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Multiresolution Analysis for Surfaces of Arbitrary Topological Type

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- Michael Lounsbery
- Alias | wavefront

- Tony DeRose
- Pixar

- Joe Warren
- Rice University

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

- Each subdivision step:
- Split
- Average

- What happens if we run it backwards?

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

Scalingfunctions:

scales & translates

Wavelet functions:

scales & translates

Scalingfunctions:

scales & translates

Wavelet functions:

scales & translates

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

- Wavelets are defined to be orthogonal to the scaling functions

- Close approximation
- Least-squares property from orthogonality
- Can rebuild exactly
- Large coefficients match areas with more information

- Efficient
- Linear time decomposition and reconstruction

Figure courtesy of Peter Schröder & Wim Sweldens

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

- Images

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

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

- 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

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

- Explicit patching methods
- Smooth
- Integrable
- No refinability

- Subdivision surfaces

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

- 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

Our wavelet:

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

- 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

Reduced: 10K triangles

Reduced: 4K triangles

Reduced: 240 triangles

Color data on the sphere

Original at 100%

Reduced to 16%

Plain image Image with mesh lines

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

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

- Can edit at different levels of detail
- (Forsey & Bartels 1988, Finkelstein et al. 1994)

Original shape Wide-scale edit Finer-scale edit

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

- Efficient
- Many potential applications