientificVisualizationVisualization

ComputerComputerGraphicsGraphics

NumericalNumericalSimulationsSimulations

CADCADCAMCAM

3D Scanning3D Scanning 3D modelers3D modelers

Numerical GeometryNumerical Geometry

1.1. VisionVisionThe data representation problemThe data representation problem

1. Background1. BackgroundDigital Geometry ProcessingDigital Geometry Processing

•A new and competitive research areaA new and competitive research area

•Converting between object representations isConverting between object representations is still an open problem still an open problem [Henri Gouraud, Malcom Sabin][Henri Gouraud, Malcom Sabin]

Need for a mathematical method that Need for a mathematical method that ‘‘understandsunderstands’ geometry’ geometry

1. Goals1. Goals

Create a « geographic coordinate system »Create a « geographic coordinate system »

u

vRI3

RI2

uu((xx,,yy,,zz))

xx((uu,,vv))

SObject space (3D) Texture space (2D)

2. Parameterization2. ParameterizationNotion of parameterizationNotion of parameterization

RI3

RI2

u

v

PiPi

ui ,vi

2. Parameterization 2. Parameterization Notion of parameterizationNotion of parameterization

Survey: [Floater 04]Survey: [Floater 04]

2. Parameterization 2. Parameterization Demo: Constrained ParameterizationDemo: Constrained Parameterization

Constrained Parameterization Constrained Parameterization [Siggraph 1998 and 2001][Siggraph 1998 and 2001]

2. Parameterization2. ParameterizationApplication: GriddingApplication: Gridding

Grid generation for flow simulatorsEarth Decision Sciences startup(Paris, Houston, Rio, Dubai)

Product: Gocad 3D modeler

3. Atlas3. AtlasNotion of atlasNotion of atlas

Conformal Map : C = || grad(u|T) - i.grad(v|T) ||2 T T

Least Squares Conformal Maps [Siggraph 2002]Least Squares Conformal Maps [Siggraph 2002]

3. Atlas3. AtlasApplication: Maya and BlenderApplication: Maya and Blender

Least Squares Conformal Maps [Siggraph 02]Least Squares Conformal Maps [Siggraph 02] Alias|WavefrontAlias|Wavefront MAYA 3D modeler MAYA 3D modeler

3. Atlas – « Tetris » 3. Atlas – « Tetris » PackingPacking

[[Nicolas Ray]Nicolas Ray] Application: Application: DirectXDirectX

Lost Area

3. Atlas – « Tetris » Packing3. Atlas – « Tetris » Packing [ [Nicolas Ray]Nicolas Ray] Application: DirectXApplication: DirectX

3. Atlas3. AtlasApplications: X-MeshApplications: X-Mesh

VSP-Technology startup

Product: X-Mesh

(Mesh Manipulation Library)

3. Atlas3. AtlasDemo: Normal-mappingDemo: Normal-mapping

3. Atlas3. AtlasApplication: Eden GamesApplication: Eden Games

Courtesy of Eden GamesCourtesy of Eden GamesAlone in the DarkAlone in the Dark

4. Periodic Global 4. Periodic Global Parameterization (PGP)Parameterization (PGP)

Create a « geographic coordinate system »Create a « geographic coordinate system »

4. PGP4. PGPThe data: a scanned meshThe data: a scanned mesh

4. PGP4. PGPProblems: arbitrary topologyProblems: arbitrary topology

How can we handle closed surfaces ?How can we handle closed surfaces ?

How can we parameterize a cylinder ?How can we parameterize a cylinder ?

Global Parameterization [Gu 2002]Global Parameterization [Gu 2002]

4. PGP4. PGP[Nicolas Ray][Nicolas Ray]

4. PGP4. PGPAffine and Complex ManifoldsAffine and Complex Manifolds

4. PGP Results4. PGP ResultsMesh-2-Spline conversion Mesh-2-Spline conversion

(demo)(demo)

4. PGP Results4. PGP ResultsRemeshingRemeshing

4. PGP Results4. PGP ResultsRemeshingRemeshing

4. PGP results4. PGP resultsRemeshingRemeshing

4. PGP4. PGPApplicationsApplications

Microsoft Research Grant:Microsoft Research Grant:Geometric IntelligenceGeometric Intelligence

ConclusionsConclusions

Digital Geometry Processing:Digital Geometry Processing: A scientific challenge:A scientific challenge:

– Solve the 3D representation problem !Solve the 3D representation problem ! Many possible industrial applicationsMany possible industrial applications

– Video-gamesVideo-games– CAD/CAM, reverse engineeringCAD/CAM, reverse engineering– Oil exploration, FEM simulationsOil exploration, FEM simulations