INTECH’ April, the 28 th 2005 Mesh Parameterization Bruno Lévy, INRIA, project ALICE INTECH’ April, the 28 th 2005 Mesh Parameterization Bruno Lévy, INRIA, project ALICE

Overview 1. Geometry in the ALICE Project: Vision, Background and Goals, Vision, Background and Goals, 2. Parameterization ( ), 3. Atlas ( ), 4. PGP [Nicolas Ray] ( …)

1.Vision From the 70’s …. To the 2000’s ….

1.Vision Computer Graphics and 3D modelling Meshesdiscrete Splinescontinuous ScientificVisualization ComputerGraphics NumericalSimulations CADCAM 3D Scanning 3D modelers Numerical Geometry

1.Vision The data representation problem

1. Background Digital 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 ‘understands’ geometry

1. Goals Create a « geographic coordinate system »

u v  RI 3 RI 2 u u ( ( x x,, y y,, z z ) ) x x ( ( u u,, v v ) ) S Object space (3D) Texture space (2D) 2. Parameterization Notion of parameterization

RI 3 RI 2 u v PiPi PiPi u i,v i 2. Parameterization Notion of parameterization Survey: [Floater 04]

2. Parameterization Demo: Constrained Parameterization Constrained Parameterization [Siggraph 1998 and 2001]

2. Parameterization Application: Gridding Grid generation for flow simulators Earth Decision Sciences startup (Paris, Houston, Rio, Dubai) Product: Gocad 3D modeler

3. Atlas Notion of atlas Conformal Map : C =  || grad(u|T) - i.grad(v|T) || 2 T  T Least Squares Conformal Maps [Siggraph 2002]

3. Atlas Application: Maya and Blender Least Squares Conformal Maps [Siggraph 02] Alias|Wavefront MAYA 3D modeler Alias|Wavefront MAYA 3D modeler

3. Atlas – « Tetris » Packing [ Nicolas Ray] Application: DirectX Lost Area

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

3. Atlas Applications: X-Mesh VSP-Technology startup Product: X-Mesh (Mesh Manipulation Library)

3. Atlas Demo: Normal-mapping

3. Atlas Application: Eden Games Courtesy of Eden Games Alone in the Dark

4. Periodic Global Parameterization (PGP) Create a « geographic coordinate system »

4. PGP The data: a scanned mesh

4. PGP Problems: arbitrary topology How can we handle closed surfaces ? How can we parameterize a cylinder ? Global Parameterization [Gu 2002]

4. PGP [Nicolas Ray]

4. PGP Affine and Complex Manifolds

4. PGP More geometry: Principal curvatures

4. PGP Integrated vector field K         .(p2-p1) Triangle integral

4. PGP Integrated vector field Edge equation Triangle equation

4. PGP Two problems What do we do for arbitrary topology ? How do we handle arbitrary vector fields ?

4. PGP Arbitrary topology cos(  ) cos(  )  sin(  )  U =         .(p2-p1) || 2

4. PGP Arbitrary Vector Fields Use local expression with rotated vectors

4. PGP Periodic Global Parameterization

4. PGP Overview of the algorithm Curvature tensor approx. [Cohen-Steiner 02] Vector field smoothing Periodic Global Parameterization Applications:Remeshing,T-Splines… Applications:Remeshing,T-Splines…

4. PGP Results Mesh-2-Spline conversion (demo)

4. PGP Results Remeshing

4. PGP results Remeshing

4. PGP Applications Microsoft Research Grant: Geometric Intelligence

Conclusions n Digital Geometry Processing: n A scientific challenge: –Solve the 3D representation problem ! n Many possible industrial applications –Video-games –CAD/CAM, reverse engineering –Oil exploration, FEM simulations