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Symmetric Box-Splines on Root Lattices

Permanent Link: http://ufdc.ufl.edu/UFE0022619/00001

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Title: Symmetric Box-Splines on Root Lattices
Physical Description: 1 online resource (125 p.)
Language: english
Creator: Kim, Min
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

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Subjects / Keywords: Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Due to their highly symmetric structure, in arbitrary dimensions root lattices are considered as efficient sampling lattices for reconstructing isotropic signals. Among the root lattices the Cartesian lattice is widely used since it naturally matches the Cartesian coordinates. However, in low dimensions, non-Cartesian root lattices have been shown to be more efficient sampling lattices. For reconstruction we turn to a specific class of multivariate splines. Multivariate splines have played an important role in approximation theory. In particular, box-splines, a generalization of univariate uniform B-splines to multiple variables, can be used to approximate continuous fields sampled on the Cartesian lattice in arbitrary dimensions. Box-splines on non-Cartesian lattices have been used limited to at most dimension three. This dissertation investigates symmetric box-splines as reconstruction filters on root lattices (including the Cartesian lattice) in arbitrary dimensions. These box-splines are constructed by leveraging the directions inherent in each lattice. For each box-spline, its degree, continuity and the linear independence of the sequence of its shifts are established. Quasi-interpolants for quick approximation of continuous fields are derived. We show that some of the box-splines agree with known constructions in low dimensions. For fast and exact evaluation, we show that and how the splines can be efficiently evaluated via their BB(Bernstein-Bezier)-forms. This relies on a technique to compute their exact (rational) BB-coefficients. As an application, volumetric data reconstruction on the FCC (Face-Centered Cubic) lattice is implemented and compared with reconstruction on the Cartesian lattice.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Min Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Peters, Jorg.
Local: Co-adviser: Entezari, Alireza.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-02-28

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Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022619:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022619/00001

Material Information

Title: Symmetric Box-Splines on Root Lattices
Physical Description: 1 online resource (125 p.)
Language: english
Creator: Kim, Min
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Due to their highly symmetric structure, in arbitrary dimensions root lattices are considered as efficient sampling lattices for reconstructing isotropic signals. Among the root lattices the Cartesian lattice is widely used since it naturally matches the Cartesian coordinates. However, in low dimensions, non-Cartesian root lattices have been shown to be more efficient sampling lattices. For reconstruction we turn to a specific class of multivariate splines. Multivariate splines have played an important role in approximation theory. In particular, box-splines, a generalization of univariate uniform B-splines to multiple variables, can be used to approximate continuous fields sampled on the Cartesian lattice in arbitrary dimensions. Box-splines on non-Cartesian lattices have been used limited to at most dimension three. This dissertation investigates symmetric box-splines as reconstruction filters on root lattices (including the Cartesian lattice) in arbitrary dimensions. These box-splines are constructed by leveraging the directions inherent in each lattice. For each box-spline, its degree, continuity and the linear independence of the sequence of its shifts are established. Quasi-interpolants for quick approximation of continuous fields are derived. We show that some of the box-splines agree with known constructions in low dimensions. For fast and exact evaluation, we show that and how the splines can be efficiently evaluated via their BB(Bernstein-Bezier)-forms. This relies on a technique to compute their exact (rational) BB-coefficients. As an application, volumetric data reconstruction on the FCC (Face-Centered Cubic) lattice is implemented and compared with reconstruction on the Cartesian lattice.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Min Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Peters, Jorg.
Local: Co-adviser: Entezari, Alireza.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022619:00001


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SYMMETRICBOX-SPLINESONROOTLATTICES By MINHOKIM ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2008 1

