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Multiresolution Analysis for Surfaces of Arbitrary Topological TypePowerPoint Presentation

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 Topological Type

- Applications
- Wavelets background
- Construction of wavelets on subdivision surfaces
- Approximation techniques
- Hierarchical editing

Subdivision surfaces Topological Type

- Each subdivision step:
- Split
- Average

- What happens if we run it backwards?

Wavelet applications Topological Type

- Surface compression
- Level of detail for animation
- Multiresolution editing of 3D surfaces

Simple wavelet example Topological Type

Simple wavelet example Topological Type

Simple wavelet example Topological Type

Simple wavelet example Topological Type

Simple wavelet example Topological Type

Simple wavelet example Topological Type

Simple wavelet example Topological Type

Scalingfunctions:

scales & translates

Wavelet functions:

scales & translates

Wavelets on surfaces Topological Type

Wavelets: subdivision run backwards Topological Type

Simple wavelet example Topological Type

Scalingfunctions:

scales & translates

Wavelet functions:

scales & translates

Nested linear spaces Topological Type

- Define linear spaces spanned by
- Hierarchy of nested spaces for scaling functions

Orthogonality Topological Type

- Wavelets are defined to be orthogonal to the scaling functions

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 Topological Type

Figure courtesy of Peter Schröder & Wim Sweldens

Wavelet applications Topological Type

- Data compression
- Functions
- 1-dimensional
- Tensor-product

- Images

- Functions
- Progressive transmission
- Order coefficients from greatest to least (Certain et al. 1996)

Constructing wavelets Topological Type

- 1. Choose a scaling function
- 2. Find an inner product
- 3. Solve for wavelets

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 Topological Type

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

Surfaces of Arbitrary Topological TypeTopological Type

- Explicit patching methods
- Smooth
- Integrable
- No refinability

- Subdivision surfaces

Scaling functions Topological Type

Computing inner products Topological Type

- Needed for constructing wavelets orthogonal to scaling functions
- For scaling functions and
- Numerically compute?

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 wavelets Topological Type

Constructing wavelets Topological Type

Constructing wavelets Topological Type

Constructing wavelets Topological Type

Constructing wavelets Topological Type

Constructing wavelets Topological Type

Constructing wavelets Topological Type

Our wavelet:

Localized approximation of wavelets Topological Type

Wavelet decomposition of surfaces Topological Type

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 Topological Type

- Scalar-based data is stored at vertices
- Treat different fields separately
- Storage
- Decomposition

- “Size” of wavelet coefficient is weighted blend

- Treat different fields separately
- Examples
- 3D data: surface geometry
- Color data: Planetary maps

Original: 32K triangles Topological Type

Reduced: 10K triangles

Reduced: 4K triangles

Reduced: 240 triangles

Color data on the sphere Topological Type

Original at 100%

Reduced to 16%

Plain image Image with mesh lines

Smooth transitions Topological Type

- Avoids jumps in shape
- Smoothly blend wavelet additions
- Linear interpolation

Remeshing Topological Type

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

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 Topological Type

- Wavelets over subdivision surfaces
- Refinable scaling functions
- Exact inner products are possible
- Locally supported wavelets

- Efficient
- Many potential applications

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