Progress in Object-Centric Machine Learning1
Slides adapted from Peter Schröder (Caltech) and Denis Zorin (NYU)
2 Multi-Res Modeling Group
3 Multi-Res Modeling Group
Geometric Modeling Surface representations
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scalable: large data on small machines solid mathematical foundation suitable for animation and simulation
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good editing semantics deep connection with wavelets
compression, solvers, speed/accuracy tradeoff, approximation properties
builtin support for LOD display very efficient
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high level control (control points) compact representation multiresolution structure
Disadvantages difficult to maintain and manage naïve rendering of large models slow
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Disadvantages heavy weight representation good editing semantics difficult limited multiresolution structure
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generalizes spline patches covers range of representations from
“pure” spline patches to “pure” meshes BUT: special connectivity (more on that
later)
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Subdivision Smooth surfaces as the limit of a sequence of successive refinements
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even at level i odd at level i
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affine combination of nearby points
-1 -1
generalizes spline patches
many others: Doo-Sabin, Butterfly, Kobbelt, Peters/Reif
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Why Subdivision? Advantages
arbitrary topology, smooth surfaces support for compression and LOD suitable for wavelet-based
numerical solvers
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little computation simple data structures integrates well with spline methods
19Subdivision for Modeling and Animation
Example: Loop scheme
Example: Loop scheme
For a “good” scheme, recursive application approximates a smooth surface
21Subdivision for Modeling and Animation
Control points vertices of the initial mesh define the surface each vertex influences a finite part of the surface
22Subdivision for Modeling and Animation
Triangular and quadrilateral subdivision
23Subdivision for Modeling and Animation
Subdivision and splines uniform splines can be computed using subdivision rules triangular spline
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Subdivision and splines For splines, the control mesh is regular
25Subdivision for Modeling and Animation
Extraordinary Vertices
Constructing the rules
Define rules for:
Constructing the rules
Smoothness and Fairness
Affine invariance
transform T
transform T
29Subdivision for Modeling and Animation
Affine invariance the coeficients of any mask should sum up to
1
displacement
i
p
p
i
a
X
SUBDIVISION ZOO
Classification of schemes
Classification criteria type of refinement rule (primal or dual) type of mesh (triangular or quad or...) approximating or interpolating
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Refinement rules Primal (vertex insertion) Dual (corner cutting)
33Subdivision for Modeling and Animation
approximation and interpolation
Advantages approximating schemes
control points on surface in-place implementation
34Subdivision for Modeling and Animation
Subdivision schemes Primal (vertex insertion) Dual (corner cutting)
InterpolatingApproximating
Boundaries and creases
special rules on and near the boundary boundary independent of the interior
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boundaries
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Modified Butterfly scheme triangular meshes, interpolating: only one rule needs larger support
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¡ 1
Modified Butterfly scheme
eigenvectors
Catmull-Clark scheme
Catmull-Clark scheme Reduction to a quadrilateral mesh
do one step of subdivision with special rules; all polygons become quads
41Subdivision for Modeling and Animation
Catmull-Clark scheme Extraordinary vertices
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Catmull-Clark scheme boundaries and creases
cubic spline (same as Loop!) need to fix rules for C1-continuity
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Kobbelt scheme quadrilateral, interpolating tensor product 4-point scheme
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Kobbelt scheme extraordinary vertices
45Subdivision for Modeling and Animation
Doo-Sabin scheme dual scheme, quadrilateral extends tensor-product biquadratic splines
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46Subdivision for Modeling and Animation
Doo-Sabin scheme after one step, all valences = 4 rule for extraordinary polygons:
®i = 1
¶
Midedge scheme dual scheme, quadrilateral extends 4-directional box spline
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Summary
InterpolatingApproximating
Limitations of subdivision
no C2 with small support decrease of smoothness with valence ripples no direct control of fairness
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Limitations of stationary subdivision
Problems with curvature continuity the only practical C2 schemes (Umlauf ) have “flat spots” at
extraordinary vertices “true” C2 has to have very large support lack of C2-continuity = nonsmooth normal changes visible for high valences
51Subdivision for Modeling and Animation
Limitations of subdivision
Loop Modified Loop
52Subdivision for Modeling and Animation
Limitations of subdivision
Limitations of subdivision
Extensions
Sederberg et al allows one to integrate NURBS with subdivision
Variational subdivision: Kobbelt, Warren slower, more complex; higher quality surfaces
CS348a: Computer Graphics -- Geometric Modeling and Processing
SubdivisionSurfaces
Geometric Modeling
Modified Butterfly scheme
Modified Butterfly scheme