Recursively generated Bspline surfaces on arbitrary topological meshes

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Recursively generated B-spline surfaces on
arbitrary topological meshes

• E Catmull and J Clark
• Computer Aided Design, 1978
• Presented by Geoffrey Lefebvre

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Outline

• Introduction to subdivision surfaces
• Definition of subdivision surfaces
• Comparison with NURBS
• Possible application of subdivisions surfaces
• Catmull Clark paper
• Overview
• Regular mesh subdivision rules
• Subdivision rules for meshes of arbitrary topology

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What is a subdivision surface?

• Smooth surface generated as the limit of a
  sequence of successively refined polyhedral
  meshes created by repeated application of the
  subdivision procedure.

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Many types of subdivision surfaces

• Face splitting technique
• Catmull-Clark Based on quads, approximative, C2
• Loop Based on triangles, approximative, C2
• Modified Butterfly Based on triangles,
  interpolative, C1
• Kobbelt Based on quads, interpolative, C1
• Vertex splitting technique
• Doo-Sabin, Midedge, C1

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Comparison with NURBS

• PRO
• Naturally handles arbitrary surface topology
• No need for trimming
• Local refinement easier to do
• No seam related problems (smoothness, visibility)
• CON
• Not parametric, harder to map textures
• Behavior not as well understood

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Possible Applications

• Use instead of NURBS in animation package
• Multi-resolution geometry compression
• Level of detail rendering

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Recursively generated B-spline surfaces on
arbitrary topological meshes (Catmull Clark,
1978)

• Paper describes a method for recursively
  generating surfaces that approximates point lying
  on a mesh of arbitrary topology.
• Generated surface is a B-spline surface
  everywhere except near extraordinary points.
• Generalization of a recursive bi-cubic B-spline
  patch subdivision algorithm.
• This algorithm was originally designed to
  recursively divide a surface patch into four
  sub-patches until the resulting patches are
  roughly the size of a pixel.

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Basic concept

• Take a standard B-spline surface patch and split
  it into 4 subpatches
• Original 16 control points mesh becomes a 25
  points mesh.
• 3 types of new control points
• face points
• edge points
• vertex points

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Control point mesh
p42
p41
p43
p44
p31
p32
p33
p34
p21
p22
p23
p24
p11
p12
p13
p14
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Control point mesh
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Control point mesh
q51
q41
q31
q21
q22
q11
q12
q13
q14
q15
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Matrix form

• B-spline patch matrix form
• S(u,v) U M G MT VT
• Where
• M is the B-spline basis matrix
• G is the set of control point
• U u3 u2 u 1
• V v3 v2 v 1

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More matrix stuff

• The subpatch 0 lt u,v lt 1/2 can be represented by
• S(u1,v1) U S M G MT ST VT
• Where
• u1 u/2
• v1 v/2
• S

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Dont worry, its almost over

• The sub-patch must still be a bi-cubic B-spline
  with its own set of control point.
• S(u,v) U M G1 MT VT
• This will be true iff
• M G1 MT S M G MT ST
• So we get
• G1 M-1 S M G MT S M-T

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Getting there

• The new set of control point is calculated from
  the old set
• G1 H G HT
• where H

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Regular control point mesh rules

• From the matrix representation we get the
  following equations
• For new face point
• For new edge point

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Regular control point mesh rules

• For new vertex points

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Arbitrary control point mesh rules

• For new face points The average of all old
  points defining the face.
• For new edge points The average of the midpoints
  of the old edge with the average of the two new
  face points of the face sharing the edge.

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Arbitrary control point mesh rules

• For new vertex point
• Where
• Q the average of the new face points of all
  faces adjacent to the old vertex point.
• R the average of the midpoints of all old edges
  incident on the old vertex point.
• S old vertex point

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Arbitrary control point mesh rules

• Each new face point is connected to the new edge
  points of the edges defining the old face.
• Each new vertex point is connected to the new
  edge points of all old edges incident on the old
  vertex point.

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An example
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An example
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An example
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An example
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Conclusion

• This paper is one of the first paper on the topic
  of subdivision surfaces.
• Paper generalize bi-cubic B-spline patch
  subdivision algorithm to mesh of arbitrary
  topology.
• Makes no formal claim of surface behavior near
  extraordinary points.