# subdivision i: the univariate settingcourses.cms.caltech.edu/.../subdivisioniunivariate.pdf ·...

of 32 /32
CS175 2005 1 Subdivision I: The Univariate Setting Peter Schröder CS175 2005 2 B-Splines (Uniform) Through repeated integration 1 1 0 B 3 (x)

Author: others

Post on 07-Jun-2020

1 views

Category:

## Documents

Embed Size (px)

TRANSCRIPT

• CS175 20051

Subdivision I:The Univariate

Setting

Peter Schröder

CS175 20052

B-Splines (Uniform)Through repeated integration

1

10

B3(x)

• CS175 20053

B-SplinesObvious properties

piecewise polynomial:

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

CS175 20054

B-SplinesRepeated convolution

box function

x

• CS175 20055

ConvolutionReminder

definition:

translation:

dilation:

CS175 20056

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

• CS175 20057

Refinability IIRefinement equation for B-splines

CS175 20058

Refinability

• CS175 20059

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

B-Spline RefinementExamples

CS175 200510

Spline Curves ISum of B-splines

curve as linear combinationcontrol points

• CS175 200511

Spline Curves IIRefine each B-spline in sum

example: linear B-spline

1/2 1/2 1

CS175 200512

Spline Curves IIIRefinement for curves

refine each B-spline in sum

refinementof control points

refinedbases

• CS175 200513

Refinement of Curves Linear operation on control points

succinctly

CS175 200514

Refinement of Curves Bases and control points

• CS175 200515

Subdivision OperatorExample

cubic splines

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

=

ΟΜΜΜΜΟΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΛΟΜΜΜΜΟ

100040006100440016004001100400600040001

81

14641

S

( )1,4,6,4,181

CS175 200516

SubdivisionApply subdivision to control points

draw successive control polygons rather than curve itself

• CS175 200517

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

CS175 200518

AnalysisSetup

polygon mapped to polygon

finite or (bi-)infinite, pij ∈ Rd

subdivision operator (linear for now)

• CS175 200519

Subdivision SchemesSome properties

affine invariancecompact supportindex invariance (topologic symmetry)local definition

CS175 200520

Subdivision OperatorProperties

compact support

affine invariance

index invariance/symmetry

• CS175 200521

Subdivision Operatorlocal definition: weights depend only on local neighborhood

Termsstationary: level independenceuniform: location independence

no boundaries (for now)

CS175 200522

Generating Functions

Subdivision operator as convolution

• CS175 200523

ExamplesSplines

linear:

higher order…

CS175 200524

Chaikin’s algorithm computes new points with weights 1/4(1, 3) and 1/4(3, 1)what happens if we change the weights?

• CS175 200525

ConvergenceHow much leeway do we have?

design of other subdivision rulesexample: 4pt scheme

establish convergenceestablish order of continuity

CS175 200526

AnalysisSimple facts

affine invariance necessary condition for uniform convergence

• CS175 200527

AnalysisConvergence

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

typicallydefine pointwise

CS175 200528

AnalysisConvergence

in max/sup norm

Theoremif then convergence of is uniform

• CS175 200529

Uniform ConvergenceProof linear spline subdivision operator

CS175 200530

Difference DecaySufficient condition

continuous limit ifanalysis by examining associated difference scheme

• CS175 200531

ExampleCubic B-splines

stencils

4 41/8

1 6 11/8

CS175 200532

ExampleCubic B-splines

differences

3 11/8

1 31/8

• CS175 200533

Difference DecayAnalysis of difference scheme

construction from the subdivision scheme itself

CS175 200534

how to show C1?divided differences must converge

check difference of divided differences

example4pt scheme

• CS175 200535

SmoothnessConsequences

4pt scheme: decay estimate

CS175 200536

Example4pt scheme

differences of divided differences

2 21/8

-1 6 -11/8

• CS175 200537

AnalysisFundamental solution

gives basis functions

CS175 200538

Fundamental SolutionProperties

refinement relation (why?)

support? non-zero coefficients:

• CS175 200539

So Far, So Good IWhat do we know now?

regular settingapproximating

B-splines

interpolating4pt scheme (Deslaurier-Dubuc)

CS175 200540

So Far, So Good IIWhat do we know now?

differencescontinuitydifferentiability

not quite general enoughcurrent setting assumes a particular parameterization

• CS175 200541

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

=

t000s000ro00qn00pmj00li00khe00gd00fc000b000a

S

More General SettingsNon-uniform

in spline case better curvessubdivision weights will vary

knot insertioninterpolation

CS175 200542

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

• CS175 200543

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…)

CS175 200544

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.

• CS175 200545

Irregular AnalysisNew tools

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

On to spectral analysis

CS175 200546

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

• CS175 200547

ExampleCubic B-spline

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

S

j

j+1

CS175 200548

NeighborhoodsWhich points influence a region?

for analysis around a point

-1 1

• CS175 200549

Subdivision MatrixInvariant neighborhood

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

CS175 200550

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

• CS175 200551

Subdivision MatrixProperties

eigen vectors of non-zero eigen values identical

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

CS175 200552

Subdivision MatrixEigen vectors and eigen functions

no overbar

• CS175 200553

Scaling RelationEigen functions scale

in neighborhood of the origin

CS175 200554

Smoothness (at Origin)Lemma for functions which scale

I

II

III

• CS175 200555

Necessary ConditionsContinuity at eigen functions

CS175 200556

Necessary ConditionsSpectrum

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

λ0 must be simple

?

• CS175 200557

Sufficient ConditionsCheck at origin

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

CS175 200558

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

• CS175 200559

ConvergenceLimit position

let j go to infinity

if λ0=1 and |λi|

• CS175 200561

More General SettingsSubtleties

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

CS175 200562

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

• CS175 200563

Affine InvarianceSanity condition

also necessary for convergence

∑ −+ = k ki2k1j

i pspdisplacement t

( )

( ) tsptpstp

1k i2k

1ji

k ki2k1j

i

434 21=

−+

−+

∑∑

+=

+=+

CS175 200564

Eigen AnalysisSummary

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

limit point: a0, tangent: a1

General setting more complicated...