Surface Compression

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Surface Compression with Geometric Bandelets
Gabriel Peyré Stéphane Mallat
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Outline

• Why Discrete Multiscale Geometry?
• Image-based Surface Processing
• Geometry in the Wavelet Domain
• Moving from 2D to 1D
• The Algorithm in Details
• Results

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Geometry of Surfaces Creation
Clay modeling
Low-poly modeling
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Geometry of Surfaces Rendering
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Geometry of SurfacesMesh processing
Robust Moving Least-squares Fitting with Sharp
Features Fleishman et Al. 05
Anisotropic Remeshing Alliez et Al. 03
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Geometry is Discrete
continous discrete multiscale geometry
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Geometry is Multiscale?
continous discrete multiscale geometry
Surface simplification Garland Heckbert 97
Normal meshes Guskov et Al. 00
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Geometry is Multiscale!

• Edge extraction is an ill-posed problem.
• Localization is not needed for compression!

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Geometry is not Defined by Sharp Features
Edge localization Ohtake et Al. 04 Ridge-valley
lines on meshes via implicit surface fitting.
Semi-sharp features DeRose et Al. 98
Subdivision Surfaces in Character Animation
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Outline

• Why Discrete Multiscale Geometry?
• Image-based Surface Processing
• Geometry in the Wavelet Domain
• Moving from 2D to 1D
• The Algorithm in Details
• Results

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Geometry images Gu et Al.
irregular mesh ? 2D array of points
r,g,b x,y,z
cut
parameterize

• No connectivity information.
• Simplify and accelerate hardware rendering.
• Allows application of image-based compression
  schemes.

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Our Functional Model
2D GIM (lit)
3D model
Uniformly regular areas Sharp features
Smoothed features
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Outline

• Why Discrete Multiscale Geometry?
• Image-based Surface Processing
• Geometry in the Wavelet Domain
• Moving from 2D to 1D
• The Algorithm in Details
• Results

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Hierarchical Cascad

• Orthogonal dilated filters cascad
• Proposition to continue the cascad.

Geometric transform
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What is a wavelet transform?
D
H
V
Wavelet transform

• Decompose an image at dyadic scales.
• 3 orientations by scales H/V/D.
• Compact representation few high coefficients.
• But still high coefficients near singularities.

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Outline

• Why Discrete Multiscale Geometry?
• Image-based Surface Processing
• Geometry in the Wavelet Domain
• Moving from 2D to 1D
• The Algorithm in Details
• Results

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Some insights about bandelets

• Moto wavelets transform is cool, re-use it!
• Goal remove the remaining high wavelet
  coefficients.
• Hope exploit the anisotropic regularity of the
  geometry.
• Tool 2D anisotropy become isotropic in 1D.

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Construction of this Reordering

• Choose a direction
• Project pointsorthogonally on
• Report values on 1D axis
• Resulting 1D signal

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Choosing the square and the direction

• Too big direction deviates from
  geometry
• How to choose 1D wavelettransform
• Too much high coefficients!

threshold T
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Choosing the Squareand the Direction

• Bad direction direction deviates
  from geometry
• Still too much high coefficients!

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Choosing the Squareand the Direction

• Correct direction direction matches
  the geometry
• Nearly no high coefficients!

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Outline

• Why Discrete Multiscale Geometry?
• Image-based Surface Processing
• Geometry in the Wavelet Domain
• Moving from 2D to 1D
• The Algorithm in Details
• Results

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The Algorithm in 10 Steps

• (1) Geometry Image
• (2) 2D Wavelet Transf.
• (3) Dyadic Subdivision
• (4) Extract Sub-square
• (5) Sample Geometry
• (6) Project Points
• (7) 1D Wavelet Transf.
• (8) Select Geometry
• (9) Output Coefficients
• (10) Build Quadtree

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The Algorithm in 10 Steps

• (1) Geometry Image
• (2) 2D Wavelet Tran
• (3) Dyadic Subdivision
• (4) Extract Sub-square
• (5) Sample Geometry
• (6) Project Points
• (7) 1D Wavelet Transf.
• (8) Select Geometry
• (9) Output Coefficients
• (10) Build Quadtree

