Multiresolution Analysis for Surfaces of Arbitrary Topological Type Michael Lounsbery Michael Lounsbery Alias | wavefront Alias | wavefront Tony DeRose Tony DeRose Pixar Pixar Joe Warren Joe Warren Rice University Rice University

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

Subdivision surfaces Each subdivision step: Each subdivision step: SplitSplit AverageAverage What happens if we run it backwards? What happens if we run it backwards?

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

Simple wavelet example

Scaling functions: scales & translates Wavelet functions: scales & translates

Wavelets on surfaces

Wavelets: subdivision run backwards

Simple wavelet example Scaling functions: scales & translates Wavelet functions: scales & translates

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

Orthogonality Wavelets are defined to be orthogonal to the scaling functions Wavelets are defined to be orthogonal to the scaling functions

Wavelet properties Close approximation Close approximation Least-squares property from orthogonalityLeast-squares property from orthogonality Can rebuild exactlyCan rebuild exactly Large coefficients match areas with more informationLarge coefficients match areas with more information Efficient Efficient Linear time decomposition and reconstructionLinear time decomposition and reconstruction

Wavelet approximation example Figure courtesy of Peter Schröder & Wim Sweldens

Wavelet applications Data compression Data compression FunctionsFunctions –1-dimensional –Tensor-product ImagesImages Progressive transmission Progressive transmission Order coefficients from greatest to least (Certain et al. 1996)Order coefficients from greatest to least (Certain et al. 1996)

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

Extending wavelets to surfaces: Why is it difficult? Translation and scaling doesn’t work Translation and scaling doesn’t work Example: can’t cleanly map a grid onto a sphereExample: can’t cleanly map a grid onto a sphere Need a more general formulation Need a more general formulation Nested spaces refinable scaling functionsNested spaces refinable scaling functions Inner productInner product

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

Surfaces of Arbitrary Topological Type Explicit patching methods Explicit patching methods SmoothSmooth IntegrableIntegrable No refinabilityNo refinability Subdivision surfaces Subdivision surfaces

Scaling functions

Computing inner products Needed for constructing wavelets orthogonal to scaling functions Needed for constructing wavelets orthogonal to scaling functions For scaling functions and For scaling functions and Numerically compute? Numerically compute?

Computing inner products is matrix of inner products at level is matrix of inner products at level Observations Observations Recurrence relation between matricesRecurrence relation between matrices Finite number of distinct entries in matricesFinite number of distinct entries in matrices Result: solve finite-sized linear system for inner product Result: solve finite-sized linear system for inner product

Constructing wavelets

Our wavelet:

Localized approximation of wavelets

Wavelet decomposition of surfaces

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

Approximating surface data Scalar-based data is stored at vertices Scalar-based data is stored at vertices Treat different fields separatelyTreat different fields separately –Storage –Decomposition “Size” of wavelet coefficient is weighted blend“Size” of wavelet coefficient is weighted blend Examples Examples 3D data: surface geometry3D data: surface geometry Color data: Planetary mapsColor data: Planetary maps

Original: 32K triangles Reduced: 10K triangles Reduced: 4K triangles Reduced: 240 triangles

Reduced to 16% Original at 100% Color data on the sphere Plain image Image with mesh lines Plain image Image with mesh lines

Smooth transitions Avoids jumps in shape Avoids jumps in shape Smoothly blend wavelet additions Smoothly blend wavelet additions Linear interpolationLinear interpolation

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

Hierarchical editing Can edit at different levels of detail Can edit at different levels of detail (Forsey & Bartels 1988, Finkelstein et al. 1994)(Forsey & Bartels 1988, Finkelstein et al. 1994) Original shape Wide-scale edit Finer-scale edit

Summary Wavelets over subdivision surfaces Wavelets over subdivision surfaces Refinable scaling functionsRefinable scaling functions Exact inner products are possibleExact inner products are possible Locally supported waveletsLocally supported wavelets Efficient Efficient Many potential applications Many potential applications