CS175 20051

Subdivision I:The Univariate

Setting

Peter Schröder

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B-Splines (Uniform)Through repeated integration

1

10

B3(x)

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B-SplinesObvious properties

piecewise polynomial:

unit integral:non-negative:partition of unity:support:

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B-SplinesRepeated convolution

box function

x

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ConvolutionReminder

definition:

translation:

dilation:

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Refinability IB-Spline refinement equation

a B-spline can be written as a linear combination of dilates and translates of itselfexample

linear B-splineand all others… 1/2 1/2

1

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Refinability IIRefinement equation for B-splines

take advantage of box refinement

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Refinability

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( )1,2,121 ( )1,4,6,4,181

B-Spline RefinementExamples

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Spline Curves ISum of B-splines

curve as linear combinationcontrol points

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Spline Curves IIRefine each B-spline in sum

example: linear B-spline

1/2 1/2 1

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Spline Curves IIIRefinement for curves

refine each B-spline in sum

refinementof control points

refinedbases

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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 OperatorExample

cubic splines

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

ΟΜΜΜΜΟΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΟΜΜΜΜΟ

100040006100440016004001100400600040001

81

14641

S

( )1,4,6,4,181

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SubdivisionApply subdivision to control points

draw successive control polygons rather than curve itself

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Summary so farSplines through refinement

B-splines satisfy refinement eq. basis refinement corresponds to control point refinementinstead of drawing curve, draw control polygonsubdivision is refinement of control polygon

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AnalysisSetup

polygon mapped to polygon

finite or (bi-)infinite, pij ∈ Rd

subdivision operator (linear for now)

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Subdivision SchemesSome properties

affine invariancecompact supportindex invariance (topologic symmetry)local definition

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Subdivision OperatorProperties

compact support

affine invariance

index invariance/symmetry

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Subdivision Operatorlocal definition: weights depend only on local neighborhood

Termsstationary: level independenceuniform: location independence

no boundaries (for now)

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Generating Functions

Subdivision operator as convolution

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ExamplesSplines

linear:

quadratic:

higher order…

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ExamplesQuadratic 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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ConvergenceHow much leeway do we have?

design of other subdivision rulesexample: 4pt scheme

establish convergenceestablish order of continuity

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AnalysisSimple facts

affine invariance necessary condition for uniform convergence

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AnalysisConvergence

define linear interpolant over given control points and associated parametric values (knot vector)

typicallydefine pointwise

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AnalysisConvergence

in max/sup norm

Theoremif then convergence of is uniform

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Uniform ConvergenceProof linear spline subdivision operator

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Difference DecaySufficient condition

continuous limit ifanalysis by examining associated difference scheme

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ExampleCubic B-splines

stencils

4 41/8

1 6 11/8

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ExampleCubic B-splines

differences

3 11/8

1 31/8

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Difference DecayAnalysis of difference scheme

construction from the subdivision scheme itself

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Higher OrdersSmoothness

how to show C1?divided differences must converge

check difference of divided differences

example4pt scheme

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SmoothnessConsequences

4pt scheme: decay estimate

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Example4pt scheme

differences of divided differences

2 21/8

-1 6 -11/8

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AnalysisFundamental solution

gives basis functions

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Fundamental SolutionProperties

refinement relation (why?)

support? non-zero coefficients:

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So Far, So Good IWhat do we know now?

regular settingapproximating

B-splines

interpolating4pt scheme (Deslaurier-Dubuc)

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So Far, So Good IIWhat do we know now?

differencescontinuitydifferentiability

not quite general enoughcurrent setting assumes a particular parameterization

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⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

t000s000ro00qn00pmj00li00khe00gd00fc000b000a

S

More General SettingsNon-uniform

in spline case better curvessubdivision weights will vary

knot insertioninterpolation

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4pt Scheme IWhere do the weights come from?

example of Deslaurier-Dubucgiven set of samples use interpolating polynomial to refine

i i+1 i+2i-1

Interpolating polynomialfor 4 successive samples

sample here

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4pt Scheme IIWeight computation

grind out interpolating polynomialresulting weights: 1/16(-1, 9, 9, -1)

Deslaurier-Dubucgeneralization of same ideahigher orders yield higher continuitytends to exhibit “ringing” (as is to be expected…)

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Deslaurier-DubucLocal polynomial reproduction

choose sk accordingly (d =1 for 4pt)

non-uniform possible, increasing smoothness, approximation power, limit for increasing d is sync fn.

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Irregular AnalysisNew tools

generating functions not applicableinstead: spectral analysis (why?)for irregular spacing only one parameter: ratio of spacing

On to spectral analysis

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AnalysisWe need a different approach

the subdivision matrixa finite submatrix representative of overall subdivision operation

based on invariant neighborhoodsstructure of this matrix key to understanding subdivision

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ExampleCubic B-spline

5 control points for 1 segment on either side of the origin

S

j

j+1

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NeighborhoodsWhich points influence a region?

for analysis around a point

-1 1

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Subdivision MatrixInvariant neighborhood

which φ(i-t) overlap the origin?tells smoothness story

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Eigen AnalysisWhat happens in the limit?

behavior in neighborhood of pointapply S infinitely many times…

suppose S has complete set of EVscontrol points

in invariantneighborhood

eigen vectors

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Subdivision MatrixProperties

eigen vectors of non-zero eigen values identical

proof by extension of yif defective, use generalized eigen vectors and values

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Subdivision MatrixEigen vectors and eigen functions

no overbar

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Scaling RelationEigen 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 ConditionsContinuity at eigen functions

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Necessary ConditionsSpectrum

must be 2-i for 0 · i · k and corresponding eigen functions must be monomialsgeneralized eigen vectors?

λ0 must be simple

?

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Sufficient ConditionsCheck at origin

eigen functions for |λ| < 2-k must be checked

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Subdivision OperatorSpectrum

necessary conditionsfor Ck must have λi=2-i for i· k

eigen functions are polynomials

generally not enough4pt scheme has 1,1/2,1/4,1/4,1/8,-1/16, -1/16

approximation properties

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ConvergenceLimit position

let j go to infinity

if λ0=1 and |λi|<1, i=1,…,n-1

example: cubic B-spline

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Geometric BehaviorMove limit point to origin

look at higher order behavior

tangent vector

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More General SettingsSubtleties

generalized eigen valuesmore subtle smoothness analysisnon-uniform subdivisioncompletely irregular subdivisionboundariesformule de commutation

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A NoteSize of subdivision matrix

for analysis need enough support to parameterize a finite neighborhood of the originfor evaluation need only enough support to zoom in on origine.g., cubic spline needs 5respectively 3 control points

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Affine InvarianceSanity condition

also necessary for convergence

∑ −+ = k ki2k1j

i pspdisplacement t

( )

( ) tsp

tpstp

1k i2k

1ji

k ki2k1j

i

434 21=

−+

−+

∑∑

+=

+=+

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Eigen AnalysisSummary

invariant neighborhood to understand behavior around pointEigen decomposition of subdivision matrix helpful

limit point: a0, tangent: a1

General setting more complicated...