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2008MinhoKim 2

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Toeveryone. 3

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ACKNOWLEDGMENTS Firstofall,Iwouldliketothankmychairadvisor,Dr.Jorg Peters.SinceIstarted toworkwithhim,hehasinspiredme,guidedmethroughallthe research,andgaveme invaluableadvice,suggestions,commentsandsupport.Ial sowouldliketoshowmy gratitudetomyco-chairadvisorDr.AlirezaEntezariforhi sguidanceandcomments duringmyresearch.Iamthankfultomysupervisorycommitte emembersfortheir mentoring.Finally,Iwouldliketogivemydeepestgratitud etomyfamilymembersand friends. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 LISTOFTABLES .....................................7 LISTOFFIGURES ....................................8 ABSTRACT ........................................10 CHAPTER 1INTRODUCTION ..................................12 1.1SymmetricBox-SplinesonRootLattices ...................12 1.2Overview ....................................14 1.3FastandStableEvaluationofBox-SplinesviatheBB-For m ........18 2LITERATUREREVIEW ..............................21 2.1Multi-dimensionalSignalProcessingandSamplingLatt ices .........21 2.2SymmetricBox-SplinesontheRootLattices .................22 2.3ReviewofExistingEvaluationTechniquesofBox-Spline s ..........23 3BACKGROUND ...................................26 3.1Notations ....................................26 3.2RootLattices ..................................27 3.3Multi-DimensionalSignalProcessing .....................34 3.4DensestSpherePackingLatticeandOptimalSamplingLat tice .......34 3.5TheBB-FormofaMultivariatePolynomial ..................37 3.6Box-Splines ...................................38 3.6.1ExamplesofBox-Splines ........................44 4SYMMETRICBOX-SPLINEONTHECARTESIANLATTICE ........46 4.1DenitionandProperties ............................46 4.2TheZP-Element ................................48 4.3The 7 -DirectionTrivariateBox-Spline ....................49 4.3.1Denition ................................49 4.3.2PolynomialStructure ..........................50 4.3.3Quasi-Interpolation ...........................53 4.3.4SplineEvaluation ............................53 5SYMMETRICBOX-SPLINEONTHE A n LATTICE ..............55 5.1DenitionandProperties ............................55 5.2The 3 -DirectionBivariateBox-SplineontheHexagonalLattice ......61 5

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5.3 C 1 ReconstructionontheFCCLattice ....................62 5.3.1The 6 -DirectionTrivariateBox-SplineontheFCCLattice .....62 5.3.2TheFCCLattice ............................63 5.3.3Quasi-Interpolation ...........................64 5.3.4ThePolynomialPiecesandEvaluationoftheSpline .........66 5.3.5Reconstruction ..............................70 5.3.6Results ..................................71 5.3.7ComputationalCost ...........................74 5.3.8ReconstructionProperties .......................76 5.3.9DiscussionandExtensions .......................76 6SYMMETRICBOX-SPLINEONTHE A n LATTICE ..............78 6.1DenitionandProperties ............................78 6.2TheSymmetricBox-Splineonthe A n Lattice ................81 7SYMMETRICBOX-SPLINEONTHE D n LATTICE ..............96 7.1DenitionandProperties ............................96 7.2The 6 -DirectionBox-SplineontheFCCLattice ...............101 8SYMMETRICBOX-SPLINEONTHE D n LATTICE ..............103 8.1DenitionandProperties ............................103 8.2The 7 -DirectionBox-SplineontheBCCLattice ...............104 9FASTANDSTABLEEVALUATIONOFBOX-SPLINESVIATHEBB-FORM 106 9.1Box-SplinewithRationalBB-Coecients ..................106 9.2Pre-processingBox-Splines ..........................108 9.2.1IndexingPolynomialPieces(Domains) ................108 9.2.2ComputingtheChangeofBasis ....................109 9.3TheSplineEvaluationAlgorithm .......................112 9.4Example .....................................114 9.4.1 6 -DirectionBox-SplineontheFCCLattice ..............114 9.5ComparisonandanApplication ........................116 REFERENCES .......................................118 BIOGRAPHICALSKETCH ................................125 6

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LISTOFTABLES Table page 1-1Symmetricbox-splinesontherootlattices .....................15 3-1Centerdensityofseveralrootlattices ........................37 4-1Verticesofdomaintetrahedraof M Z 3 ........................51 4-2BB-coecientsof M Z 3 ................................52 4-3Orthogonaltransformationsfordomaintetrahedraof M Z 3 ............54 5-1Twotypesofdomaintetrahedraof M fcc .......................67 5-2BB-coecientsof M fcc ................................69 5-3Comparisonofrenderingtimebetweentwo C 1 reconstructionmethods .....75 5-4Reconstructionpropertiesofthreereconstructionmet hods ............76 6-1Box-splinespacesrelatedbychangeofvariables ..................82 9-1Verticesofthedomaintetrahedraof M e fcc .....................115 9-2Evaluationtime ....................................116 7

