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Distance Between a Catmull-Clark Subdivision Surface and Its Limit Mesh. Zhangjin Huang, Guoping Wang Peking University, China. Generalization of uniform bicubic B-spline surface continuous except at extraordinary points The limit of a sequence of recursively refined control meshes. - PowerPoint PPT Presentation
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Distance Between a Catmull-Clark Subdivision Surf
ace and Its Limit Mesh
Zhangjin Huang, Guoping Wang
Peking University, China
Generalization of uniform bicubic B-spline surface continuous except at extraordinary points
The limit of a sequence of recursively refined control meshes
Catmull-Clark subdivision surface (CCSS)
initial mesh step 1 limit surface
2C
CCSS patch: regular vs. extraordinary
Assume each mesh face in the control mesh a quadrilateral at most one extraordinary point (valence n is not 4)
An interior mesh face in the control mesh → a surface patch in the limit surface Regular patch: bicubic B-spline patches, 16 control points Extraordinary patch: not B-spline patches, 2n+8 control points
Control mesh Limit surface
SF
Blue: regular
Red: extraordinary
Control mesh approximation and error Control mesh is a piecewise linear app
roximation to a CCSS
Approximation error: the maximal distance between a CCSS and the control mesh
Distance between a CCSS patch and its mesh face (or control mesh) is defined as
is unit square is Stam’s parametrization of over is bilinear parametrization of over ( , )u vS
S
( , )u vFS
F
F
( , )max ( , ) ( , )u v
u v u v
S F
[0,1] [0,1]
Distance bound for control mesh approximation The distance between a CCSS patch and its control
mesh is bounded as [Cheng et al. 2006]
is a constant that only depends on valence n is the the second order norm of 2n+8 control points of For regular patches,
S
( )C nM
S
1(4)3
C
( , )max ( , ) ( , ) ( )u v
u v u v C n M
S F
Limit mesh approximation
An interior mesh face → a limit face → a surface patch We derive a bound on the distance between a patch and
its limit face (or limit mesh) as
means that the limit mesh approximates a CCSS better than the control mesh
F FS
Limit mesh : push the control points to their limit positions. It inscribes the limit surface
( ) ( )n C n
S
F
( , )max ( , ) ( , ) ( )u v
u v u v n M
S F
Regular patches: how to estimate
Regular patch is a bicubic B-spline patch:
Limit face , then
It is not easy to estimate directly!
S3 3
3 3,
0 0
( , ) ( ) ( ),( , )i j i ji j
u v N u N v u v
S p
( , ) ( , )u v u vS F
1,1 2,1 2,2 1,2{ , , , }F p p p p
1,1 2,1 2,2 1,2( , ) (1 )[(1 ) ] [(1 ) ]u v v u u v u u F p p p p( , ) ( , )u v u vS F
S
F
1,1p
1,2p2,1p
2,2p
1,1p
2,1p1,2p
Transformation into bicubic Bézier forms
Both and can be transformed into bicubic Bézier form:3 3
3 3,
0 0
( , ) ( ) ( )i j i ji j
u v B u B v
S b
S
( , )u vS ( , )u vF
3 33 3
,
0 0
( , ) ( ) ( )i j i ji j
u v B u B v
F b
S
F
Regular patches: distance bound
Core idea: Measure through measuring
S
F
1,2b
1,2b
3 33 3
,,0 0
( , ) ( , ) ( ) ( ) ( )i ji j i ji j
u v u v B u B v
S F b b
3 33 3
,,0 0
( ) ( )i ji j i ji j
B u B v
b b
( , ) ( , )u v u vS F
,, i ji j b b
Regular patches: distance bound (cont.) Bound with the second order norm , it
follows that
Distance bound function of with respect to is
