EGRW 2002 Signal-Specialized Signal-Specialized Parametrization Parametrization 1,2 Pedro V. Sander Pedro V. Sander1,2 John John Snyder Snyder11 22 Steven J. Gortler Steven J. Gortler Hugues Hugues Hoppe Hoppe11 Microsoft Microsoft Research Research11 Harvard Harvard University University22 Motivation Motivation Powerful Powerful rasterization

rasterization hardware hardware (GeForce3,) (GeForce3,) multi-texturing, multi-texturing, programmable programmable Many Many types types of of signals: signals: texture texture map map bump bump map map displacement displacement map map irradiance irradiance transfer transfer

(color) (color) (normal) (normal) (geometry) (geometry) (spherical (spherical harmonics) harmonics) Texture Texture mapping: mapping: two two scenarios scenarios Authoring: map a texture image onto a surface normal normal map map normal normal signal signal Sampling: store an existing surface signal Goal (128x128 texture)

Geometry-based parametrization Signal-specialized parametrization demo demo Previous Previous work: work: Signal-independent Signal-independent parametrization parametrization Angle-preserving Angle-preservingmetrics metrics Eck Ecketetal. al.1995 1995 Floater Floater1997 1997

Hormann Hormannand andGreiner Greiner1999 1999 Hacker Hackeretetal. al.2000 2000 Other Othermetrics metrics Maillot Maillotetetal. al.1993 1993 Levy Levyand andMallet Mallet1998 1998

Sander Sanderetetal. al.2001 2001 Previous Previous work: work: Signal-specialized Signal-specialized parametrization parametrization Terzopoulos Terzopoulos and and Vasilescu Vasilescu 1991 1991 Approximate Approximate 2D 2D image image with with warped warped grid. grid.

Hunter Hunter and and Cohen Cohen 2000 2000 Compress Compress image image as as set set of of texture-mapped texture-mapped rectangles. rectangles. Sloan Sloan et et al. al. 1998 1998 Warp Warp texture texture domain domain onto onto itself. itself. Parametrization Parametrization linear map 2D texture domain surface in 3D singular values: ,

Parametrization Parametrization linear map T 2D texture domain surface in 3D singular values: , length-preserving length-preserving ((isometric isometric)) angle-preserving angle-preserving ((conformal conformal)) area-preserving area-preserving == == 11 == == 11

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

signal geometric geometric stretch: stretch: E Eff == ff22 ++ ff22 == tr(M tr(Mff)) signal signal stretch: stretch: E Ehh == hh22 ++ hh22 == tr(M tr(Mhh)) Integrated Integrated metric metric tensor tensor (IMT) (IMT) computed computed over over each each triangle triangle using using numerical numerical integration.

integration. 2x2 2x2 symmetric symmetric matrix matrix recomputed recomputed for for affinely affinely warped warped triangle triangle using using simple simple transformation transformation rule. rule. No No need need to to reintegrate reintegrate the the signal. signal. D D

Signal e h h Mh = JeT Mh Je Deriving Deriving signal signal stretch stretch Taylor Taylor expansion expansion to to signal signal approximation approximation error error locally locally constant constant reconstruction reconstruction asymptotically asymptotically dense dense sampling

sampling original reconstructed signal approximation error Boundary Boundary optimization optimization Optimize Optimize boundary boundary vertices vertices Texture Texture domain domain grows grows to to infinity. infinity. Solution Solution Multiply Multiply by by domain domain area

area (scale (scale invariant): invariant): E Ehh== E Ehh ** area(D) area(D) == tr(M tr(Mhh(S)) (S)) ** area(D) area(D) Fixed boundary Optimized boundary Boundary Boundary optimization optimization Grow Grow to to bounding bounding square/rectangle: square/rectangle: Minimize Minimize E Ehh Constrain Constrain vertices vertices to to stay stay inside

inside bounding bounding square. square. Optimized boundary Bounding square boundary Floater Floater Geometric Geometric stretch stretch Signal Signal stretch stretch Hierarchical Hierarchical Parametrization Parametrization algorithm algorithm Advantages: Advantages:

Faster. Faster. Finds Finds better better minimum minimum (nonlinear (nonlinear metric). metric). Algorithm: Algorithm: Construct Construct PM PM.. Parametrize Parametrize coarse-to-fine. coarse-to-fine. demo demo Iterated Iterated multigrid multigrid strategy strategy Problem: Problem:

Coarse Coarse mesh mesh does does not not capture capture signal signal detail. detail. Traverse Traverse PM PM fine-to-coarse. fine-to-coarse. For For each each edge edge collapse, collapse, sum sum up up metric metric tensors tensors and and store store them them at at each each face. face.

Traverse Traverse PM PM coarse-to-fine. coarse-to-fine. Optimize Optimize signal-stretch signal-stretch of of introduced introduced vertices vertices using using the the stored stored metric metric tensors. tensors. Repeat Repeat last last 22 steps steps until until convergence. convergence. Use Use bounding bounding rectangle

rectangle optimization optimization on on last last iteration. iteration. Results Results (64x64 texture) Scanned Scanned Color Color Geometric stretch Signal stretch Painted Painted Color Color Geometric stretch Signal stretch 128x128 texture - multichart Precomputed Precomputed Radiance Radiance Transfer

Transfer Geometric stretch Signal stretch 25D signal 256x256 texture from [Sloan et al. 2002] Normal Normal Map Map demo demo Geometric stretch Signal stretch 128x128 texture - multichart Summary Summary Many Many signals signals are are unevenly

unevenly distributed distributed over over area area and and direction. direction. Signal-specialized Signal-specialized metric metric Integrates Integrates signal signal approximation approximation error error over over surface surface Each Each mesh mesh face face is is assigned assigned an an IMT. IMT. Affine Affine transformation

transformation rules rules can can exactly exactly transform transform IMTs. IMTs. Hierarchical Hierarchical parametrization parametrization algorithm algorithm IMTs IMTs are are propagated propagated fine-to-coarse. fine-to-coarse. Mesh Mesh is is parametrized parametrized coarse-to-fine. coarse-to-fine. Boundary Boundary can can be be optimized optimized during during the the process. process.

Significant Significant increase increase in in quality quality for for same same texture texture size. size. Texture Texture size size reduction reduction up up to to 4x 4x for for same same quality. quality. Future Future work work

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