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LISTOFFIGURES Figure page 1-1ConstructionofZP-element .............................16 1-2Shiftsoflinearbox-splineonthe A n latticebyorthogonalprojectionofaslab .17 1-3Isosurfacesbeforeandaftercorrection .......................19 3-1Symmetrygroupofthehexagonallattice ......................29 3-2Invalidnitereectiongroups ............................30 3-3Exampleofreectiongroup .............................31 3-4Exampleoflabelingconvention ...........................31 3-5Non-crystallographicrootsystem ..........................33 3-6Reconstructionandsamplingdensity ........................35 3-7Spherepackingfordierentlatticeswithrespecttothe irdensity .........36 3-8Packingdensityofrootlattices ...........................37 3-9Geometricdenitionofbox-spline ..........................38 4-1ZP-element ......................................48 4-2 7 -directionbox-spline M Z 3 ..............................49 4-3Knotplanesof M Z 3 intheVoronoicell .......................50 4-4Domaintetrahedraof M Z 3 intheVoronoicell ...................50 4-5Levelsetsof M Z 3 ...................................51 4-6`Standard'domaintetrahedronof M Z 3 .......................52 5-1Rootsystem A 3 ....................................55 5-2Coxeterdiagramof A n ................................56 5-3Geometricconstructionof A n in R n .........................58 5-4 6 -directionbox-splineontheFCClattice, M fcc ...................62 5-5FindingnearestFCClatticepoint ..........................64 5-6Octet-truss ......................................66 5-7Levelsetsof M fcc ...................................67 8

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5-8Twotypesofdomaintetrahedraof M fcc .......................68 5-9Benchmarkdatasets .................................72 5-10ErrorimagesofMarschner-Lobbtestdataset ....................73 5-11ReconstructionofMarschner-Lobbdataset .....................74 5-12ReconstructionofCarpdataset ...........................75 6-1Geometricconstructionof A n in R n .........................79 6-2 4 -directiontrivariatebox-spline M A n ........................80 6-3Relationbetweenthreelinearbox-splines ......................84 6-4Comparisonofthreelinearbox-splines .......................84 6-5Kuhntriangulationfor n =3 ............................87 6-6Twotypesofsupportdecomposition ........................89 7-1Coxeterdiagramof D n ................................96 7-2Coxeterdiagramof D 4 ................................97 8-1 7 -directionbox-splineontheBCClattice, M D 3 ..................104 9-1Indexingapolynomialdomain ............................109 9-2Knotplanesof M e fcc .................................115 9-3Polynomialpiecesof M e fcc ..............................115 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SYMMETRICBOX-SPLINESONROOTLATTICES By MinhoKim August2008 Chair:JorgPetersMajor:ComputerEngineering Duetotheirhighlysymmetricstructure,inarbitrarydimen sionsrootlatticesare consideredasecientsamplinglatticesforreconstructin gisotropicsignals.Amongthe rootlatticestheCartesianlatticeiswidelyusedsinceitn aturallymatchestheCartesian coordinates.However,inlowdimensions,non-Cartesianro otlatticeshavebeenshownto bemoreecientsamplinglattices. Forreconstructionweturntoaspecicclassof multivariatesplines .Multivariate splineshaveplayedanimportantroleinapproximationtheo ry.Inparticular,box-splines, ageneralizationofunivariateuniformB-splinestomultip levariables,canbeusedto approximatecontinuouseldssampledontheCartesianlatt iceinarbitrarydimensions. Box-splinesonnon-Cartesianlatticeshavebeenusedlimit edtoatmostdimensionthree. Thisdissertationinvestigatessymmetricbox-splinesasr econstructionltersonroot lattices(includingtheCartesianlattice)inarbitrarydi mensions.Thesebox-splinesare constructedbyleveragingthedirectionsinherentineachl attice.Foreachbox-spline,its degree,continuityandthelinearindependenceoftheseque nceofitsshiftsareestablished. Quasi-interpolantsforquickapproximationofcontinuous eldsarederived.Weshowthat someofthebox-splinesagreewithknownconstructionsinlo wdimensions. Forfastandexactevaluation,weshowthatandhowthespline scanbeeciently evaluatedviatheirBB(Bernstein-Bezier)-forms.Thisre liesonatechniquetocompute theirexact(rational)BB-coecients.Asanapplication,v olumetricdatareconstructionon 10

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theFCC(Face-CenteredCubic)latticeisimplementedandco mparedwithreconstruction ontheCartesianlattice. 11

