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Subdivision Depth ComputationSubdivision Depth Computationfor for

Catmull-Clark Subdivision SurfacesCatmull-Clark Subdivision Surfaces

Fuhua (Frank) Cheng

University of Kentucky, Lexington, KY

Junhai Yong

Tsinghua University, Beijing, China

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Outline:

1. Introduction

2. Subdivision depth computation for regular

CCSS patches

3. For extra-ordinary CCSS patches

4. Examples

5. Concluding Remarks

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Introduction:Introduction:

Recent focusRecent focus: Subdivision Surfaces

The de-facto standard for generating freeform curves and surfaces of arbitrary topology

(in visualization and animation application)

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Graphical Modeling

Games

Animation

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• Pixar’s Renderman

• Alias|Wavefront’s Maya

• Nichimen’s Mirai

• Newtek’s Lightwave 3D

Used as primary representation scheme in:

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What is a

subdivision surface?

Triangular

Loop Butterfly

QuadrilateralCatmull-Clark Doo-Sabin

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• Repeatedly refining the given control mesh to get M0,M1,M2,M3, limit surface (subdivision surface)

M0

M1

M2

M3

S = M∞

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What so special?

One piece representation™

(arbitrary topology)

Multi-resolution(Scalability)

Code Simplicit

y

Covers both polygon form and

surface form(Uniformity)

Numerical stability

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Why is One Piece Representation™ Good?

Data Management: Simpler

Rendering: More efficient

Machining: More precise

Animation: Crack free

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Graphic Modeling

Games

Animation

What is missing?CAD/CAM

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Why?

1. No parameterization2. No error control3. No adaptive tessellation

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• Without error controlNo CAD/CAM applications

• Without parameterizationDifficult to perform picking, rendering,

texture mapping

• Without adaptive tessellationToo expensive to use

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What is What is errorerror control? control?

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

M0

M1

M2

Mn

S = M∞

Cross-Sectional View

Limit Surface

εε

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Subdivision Depth Computation

For Regular CCSS Patches:

2nd Order Forward Difference

]2

C A - B [ 2 C - A- B 2

B

A C

2

C A

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] VVV2 , VVV2 [ max6

1 )t(C)t(L 312261

Cubic B-spline Curve Segment

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Bicubic B-spline Surface Patch

]uV V u)-(1 [ v ] V u V u)-(1 [ v)-(1 v)(u,L 22122111

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Bicubic B-spline Surface Patch

VVV2 , VVV2 maxM

M3

1 )v,u(S)v,u(L

1j,i1j,ij,ij,1ij,1ij,i

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Recurrence Formula

0k

kmn

kmn

kmn

M )4

1(

3

1

M3

1

)v,u(S)v,u(L

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)3

M(log k 4

Hence, for

We need

M )4

1(

3

1 k

Subdivision Depth

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Subdivision Depth Computation

For Extra-Ordinary CCSS Patches:

Ω - Partition based approach

Extra-ordinary point

(0,0)

1

12

13

11

23

222

1

33323

1

1v

u

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Subdivision of a Extra-ordinary patch

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13

12

11 S S S subpatches of vertices Control ,,

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Two Regions:– Vicinity of Extra-Ordinary point– The remaining (regular) regions

(0,0)

1

12

13

11

23

222

1

33323

1

1v

u

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have e

n D max

if Then

k

)v,u(L)v,u(S

w

n

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n n when

n find Given

)v,u(L)v,u(S

,.t.s

0,0

0n

n0

For vicinity of Extra-Ordinary point:

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3 2, 1, b

k

D find

n k 1 each or

)v,u(L)v,u(S

.t.s

f

kb

kb

For the remaining part :

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S of norm orderfirst :G

n on dependingconstant

) 2

7G ( log

n finding in is key the , Therefore

:

n

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Examples:

d

0.25 1

0.2 3

0.1 9

0.01 24

0.001 40

0.0001 56

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Concluding Remarks:

1. Provides precision/error control for all tessellation based application of CCSSs2. Possible disadvantage: might generate relatively large subdivision depth for vicinity of an extra-ordinary vertex already flat

enough3. Possible improvement: use second order norm for extra-ordinary patches

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Acknowledgement:

• Research work presented here is supported by NSF (DMS-0310645,

DMI-0422126).