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### 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”)

Papers

- Wavelets on Irregular Point Setsby 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 Meshesby Guskov, Sweldens and Schroder(Siggraph 1999)
- Multiresolution Hierarchies On Unstructured Triangle Meshes by Kobbelt, Vorsatz, and Seidel (Compu. Geometry: Theory and Applications, 1999)

Outline

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

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

The Lifting Scheme

Split

The Lifting Scheme

Predict

The Lifting Scheme

Predict

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

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).

Wavelets on Spheres

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

Topological Earth Data

- Data is not smooth
- All bases performed equally poorly.

(picture from “Spherical Wavelets”)

15,000 coefficients 190,000 coefficients

Spherical Function: BRDF

- Face Based methods are terrible (Haar-based)
- Lifting doesn’t significantly help Butterfly.
- Linear does better than Quadratic.

19, 73, 205 coefficients

(pictures from “Spherical Wavelets”)

Up-Sampling Problems

- Smooth interpolating polynomials
- over-shooting
- added undulations.
- Linear interpolation isn’t smooth, but results are more intuitive.

Up-Sampling Problems

- Similar problems can occur on surfaces.

(picture from “Multiresolution Hierarchies On Unstructured Triangle Meshes”)

Wavelets on Spheres

- Lessons:
- Prediction is hard for arbitrary data sampling
- Maybe lifting isn’t necessary for very smooth subdivision schemes?
- Spheres are Special:
- Clearly defined DC.(??zeroth order rep, smooth rep??)
- Can easily make semi-regular mesh.

Outline

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

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)

Lifting is hard

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

- Can’t use butterfly sub-division.
- (picture from “Multiresolution Signal Processing For Meshes”)

Guskov et al.

- Need Smoother as part of algorithm

Guskov et al.

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

Guskov et al.

- Collapse the Edge

Guskov et al.

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

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”)

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”)

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

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”)

Are these wavelets?

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

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.

Outline

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

Editing

- Replacing conventional surface editing. (NURBS)

- (picture from “Multiresolution Signal Processing For Meshes” ,
- “Multiresolution Hierarchies On Unstructured Triangle Meshes”)

Compression

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

- (picture from “Normal Mesh Compression”)

Remeshing

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

- (picture from “Consistent Mesh Parameterizations”)

An Opportunity

- Analysis of the wavelet coefficients

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)

Feature Detection

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

- (picture from “Multiresolution Signal Processing For Meshes” )

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.

Solving PDEs

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

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.

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.

To Do List

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

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