Bicubic G1 interpolation of arbitrary quad meshes using a 4-split

S. Hahmann G.P. Bonneau B. Caramiaux

CAI Hongjie

Mar. 20, 2008

Geometric Modeling and Processing 2008

Authors

Stefanie Hahmann

• Main Posts Professor at Institut National Polytechnique

de Grenoble (INPG), France

Researcher at Laboratorie Jean Kuntzmann (LJK)

• Research CAGD

Geometry Processing

Scientific Visualization

Authors

Georges-Pierre Bonneau

• Main Posts Professor at Université

Joseph Fourier

Researcher at LJK

• Research CAGD

Visualization

Outline• Applications of surface modeling

• Background Subdivision surface Global tensor product surface Locally constructed surface

• Circulant Matrices

• Vertex Consistency Problem

• Surface Construction by Steps

Applications of Surface Modeling

• Medical imaging

• Geological modeling

• Scientific visualization

• 3D computer graphic animation

A peep of HD 3D Animation

From Appleseed EX Machina (2007)

Subdivision Surface

Doo-Sabin 细分方法

Catmull-Clark 细分方法

Loop 细分方法

Butterfly 细分方法

From PhD thesis of Zhang Jinqiao

Locally Constructed Surface

From S. Hahmann, G.P. Bonneau. Triangular G1 interpolation by 4-splitting domain triangles

Circulant Matrices

• Definition: A circulant matrix M is of the form

• Remark: Circulant matrix is a special case of Toeplitz matrix

0 1 2 2 1

1 0 1 3 2

1 2 3 1 0

n n

n n n

n n n

a a a a a

a a a a aM

a a a a a

Circulant Matrices

• Property: Let f(x)=a0+a1x +…+ an-1xn-1,

then eigenvalues, eigenvectors and determinant of M are

Eigenvalues:

Eigenvectors:

Determinant:

2 /i ne

( ), 0,..., 1.kk f k n

2 ( 1) T(1, , ,..., )k k n kk v

1

0

det( ) ( )n

k

k

f

M

Examples of Circulant Matrices• Determine the singularity of

Solution: f(x)=0.5+0.5xn-1,

0.5 0 0 0.5

0.5 0.5 0 0

0 0 0.5 0

0 0 0.5 0.5n n

T

2 /i ne 2 ( 1) /det( ) 0 ( ) 0.5(1 ) 0, 0,..., 1

2 ( 1) / 2 1,

2 |

k k n i nk f e k n

k n n m m

n

T

Examples of Circulant Matrices• Compute the determinant of

• Compute the rank of

0

0

0

0.5 0 0 0.5

0.5 0.5 0 0

0.5 0 0 0.5n n

A1

0

0

det( ) (cos(2 / ) )n

k

k n

A

1 cos(2 / ) cos(4 / ) cos(2( 1) / )

cos(2 / ) 1 cos(2 / ) cos(2( 2) / )

cos(2( 1) / ) cos(2( 2) / ) cos(2( 3) / ) 1n n

n n n n

n n n n

n n n n n n

A

( ) 2rank A

Vertex Consistency Problem• For C2 surface assembling

If G1 continuity at boundary is satisfied,

then , i i iv scalar functions and such that1

1 1

( ) ( ,0) ( ) ( ,0) ( ) (0, ),

[0,1], 1,..., .

i i i

i i i i i i i i ii i i

i

u u v u u u uu u u

u i n

S S S

Vertex Consistency Problem

• Twist compatibility for C2 surface

then

2 2

1 1

(0,0) (0,0)i i

i i i iu u u u

S S

2 2 1

1 1

2'

1' '

1 1

(0) (0,0) (0) (0,0)

(0) (0,0) (0) (0,0)

(0) (0,0) (0) (0,0)

i i

i ii i i i

i i

i ii i i

i i

i ii i

vu u u u

u u u

vu u

S S

S S

S S

Vertex Consistency Problem

• Matrix form

It is generally unsolvable when n is even

2 111 1

1 2 22 2

2 111 1

1

(0) 0 0 (0)

(0) (0) 0 0

0 (0) (0) 0

0 0 (0) (0)nn n

n n nn

v

u uv

v

v u u

rS

r

rS

r

2 1' ' '

1 1

(0) (0,0) (0) (0,0) (0) (0,0) (0) (0,0)i i i i

i i i i ii i i i i

vu u u u u

S S S S

r

Sketch of the Algorithm• Given a

quad mesh

• To find 4

interpolated bi-cubic

tensor surfaces for

each patch with

G1 continuity at

boundary

Preparation: Simplification

• Simplification of G1 continuity condition1

1 1

( ) ( ,0) ( ) ( ,0) ( ) (0, ),

[0,1], 1,..., .

i i i

i i i i i i i i ii i i

i

u u v u u u uu u u

u i n

S S S

( ) ( ) 1/ 2, [0,1], i i i i iv u u u Let then

1

1 1

1 1( ) ( ,0) ( ,0) (0, ),

2 2

[0,1], 1,..., .

i i i

i i i i ii i i

i

u u u uu u u

u i n

S S S

Choice of

• Let be constant, depended only on n (the order of vertex v)

