Signal-Specialized Parametrization

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Signal-SpecializedParametrization
EGRW 2002
Steven J. Gortler2 Hugues Hoppe1
Pedro V. Sander1,2 John Snyder1

• Microsoft Research1
• Harvard University2

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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)

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Texture mapping two scenarios
Authoring map a texture image onto a surface
normal map
normal signal
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Goal
(128x128 texture)
Geometry-based parametrization
Signal-specialized parametrization
demo
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Previous workSignal-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

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Previous workSignal-specialized parametrization

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

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Parametrization
2D texture domain
surface in 3D
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Parametrization
2D texture domain
surface in 3D

• length-preserving (isometric) ? G 1
• angle-preserving (conformal) ? G
• area-preserving ? G 1

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Geometric stretch metric
2D texture domain
surface in 3D
Geometric stretch ?2 G2 tr(M(T)) where
metric tensor M(T) J(T)T J(T) E(S) surface
integral of geometric stretch
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Signal stretch metric
domain
surface
f
h
g
signal

• geometric stretch Ef ?f2 Gf2 tr(Mf)
• signal stretch Eh ?h2 Gh2 tr(Mh)

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Integrated metric tensor (IMT)

• computed over each triangle using numerical
  integration.
• 2x2 symmetric matrix
• recomputed for affinely warped triangle using
  simple transformation rule. No need to
  reintegrate the signal.

D
D
Signal
h
e
h
Mh JeT Mh Je
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Deriving signal stretch

• Taylor expansion to signal approximation error
• locally constant reconstruction
• asymptotically dense sampling

original
reconstructed
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Boundary optimization

• Optimize boundary vertices Texture domain
  grows to infinity.
• Solution Multiply by domain area (scale
  invariant) Eh Eh area(D) tr(Mh(S))
  area(D)

Fixed boundary
Optimized boundary
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Boundary optimization

• Grow to bounding square/rectangle Minimize
  Eh Constrain vertices to stay inside bounding
  square.

Optimized boundary
Bounding square boundary
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Floater
Geometric stretch
Signal stretch
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Hierarchical Parametrization algorithm

• Advantages
• Faster.
• Finds better minimum (nonlinear metric).
• Algorithm
• Construct PM.
• Parametrize coarse-to-fine.

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Iterated multigrid strategy

• ProblemCoarse 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.

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Results
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(64x64 texture)
ScannedColor
Geometric stretch
Signal stretch
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Painted Color
Geometric stretch
Signal stretch
128x128 texture - multichart
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Precomputed Radiance Transfer
Geometric stretch
Signal stretch
25D signal 256x256 texture from Sloan et al.
2002
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Normal Map
demo
Geometric stretch
Signal stretch
128x128 texture - multichart
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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.

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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.