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Signal-Specialized Parametrization Microsoft Research 1 Harvard University 2 Microsoft Research 1 Harvard University 2 Steven J. Gortler 2 Hugues Hoppe 1 Steven J. Gortler 2 Hugues Hoppe 1 Pedro V. Sander 1,2 John Snyder 1 Pedro V. Sander 1,2 John Snyder 1 EGRW 2002

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

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

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(128x128 texture) Geometry-basedparametrization Signal-specializedparametrization demo GoalGoal

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

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

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ParametrizationParametrization 2D texture domain surface in 3D linear map singular values: γ, Γ

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ParametrizationParametrization length-preserving (isometric) γ = Γ = 1 length-preserving (isometric) γ = Γ = 1 angle-preserving (conformal) γ = Γ angle-preserving (conformal) γ = Γ area-preserving γ Γ = 1 area-preserving γ Γ = 1 length-preserving (isometric) γ = Γ = 1 length-preserving (isometric) γ = Γ = 1 angle-preserving (conformal) γ = Γ angle-preserving (conformal) γ = Γ area-preserving γ Γ = 1 area-preserving γ Γ = 1 2D texture domain surface in 3D linear map TT singular values: γ, Γ

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

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Signal stretch metric f hg n geometric stretch: E f = γ f 2 + Γ f 2 = tr(M f ) n signal stretch: E h = γ h 2 + Γ h 2 = tr(M h ) n geometric stretch: E f = γ f 2 + Γ f 2 = tr(M f ) n signal stretch: E h = γ h 2 + Γ h 2 = tr(M h ) domainsurface signal

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l Taylor expansion to signal approximation error n locally constant reconstruction n asymptotically dense sampling l Taylor expansion to signal approximation error n locally constant reconstruction n asymptotically dense sampling Deriving signal stretch signal approximation error originalreconstructed

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Integrated metric tensor (IMT) l 2x2 symmetric matrix l computed over each triangle using numerical integration. l recomputed for affinely warped triangle using simple transformation rule. No need to reintegrate the signal. l 2x2 symmetric matrix l computed over each triangle using numerical integration. l recomputed for affinely warped triangle using simple transformation rule. No need to reintegrate the signal. he h´h´h´h´ D´D´D´D´ D Signal M h´ = J e T M h J e

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Boundary optimization l Optimize boundary vertices Texture domain grows to infinity. l Solution Multiply by domain area (scale invariant): E h ´= E h * area(D) = tr(M h (S)) * area(D) l Optimize boundary vertices Texture domain grows to infinity. l Solution Multiply by domain area (scale invariant): E h ´= E h * area(D) = tr(M h (S)) * area(D) Fixed boundary Optimized boundary

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Boundary optimization l Grow to bounding square/rectangle: Minimize E h Constrain vertices to stay inside bounding square. Optimized boundary Bounding square boundary

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Geometric stretch FloaterFloater Signal stretch

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Geometric stretch Signal stretch

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Hierarchical Parametrization algorithm l Advantages: n Faster. n Finds better minimum (nonlinear metric). l Algorithm: n Construct PM. n Parametrize coarse-to-fine. l Advantages: n Faster. n Finds better minimum (nonlinear metric). l Algorithm: n Construct PM. n Parametrize coarse-to-fine. demo

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

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ResultsResults

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Geometric stretch Signal stretch (64x64 texture) Scanned Color

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Painted Color 128x128 texture - multichart Geometric stretch Signal stretch

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

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

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

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

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