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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

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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

  • Applications

  • Wavelets background

  • Construction of wavelets on subdivision surfaces

  • Approximation techniques

  • Hierarchical editing


Subdivision surfaces l.jpg

Subdivision surfaces

  • Each subdivision step:

    • Split

    • Average

  • What happens if we run it backwards?


Wavelet applications l.jpg

Wavelet applications

  • Surface compression

  • Level of detail for animation

  • Multiresolution editing of 3D surfaces


Simple wavelet example l.jpg

Simple wavelet example


Simple wavelet example6 l.jpg

Simple wavelet example


Simple wavelet example7 l.jpg

Simple wavelet example


Simple wavelet example8 l.jpg

Simple wavelet example


Simple wavelet example9 l.jpg

Simple wavelet example


Simple wavelet example10 l.jpg

Simple wavelet example


Simple wavelet example11 l.jpg

Simple wavelet example

Scalingfunctions:

scales & translates

Wavelet functions:

scales & translates


Wavelets on surfaces l.jpg

Wavelets on surfaces


Wavelets subdivision run backwards l.jpg

Wavelets: subdivision run backwards


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Simple wavelet example

Scalingfunctions:

scales & translates

Wavelet functions:

scales & translates


Nested linear spaces l.jpg

Nested linear spaces

  • Define linear spaces spanned by

  • Hierarchy of nested spaces for scaling functions


Orthogonality l.jpg

Orthogonality

  • Wavelets are defined to be orthogonal to the scaling functions


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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


Wavelet approximation example l.jpg

Wavelet approximation example

Figure courtesy of Peter Schröder & Wim Sweldens


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Wavelet applications

  • 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

  • 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: 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


Refinability l.jpg

Refinability

  • 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 Type

  • Explicit patching methods

    • Smooth

    • Integrable

    • No refinability

  • Subdivision surfaces


Scaling functions l.jpg

Scaling functions


Computing inner products l.jpg

Computing inner products

  • Needed for constructing wavelets orthogonal to scaling functions

  • For scaling functions and

  • Numerically compute?


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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


Constructing wavelets27 l.jpg

Constructing wavelets


Constructing wavelets28 l.jpg

Constructing wavelets


Constructing wavelets29 l.jpg

Constructing wavelets


Constructing wavelets30 l.jpg

Constructing wavelets


Constructing wavelets31 l.jpg

Constructing wavelets


Constructing wavelets32 l.jpg

Constructing wavelets


Constructing wavelets33 l.jpg

Constructing wavelets

Our wavelet:


Localized approximation of wavelets l.jpg

Localized approximation of wavelets


Wavelet decomposition of surfaces l.jpg

Wavelet decomposition of surfaces


Surface approximation l.jpg

Surface approximation

  • 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

  • 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

Reduced: 10K triangles

Reduced: 4K triangles

Reduced: 240 triangles


Slide39 l.jpg

Color data on the sphere

Original at 100%

Reduced to 16%

Plain image Image with mesh lines


Smooth transitions l.jpg

Smooth transitions

  • Avoids jumps in shape

  • Smoothly blend wavelet additions

    • Linear interpolation


Remeshing l.jpg

Remeshing

  • We assume simple base mesh

  • Difficult to derive from arbitrary input

    • Eck et al. (1995) addresses


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Hierarchical editing

  • 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

  • Wavelets over subdivision surfaces

    • Refinable scaling functions

    • Exact inner products are possible

    • Locally supported wavelets

  • Efficient

  • Many potential applications


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