Signal-Specialized Parametrization

1. EGRW 2002 Signal-SpecializedParametrization Microsoft Research1 Harvard University2 Pedro V. Sander1,2 John Snyder1 Steven J. Gortler2 Hugues Hoppe1

2. Motivation Powerful rasterization hardware (GeForce3,…) • multi-texturing, programmable Many types of signals: • texture map (color) • bump map (normal) • displacement map (geometry) • irradiance transfer (spherical harmonics) • …

3. Sampling: store an existing surface signal Texture mapping: two scenarios Authoring: map a texture image onto a surface normal map normal signal

4. (128x128 texture) Goal Geometry-based parametrization Signal-specialized parametrization demo

5. Previous work:Signal-independent parametrization • Angle-preserving metrics • Eck et al. 1995 • Floater 1997 • Hormann and Greiner 1999 • Hacker et al. 2000 • Other metrics • Maillot et al. 1993 • Levy and Mallet 1998 • Sander et al. 2001

6. Previous work:Signal-specialized parametrization • Terzopoulos and Vasilescu 1991Approximate 2D image with warped grid. • Hunter and Cohen 2000Compress image as set of texture-mapped rectangles. • Sloan et al. 1998Warp texture domain onto itself.

7. linear map singular values: γ , Γ g G Parametrization 2D texture domain surface in 3D

8. T linear map singular values: γ , Γ Parametrization • length-preserving (isometric) γ = Γ= 1 • angle-preserving (conformal) γ = Γ • area-preserving γΓ= 1 2D texture domain surface in 3D

9. T linear map singular values: γ , Γ high stretch! Geometric stretch metric 2D texture domain surface in 3D Geometric stretch = γ2 + Γ2 = tr(M(T)) where metric tensor M(T) = J(T)T J(T) E(S) = surface integral of geometric stretch

10. Signal stretch metric domain surface f h g signal • geometric stretch: Ef= γf2 + Γf2 = tr(Mf) • signal stretch: Eh = γh2 + Γh2 = tr(Mh)

11. signal approximation error Deriving signal stretch • Taylor expansion to signal approximation error • locally constant reconstruction • asymptotically dense sampling original reconstructed

12. Integrated metric tensor (IMT) • 2x2 symmetric matrix • computed over each triangle using numerical integration. • recomputed for affinely warped triangle using simple transformation rule.No need to reintegrate the signal. D D´ Signal e h h´ Mh´ = JeT Mh Je

13. Boundary optimization • Optimize boundary verticesTexture domain grows to infinity. • SolutionMultiply by domain area (scale invariant): Eh´= Eh * area(D) = tr(Mh(S)) * area(D) Fixed boundary Optimized boundary

14. Boundary optimization • Grow to bounding square/rectangle: Minimize EhConstrain vertices to stay inside bounding square. Optimized boundary Bounding square boundary

15. Floater Geometric stretch Signal stretch

16. Geometric stretch Signal stretch

17. demo Hierarchical Parametrization algorithm • Advantages: • Faster. • Finds better minimum (nonlinear metric). • Algorithm: • Construct PM. • Parametrize “coarse-to-fine”.

18. Iterated multigrid strategy • Problem:Coarse mesh does not capture signal detail. • Traverse PM fine-to-coarse. For each edge collapse, sum up metric tensors and store them at each face. • Traverse PM coarse-to-fine. Optimize signal-stretch of introduced vertices using the stored metric tensors. • Repeat last 2 steps until convergence. • Use bounding rectangle optimization on last iteration.

19. Results

20. (64x64 texture) ScannedColor Geometric stretch Signal stretch

21. Painted Color Geometric stretch Signal stretch 128x128 texture - multichart

22. Precomputed Radiance Transfer Geometric stretch Signal stretch 25D signal – 256x256 texture from [Sloan et al. 2002]

23. Normal Map demo Geometric stretch Signal stretch 128x128 texture - multichart

24. Summary • Many signals are unevenly distributed over area and direction. • Signal-specialized metric • Integrates signal approximation error over surface • Each mesh face is assigned an IMT. • Affine transformation rules can exactly transform IMTs. • Hierarchical parametrization algorithm • IMTs are propagated fine-to-coarse. • Mesh is parametrized coarse-to-fine. • Boundary can be optimized during the process. Significant increase in quality for same texture size. Texture size reduction up to 4x for same quality.

25. Future work • Metrics for locally linear reconstruction. • Parametrize for specific sampling density. • Adapt mesh chartification to surface signal. • Propagate signal approximation error through rendering process. • Perceptual measures.