Multiresolution analysis for surfaces of arbitrary topological type
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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 l.jpg
Multiresolution Analysis for Surfaces of Arbitrary Topological Type

  • Michael Lounsbery

    • Alias | wavefront

  • Tony DeRose

    • Pixar

  • Joe Warren

    • Rice University


Overview l.jpg
Overview Topological Type

  • Applications

  • Wavelets background

  • Construction of wavelets on subdivision surfaces

  • Approximation techniques

  • Hierarchical editing


Subdivision surfaces l.jpg
Subdivision surfaces Topological Type

  • Each subdivision step:

    • Split

    • Average

  • What happens if we run it backwards?


Wavelet applications l.jpg
Wavelet applications Topological Type

  • Surface compression

  • Level of detail for animation

  • Multiresolution editing of 3D surfaces


Simple wavelet example l.jpg
Simple wavelet example Topological Type


Simple wavelet example6 l.jpg
Simple wavelet example Topological Type


Simple wavelet example7 l.jpg
Simple wavelet example Topological Type


Simple wavelet example8 l.jpg
Simple wavelet example Topological Type


Simple wavelet example9 l.jpg
Simple wavelet example Topological Type


Simple wavelet example10 l.jpg
Simple wavelet example Topological Type


Simple wavelet example11 l.jpg
Simple wavelet example Topological Type

Scalingfunctions:

scales & translates

Wavelet functions:

scales & translates


Wavelets on surfaces l.jpg
Wavelets on surfaces Topological Type



Simple wavelet example14 l.jpg
Simple wavelet example Topological Type

Scalingfunctions:

scales & translates

Wavelet functions:

scales & translates


Nested linear spaces l.jpg
Nested linear spaces Topological Type

  • Define linear spaces spanned by

  • Hierarchy of nested spaces for scaling functions


Orthogonality l.jpg
Orthogonality Topological Type

  • Wavelets are defined to be orthogonal to the scaling functions


Wavelet properties l.jpg
Wavelet properties Topological Type

  • Close approximation

    • Least-squares property from orthogonality

    • Can rebuild exactly

    • Large coefficients match areas with more information

  • Efficient

    • Linear time decomposition and reconstruction


Wavelet approximation example l.jpg
Wavelet approximation example Topological Type

Figure courtesy of Peter Schröder & Wim Sweldens


Wavelet applications19 l.jpg
Wavelet applications Topological Type

  • Data compression

    • Functions

      • 1-dimensional

      • Tensor-product

    • Images

  • Progressive transmission

    • Order coefficients from greatest to least (Certain et al. 1996)


Constructing wavelets l.jpg
Constructing wavelets Topological Type

  • 1. Choose a scaling function

  • 2. Find an inner product

  • 3. Solve for wavelets


Extending wavelets to surfaces why is it difficult l.jpg
Extending wavelets to surfaces: Topological TypeWhy 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


Refinability l.jpg
Refinability Topological Type

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


Surfaces of arbitrary topological type l.jpg
Surfaces of Arbitrary Topological TypeTopological Type

  • Explicit patching methods

    • Smooth

    • Integrable

    • No refinability

  • Subdivision surfaces


Scaling functions l.jpg
Scaling functions Topological Type


Computing inner products l.jpg
Computing inner products Topological Type

  • Needed for constructing wavelets orthogonal to scaling functions

  • For scaling functions and

  • Numerically compute?


Computing inner products26 l.jpg
Computing inner products Topological Type

  • 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


Constructing wavelets27 l.jpg
Constructing wavelets Topological Type


Constructing wavelets28 l.jpg
Constructing wavelets Topological Type


Constructing wavelets29 l.jpg
Constructing wavelets Topological Type


Constructing wavelets30 l.jpg
Constructing wavelets Topological Type


Constructing wavelets31 l.jpg
Constructing wavelets Topological Type


Constructing wavelets32 l.jpg
Constructing wavelets Topological Type


Constructing wavelets33 l.jpg
Constructing wavelets Topological Type

Our wavelet:




Surface approximation l.jpg
Surface approximation Topological Type

  • 1. Select subset of wavelet coefficients

  • 2. Add them back to the base mesh

  • Selection strategies

    • All coefficients >e

    • guarantee


Approximating surface data l.jpg
Approximating surface data Topological Type

  • 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


Slide38 l.jpg

Original: 32K triangles Topological Type

Reduced: 10K triangles

Reduced: 4K triangles

Reduced: 240 triangles


Slide39 l.jpg

Color data on the sphere Topological Type

Original at 100%

Reduced to 16%

Plain image Image with mesh lines


Smooth transitions l.jpg
Smooth transitions Topological Type

  • Avoids jumps in shape

  • Smoothly blend wavelet additions

    • Linear interpolation


Remeshing l.jpg
Remeshing Topological Type

  • We assume simple base mesh

  • Difficult to derive from arbitrary input

    • Eck et al. (1995) addresses


Hierarchical editing l.jpg
Hierarchical editing Topological Type

  • Can edit at different levels of detail

    • (Forsey & Bartels 1988, Finkelstein et al. 1994)

Original shape Wide-scale edit Finer-scale edit


Summary l.jpg
Summary Topological Type

  • Wavelets over subdivision surfaces

    • Refinable scaling functions

    • Exact inner products are possible

    • Locally supported wavelets

  • Efficient

  • Many potential applications