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Mesh Parametrization with Butterfly Subdivision Scheme

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Mesh Parametrization with Butterfly Subdivision Scheme

CGIV, 26~29 July 2004, Penang, Malaysia

Graduate School of Software

Dongseo University, Busan, Korea

Byung Gook Lee and Chung Jae Lim

Nam Woo Kim, [email protected]

Parametrization of a surface triangulation S(G;X) is any planar triangulation P which is isomorphic to G

Subdivision is a very powerful paradigm for the generation of smooth curves and surfaces of arbitrary topology

This paper proposes an approach of parametrization using subdivision technique to decrease the distortion

(0,1)

(1,1)

(0,0)

(1,0)

The method how to map a bitmap image onto the 3D model

Improvement of reality by using even for a small number of vertices

Bitmap

Texture

coordinate

(s,t)

v

v

z

t

y

s

x

parametrization in 2D

triangular mesh in 3D

- Embedding 3D mesh to 2D parameter space
- Requirements
- Distortion minimization
- One-to-one mapping

Determine shape of parameter space

Map boundary vertices onto a convex polygon

Determine coefficients for the inner vertices

To set each u to be a convex combination of its neighbours

Solve a linear system Ax = b

Benefit : simple and fast, one-to-one embedding

Drawback : high distortions in small mesh

1-ring neighborhood

in parametric space

parameterization

with rectangular boundary

3D mesh

0

~

~

a

a

=

~

~

~

~

b

b

0

0

+

+

a

a

A

a‘

R2

b‘

B

x1, x2, x3, ... , xn:Interior points

xn+1, xn+2, xn+3, ... , xN: Boundary points

a‘+b‘

a : Distance between A and B in 3D mesh

b : Distance between B and C in 3D mesh

Uniform Parametrization [Tutte 63]

Shape-preserving Parametrization [Floater 97]

=

(

)

¸

d

f

l

l

E

i

j

1

(

)

¸

E

2

i

j

0

(

)

=

o

r

a

¸

>

2

E

=

i

j

0

i

j

i

2

i

j

;

=

i

j

;

f

g

h

F

i

1

2

3

;

2

N

N

o

r

e

a

c

n

;

;

;

:

:

:

;

X

X

¸

¸

1

u

u

=

=

i

j

i

i

j

j

j

j

1

1

=

=

P5

P3

P4

y

P

P2

P6

P1

x

- Uniform Parametrization [Tutte 63]
- ui is the barycentre of its neighbours

di: The number of neighbours ui

- Shape-preserving parametrization [Floater97]
- Conformal mapping of 1-ring neighborhood
- Average of barycentric coordinates

1-ring neighborhood

in 3D

conformal mapping

onto 2D

averaging

barycentric coord.

d

d

i

i

X

X

¸

¸

P

P

1

=

=

k

i

j

i

j

;

:

k

k

;

;

(

)

P

P

P

a

r

e

a

k

k

1

1

2

3

=

=

;

;

k

k

k

k

(

)

¸

P

P

1

¡

¡

¸

¸

¸

P

P

P

P

+

+

x

x

=

=

i

j

k

j

i

=

;

i

j

i

j

i

j

1

2

3

1

(

)

k

P

P

P

;

1

2

3

a

r

e

a

;

;

;

1

2

3

;

;

(

)

(

)

=

(

)

µ

P

P

P

2

2

a

n

g

¼

a

n

g

x

x

x

=

k

k

k

k

j

i

j

i

1

1

+

+

(

)

P

P

P

;

;

;

;

a

r

e

a

1

3

;

;

¸

=

i

j

;

2

(

)

P

P

P

;

a

r

e

a

1

2

3

;

;

(

)

P

P

P

a

r

e

a

1

2

;

;

¸

=

i

j

;

3

(

)

P

P

P

;

a

r

e

a

1

2

3

;

;

Step 1 : A vertex is selected and sub triangles are created according to the neighbours. These drawings have to satisfy the below conditions:

P

P

P

P

(

)

(

)

l

l

l

1

+

r

r

d

d

d

i

1

i

i

X

X

X

¸

k

d

1

P

P

¢

¢

¢

¹

1

0

¸

=

=

k

l

i

j

i

¹

¹

¹

=

;

=

;

;

:

k

k

l

k

k

l

k

l

d

;

;

;

:

:

;

;

;

i

l

1

=

k

k

1

1

=

=

Step 2 : If di≥3 rotation has to be done in order to draw triangles. In each rotation, a straight line is drawn from the originated vertex through middle point until it meets the sub mesh’s boundary. This will results a triangle as shown below

2w

-w

-w

1/2

1/2

-w

2w

-w

Subdivision defines a smooth curve or surface as the limit of a sequence of successive refinements

It is a subdivision scheme for surfaces which used triangular topology

Subdivision and Butterfly schemes

Approach 1

Run

Butterfly

Subdivision

Schemes

Initial mesh

Compute

coefficients i,j

(inner vertices +

boundary vertices)

parametrization

Approach 2

Compute

coefficients i,j

(inner vertices +

boundary vertices)

Initial mesh

Run

Butterfly

Subdivision

Schemes

parametrization

- M1=SM0
- Approach 1 is expensive
- Why? : more computational time and memory to solve the linear system

- Approach 2 is easy to calculate the coefficients
- We have to compare these two approaches with an appropriate distortion measure

# of vertices of M1= # of vertices of M0 + # of edges of M0