Subdivision I:
The Univariate Setting
Peter Schröder
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B-Splines (Uniform)
Through repeated integration
1
0
1
B3(x)
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B-Splines
Obvious properties

piecewise polynomial:
unit integral:
 non-negative:
 partition of unity:
 support:

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B-Splines
Repeated convolution

box function
x
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Convolution
Reminder

definition:

translation:

dilation:
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Refinability I
B-Spline refinement equation
a B-spline can be written as a
linear combination of dilates and
translates of itself
 example



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linear B-spline
and all others…
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1/2
1/2
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Refinability II
Refinement equation for Bsplines

take advantage of box refinement
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Refinability
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B-Spline Refinement
Examples
1 2 1, 2, 1
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1 8 1, 4, 6, 4, 1
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Spline Curves I
Sum of B-splines

curve as linear combination
control points
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Spline Curves II
Refine each B-spline in sum

example: linear B-spline
1/2
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1/2
1
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Spline Curves III
Refinement for curves

refine each B-spline in sum
refined
bases
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refinement
of control points
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Refinement of Curves
Linear operation on control points

succinctly
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Refinement of Curves
Bases and control points
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Subdivision Operator
Example

cubic splines









1
S  
8










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
1
4
6
4
1
0
0
0
0
0
0


0
0
1
4
6
4
1
0
0
0
0


0
0
0
0
1
4
6
4
1
0
0


0
0
0
0
0
0
1
4
6
4
1




















1 8 1, 4, 6, 4, 1
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Subdivision
Apply subdivision to control
points

draw successive control
polygons rather than curve itself
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Summary so far
Splines through refinement
B-splines satisfy refinement eq.
 basis refinement corresponds to
control point refinement
 instead of drawing curve, draw
control polygon
 subdivision is refinement of
control polygon

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Analysis
Setup

polygon mapped to polygon


finite or (bi-)infinite, pij 2 Rd
subdivision operator (linear for
now)
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Subdivision Schemes
Some properties
affine invariance
 compact support
 index invariance (topologic
symmetry)
 local definition

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Subdivision Operator
Properties

compact support

affine invariance

index invariance/symmetry
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Subdivision Operator

local definition: weights depend
only on local neighborhood
Terms
stationary: level independence
 uniform: location independence


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no boundaries (for now)
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Generating Functions
Subdivision operator as
convolution
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Examples
Splines

linear:

quadratic:

higher order…
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Examples
Quadratic splines
Chaikin’s algorithm computes
new points with weights 1/4(1, 3)
and 1/4(3, 1)
 what happens if we change the
weights?

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Convergence
How much leeway do we have?

design of other subdivision rules

example: 4pt scheme
establish convergence
 establish order of continuity

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Analysis
Simple facts

affine invariance necessary
condition for uniform
convergence
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Analysis
Convergence

define linear interpolant over
given control points and
associated parametric values
(knot vector)
typically
 define pointwise

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Analysis
Convergence

in max/sup norm
Theorem

if
then convergence of
is uniform
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Uniform Convergence
Proof
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linear spline subdivision operator
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Difference Decay
Sufficient condition
continuous limit if
 analysis by examining associated
difference scheme

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Example
Cubic B-splines

1/8
stencils
4 4
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1/8
1
6
1
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Example
Cubic B-splines

1/8
differences
3 1
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1/8
1
3
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Difference Decay
Analysis of difference scheme

construction from the subdivision
scheme itself
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Higher Orders
Smoothness
how to show C1?
 divided differences must
converge



check difference of divided
differences
example

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4pt scheme
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Smoothness
Consequences

4pt scheme: decay estimate
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Example
4pt scheme

1/8
differences of divided differences
2 2
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1/8
-1 6
-1
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Analysis
Fundamental solution

gives basis functions
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Fundamental Solution
Properties

refinement relation (why?)

support? non-zero coefficients:
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So Far, So Good I
What do we know now?
regular setting
 approximating



