Wavelets on Surfaces

1
Wavelets on Surfaces
In partial fulfillment of the Area Exam
doctoral requirements

• By Samson Timoner
• May 8, 2002
• (picture from Wavelets on Irregular Point Sets)

2
Papers

• Wavelets on Irregular Point Sets by Daubechies
  Guskov, Schroder and Sweldens (Trans. R. Soc.
  1999)
• Spherical Wavelets Efficiently Representation
  Functions on the Sphere by Schroder and Sweldens
• The Lifting Scheme Construction of second
  generation wavelets by Sweldens

• Multiresolution Signal Processing For Meshes by
  Guskov, Sweldens and Schroder (Siggraph 1999)
• Multiresolution Hierarchies On Unstructured
  Triangle Meshes by Kobbelt, Vorsatz, and Seidel
  (Compu. Geometry Theory and Applications, 1999)

3
Outline

• Wavelets
• The Lifting Scheme
• Extending the Lifting Scheme
• Application Wavelets on Spheres
• Wavelets on Triangulated Surfaces
• Applications

4
Wavelets

• Multi-resolution representation.
• Basis functions (low pass filter).
• Detail Coefficients (high pass filter).
• We have bi-orthogonality between the detail
  coefficients and the basis-coefficients
• Vanishing Moments

5
The Lifting Scheme

• Split

6
The Lifting Scheme

• Predict

7
The Lifting Scheme

• Predict

8
The Lifting Scheme

• Update

1/8-1,2,6,2,-1, ½-1,2,-1
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The Lifting Scheme

• Introduced Prediction
• Translated and Scaled one filter.
• We have bi-orthogonality between the detail
  coefficients and the basis-coefficients
• 2 Vanishing Moments (mean and first)
• More Details

10
Irregularly Sampled Points
Split
Predict
Update
11
Irregularly Sampled Points

• Filters are no longer translations of each
  other.
• Detail coefficients indicate different
  frequencies.
• Perhaps it is wiser not to select every other
  point?
• You can show bi-orthogonality(by vanishing
  moments).

12
Wavelets on Spheres

• Sub-division on edges
• Same steps
• Split
• Predict
• Update

13
Topological Earth Data
15,000 coefficients 190,000 coefficients

• Data is not smooth
• All bases performed equally poorly.
• (picture from
  Spherical Wavelets)

14
Spherical Function BRDF
19, 73, 205 coefficients (pictures from
Spherical Wavelets)

• Face Based methods are terrible (Haar-based)
• Lifting doesnt significantly help Butterfly.
• Linear does better than Quadratic.

15
Up-Sampling Problems

• Smooth interpolating polynomials
• over-shooting
• added undulations.
• Linear interpolation isnt smooth, but results
  are more intuitive.

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Up-Sampling Problems

• Similar problems can occur on surfaces.
• (picture from Multiresolution Hierarchies On
  Unstructured Triangle Meshes)

17
Wavelets on Spheres

• Lessons
• Prediction is hard for arbitrary data sampling
• Maybe lifting isnt necessary for very smooth
  subdivision schemes?
• Spheres are Special
• Clearly defined DC.(??zeroth order rep, smooth
  rep??)
• Can easily make semi-regular mesh.

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Outline

• Wavelets The Lifting Scheme
• Wavelets on Triangulated Surfaces
• Up-sampling problems
• Applications

19
Triangulated Surfaces

• It is not clear how to design updates that make
  the wavelet transform numerically stable.
  (Wavelets on Irregular Point Sets)
• It is difficult to design filters which after
  iteration yield smooth surfaces. (Wim Sweldens
  in personal communication)

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Lifting is hard

• Prediction step is hard.
• If you zero detail coefficients, you should get a
  fair surface.

• Cant use butterfly sub-division.
• (picture from Multiresolution Signal Processing
  For Meshes)

21
Guskov et al.

• Need Smoother as part of algorithm

22
Guskov et al.

• Point Selection
• Choose Smallest Edge
• Remove one vertex in each level

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Guskov et al.

• Collapse the Edge

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Guskov et al.

