3D Modeling Subdivision Surfaces

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3D ModelingSubdivision Surfaces
Geris Game (1998) Pixar Animation Studios
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Modeling

• How do we ...
• Represent 3D objects in a computer?
• Construct such representations quickly and/or
  automatically with a computer?
• Manipulate 3D objects with a computer?
• 3D Representations provide the foundations for
  Computer Graphics, Computer-Aided Geometric
  Design, Visualization, and Robotics
• Different methods for different object
  representations

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3D Object Representations

• Raw data
• Point cloud
• Range image
• Polygon soup
• Surfaces
• Mesh
• Subdivision
• Parametric
• Implicit

• Solids
• Voxels
• BSP tree
• CSG

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Point Cloud

• Unstructured set of 3D point samples

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Range Image

• Set of 3D points mapping to pixels of depth image
  
• Acquired from range scanner

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Polygon Soup

• Unstructured set of polygons

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3D Object Representations

• Raw data
• Point cloud
• Range image
• Polygon soup
• Surfaces
• Mesh
• Subdivision
• Parametric
• Implicit

• Solids
• Voxels
• BSP tree
• CSG

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Mesh

• Connected set of polygons (usually triangles)
• May not be closed

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Parametric Surface

• Tensor product spline patchs
• Careful constraints to maintain continuity

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Subdivision Surface

• Coarse mesh subdivision rule
• Define smooth surface as limit of sequence of
  refinements

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Implicit Surface

• Points satisfying F(x,y,z) 0

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3D Object Representations

• Raw data
• Point cloud
• Range image
• Polygon soup
• Surfaces
• Mesh
• Subdivision
• Parametric
• Implicit

• Solids
• Voxels
• BSP tree
• CSG

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Voxels

• Uniform grid of volumetric samples
• Acquired from CAT, MRI, etc.

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BSP Tree

• Binary space partition with solid cells labeled
• Constructed from polygonal representations

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CSG

• Hierarchy of boolean set operations (union,
  difference, intersect) applied to simple shapes

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Subdivision Surface

• Subdivision methods refine a (simple) control
  polygon such that in the limit it converges to a
  smooth curve.
• A multitude of different curve and surface types
  can be described by subdivision.

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Subdivision Surfaces

• Approach Limit Curve Surface through an Iterative
  Refinement Process.

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Subdivision in 3D

• Same approach works in 3D

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Problems with Splines

• Traditionally spline patches (NURBS) have been
  used in production for character animation.
• Difficult to stitch together
• Maintaining continuity is hard
• Trimming boundaries is hard
• Difficult to model objects with complex topology

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

• Subdivision surfaces are used for
• Geris hands and head.
• Clothes Jacket, Pants, Shirt.
• Tie and Shoes.
• Movie

(Geris Game, Pixar 1998)
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Why Subdivision Surfaces?

• Subdivision methods have a series of interesting
  properties
• Applicable to meshes of arbitrary topology
  (non-manifold meshes).
• Scalability, level-of-detail.
• Numerical stability.
• Simple implementation.
• Compact support.
• Affine invariance.
• Continuity.

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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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Subdivision in 1-D

• Simplest example piecewise linear subdivision

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Chaikens Algorithm
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Key Questions in Subdivision

• How refine mesh?
• Topology change
• Where to place new vertices
• Provable properties about limit surface

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Loop Subdivision Scheme

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

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Loop Subdivision Scheme

• How position new vertices?
• Choose locations for new vertices as weighted
  average of original vertices in local
  neighborhood
• What if vertex does not have degree 6?

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Loop Subdivision SchemeRule for extraordinary
vertices
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Loop Subdivision Scheme

• How to choose
• Analyze properties of limit surface
• Interested in continuity of surface and
  smoothness
• Involves calculating eigenvalues of matrices
• Original
• Warren

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Loop Subdivision Boundaries

• Subdivision Mask for Boundary Conditions

Vertex Rule
Edge Rule
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Subdivision as Matrices

• Subdivision can be expressed as a matrix Smask of
  weights w.
• Smask is very sparse
• Never Implement this way!
• Allows for analysis
• Curvature
• Limit Surface

Smask Weights
Old Control Points
New Points
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Loop Subdivision Scheme

• Works on triangular meshes
• Is an Approximating Scheme
• Guaranteed to be smooth everywhere except at
  extraordinary vertices.

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Subdivision Schemes

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

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Catmull-Clark Subdivision
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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 97)
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Subdivision Surfaces for Compression
?
Progressive Geometry Compression Andrei
Khodakovsky, Peter Schröder and Wim Sweldens
(SIGGRAPH 2000)
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Subdivision Surface Summary

• Advantages
• Simple method for describing complex surfaces
• Relatively easy to implement
• Arbitrary topology
• Local support
• Guaranteed continuity
• Multi-resolution
• Difficulties
• Intuitive specification
• Parameterization
• Intersections