Diagonal By symmetry,
,, i ji j b b M
1( , ) ( , ) ( (1 ) (1 ))
2u v u v u u v v M S F
( , )u vS ( , )u vF
1B( , ) ( (1 ) (1 )),( , )
2u v u u v v u v
( , ) 0 1
1max B( , ) maxD( )
4u v tu v t
D( ) B( , ) (1 ),0 1t t t t t t
Regular patch: distance bound (cont.) Theorem 1 The distance between a regular CCSS
patch and the corresponding limit face is bounded by
The distance between a regular patch and its corresponding mesh face is bounded as [Cheng et al 2006]
S F
( , )
1max ( , ) ( , )
4u vu v u v M
S F
S
F
( , )
1max ( , ) ( , )
3u vu v u v M
S F
Extraordinary patches: parametrization
An extraordinary patch can be partitioned into an infinite sequence of uniform bicubic B-spline patches
Partition the unit square into tiles Stam’s parametrization:
Transformation maps the tile onto the unit square
{ }, 1, 1,2,3km k m S
S
{ }, 1, 1,2,3km k m
( , ) ( , ) ( ( , ))km
k k km m mu v u v t u v
S S S
kmt
km
12S
22S
11S
21S
23S
13S
Extraordinary patches: distance bound Limit face can be partitioned into bilinear subfaces defined o
ver :
Similar to the regular case, for
By solving 16 constrained minimization problems, we have
( , )u vFkm
( , ) kmu v
( , ) ( ( , ))km
kk
m mu v u v
F F t
( , ) ( , ) ( , ) ( , )k km mu v u v u v u v S F S F
3 3
3 3,,
0 0
( ) ( )i ji j i ji j
B u B v
b b
0 , 3i j ,, , ,i ji j i jc M b b
Extraordinary patch: distance bound function Thus
is the distance bound function of with respect to :
The distance bound function of with respect to is defined as:
Diagonal
By symmetry,
( , ) ( , ) B ( , ) ,( , )k k km m mu v u v u v M u v S F
B ( , )km u v ( , )k
m u vS ( , )k
m u vF
( , )u vS ( , )u vF
B( , ) B ( ( , )), 1, 1,2,3km
k km mu v u v k m
t
3 33 3
,0 0
B ( , ) ( ) ( )km i j i j
i j
u v c B u B v
D( ) B( , ),0 1t t t t
( , ) 0 1max B( , ) maxD( ) : ( )u v t
u v t n
Extraordinary patches: distance bound Theorem 2 The distance between an extra-ordinary C
CSS patch and the corresponding limit face is bounded by
has the following properties:
, attains its maximum in
, attains its maximum in
Only needed to consider 2 subpatches and
S
F
( , )max ( , ) ( , ) ( )u v
u v u v n M
S F
D( ),0 1t t
3n D( )t1 1,4 2
4n D( )t1,12
12S
22S
Extraordinary patches: bound constant ,
, strictly decreases as n increases
( )n
4n 1( ) (4)
4n
3 48n ( )n
Comparison of bound constants First two lines are for control mesh approx. Last line are for limit mesh approximation , ,
n 3 4 5 6 7 8 9 10
Cheng et al. 06
0.784814 0.333333 0.574890 0.642267 0.527357 0.582436 0.510181 0.678442
Huang et al. 06
0.784314 0.333333 0.574890 0.549020 0.527357 0.424242 0.510181 0.519591
Limit mesh
0.258146 0.25 0.243129 0.237454 0.232761 0.228848 0.225549 0.222738
( )C n
( )n
( ) ( )n C n 3n
4n 1 1( ) (4) (4) ( )
4 3n C C n
Application to adaptive subdivison
Error tolerance
Frog model Car modelControl mesh Limit mesh Control mesh Limit mesh
0.1 25,642 14,527 14,717 8,285
0.05 43,864 26,482 24,905 14,375
0.01 172,159 113,713 97,704 58,805
The number of faces decreases by about 30%
Application to CCSS intersection
Conclusion Propose an approach to derive a bound on the
distance between a CCSS and its limit mesh Our approach can be applied to other spline based
subdivision surfaces Show that a limit mesh may approximate a CC
SS better than the corresponding control mesh
Thank you!