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Signal-Specialized Parametrization

This research explores signal-specialized parametrization techniques for texture mapping, addressing issues of optimization and signal approximation errors. The study delves into geometric stretch, signal stretch, hierarchical parametrization algorithms, and boundary optimization, enhancing texture quality and reducing file sizes. Results demonstrate significant advancements in texture mapping quality and efficiency.

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Signal-Specialized Parametrization

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  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.

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