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
Multiresolution Analysis for Surfaces of Arbitrary Topological Type
  • Michael Lounsbery
    • Alias | wavefront
  • Tony DeRose
    • Pixar
  • Joe Warren
    • Rice University
overview
Overview
  • Applications
  • Wavelets background
  • Construction of wavelets on subdivision surfaces
  • Approximation techniques
  • Hierarchical editing
subdivision surfaces
Subdivision surfaces
  • Each subdivision step:
    • Split
    • Average
  • What happens if we run it backwards?
wavelet applications
Wavelet applications
  • Surface compression
  • Level of detail for animation
  • Multiresolution editing of 3D surfaces
simple wavelet example11
Simple wavelet example

Scalingfunctions:

scales & translates

Wavelet functions:

scales & translates

simple wavelet example14
Simple wavelet example

Scalingfunctions:

scales & translates

Wavelet functions:

scales & translates

nested linear spaces
Nested linear spaces
  • Define linear spaces spanned by
  • Hierarchy of nested spaces for scaling functions
orthogonality
Orthogonality
  • Wavelets are defined to be orthogonal to the scaling functions
wavelet properties
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
Wavelet approximation example

Figure courtesy of Peter Schröder & Wim Sweldens

wavelet applications19
Wavelet applications
  • Data compression
    • Functions
      • 1-dimensional
      • Tensor-product
    • Images
  • Progressive transmission
    • Order coefficients from greatest to least (Certain et al. 1996)
constructing wavelets
Constructing wavelets
  • 1. Choose a scaling function
  • 2. Find an inner product
  • 3. Solve for wavelets
extending wavelets to surfaces why is it difficult
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
Refinability
  • A coarse-level scaling function may be defined in terms of finer-level scaling functions
surfaces of arbitrary topological type
Surfaces of Arbitrary Topological Type
  • Explicit patching methods
    • Smooth
    • Integrable
    • No refinability
  • Subdivision surfaces
computing inner products
Computing inner products
  • Needed for constructing wavelets orthogonal to scaling functions
  • For scaling functions and
  • Numerically compute?
computing inner products26
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
surface approximation
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
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

Original: 32K triangles

Reduced: 10K triangles

Reduced: 4K triangles

Reduced: 240 triangles

slide39

Color data on the sphere

Original at 100%

Reduced to 16%

Plain image Image with mesh lines

smooth transitions
Smooth transitions
  • Avoids jumps in shape
  • Smoothly blend wavelet additions
    • Linear interpolation
remeshing
Remeshing
  • We assume simple base mesh
  • Difficult to derive from arbitrary input
    • Eck et al. (1995) addresses
hierarchical editing
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
Summary
  • Wavelets over subdivision surfaces
    • Refinable scaling functions
    • Exact inner products are possible
    • Locally supported wavelets
  • Efficient
  • Many potential applications
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