Subdivision Surfaces

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Subdivision Surfaces
CS 230
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Introduction

• 3D Subdivision example

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Introduction
Geris Game (1998) Pixar Animation Studios
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Goals of Subdivision Surfaces
Introduction

• How do we represent curved surfaces in the
  computer?
• Efficiency of Representation
• Continuity
• Affine Invariance
• Efficiency of Rendering
• Why use subdivision rather than Bezier patches?

vs
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Introduction

• Create Curve Surface through an Iterative
  Refinement Process.

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Introduction Chaikens Algorithm
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Surface evolution
Surface evolution with subdivision level
Limit surface
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Types of Subdivision

• Interpolating Schemes
• Limit Surfaces/Curve will pass through original
  set of data points.
• Approximating Schemes
• Limit Surface will not necessarily pass through
  the original set of data points.

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Types of Subdivision

• Approximation Schemes
• Catmull-Clark Subdivision
• Doo-Sabin Subdivision
• Loop Subdivision

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Ordinary and Extraordinary
Catmull-Clark Subdivision Valence 4
Loop Subdivision Valence 6

• Subdividing a mesh does not add extraordinary
  vertices.
• Subdividing a mesh does not remove extraordinary
  vertices.
• How should extraordinary vertices be handled?
• Make up rules for extraordinary vertices that
  keep the surface smooth.

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Catmull-Clark Subdivision
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Catmull-Clark Subdivision
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Modeling with Catmull-Clark

• Subdivision produces smooth continuous surfaces.
  
• How can sharpness and creases be controlled in
  a modeling environment?
• ANSWER Define new subdivision rules for
  creased edges and vertices.

• Tag Edges sharp edges.
• If an edge is sharp, apply new sharp subdivision
  rules.
• Otherwise subdivide with normal rules.

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Doo-Sabin Subdivision
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Doo-Sabin Subdivision
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Loop Subdivision

• Works on triangular meshes
• Guaranteed to be smooth everywhere except at
  extraordinary vertices.

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Loop Subdivision
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Subdivision Boundaries

• Subdivision for Boundary Conditions

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Loop Subdivision
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Comparison
Catmull- Clark Doo- Sabin Loop
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Subdivision in a production environment.

• Traditionally spline patches (NURBS) have been
  used in production for character animation.
• Difficult to control spline patch density in
  character modelling.

Subdivision in Character Animation Tony Derose,
Michael Kass, Tien Troung (SIGGRAPH 98)
(Geris Game, Pixar 1998)
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Sharp Edges

• Tag Edges as sharp or not-sharp
• n 0 not sharp
• n gt 0 sharp
• During Subdivision,
• if an edge is sharp, use sharp subdivision
  rules. Newly created edges, are assigned a
  sharpness of n-1.
• If an edge is not-sharp, use normal smooth
  subdivision rules.

IDEA Edges with a sharpness of n do not get
subdivided smoothly for n iterations of the
algorithm.
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Other Subdivision Schemes

• Complex data structures required to perform
  subdivision.
• Every polygon ( triangle, quad, ..) must know its
  neighbors
• Every vertex must know its neighbors
• Can we do something simpler?
• Use vertex normal information to help guess
  about neighboring polygons.
• Subdivide based on the normals.

subdivision
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PN Triangles

• Interpolating Scheme.
• Geom Normal

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Local Subdivision (PN triangles)
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Local Subdivision (PN triangles)

• Defined from triangular bezier patches.

Curved PN Triangles Alex Vlachos Joerg
Peters Chas Boyd Jason Mitchel
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Local Subdivision (PN triangles)
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Subdivision

• Advantages
• Easy to implement
• No complex data structures
• Easy to integrate into existing graphics
  applications
• Hardware amenable
• ATI Radeon 8500
• Looks good
• Disadvantages
• No guarantees on higher level continuity.
• Is limited in the amount of curvature it can
  provide.
• In some sense it is a hack and not as correct.

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Adaptive Subdivision

• Not all regions of a model need to be subdivided.
• Idea Use some criteria and adaptively subdivide
  mesh where needed.
• Curvature
• Screen size ( make triangles lt size of pixel )
• View dependence
• Distance from viewer
• Silhouettes
• In view frustum
• Careful! Must ensure that cracks arent made

crack
subdivide
View-dependent refinement of progressive meshes
Hugues Hoppe. (SIGGRAPH 87)
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Subdivision Surfaces for Compression
?
(Refinement)-1
Progressive Geometry Compression Andrei
Khodakovsky, Peter Schröder and Wim Sweldens
(SIGGRAPH 2000)
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Meshing from Particle Systems

• Witkin Heckbert Siggraph 94
• Constrain particle system to implicit surface
  (Implicit surface f 0 becomes constraint
  surface C 0)
• Particles exert repulsion forces onto each other
  to spread out across surface
• Particles subdivide to fill open gaps
• Particles commit suicide if overcrowded
• Display particle as oriented disk
• Constrain implicit surface to particles!

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Meshing Particles

• Stander Hart Siggraph 97
• Use particles as vertices
• Connect vertices into mesh
• Track/find critical points
• of function to detect changes
• in topology of implicit surface

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Shrinkwrapping

• Look at family of surfaces f -1(s) for s gt 0
• For s large, f -1(s) spherical
• Polygonize sphere
• Reduce s to zero
• Vertices to track surface
• Subdivide polygons as
• necessary when curvature
• increases

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Conclusions and Future Work

• Currently there is no standard data structure for
  handling general Subdivision Surface schemes
• Hardware (GPU) implementations do not exist
• Subdivision of higher dimensional surfaces