1

What is subdivision based

representation?

Subdivision Surfaces

15.12 Subdivision Surfaces

2

What is so special?

One piece

representation™

(arbitrary topology)

Multi-resolution

(Scalability)

Code

Simplicit

y

Covers both

polygon form and

surface form

(Uniformity)

Numerical

stability

3

One piece

representation™

4

Multi-resolution

(Scalability)

5

Covers both polygon form and surface form

(Uniformity of representation)

6

So, just what is a

subdivision surface?

Triangular

Loop Butterfly

QuadrilateralCatmull-Clark Doo-Sabin

7

Basic Concept (Catmull-Clark Scheme):

: vertices from mesh M0

, , : vertices to be generated for M1

e5

0

e4

0

e3

0

e2

0

e1

0

v0

Around a vertex v of degree 5

8

Basic Concept (Catmull-Clark Scheme):

Generating new face points

Face point: centroid of each face

f5

1f4

1

f3

1

f2

1f11

v0

: face point

9

Basic Concept (Catmull-Clark Scheme):

Generating new edge points

f5

1f4

1

f3

1

f2

1f11

v0

: edge point

e2

1

e11

e5

1

e3

1

e4

1

4

11

1

001 iiii

ffeve

10

Basic Concept (Catmull-Clark Scheme):

Generating new vertex points

f5

1f4

1

f3

1

f2

1f11

v0

: vertex point

e2

1

e11

e5

1

e3

1

e4

1

12

0

2

01 112ii f

ne

nv

n

nv

v1

11

Basic Concept (Catmull-Clark Scheme):

Forming new edges

f5

1f4

1

f3

1

f2

1f11

v0

: vertex point

e2

1

e11

e5

1

e3

1

e4

1

v1

12

• Repeatedly refining the control meshes,

one gets M0,M1,M2,M3, limit

surface (subdivision surface)

M0

M1

M2

M3

S = M∞

13

Modeling made much easier. Why?• No restrictions on the topology of the control points

• Local refinement is possible

NURBS Catmull-Clark

NURBS Catmull-Clark

14

Example of control meshes of

Catmull-Clark subdivision surfaces

15

Can model any kind of special features

(by modifying the subdivision rules)

16

Most importantly, can represent any shape with

just one surface

(one piece representation™)

One Piece Multi-Piece

Solid Modeling

17

Is One Piece Representation™ Good?

Data Management: Simpler

Rendering: More efficient

Machining: More precise

Animation: Crack free

18

Does this mean

the solid modeling area

is no longer needed?

19

What is subdivision based representation?

Subdivision Surfaces

?CAD/CAM

20

What is missing?

1. No parameterization

2. No error control

3. No adaptive tessellation

21

• Without error controlNo CAD/CAM applications

• Without parameterizationDifficult to perform picking, rendering, texture mapping

• Without adaptive tessellationToo expensive to use

22

A major breakthrough occurred in 1998

• Jos Stam

• Parameterization of Catmull-Clark

Subdivision Surfaces

• 1998

23

Work on Subdivision Surface

Parameterization

1. J. Stam (1998)

2. D. Zorin, D. Kristjansson (2002)

3. S. Lai, F. Cheng (2005)

24

J. Stam Lai/Cheng

(Discrete Fourier Transform)

The Extended Subdivision Diagram

Parameterization

25

Applications of the new

parameterization technique

• Surface Evaluation

• Texture Mapping

• Boolean Operations

• Surface Trimming

• Adaptive Tessellation

• Animation

26

Surface Evaluation

Fast, Exact Rendering

27

Texture Mapping1

1: Lai and Cheng, 2005

28

Texture Mapping1

1: Lai and Cheng, 2005

29

Texture Mapping1

1: Lai and Cheng, 2005

30

Boolean Operations2

2: Lai and Cheng, 2005

31

Surface Trimming2

2: Lai and Cheng, 2005

32

Adaptive Tessellation3

3: Lai and Cheng, 2005

33

What is error control?

34

Error Control: Given ε > 0, when would ║Mn - S║< ε ?

M0

M1

M2

Mn

S = M∞

Cross-Sectional View

Limit Surface

ε

35

What metric should we use to assess

║Mn - S║ for an extra-ordinary patch?

36

A solution is finally available…

• F. Cheng, G. Chen, J. Yong

• Subdivision Depth Computation for

Catmull-Clark Subdivision Surfaces

• 2005

37

This work is also important for adaptive subdivision5.

5: J. Yong, F. Cheng, (2004)

Control Mesh Limit Surface Uniform Subdivision Adaptive Subdivision

38

Basic Idea: Use unbalanced subdivision6 to

provide smooth transition between areas with

different densities

6: F. Cheng, J. Jaromczyk et al (1989)

2

0

0

0

12/9/2014 39University of Kentucky

Adaptive subdivision:input: a piecewise surface P and a subdivision level

assignment S

output: a triangular linear approximaiton P** of P

Three phases:Phase 1: define a label for each vertex of P

Phase 2: generate a gradrilateral subdivision mesh P*

of P

Phase 3: convert P* to a triangular linear approximation

P** of P

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Phase 1:

/* F { f | f is a patch of P } */

for each vertex v of P do

L (v ) := max( {1} { S(f ) | f F, v is a vertex of f } )

12/9/2014 41University of Kentucky

Phase 2:

1. for each vertex v of P do

LABEL(v ) := L(v );

2. for each patch f of P do

Subdivide(f );

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Subdivide(f : quadrilateral surface patch);

if (LABEL(v ) > 0 for more than one vertex of f )

then

balanced_sub( );

for i :=1 to 4 do

subdivide( );

else if (LABEL(v ) > 0 for only one vertex of f )

then

unbalanced_sub( );

for i :=1 to 3 do

subdivide( );

if

if

4321 ,,,, fffff

321 ,,, ffff

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46

Example of adaptive subdivision

Significant savings

47

• Pixar’s Renderman

• Alias|Wavefront’s Maya

• Nichimen’s Mirai

• Newtek’s Lightwave 3D

Subdivision surfaces have already

been used in

http://www.pixar.com/index.html

48

Question:

“Is subdivision the representation scheme for

future visualization & animation applications?”

49

The End

50

Acknowledgement:

• Research work presented here is supported

by NSF (DMS-0310645, DMI-0422126).

• Some datasets are taken from P. Schroeder,

D. Zorin, L. Kobbelt, H. Hoppe.

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