• Prediction
• Re-introduce the Edge.
• Minimize Dihedral Angles
• Detail Vector Difference vector
• (tangent plane coordinates)

25
Guskov et al.

• Quasi-Update
• Smooth surrounding
• points (minimize
• dihedral angles)

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Guskov et al.

• Rough order of spatial frequencies.
• Detail coefficients look meaningful.
• Simple Smoothing No overshooting errors.
• No Guarantee of vanishing moments.
• No Guarantee of bi-orthogonality.
• (picture from Multiresolution Signal Processing
  For Meshes)

27
Guskov et al.

• Editing

(picture from Multiresolution Signal Processing
For Meshes)
28
Kobbelt et al.

• Double Laplacian Smoother (thin plate energy
  bending minimization).
• Solving PDE is slow!
• Instead, solve hierarchically.
• (picture from Multiresolution Hierarchies On
  Unstructured Triangle Meshes)

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Kobbelt et al.

• Many vertices in each step (smallest edges first)
• Prediction Step location to minimize smoothing.
• Detail Perpendicular vector to local coordinate
  system.
• Update Smooth surrounding points

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Kobbelt et al.

• Rough order of spatial frequencies.
• Fast O(mn) with m levels, n verticies.
• Many coefficients.
• Bi-orthogonality?
• Locality of filters?
• (picture from Multiresolution Hierarchies On
  Unstructured Triangle Meshes)

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Are these wavelets?

• Mathematically No.
• Bi-orthogonality
• Too many coefficients.

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Is this representation useful?

• Patches do not wiggle they remain in roughly the
  same position during down-sampling.
• Smooth regions stay smooth.
• Small detail coefficients.
• Meaningful detail coefficients.

33
Outline

• Wavelets The Lifting Scheme
• Wavelets on Triangulated Surfaces
• Applications
• Existing
• Opportunities for new research

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Editing

• Replacing conventional surface editing. (NURBS)

(picture from Multiresolution Signal Processing
For Meshes , Multiresolution Hierarchies On
Unstructured Triangle Meshes)
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Feature Enhancement

• For show only.

(picture from Multiresolution Signal Processing
for Meshes)
36
Compression

• 549 Bytes(54e-4) 1225 Bytes(20e-4) 3037
  Bytes(8e-4) 18111 Bytes(1.7e-4) Original

(picture from Normal Mesh Compression)
37
Remeshing

• Go to low-resolution (to keep topology) and then
  sub-divide to restore original detail.

(picture from Consistent Mesh Parameterizations)
38
An Opportunity

• Analysis of the wavelet coefficients

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Statistics across Meshes

• Use identical
• Triangulations across objects.
• Look at statistics on detail coefficients rather
  than on points.
• No global alignment problems.
• No local alignment problems.

(I generated these images)
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Feature Detection

• Should be able to find signature hierarchical
  detail coefficients.
• Hard with different triangulations.

(picture from Multiresolution Signal Processing
For Meshes )
41
Acknowledgements

• Professor White for suggesting the topic.
• Wim Sweldens for responding to my e-mails.
• Mike Halle and Steve Pieper for providing
  background information on the graphics community.
• Thank you all for coming today.

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The Lifting Scheme

• Mathematics

Low Pass Filter 1/8(-1,2,6,2,-1) High Pass
Filter ½(-1,2,1)
Back
43
Solving PDEs

• Roughly, one can change the update and prediction
  step to have vanishing moments in the new
  orthogonality relationship.

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Guskov et al.

• Remove vertices in smoothest regions first.
• Half-Edge Collapse to remove one vertex
• Add vertex in, minimizing second order
  difference.
• Smooth neighbors using same minimization
• Detail coefficients are the movements between
  initial locations and final locations.

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Kobbelt et al.

• Select a fraction of the vertices.
• Do half-edge collapses to remove the vertices.
• Find a local parameterization around each vertex.
• Add the vertex back in, minimizing the bending
  energy of the surface (Laplacian).
• The detail vector is given by the coordinates of
  the point in the local coordinate system and a
  perpendicular height.

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To Do List

• Check Sphere coefficients
• Sweldons Quote change to published quote.
• Edit Guskov et al
• Compression Page comments underneath.