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The Algorithm in 10 Steps

• (1) Geometry Image
• (2) 2D Wavelet Transf.
• (3) Dyadic Subdivis
• (4) Extract Sub-square
• (5) Sample Geometry
• (6) Project Points
• (7) 1D Wavelet Transf.
• (8) Select Geometry
• (9) Output Coefficients
• (10) Build Quadtree

Zoom on D
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The Algorithm in 10 Steps

• (1) Geometry Image
• (2) 2D Wavelet Transf.
• (3) Dyadic Subdivis
• (4) Extract Sub-sq
• (5) Sample Geometry
• (6) Project Points
• (7) 1D Wavelet Transf.
• (8) Select Geometry
• (9) Output Coefficients
• (10) Build Quadtree

Sub-square
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The Algorithm in 10 Steps

• (1) Geometry Image
• (2) 2D Wavelet Transf.
• (3) Dyadic Subdivision
• (4) Extract Sub-square
• (5) Sample Geometr
• (6) Project Points
• (7) 1D Wavelet Transf.
• (8) Select Geometry
• (9) Output Coefficients
• (10) Build Quadtree

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The Algorithm in 10 Steps

• (1) Geometry Image
• (2) 2D Wavelet Transf.
• (3) Dyadic Subdivision
• (4) Extract Sub-square
• (5) Sample Geometry
• (6) Project Points
• (7) 1D Wavelet Transf.
• (8) Select Geometry
• (9) Output Coefficients
• (10) Build Quadtree

1D Signal
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The Algorithm in 10 Steps

• (1) Geometry Image
• (2) 2D Wavelet Transf.
• (3) Dyadic Subdivision
• (4) Extract Sub-square
• (5) Sample Geometry
• (6) Project Points
• (7) 1D Wavelet Tran
• (8) Select Geometry
• (9) Output Coefficients
• (10) Build Quadtree

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The Algorithm in 10 Steps

• (1) Geometry Image
• (2) 2D Wavelet Transf.
• (3) Dyadic Subdivision
• (4) Extract Sub-square
• (5) Sample Geometry
• (6) Project Points
• (7) 1D Wavelet Transf.
• (8) Select Geometry
• (9) Output Coefficients
• (10) Build Quadtree

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The Algorithm in 10 Steps

• (1) Geometry Image
• (2) 2D Wavelet Transf.
• (3) Dyadic Subdivision
• (4) Extract Sub-square
• (5) Sample Geometry
• (6) Project Points
• (7) 1D Wavelet Transf.
• (8) Select Geometry
• (9) Output Coefficie
• (10) Build Quadtree

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The Algorithm in 10 Steps

• (1) Geometry Image
• (2) 2D Wavelet Transf.
• (3) Dyadic Subdivision
• (4) Extract Sub-square
• (5) Sample Geometry
• (6) Project Points
• (7) 1D Wavelet Transf.
• (8) Select Geometry
• (9) Output Coefficients
• (10) Build Quadtree

Zoom on D

• Dont use every dyadic square
• Compute an optimal segmentation into squares.
• Fast pruning algorithm (see paper).

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What does bandelets look like?

• Transform decomposition on an orthogonal basis.
• Basis functions are elongated bandelets.
• The transform adapts itself to the geometry.

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Transform Coding in a Bandelet Basis

• Bandelet coefficients are quantized and entropy
  coded.
• Quadtree segmentation and geometry is coded.
• Possibility to use more advanced image coders
  (e.g. JPEG2000).

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Outline

• Why Discrete Multiscale Geometry?
• Image-based Surface Processing
• Geometry in the Wavelet Domain
• Moving from 2D to 1D
• The Algorithm in Details
• Results

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ResultsSharp features
Original
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ResultsMore complex features
Original
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Blurred Features
Original
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Spherical Geometry Images
Original
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Spherical Geometry Images
Original
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Conclusion

• Approach re-use wavelet expansion.
• Contribution bring geometry into the multiscale
  framework.
• Results improvement over wavelets even for
  blurred features.
• Extension other maps (normals, BRDF, etc.) and
  other processings (denoising, deblurring, etc.).