Subdivision
S
bdi i i Schemes
S h
for
f
Geometric Modelling
Kai Hormann
Ka
o a
University of Lugano
Outline
ƒ Sep 5 – Subdivision as a linear process
ƒ basic concepts, notation, subdivision matrix
ƒ Sep 6 – The Laurent polynomial formalism
ƒ algebraic approach, polynomial reproduction
ƒ Sep 7 – Smoothness analysis
ƒ Hölder regularity of limit by spectral radius method
ƒ Sep 8 – Subdivision surfaces
ƒoverview of most important
p
schemes & p
properties
p
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
The functional setting
ƒ initial data
ƒ mask
ƒ refinement rule
ƒ parameter values
ƒ piecewise linear functions
ƒ limit function
ƒ consider initial data
ƒ basic limit function
ƒ by linearity of the scheme
2
with
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Basic limit function
ƒ Examples
ƒ polygon subdivision
ƒ Chaikin’s corner cutting
ƒ cubic B-spline scheme
ƒ interolating 4-point scheme
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Convergence
ƒ if the sequence
of piecewise linear
g (uniformly)
(
y) for any
y initial
functions converges
data, then the scheme Sa is called convergent
ƒ the limit is necessarily a continuous function
ƒ necessary conditions for Sa to be convergent
ƒ even/odd coefficients of the mask sum to 1
⇔
a(z) = (1+ z )b(z)
and
b(1) = 1
ƒ 1 is the single dominant eigenvalue of the local
subdivision matrix
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Convergence
ƒ Example
ƒ scheme with mask a = [ –7,
7, 7, 16, 16, 7, –7
7 ] / 16
ƒ even/odd coefficients sum to 1
ƒ local
l
l subdivision
bdi i i matrix
ti
- eigenvalues:
ƒ necessary conditions for convergence satisfied
ƒ still, the scheme does not converge
g
ƒ more analysis needed!
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Convergence
ƒ Theorem
the scheme Sa converges, if and only if the scheme
Sb is contractive, i.e.
for any initial data
ƒ remember: Sb is the scheme for the differences
ƒ the scheme Sb is contractive, if
ƒ and
d that
th t iis th
the case if
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Contractivity
ƒ Examples
ƒ general primal 3-point
3 point a = [ w, ½, 1
1– 2 w, ½, w ]
- difference scheme b = [ w, ½ – w, ½ – w, w ]
- || b || = | w | + | ½ – w | < 1
for
w ∈ (–¼,¾)
- Sb is contractive, hence the scheme converges
ƒ scheme with mask a = [ –7, 7, 16, 16, 7, –7 ] / 16
- difference scheme b = [ –7, 14, 2, 14, –7 ] / 16
- || b || = max ( 7 + 2 + 7, 14 + 14 ) / 16 = 7 / 4 > 1
- Sb is not contractive, but …
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Contractivity
ƒ Example
ƒ scheme with mask a = [ –1,
1, 1, 8, 8, 1, –1
1 ]/8
- difference scheme b = [ –1, 2, 6, 2, –1 ] / 8
- || b || = max ( 1+ 6 + 1,
1 2+2)/8 = 1
- Sb is not contractive, but …
ƒ consider
id 2 steps
t
off the
th scheme,
h
i.e.
i the
th scheme
h
with symbol b(z) b(z2)
- maskk b2 = [ 1,
1 –2,
2 –8,
8 2,
2 7,
7 16,
16 32,
32 16,
16 7,
7 2,
2 –8,
8 –2,
2 1] / 64
- || b2 || = max (1+7+7+1, 2+16+2, 8+32+8) / 64 < 1
8
is contractive, hence the scheme converges
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Convergence
ƒ Theorem
the scheme Sa converges, if and only if the scheme
Sb is contractive
ƒ the scheme Sb is contractive
contractive, if
for some `>0, with
where are the coefficients of the scheme
with symbol b`(z)
(z)= b(z)b(z2) ··· b(z2`−1)
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Smoothness
ƒ Theorem
if the scheme Sb converges, then the limit curves of
the scheme Sa with symbol
are C m-continuous
continuous
ƒ Sb is the scheme for the m-th divided differences
and
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Smoothness
ƒ Example
ƒ4
4-point
point scheme
- symbol:
- mask of Sb : b = [ –1,
1 1,
1 8,
8 8,
8 1,
1 –1
1 ]/8
- this scheme converges (see above)
- the limit curves of the 4
4-point
point scheme are C 1-continuous
continuous
ƒ to check C 2-continuity, consider
- but
b t c = [ –1,
1 3,
3 3,
3 –1]
1] / 4 ⇒ || c || = 1
and c2 = [ 1, –3, –6, 10, 6, 6, 10, –6, –3, 1] / 16 ⇒ || c2 || = 1
- likewise for c` ⇒ Sc not contractive ⇒ no C 2-continuity
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Hölder regularity
ƒ Definition
a function φ is called Hölder regular of order n + α
(n ∈ N, 0 < α ≤ 1), if it is n times continuously
differentiable and φ(n) is Lipschitz of order α, i.e.