B-splines
interpolating

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4pt scheme (Deslaurier-Dubuc)
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So Far, So Good II
What do we know now?

differences



continuity
differentiability
not quite general enough

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current setting assumes a
particular parameterization
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More General Settings
Non-uniform
in spline case better curves
 subdivision weights will vary



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knot insertion
interpolation
a

b
c

d

e
S  0

0
0

0
0

0
0 0 0

0 0 0
f 0 0

g 0 0

h k 0
i l 0

j m p
0 n q

0 o r
0 0 s

0 0 t 
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4pt Scheme I
Where do the weights come
from?
example of Deslaurier-Dubuc
 given set of samples use
sample
here
interpolating
polynomial
to refine

Interpolating polynomial
for 4 successive samples
i-1
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i
i+
1
i+
2
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4pt Scheme II
Weight computation

grind out interpolating polynomial


resulting weights: 1/16(-1, 9, 9, -1)
Deslaurier-Dubuc
generalization of same idea
 higher orders yield higher
continuity
 tends to exhibit “ringing” (as is to
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
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Deslaurier-Dubuc
Local polynomial reproduction

choose sk accordingly (d =1 for
4pt)
non-uniform possible, increasing
smoothness, approximation
power, limit for increasing d is
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sync fn.

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Irregular Analysis
New tools
generating functions not
applicable
 instead: spectral analysis (why?)
 for irregular spacing only one
parameter: ratio of spacing

On
to
spectral
analysis
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Analysis
We need a different approach

the subdivision matrix

a finite submatrix representative of
overall subdivision operation
based on invariant
neighborhoods
 structure of this matrix key to
understanding subdivision

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Example
Cubic B-spline

5 control points for 1 segment on
either side of the origin
j
S
j+1
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Neighborhoods
Which points influence a region?

for analysis around a point
-1
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Subdivision Matrix
Invariant neighborhood

which f(i-t) overlap the origin?

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tells smoothness story
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Eigen Analysis
What happens in the limit?

behavior in neighborhood of point

apply S infinitely many times…
suppose S has complete set of
eigen vectors
EVs
control
points

in invariant
neighborhood
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Subdivision Matrix
Properties

eigen vectors of non-zero eigen
values identical


proof by extension of y
if defective, use generalized
eigen vectors and values
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Subdivision Matrix
Eigen vectors and eigen
functions
no overbar
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Scaling Relation
Eigen functions scale
in neighborhood
of the origin
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Smoothness (at Origin)
Lemma for functions which scale
I
II
III
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Necessary Conditions
Continuity at eigen functions
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Necessary Conditions
Spectrum
must be 2-i for 0 · i · k and
corresponding eigen functions
must be monomials
 generalized eigen vectors?


l0 must be simple
?
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Sufficient Conditions
Check at origin

eigen functions for |l| < 2-k must
be checked
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Subdivision Operator
Spectrum

necessary conditions

for Ck must have li=2-i for i· k


generally not enough


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eigen functions are polynomials
4pt scheme has 1,1/2,1/4,1/4,1/8,-1/16,
-1/16
approximation properties
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Convergence
Limit position

let j go to infinity

if l0=1 and |li|<1, i=1,…,n-1

example: cubic B-spline
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Geometric Behavior
Move limit point to origin

look at higher order behavior
tangent vector
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More General Settings
Subtleties
generalized eigen values
 more subtle smoothness analysis
 non-uniform subdivision
 completely irregular subdivision
 boundaries
 formule de commutation

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A Note
Size of subdivision matrix
for analysis need enough support
to parameterize a finite
neighborhood of the origin
 for evaluation need only enough
support to zoom in on origin
 e.g., cubic spline needs 5
respectively 3 control points

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Eigen Analysis
Summary
invariant neighborhood to
understand behavior around point
 Eigen decomposition of
subdivision matrix helpful


limit point: a0, tangent: a1
General setting more
complicated...
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