Subdivision Schemes

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Subdivision Schemes

• Lee Byung-Gook
• Dongseo Univ.
• http//kowon.dongseo.ac.kr/lbg/

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What is Subdivision?

• Subdivision is a process in which a
  poly-line/mesh is recursively refined in order to
  achieve a smooth curve/surface.
• Two main groups of schemes
• Approximating - original vertices are moved
• Interpolating original vertices are unaffected

Is the scheme used here interpolating or
approximating?
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Geris Game
Frame from Geris Game by Pixar
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Subdivision Curves

• How do You Make a Smooth Curve?

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Chaikins Algorithm
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Corner Cutting
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Corner Cutting
3 1
1 3
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Corner Cutting
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Corner Cutting
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Corner Cutting
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Corner Cutting Limit Curve
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Corner Cutting
The limit curve Quadratic B-Spline Curve
A control point
The control polygon
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Linear B-spline Subdivision Scheme
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Quadratic B-spline Subdivision Scheme
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Cubic B-spline Subdivision
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N. Dyn 4-points Subdivision Scheme
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4-Point Scheme
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4-Point Scheme
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4-Point Scheme
1 1
1 1
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4-Point Scheme
1 8
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4-Point Scheme
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4-Point Scheme
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4-Point Scheme
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4-Point Scheme
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4-Point Scheme
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4-Point Scheme
A control point
The limit curve
The control polygon
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Comparison

• Non interpolatory subdivision schemes
• Corner Cutting

• Interpolatory subdivision schemes
• The 4-point scheme

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Theoretical Questions

• Given a Subdivision scheme, does it converge for
  all polygons?
• If so, does it converge to a smooth curve?
• Better?
• Does the limit surface have any singular points?
• How do we compute the derivative of the limit
  surface?

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Subdivision Surfaces
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Subdivision Surfaces

• Geris hand as a piecewise smooth Catmull-Clark
  surface

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Subdivision Surfaces

• A surface subdivision scheme starts with a
  control net (i.e. vertices, edges and faces)
• In each iteration, the scheme constructs a
  refined net, increasing the number of vertices by
  some factor.
• The limit of the control vertices should be a
  limit surface.
• a scheme always consists of 2 main parts
• A method to generate the topology of the new net.
• Rules to determine the geometry of the vertices
  in the new net.

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General Notations

• There are 3 types of new control points
• Vertex points - vertices that are created in
  place of an old vertex.
• Edge points - vertices that are created on an old
  edge.
• Face points vertices that are created inside an
  old face.
• Every scheme has rules on how (if) to create any
  of the above.
• If a scheme does not change old vertices (for
  example - interpolating), then it is viewed
  simply as if

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Doo-Sabin Subdivision
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Doo-Sabin Subdivision
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Doo-Sabin Subdivision
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Doo-Sabin Subdivision
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Doo-Sabin subdivision
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Catmull-Clark Subdivision
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Catmull-Clark Subdivision
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Catmull-Clark Subdivision
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Catmull-Clark Subdivision
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Catmull-Clark Subdivision

• The mesh is the control net of a tensor product
  B-Spline surface.
• The refined mesh is also a control net, and the
  scheme was devised so that both nets create the
  same B-Spline surface.
• Uses face points, edge points and vertex points.
• The construction is incremental
• First the face points are calculated,
• Then using the face points, the edge points are
  computed.
• Finally using both face and edge points, we
  calculate the vertex points.

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Catmull-Clark - results
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Catmull-Clark - results
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Catmull-Clark - results
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Catmull-Clark - results
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Catmull-Clark - results
Catmull-Clark Scheme results in a surface which
is almost everywhere
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Loop Subdivision

• Based on a triangular mesh
• Loops scheme does not create face points

Vertex points
New face
Old face
Edge points
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Loop Subdivision

• How Position New Vertices?
• Choose locations for new vertices as weighted
  average of original vertices in local neighborhood

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Loop Subdivision

• Every new vertex is a weighted average of old
  ones.
• The list of weights is called a Stencil
• Is this scheme approximating or interpolating?

The rule for vertex points
1
1
The rule for edge points
1
1
1
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Loop Subdivision

• How Refine Mesh?
• Refine each triangle into 4 triangles by
  splitting each edge and connecting new vertices

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Box Spline
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Loop - Results
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Loop - Results
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Loop - Results
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Loop - Results
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Loop - Results
Loops scheme results in a limit surface which is
of continuity everywhere except for a
finite number of singular points, in which it is
.
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Butterfly Scheme

• Butterfly is an interpolatory scheme.
• Topology is the same as in Loops scheme.
• Vertex points use the location of the old vertex.
• Edge points use the following stencil

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Butterfly - results
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Butterfly - results
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Butterfly - results
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Butterfly - results
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Butterfly - results
The Butterfly Scheme results in a surface which
is but is not differentiable twice anywhere.
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Comparison
Butterfly
Loop
Catmull-Clark
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Kobblet sqrt(3) Subdivision
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Kobblet sqrt(3) Subdivision
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Kobblet sqrt(3) Subdivision
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Subdivision Schemes
Loop
Butterfly
Catmull-Clark
Doo-Sabin
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Subdivision Schemes
Loop
Butterfly
Catmull-Clark
Doo-Sabin
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Advantages Difficulties

• Advantages
• Simple method for describing complex surfaces
• Relatively easy to implement
• Arbitrary topology
• Smoothness guarantees
• Multiresolution
• Difficulties
• Intuitive specification
• Parameterization
• Intersections

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Subdivision Schemes

• There are different subdivision schemes
• Different methods for refining topology
• Different rules for positioning vertices
• Interpolating versus approximating

Face Split
Vertex Split
Quad. Meshes
Triangular Meshes
Doo-Sabin, Midege (C1)
Catmull-Clark (C2)
Loop (C2)
Approximating
Biquartic (C2)
Kobbelt (C1)
Butterfly (C1)
Interpolating