for all x and h and some constant c
p
of order 1 is
ƒ remember: a function that is Lipschitz
not necessarily differentiable
g
y of order n + 1 is weaker than
ƒ Hölder regularity
being n + 1 times differentiable
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Lower bound on Hölder regularity
ƒ Theorem
the scheme Sa with symbol
generates limit curves with Hölder regularity
r ≥ m – log
l 2 ( || b` || ) / ` for
f any `
ƒ Examples
ƒ 4-point scheme
- m = 4,
4 b = [ –1,
–1 4,
4 –1] ⇒ r ≥ 4 – log2 ( 4 ) = 2
ƒ cubic B-spline scheme
- m = 4, b = [ 2 ] ⇒ r ≥ 4 – log2 ( 2` ) / ` = 3
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Lower bound on Hölder regularity
ƒ Example
ƒ general primal 3-point
3 point
ƒ m = 2, ` = 1, b = [ 4w, 2 – 8w, 4w ] ⇒ r ≥ 2 – log2 ( || b || )
ƒ m = 2,
2 ` = 2,
2
b2 = [ 16w2 , 8w(1– 4w),
8w(1– 2w), 4(1– 4w)2 , … ]
⇒ r ≥ 2 – log2 ( || b2 || ) / 2
ƒ larger `, …
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Support
ƒ suppose the mask a is supported on [0,N],
i.e. ai = 0 for i<0 and i>N
ƒ all masks are of this kind after an index shift
ƒ refine the initial data f 0 = (…,0,1,0,…)
( 01 0 )
ƒ remember:
ƒ hence, f 1 is supported on [ 0, N ]
ƒ likewise,, f j is supported
pp
on [ 0,, ( 2 j – 1)) N ]
ƒ assume primal parameterization
ƒ the ssupport
pport of the basic limit ffunction
nction φa is [ 0,
0 N]
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Support
ƒ for arbitrary initial data f 0, the limit function is
ƒ assume the support of φa is [0,N],
[0 N] then the
values of the limit function
ƒ on [ 0,1]
0 1] are determined by the N control points
ƒ on [ 0, ½ ] are determined by
ƒ on [ ½,, 1 ] are determined by
y
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Support
ƒ Example
ƒ cubic B-spline
p
scheme with N = 4
A0
u0
A1
ƒ
determines
on [ 0,1]
ƒ A0 u0 dete
determines
es tthe
e values
a ues o
on [ 0, ½ ]
ƒ A1 u0 determines the values on [ ½, 1 ]
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Another local limit analysis
ƒ suppose the mask a of a convergent scheme
is supported
pp
on [0,N]
ƒ consider the local N × N subdivision matrices A0 , A1
ƒ take any x ∈ [ 0,1] with binary representation
x = 0.i1 i2 i3 i4 …
ƒ then the limit value
with ik ∈ { 0,1}
is given (N times) by
… Ai4 Ai3 Ai2 Ai1 u0
with
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Joint spectral radius
ƒ Definition
the joint spectral radius of two matrices A0 , A1 is
ƒ is bounded by the spectral radii and the norms of
A0 and A1
ƒ does not dependent on the chosen matrix norm
ƒ is usually very hard to determine exactly
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Hölder regularity
ƒ Theorem
the scheme Sa with symbol
generates limit curves with Hölder regularity
r = m – log
l 2 ( μ ) , where
h
μ is
i th
the jjoint
i t spectral
t l radius
di
of the local matrices B0 , B1 from the scheme Sb
ƒ in practice, the lower and upper bounds on μ are
used to g
get upper
pp and lower bounds on r
ƒ the lower bound then is the same as before,
because
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Hölder regularity
ƒ Example
ƒ cubic B
B-spline
spline scheme
- b = [ 2 ] ⇒ B0 = B1= ( 2 ) ⇒ μ = 2 ⇒ r = 4 – log2 ( 2) = 3
- scheme
h
gives
i
C 2 limit
li it curves, whose
h
second
dd
derivatives
i ti
are Lipschitz of order 1; sometimes called C 3 – ²
ƒ 4-point
4
i t scheme
h
- b = [ –1, 4, –1] ⇒
- ||B0 || = ||B1 || = ρ(B0 ) = ρ(B2 ) = 4 = μ ⇒ r = 4 – log2 ( 4 ) = 2
- scheme gives C 3 – ² limit curves
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Hölder regularity
ƒ Example
ƒ dual 4
4-point
point scheme
- evaluate local cubic interpolant in a dual fashion
- a = [ –5,
5 –7,
7 35,
35 105,
105 105,
105 35,
35 –7,
7 –5
5 ] / 128
- divide m = 5 times by (1+ z ) / 2 ⇒ b = [ –5, 18, –5 ] / 4
- ||B
|| 0 || = ||B
|| 1 || = ρ(B
( 0 ) = ρ(B
( 1 ) = 4.5
4 5 = μ ⇒ r = 5 – log
l 2 ( 4.5
4 5)
- scheme gives C 2.83 limit curves
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Summary
ƒ basic limit function φa and support size
ƒ if all but N + 1 consecutive mask coefficients are
zero, then N is the support size of the mask and
the basic limit function
ƒ a scheme Sa converges if the difference
scheme Sb is contractive
ƒ the norm of b or the `-iterated scheme b` is less
than one
ƒ a scheme is C m-continuous, if the scheme for
the m-th
th divided differences converges
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Summary
ƒ the norm of b` leads to a lower bound on the
Hölder regularity
g
y of the limit functions
ƒ lower and upper bound are given by joint
spectral radius analysis
ƒ given a scheme Sa , divide a(z) by as many factors
(1+ z ) / 2 as possible
possible, say m such factors
ƒ for the remaining scheme Sb with support size N,
consider the local N × N subdivision matrices B0 , B1
ƒ determine the joint spectral radius μ = ρ(B0 , B1)
ƒ Hölder regularity of limit curves is r = m – log2 ( μ )
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Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei