Subdivision Surfaces

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Subdivision Surfaces

• Computational Geometry Presentation
• Adam Lake

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Motivation

• Scalability across machines
• Speed of evaluation
• LOD in a scene
• Topological restrictions of NURBS surfaces
• Planes, Cylinders, and Torii

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Motivation

• Trimming is expensive and prone to numerical
  error
• It is difficult to maintain smoothness at seams
  of patchwork.
• Example hiding seams in Woody (Toy Story)
  DeRose98
• Advantage of NURBS Trimmed NURBS are readily
  available in commercial systems!!

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Subdivisions of Talk

• 2D and 3D Examples
• Give data structure for geometry
• Define B-spline curve
• Derive splitting matrix for bicubics
• Define arbitrary topology splitting rules
• Define extraordinary vertices, Doo-Sabin surfaces

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2D subdivision curve example

• De Casteljau curve Foley90

P2
P3
P4
P1
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2D subdivision curve example

• De Casteljau curve Foley90

H(P2P3)/2
P2
P3
R2(HR3)/2
L3(L2H)/2
L2(P1P2)/2
L4R1(L3R2)/2
R3(P3P4)/2
P4R4
P1L1
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3D subdivision surface example
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Major categories of subdivision surfaces

• Catmull-Clark Catmull78
• Doo-Sabin Doo78
• Loop Loop87
• Butterfly scheme Dyn90

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Subdivision Surface Data Structure
A mesh is a pair (K,V) where K is a simplical
complex representing connectivity of the
vertices, edges, and faces. Determines
topological type of the mesh. Vv1,v2,v3vn,
vi in R3 is a set of vertex positions defining
the shape of the mesh in R3. Example V0,0,0
,1,0,0,0,0,1 Simplical complex K vertices
1,2,3 edges 1,2,2,3,1,3 faces
1,2,3
2
1
3
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B-Spline Patch Splitting
The Bicubic B-Spline Patch can be expressed as
Where
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B-Spline Patch Splitting

• Now, consider a subdivision scheme for
  0ltu,vlt1/2 and introduce a matrix
• In order to obtain the same patch we must have

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B-Spline Patch Splitting

• Assuming M is invertible

This is known as the splitting matrix. Note
Mistake in original paper!!!
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B-Spline Patch Splitting
p31
p21
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B-Spline Patch Splitting
p11
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Arbitrary Topology

• 3 types of points
• Face
• Edge
• Vertex
• Edges formed by 2 rules
• Subdivision surface is the LIMIT surface

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Arbitrary Topology

• New face points
• Average of all old points defining the face.
• Example

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Arbitrary Topology

• New edge points
• Average midpoints of old edge with average of the
  two new face points of faces sharing edge
• Example

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Arbitrary Topology

• New vertex points

• Qaverage of the new face points of all faces
  adjacent to the old vertex point
• Raverage of the midpoints of all old edges
  incident on the old vertex points
• Sold vertex point

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Arbitrary Topology

• New edges formed by
• Connecting each new face point to new edge points
  of edges defining the old face
• Connecting each new vertex point to the new
  points of all old edges incident to old vertex
  point
• Example

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Extraordinary points

• Previous method generates continuous surfaces
  except at extraordinary points
• Extraordinary points have valence other than 4

N5
N3
Extraordinary points
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Doo-Sabin Surfaces

• Doo-Sabin showed via Eigenanalysis that at
  extraordinary points the surface is discontinuous
  and proposed modified rules to maintain
  continuity in the surface Doo78.
• The improved formulation is known as Doo-Sabin
  surfaces.

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Bi-quadratic surfaces

• Applying the method for bi-cubic can also be used
  to derive bi-quadratic surfaces.

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Recap

• So far
• 2D and 3D examples
• Data structure for geometry
• Defined B-spline curve
• Derived the splitting matrix for bicubics
• Defined arbitrary topology splitting rules
• Defined extraordinary vertices, Doo Sabin surfaces

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Next

• Subdivision surfaces in Character Animation
  DeRose98
• Non-uniform Subdivision Surfaces Sederberg98
• Exact Evaluation of Catmull Clark surfaces at
  arbitrary parameter values Stam98
• MAPS Multiresolution Adaptive Parameterization
  of Surfaces Lee98

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Subdivision Surfaces in character animation
DeRose98

• Used for first time in Geris game to overcome
  topological restriction of NURBS
• Modeled Geris head, hands, jacket, pants, shirt,
  tie, and shoes
• Developed cloth simulation methods
• Method to construct scalar fields enabling use of
  programmable shaders

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Why Catmull-Clark?

• Quads are better than tris at capturing the
  symmetries of natural and man-made objects. Tube
  like surfaces (arms, legs, fingers) are easier to
  model.
• Generalize uniform tensor product cubic
  B-splines, makes it easier to use in-house and
  commercial systems (Renderman and
  Alias-Wavefront).