• Specialize G1 continuity condition at ui=0, then

• Non-trivial solution require

10

10

0

(0,0)0.5 0 0 0.5

0.5 0.5 0 0

(0,0)0.5 0 0 0.5n

n

u

u

0

S

S

( )i iu

0 1 ': (0), : (0)i i

10 0

0

det( ) (cos(2 / ) ) 0, cos(2 / )n

k

k n n

choose

Choice of

• Determine

(1)i

( )i iu

1

1 1

1 1 1 1

2 1 1cos( ) (0,0) (0,0) (0,0),

2 2

1 , ,

i i i

i i i i

i i i i i i

n u u u

u u u u u u

S S S

(1)i

1

1 1

2 1 1cos( ) (1,0) (0,0) (0,0),

2 2

i i i

i i i in u u u

S S S

2(1) cos( )i

in

2 1cos (1 2 ) [0, ]

2 ( )

2 1cos (2 1) [ ,1]

2

i i

i i

i ii

u un

uu u

n

choose

ni is the order of vi

Step 1:Determine Boundary Curve

• Differentiate G1 continuity equation and specialize at ui=0, then

• Matrix form

2 2 1 21 0

21 1

1 ' 0

1 1(0,0) (0,0) (0) (0,0) (0,0)

2 2

(0), (0) cos(2 / )

i i i i

i i i i i i

i i

u u u u u u

n

S S S S

where

1 01 2 Tt d d

1 2

1

, (0,0)i

ii i

n

u u

t

St t

tT T1 2 1 2

1 2 2 21 1(0,0) (0,0)

, n n

n nu u u u

S S S S

d d

Examples of Circulant Matrices• Determine the singularity of

Solution: f(x)=0.5+0.5xn-1,

0.5 0 0 0.5

0.5 0.5 0 0

0 0 0.5 0

0 0 0.5 0.5n n

T

2 /i ne 2 ( 1) /det( ) 0 ( ) 0.5(1 ) 0, 0,..., 1

2 ( 1) / 2 1,

2 |

k k n i nk f e k n

k n n m m

n

T

Step 1:Determine Boundary Curve

• Differentiate G1 continuity equation and specialize at ui=0, then

• Matrix form

2 2 1 21 0

21 1

1 ' 0

1 1(0,0) (0,0) (0) (0,0) (0,0)

2 2

(0), (0) cos(2 / )

i i i i

i i i i i i

i i

u u u u u u

n

S S S S

where

1 01 2 Tt d d

1 2

1

, (0,0)i

ii i

n

u u

t

St t

tT T1 2 1 2

1 2 2 21 1(0,0) (0,0)

, n n

n nu u u u

S S S S

d d

Step 1:Determine Boundary Curve

• Notations

• Selection of d1,d2

T1,0 ,0 , 0,1,2.n

k k k k b b b

T

1 n p v v

33

,00

( ,0) (2 ), [0,0.5].i ii k k i i

k

u B u u

S b

1 11 1 1 0

2 ( )[ (0,0)] 6( ) 6 , cos

ini ij

i

j iB B

u n n

S

d b b p

22 1

2 1 2 1 0 2 02

2[ (0,0)] 24( - 2 ) 24[ ( 2 ) ]

3

ini

i

B Bu

S

d b b b b p

Step 2:Twist Computations

• d1,d2 is in the image of T

• Determine the twist

• Determine

1 1 2 2, d Td d Td

1 01 2 Tt d d

1 01 2 t d d

11ib

2

11 10 01 001

(0,0) 36( )i

i i i ii

i iu u

S

t b b b b

Change of G1 Conditions

• From

• To

1

1 1

1 1( ) ( ,0) ( ,0) (0, )

2 2

i i i

i i i i ii i i

u u u uu u u

S S S

1

1

1

( ,0) ( ) ( ,0) ( ) ( )

( ,0) ( ) ( ,0) ( ) ( )

i i

i i i i i i i ii i

i i

i i i i i i i ii i

u u u u uu u

u u u u uu u

S SV

S SV

( ) ( ) i i i iu u Vand to be determined

Step 3: Edge Computations• Determine

• Determine Vi(ui)

where

V0,V1 are two n×n matrices determined by

G1 condition

( )i iu( ) sin(2 / )(1 ) sin(2 / ) , [0,1]i i i i i iu n u n u u

22

0

( ) (2 ), [0,0.5]ii i j j i i

j

u B u u

V v

1 T 0 1 T 10 0 1 1 2 1 1

1[ ,..., ] , [ ,..., ] , ( )

2n n i i k v v V p v v V p v v v

Step 3: Edge Computations

• Determine 21 31,i ib b

33

,1 ,001

( ,0) 6( ) (2 )

1( ) ( ,0) ( ) ( ), [0, ]

2

ii i

i j j j iji

i

i i i i i i i ii

u B uu

u u u u uu

Sb b

SV

Step 4: Face Computations

• C1 continuity between inner micro faces

• We choose A1,A2,A3,A4 as dof.

1

2i i

i

A A

B

2

1 2 3 4

2

4

i i

B BC

A A A A

Results

Results

Conclusions

• Suited to arbitrary topological quad mesh

• Preserved G1 continuity at boundary

• Given explicit formulas

• Low degrees (bi-cubic)

• Shape parameters control is available

Reference• S. Hahmann, G.P. Bonneau, B. Caramiaux

Bicubic G1 interpolation of arbitrary quad meshes using a 4-split

• S. Hahmann, G.P. Bonneau

Triangular G1 interpolation by 4-splitting domain triangles

• Charles Loop

A G1 triangular spline surface of arbitrary topological type

• S. Mann, C. Loop, M. Lounsbery, et al

A survey of parametric scattered data fitting using triangular interpolants

Thanks!

Q&A