1 Wavelets on Surfaces By Samson Timoner May 8, 2002 (picture from “Wavelets on Irregular Point Sets”) In partial fulfillment of the “Area Exam” doctoral.

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Wavelets on Surfaces

By Samson Timoner

May 8, 2002

(picture from “Wavelets on Irregular Point Sets”)

In partial fulfillment of the “Area Exam” doctoral requirements

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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)

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Outline

• Wavelets– The Lifting Scheme– Extending the Lifting Scheme– Application: Wavelets on Spheres

• Wavelets on Triangulated Surfaces

• Applications

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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

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The Lifting Scheme

Split

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The Lifting Scheme

Predict

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The Lifting Scheme

Predict

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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

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Irregularly Sampled PointsSplit Predict

Update

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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).

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Wavelets on Spheres• Sub-division on edges

• Same steps– Split– Predict– Update

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Topological Earth Data

• Data is not smooth

• All bases performed equally poorly. (picture from “Spherical Wavelets”)

15,000 coefficients 190,000 coefficients

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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”)

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• Smooth interpolating polynomials– over-shooting – added undulations.

• Linear interpolation isn’t smooth, but results are more intuitive.

Up-Sampling Problems

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• Similar problems can occur on surfaces.

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

Up-Sampling Problems

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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.

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Outline

Wavelets: The Lifting Scheme

• Wavelets on Triangulated Surfaces– Up-sampling problems

• Applications

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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.

•Can’t use butterfly sub-division.

(picture from “Multiresolution Signal Processing For Meshes”)

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Guskov et al.

• Need Smoother as part of algorithm

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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)

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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”)

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Guskov et al.

• Editing

(picture from “Multiresolution Signal Processing For Meshes”)

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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.

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Outline

Wavelets: The Lifting SchemeWavelets 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”)

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Compression

(picture from “Normal Mesh Compression”)

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

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Remeshing

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

(picture from “Consistent Mesh Parameterizations”)

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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” )

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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

...

...

...

...

...

...

j,2k j,2k+1 j,2k+2

S D S

S S

-1/2-1/2 -1/2 -1/2

1/41/4 1/4 1/4

D j-1,k j,2k+2 j,2k

j-1,k j-1,k j-1,k+1

S S S

Low Pass Filter: 1/8(-1,2,6,2,-1)

High Pass Filter: ½(-1,2,1)Back

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Solving PDEs

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

DxDpLk -DL

VuVx

VuVxuLxVVL

uufuLx

fLx

222

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)(, :Examples

,Let

,, *, if

,,

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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.