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Hybrid subdivision

• Hoppe wrote about smooth surfaces with infinitely
  sharp creases.
• DeRose generalizes this to semi-sharp creases
• Select certain vertices to use rules to subdivide
  to a specific level, then switch to another
  subdivision scheme applied to the limit.
• Sharp at coarse scales, smooth at finer scales
• Calls this hybrid subdivision

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Subdivision Surfaces in character animation

• Implemented in Renderman
• Shows subdivision surfaces can be used in high
  end rendering

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Non-uniform subdivision surfaces Sederberg98
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Non-uniform subdivision surfaces Sederberg98
Cannot be represented as NURB nor uniform
Catmull Clark

• For a non-uniform surface, each vertex is
  assigned a knot spacing that may be different
  from each edge radiating from it.
• If all knot intervals are equal, back to regular
  Catmull-Clark or Doo-Sabin.

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Non-uniform subdivision surfaces
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2D Non-uniform subdivision surfaces
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3D Non-uniform subdivision surfaces
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Exact Evaluation of Catmull-Clark Surfaces at
arbitrary parameter values Stam98
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Exact Evaluation of Catmull-Clark Surfaces at
arbitrary parameter values Stam98

• Disproves belief that Catmull Clark surfaces
  cannot be evaluated w/o explicit subdividing.
• Uses a set of eigenbasis functions and derive
  analytical expressions for these functions.
• Cost comparable to that of a bi-cubic B-spline.
• Allows algorithms developed for parametric
  surfaces to be applied to Catmull-Clark surfaces.

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First high resolution plots around regions of
high curvature
Extraordinary point in center. Patches using
technique are in blue. Derivative information is
computed and displayed in model on right.
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MAPS Multiresolution Adaptive Parameterization
of Surfaces Lee98
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MAPS Multiresolution Adaptive Parameterization
of Surfaces Lee98
Note Great computational geometry paper!!
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Step 1A scanned input mesh

• Acquired via laser range scanning or MRI
  volumetric imaging followed by isosurface
  extraction
• Arbitrary topology
• Irregular structure
• Tremendous size

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Step 2 Via mesh simplification, obtain the
parameter or base domain

• During simplification, building a topological
  representation for O(nlgn) multiresolution
  representation.

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Step 2 Via mesh simplification, obtain the
parameter or base domain
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Step 3 Assign triangles in original mesh to
their base domain triangle (during 1-gt2)


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Step 4 Adaptive Remeshing with subdivision
connectivity (epsilon 1)
Uniform mesh obtained via Loop scheme Interested
in smooth parameterizations, not simplification!!
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Step 4 (detail)Motivation

• Assume a smooth parameterization within each
  subdomain (see paper for details)
• One way to obtain global smoothness would be to
  minimize a global smoothness functional.
• Requires PDE solver.
• Found that this is needlessly cumbersome.
• Instead, use simpler and cheaper smoothing
  technique based on Loop subdivision.

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Step 4 (detail)Modified Loop Scheme

• If all points of stencil needed for computing new
  point or smoothing old point are inside same
  triangle of base domain, add Loop weights and new
  points will be in the same face
• If stencil stretches across 2 faces, flatten
  using hinge map at common edge.
• If stencil stretches across multiple faces, use a
  conformal flattening strategy (see paper).

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Step 4 (details)
Smoothed mesh using modified Loop simplification
scheme
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Step 5 Multiresolution editing
Smooth parameterization allows efficient mesh
modification. Texture coordinates and polygons
nicely behaved.
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Summary

• Who developed them?
• Catmull-Clark
• Doo-Sabin
• Loop
• Dyn (butterfly)
• Recent work
• DeRose, Sederberg, Lee, Stam

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Summary

• What are they?
• Mesh based representation of geometry
• Defined recursively
• Many levels of fidelity

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Summary

• Where can they be used?
• Animation
• CAD modeling
• Game engines
• more?

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Summary

• Why are they useful?
• Scalable
• LODs
• Efficient to implement

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Summary

• How to IMPLEMENT!!

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References

• DeRose, Tony, Michael Kass, and Tien Truong.
  Subdivision Surfaces in Character Animation.
  SIGGRAPH 98.
• Clark, E., and J. Clark. Recursively generated
  B-spline surfaces on arbitrary topological
  meshes. Computer Aided Geometric Design, Vol.
  10, No. 6, 1978.
• Stam, Joe. Exact Evaluation of Catmull-Clark
  Subdivision Surfaces At Arbitrary Parameter
  Values. SIGGRAPH 98.
• Lee, Aaron W.F., Wim Swelders, Peter Schroeder,
  Lawrence Cowsar, David Dobkin. MAPS
  Multiresolution Adaptive Parameterization of
  Surrfaces. SIGGRAPH 98.
• Sederberg, Thomas W., David Sewell, Malcolm
  Sabin. Non-Uniform Recursive Subdivision
  Surfaces. SIGGRAPH 98.
• Foley, James D., Andries van Dam, Steven Feiner,
  John Hughes. Computer Graphics Principles and
  Practice, 2nd Edition. Pages 507-510. 1990.
• Loop, C.T. Smooth Subdivision Surfaces Based on
  Triangles. M.S.Thesis. Department of
  Mathematics. University of Utah. August 1987.
• Doo, D. and M. Sabin. Behavior of Recursive
  Division Surfaces Near Extraordinary Points.
  Computer-Aided Design. Vol. 10, No. 6, 1978.
• Dyn, Nyra, David Leven, John Gregory. A
  butterfly subdivision scheme for surface
  interpolation with tension control. ACM
  Transactions on Graphics, 9(2)160-169, April
  1990.