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Multiresolution Analysis for Surfaces of Arbitrary Topological Type

Multiresolution Analysis for Surfaces of Arbitrary Topological Type. Michael Lounsbery Alias | wavefront Tony DeRose Pixar Joe Warren Rice University. Overview. Applications Wavelets background Construction of wavelets on subdivision surfaces Approximation techniques

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Multiresolution Analysis for Surfaces of Arbitrary Topological Type

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  1. Multiresolution Analysis for Surfaces of Arbitrary Topological Type • Michael Lounsbery • Alias | wavefront • Tony DeRose • Pixar • Joe Warren • Rice University

  2. Overview • Applications • Wavelets background • Construction of wavelets on subdivision surfaces • Approximation techniques • Hierarchical editing

  3. Subdivision surfaces • Each subdivision step: • Split • Average • What happens if we run it backwards?

  4. Wavelet applications • Surface compression • Level of detail for animation • Multiresolution editing of 3D surfaces

  5. Simple wavelet example

  6. Simple wavelet example

  7. Simple wavelet example

  8. Simple wavelet example

  9. Simple wavelet example

  10. Simple wavelet example

  11. Simple wavelet example Scalingfunctions: scales & translates Wavelet functions: scales & translates

  12. Wavelets on surfaces

  13. Wavelets: subdivision run backwards

  14. Simple wavelet example Scalingfunctions: scales & translates Wavelet functions: scales & translates

  15. Nested linear spaces • Define linear spaces spanned by • Hierarchy of nested spaces for scaling functions

  16. Orthogonality • Wavelets are defined to be orthogonal to the scaling functions

  17. Wavelet properties • Close approximation • Least-squares property from orthogonality • Can rebuild exactly • Large coefficients match areas with more information • Efficient • Linear time decomposition and reconstruction

  18. Wavelet approximation example Figure courtesy of Peter Schröder & Wim Sweldens

  19. Wavelet applications • Data compression • Functions • 1-dimensional • Tensor-product • Images • Progressive transmission • Order coefficients from greatest to least (Certain et al. 1996)

  20. Constructing wavelets • 1. Choose a scaling function • 2. Find an inner product • 3. Solve for wavelets

  21. Extending wavelets to surfaces: Why is it difficult? • Translation and scaling doesn’t work • Example: can’t cleanly map a grid onto a sphere • Need a more general formulation • Nested spaces <-> refinable scaling functions • Inner product

  22. Refinability • A coarse-level scaling function may be defined in terms of finer-level scaling functions

  23. Surfaces of Arbitrary Topological Type • Explicit patching methods • Smooth • Integrable • No refinability • Subdivision surfaces

  24. Scaling functions

  25. Computing inner products • Needed for constructing wavelets orthogonal to scaling functions • For scaling functions and • Numerically compute?

  26. Computing inner products • is matrix of inner products at level • Observations • Recurrence relation between matrices • Finite number of distinct entries in matrices • Result: solve finite-sized linear system for inner product

  27. Constructing wavelets

  28. Constructing wavelets

  29. Constructing wavelets

  30. Constructing wavelets

  31. Constructing wavelets

  32. Constructing wavelets

  33. Constructing wavelets Our wavelet:

  34. Localized approximation of wavelets

  35. Wavelet decomposition of surfaces

  36. Surface approximation • 1. Select subset of wavelet coefficients • 2. Add them back to the base mesh • Selection strategies • All coefficients >e • guarantee

  37. Approximating surface data • Scalar-based data is stored at vertices • Treat different fields separately • Storage • Decomposition • “Size” of wavelet coefficient is weighted blend • Examples • 3D data: surface geometry • Color data: Planetary maps

  38. Original: 32K triangles Reduced: 10K triangles Reduced: 4K triangles Reduced: 240 triangles

  39. Color data on the sphere Original at 100% Reduced to 16% Plain image Image with mesh lines

  40. Smooth transitions • Avoids jumps in shape • Smoothly blend wavelet additions • Linear interpolation

  41. Remeshing • We assume simple base mesh • Difficult to derive from arbitrary input • Eck et al. (1995) addresses

  42. Hierarchical editing • Can edit at different levels of detail • (Forsey & Bartels 1988, Finkelstein et al. 1994) Original shape Wide-scale edit Finer-scale edit

  43. Summary • Wavelets over subdivision surfaces • Refinable scaling functions • Exact inner products are possible • Locally supported wavelets • Efficient • Many potential applications

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