Signal-Specialized Parametrization

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

• Microsoft Research1
• Harvard University2

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
Texture mapping two scenarios
Authoring map a texture image onto a surface
normal map
normal signal
4
Goal
(128x128 texture)
Geometry-based parametrization
Signal-specialized parametrization
demo
5
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

6
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.

7
Parametrization
2D texture domain
surface in 3D
8
Parametrization
2D texture domain
surface in 3D

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

9
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
10
Signal stretch metric
domain
surface
f
h
g
signal

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

11
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
h
e
h
Mh JeT Mh Je
13
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
14
Boundary optimization

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

Optimized boundary
Bounding square boundary
15
Floater
Geometric stretch
Signal stretch
16
Geometric stretch
Signal stretch
17
Hierarchical Parametrization algorithm

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

18
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.

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.