subdivision schemes
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Subdivision SchemesSubdivision Schemes
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
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 movedInterpolating – original vertices are unaffected
Is the scheme used here interpolating or approximating?Is the scheme used here interpolating or approximating?
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Why Subdivision?Frame from “Geri’s Game” by PixarFrame from “Geri’s Game” by Pixar
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
LOD
Compression
Smoothing
Why Subdivision?
424Kb424Kb
1Kb1Kb52Kb52Kb
13.3M13.3Mbb
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Corner Cutting
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Corner Cutting
1 : 33
: 1
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Corner Cutting
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Corner Cutting
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Corner Cutting
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Corner Cutting – Limit Curve
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Corner Cutting
The control polygon
The limit curve – Quadratic B-Spline Curve
A control point
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
4-Point Scheme
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
4-Point Scheme
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
4-Point Scheme
1 :
1
1 :
1
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
4-Point Scheme
1 :
8
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
4-Point Scheme
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
4-Point Scheme
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
4-Point Scheme
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
4-Point Scheme
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
4-Point Scheme
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
4-Point Scheme
The control polygon
The limit curveA control point
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Comparison
Non interpolatory subdivision schemes
• Corner Cutting
Interpolatory subdivision schemes
• The 4-point scheme
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
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?
1C2C
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Surface subdivision 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.
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
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
oldi
newi VV
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Loop’s Subdivision - topology Based on a triangular mesh Loop’s scheme does not
create face points
Old face
New face
Vertex points
Edge points
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Loop’s subdivision – stencil Every new vertex is a weighted average of old ones. The list of weights is called a Stencil Is this scheme approximating or interpolating?
3 3
1
1
1
1
1
1
1
nw
n
nnwn
22cos2340
64
n – vertex point’s valence
The rule for vertex points The rule for edge points
833 lkj VVVV
V i
new
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Loop - Results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Loop - Results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Loop - Results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Loop - Results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Loop - Results
Loop’s scheme results in a limit surface which is of continuity everywhere except for a finite number of singular points, in which it is .
2C
1C
Behavior of the subdivision along edges
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Butterfly Scheme Butterfly is an interpolatory scheme. Topology is the same as in Loop’s scheme. Vertex points use the location of the old vertex. Edge points use the following stencil:
-1
-1
-1
-1
88
2
2
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Butterfly - results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Butterfly - results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Butterfly - results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Butterfly - results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Butterfly - results
The Butterfly Scheme results in a surface which is but is not differentiable twice anywhere.
1C
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Catmull-Clark 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.
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Catmull-Clark
1
1
11
1
First, all the face points are calculated
Step 1
1 1
1
1
Then the edge points are calculated using the values of the face points and the
original vertices
Step 2Last, the vertex points are calculated using the values of the face and edge points
and the original vertex
Step 3
22
21
212
1
1
nw
)2( nnwn
n - the vertex valence
1
Face pointsFace pointsEdge pointsEdge points
Vertex pointsVertex points
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Connecting The Dots After Computing the new points, new edges are formed by:
connecting each new face point to the new edge points of the edges defining the old face.Connecting each new vertex point to the new edge points of all old edges incident on the old vertex point.
GoneGone
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Catmull-Clark - results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Catmull-Clark - results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Catmull-Clark - results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Catmull-Clark - results
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Catmull-Clark - results
Catmull-Clark Scheme results in a surface which is almost everywhere2C
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
LoopLoop
Catmull-Catmull-ClarkClark
ButterflyButterfly
Comparison
Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion
Pros and Cons Pros
A single mesh defines the whole modelSimple local rules
Easy to implement.Numerical Stability Easy to generate sharp feature withLod, Compression etc
ConsEvaluating a single point on the surface is hard
Not suitable for CAGDMesh topology has great influence on the over all shape.May become expensive in term of rendering
Global subdivision
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