subdivision schemes

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?


Why Subdivision?Frame from “Geri’s Game” by PixarFrame from “Geri’s Game” by Pixar


LOD

Compression

Smoothing

Why Subdivision?

424Kb424Kb

1Kb1Kb52Kb52Kb

13.3M13.3Mbb


Corner Cutting


Corner Cutting

1 : 33

: 1


Corner Cutting


Corner Cutting – Limit Curve


Corner Cutting

The control polygon

The limit curve – Quadratic B-Spline Curve

A control point


4-Point Scheme


4-Point Scheme

1 :

1

1 :

1


4-Point Scheme

1 :

8


4-Point Scheme


4-Point Scheme

The control polygon

The limit curveA control point


Comparison

Non interpolatory subdivision schemes

• Corner Cutting

Interpolatory subdivision schemes

• The 4-point scheme


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


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.


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


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


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


Loop - Results


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


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


Butterfly - results


Butterfly - results

The Butterfly Scheme results in a surface which is but is not differentiable twice anywhere.

1C


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.


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


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


Catmull-Clark - results


Catmull-Clark - results

Catmull-Clark Scheme results in a surface which is almost everywhere2C


LoopLoop

Catmull-Catmull-ClarkClark

ButterflyButterfly

Comparison


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

subdivision schemes

Documents

control vertices

vertex points vertices

limit surface

edge points vertices

face points vertices

old vertices

interpolatory scheme

control net