Wavelets on surfaces
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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 Sets by Daubechies Guskov, Schroder and Sweldens (Trans. R. Soc. 1999)

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Wavelets on surfaces

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

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

Outline

  • Wavelets

    • The Lifting Scheme

    • Extending the Lifting Scheme

    • Application: Wavelets on Spheres

  • Wavelets on Triangulated Surfaces

  • Applications


Wavelets

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

The Lifting Scheme

Split


The lifting scheme1

The Lifting Scheme

Predict


The lifting scheme2

The Lifting Scheme

Predict


The lifting scheme3

The Lifting Scheme

Update

1/8{-1,2,6,2,-1}, ½{-1,2,-1}


The lifting scheme4

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

Irregularly Sampled Points

Split

Predict

Update


Irregularly sampled points1

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

Wavelets on Spheres

  • Sub-division on edges

  • Same steps

    • Split

    • Predict

    • Update


Topological earth data

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

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

Up-Sampling Problems

  • Smooth interpolating polynomials

    • over-shooting

    • added undulations.

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


Up sampling problems1

Up-Sampling Problems

  • Similar problems can occur on surfaces.

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


Wavelets on spheres1

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.


Outline1

Outline

  • Wavelets: The Lifting Scheme

  • Wavelets on Triangulated Surfaces

    • Up-sampling problems

  • Applications


Triangulated surfaces

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

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

Guskov et al.

  • Need Smoother as part of algorithm


Guskov et al1

Guskov et al.

  • Point Selection

  • Choose Smallest Edge

  • Remove one vertex in each level


Guskov et al2

Guskov et al.

  • Collapse the Edge


Guskov et al3

Guskov et al.

  • Prediction

  • Re-introduce the Edge.

  • Minimize Dihedral Angles

  • Detail Vector: Difference vector

  • (tangent plane coordinates)


Guskov et al4

Guskov et al.

Quasi-Update

  • Smooth surrounding

    points (minimize

    dihedral angles)


Guskov et al5

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”)


Guskov et al6

Guskov et al.

  • Editing

(picture from “Multiresolution Signal Processing For Meshes”)


Kobbelt et al

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 al1

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 al2

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

Are these wavelets?

  • Mathematically: No.

    • Bi-orthogonality

    • Too many coefficients.


Is this representation useful

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.


Outline2

Outline

  • Wavelets: The Lifting Scheme

  • Wavelets on Triangulated Surfaces

  • Applications

    • Existing

    • Opportunities for new research


Editing

Editing

  • Replacing conventional surface editing. (NURBS)

  • (picture from “Multiresolution Signal Processing For Meshes” ,

  • “Multiresolution Hierarchies On Unstructured Triangle Meshes”)


Feature enhancement

Feature Enhancement

  • “For show only.”

  • (picture from “Multiresolution Signal Processing for Meshes”)


Compression

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

Remeshing

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

  • (picture from “Consistent Mesh Parameterizations”)


An opportunity

An Opportunity

  • Analysis of the wavelet coefficients


Statistics across meshes

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

Feature Detection

  • Should be able to find signature hierarchical detail coefficients.

  • Hard with different triangulations.

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


Acknowledgements

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.


The lifting scheme5

The Lifting Scheme

Mathematics

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

High Pass Filter: ½(-1,2,1)

Back


Solving pdes

Solving PDEs

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


Guskov et al7

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 al3

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

To Do List

  • Check Sphere coefficients

  • Sweldons Quote: change to published quote.

  • Edit Guskov et al

  • Compression Page: comments underneath.


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