Texture-Mapping Progressive Meshes

1
Texture-MappingProgressive Meshes
SIGGRAPH 2001
Pedro V. SanderSteven J. Gortler
John SnyderHugues Hoppe
Harvard University
Microsoft Research
2
Texture-MappingProgressive Meshes
SIGGRAPH 2001
Pedro V. SanderSteven J. Gortler
John SnyderHugues Hoppe
Harvard University
Microsoft Research
3
69,000faces
15,000faces
600faces
progressive mesh
4
69,000faces
15,000faces
600faces
progressive mesh
600faces
simplified mesh normal mapConveys detail of
original geometry
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Texture mapping
Authoring map a texture image onto a surface
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Texture mapping
Authoring map a texture image onto a surface
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Our problem

• Sample the surface signal into a texture (e.g.
  normal, displacement, BRDF, )
• Goals
• single texture for entire PM sequence
• quality metrics
• minimize appearance changes over PM
• efficiently distribute the texture samples

demo
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Our problem

• Sample the surface signal into a texture (e.g.
  normal, displacement, BRDF, )
• Goals
• single texture for entire PM sequence
• quality metrics
• minimize appearance changes over PM
• efficiently distribute the texture samples

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Simple approach chart-per-face
Soucy 96, Cignoni 98, Sander 00
500 faces
atlas of 500 triangles

• Define texture for single-LOD mesh.
• Cannot use texture for any simpler mesh!

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Our approach multi-face charts

• Partition mesh into charts.
• Simplify respecting chart topology. Cohen 98
• Same texture still applicable.

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Chart constraint 1Faces cannot span chart
boundaries
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Chart constraint 2Texture boundaries must be
straight
fine mesh
coarse mesh
texture map
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Our problem

• Sample the surface signal into a texture (e.g.
  normal, displacement, BRDF, )
• Goals
• single texture for entire PM sequence
• quality metrics
• minimize appearance changes over PM
• efficiently distribute the texture samples

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Parametrization quality metrics

• (1) Minimize texture deviation

(stricter than geometric error)
Cohen et al 98
demo
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Parametrization quality metrics

• (2) Minimize texture stretch

undersampling
high stretch ?
low stretch ?
2D texture
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Parametrization quality metrics

• (2) Minimize texture stretch

blurring
high stretch ?
low stretch ?
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Contributions Texture mapping PMs

• Chartification algorithm (considers
  simplification quality)
• Texture stretch metric (penalizes
  undersampling)
• Parametrization algorithm (minimizes stretch)
• PM optimization

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Approach
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
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Approach
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
20
Approach
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
21
Approach
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
22
Approach
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
23
Approach
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
24
Approach
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
25
Approach
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
26
Approach
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
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Approach Details
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
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Partition chart merging

• Assign each face to its own region.
• Merge regions in greedy fashion based on
• planarity distance2 to best-fitting plane
• compactness perimeter length2
• Preserves mesh connectivity.
• Maillot 93, Eck 95, Lee 98, Garland 01

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Partition boundary straightening

• Improves parametrization (boundary will be
  straight in texture domain)

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Approach Details
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
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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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Stretch-minimizing parametrization
2D texture domain
surface in 3D
L8(T) G L2(T) v(?2 G2)/2
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Stretch-minimization algorithm
demo

• Start with uniform parametrization.
• Perform several optimization iterations
• for each vertex, try random line searches.

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Parametrization example
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Comparison
Conformal parametrization ( MIPS, Floater) L2
2.28 L? 10.07
L2 stretch minimization L2 1.22 L? 2.13
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Comparison
Uniform parametrization L2 2.60 L? 12.52
L2 stretch minimization L2 1.22 L? 2.13
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Comparison
Area-preserving parametrization L2 1.57 L?
4.19
L2 stretch minimization L2 1.22 L? 2.13
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Example of stretch minimization
demo
ignoring stretch
minimizing stretch
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Approach Details
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images

• Half-edge collapses ordered by deviation
• Constrained simplification

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Approach Details
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
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Parametrization optimization

• Min ?M in PM stretch(M) deviation(M)
• Improves deviation over entire range.
• Improves stretch at coarser LODs(stretch was
  ignored during simplification).

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Approach Details
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
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Pack chart polygons

• NP-Hard problem.
• We designed a heuristic.

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Packing Heuristic

• Calculate the minimum bounding rectangle.
• Rotate chart to make rectangle vertical.

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Chart placement

• Sort chart rectangles by height.
• Sequentially place left-to-right and
  right-to-left.

Igarashi 01
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Approach Details
(1) partition original mesh into charts(2)
parametrize charts(3) resize chart polygons(4)
simplify mesh(5) optimize parametrization(6)
pack chart polygons(7) sample texture images
mipmap artifacts!
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Results(Measurements)

• Scale charts to meet low-stretch requirement.
• Stretch efficiency3D surface area / 2D chart
  area
• Packing efficiency2D chart area / texture domain
  area
• Texture efficiencystretch efficiency packing
  efficiency3D area / texture domain area

/
/
/
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Results

• Efficiencies on fine meshes

Models bunny parasaur horse hand
faces in Mn 69,630 43,866 96,956 60,856
charts 75 75 120 60
uniform param. stretch efficiency 0.63 0.003 0.61 0.11
our stretch efficiency 0.84 0.63 0.80 0.68
packing efficiency 0.67 0.63 0.70 0.62
texture efficiency 0.56 0.40 0.56 0.42
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Results across PM
stretch
deviation
demo
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Demos
demo
demo
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Summary

• Automatic PM parametrization scheme.
• Optimizes both deviation and stretch.
• Novel stretch metric prevents undersampling at
  all locations and in all directions.
• Robust parametrization algorithm.

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Future work

• Use hierarchical parametrization.
• Constrain anisotropy.
• Consider content of texture signal.
• Address mip-mapping problems.