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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)

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  • CS175 20051

    Subdivision I:The Univariate


    Peter Schröder

    CS175 20052

    B-Splines (Uniform)Through repeated integration




  • CS175 20053

    B-SplinesObvious properties

    piecewise polynomial:

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

    CS175 20054

    B-SplinesRepeated convolution

    box function


  • CS175 20055





    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


  • CS175 20057

    Refinability IIRefinement equation for B-splines

    take advantage of box refinement

    CS175 20058


  • 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


  • CS175 200513

    Refinement of Curves Linear operation on control points


    CS175 200514

    Refinement of Curves Bases and control points

  • CS175 200515

    Subdivision OperatorExample

    cubic splines









    ( )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


    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




    higher order…

    CS175 200524

    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?

  • 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


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

    typicallydefine pointwise

    CS175 200528


    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


    4 41/8

    1 6 11/8

    CS175 200532

    ExampleCubic B-splines


    3 11/8

    1 31/8

  • CS175 200533

    Difference DecayAnalysis of difference scheme

    construction from the subdivision scheme itself

    CS175 200534

    Higher OrdersSmoothness

    how to show C1?divided differences must converge

    check difference of divided differences

    example4pt scheme

  • CS175 200535


    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


    interpolating4pt scheme (Deslaurier-Dubuc)

    CS175 200540

    So Far, So Good IIWhat do we know now?


    not quite general enoughcurrent setting assumes a particular parameterization

  • CS175 200541






    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




    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




  • 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


    k ki2k1j


    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...