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CHAPTER1 INTRODUCTION 1.1SymmetricBox-SplinesonRootLattices Signalprocessingisacoreareaincomputergraphicsandvis ion.Givendiscrete samplesona samplinglattice ,theoriginalsignalisreconstructedbyrecoveringitspri mal frequencywithaproper reconstructionlter .Applicationsincludereconstructionof volumetricdatasetfromCT(ComputedTomography)orMRI(Ma gneticResonance Imaging)datainmedicalimaging:Heretheisosurface(orle velset)ofthereconstructed eldistobeextractedandvisualizedeitherimplicitly(e. g.,viaray-casting)orexplicitly (e.g.,viamarchingcubeextraction).Indimensiononether eisonlyonetypeofsampling lattice.Hencethereconstructionlteristheonlyfactort hataectsthequalityofthe reconstruction.Butinhigherdimensions,thereareinnit elymanysamplinglattices, andthechoiceofsamplinglatticeplaysasmuchimportantro leasthelter.Whilethe Cartesianlatticehasbeenthemostpopularsamplinglattic esinceitnaturallymatches theCartesiancoordinates,therearemanyotheralternativ echoices.Technically,thebest samplinglatticedependsontheindividualinputsignal.Bu titisnotpracticaltouse dierentsamplinglatticeforeachinputsignal,andusuall ywecannotpredictorknowthe inputsignaltobereconstructedbeforehand.Thereforethe bestsamplinglatticeischosen basedongenericassumptionsontheinputsignal.Astandard assumptionisthatthe inputsignalis band-limited anditsspectrumis isotropic .Underthisassumption,Petersen andMiddleton[ 82 ]observedthatndingthe optimalsamplinglattice ,onwhichthe originalsignalcanbereconstructedwithoutaliasingwith lowestdensity,isthedualofthe solutiontothedensestspherepackingproblem.Althoughno tsolvedforalldimensions, someofthedensestspherepackinglatticesareknownandthe rootlattices appeartobe theanswersinmanydimensions.Conversely,theCartesianl attice,whichisalsoaroot lattice,showsverypoorsamplingeciencycomparedtoothe rrootlattices.Forexample, indimensionsthree,wecanreconstructthesameisotropicb and-limitedsignalontheBCC 12

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latticewithonly70.7%samplesoftheCartesianlattice.In dimensionstwoandthree, specicrootlatticeshavebeenanalyzed,butnotsoinhighe rdimensionsapparentlydue tothelackofecientsymmetricreconstructionlters.Onl yontheCartesianlattice, thetensor-productB-splinesareeasilyavailablereconst ructionlterssincetheycanbe reducedtotensoredunivariateconstructions. Onothersamplinglattices, multivariatesplines shouldyieldmoregeneralreconstruction lters.Multivariatesplineshaveplayedanimportantrole inthe approximationtheory Inparticularbox-splines,ageneralizationof(univariat e)uniformB-splinestomultiple variables,havebeenwidelyusedinmanyareas.Intheformof piecewisepolynomials denedbyconsecutivedirectionalconvolutions,box-spli nesusuallypossessrelativelyhigh continuitygiventheirlow(total)degree(Figure 1-1 ),especiallywhencomparedtothe tensor-productB-splines.Moreover,thefactthatasetof( integer)directionsdenea box-splinegivesmuchexibilitytoconstructthem.Inthef ormofa cardinalspline ,a weightedsumoftheshiftsofabox-splineontheCartesianla ttice,theycanbeusedto approximate(orreconstruct)acontinuouseld.Oftenabox -splineonanon-Cartesian latticecanberelatedtoabox-splineontheCartesianlatti ceviachange-of-variables. Indimensiontwoandthree,specicinstancesofnon-Cartes ianrootlatticeswith symmetricbox-splinesasreconstructionltershavebeens howntobeeective.In thisdissertation,Iproposeafamilyofsymmetricbox-spli nesonsomerootlattices (includingCartesianlattice)inarbitrarydimensions.Am ongtherootlattices,Iconsider the irreducible (Section 3.2 )onesdenedinarbitrarydimensions:theCartesian, A n (and itsdual A n )and D n (anditsdual D n )lattices(seeTable 1-1 ).Theremainingirreducible rootlatticesare: E 6 E 7 E 8 andtheirduals.Ishowthatmanyofthebox-splinesproposed hereagreewithknownconstructions.Ineachrootlattice,a symmetricbox-splineis denedbyleveragingtheinherentdirectionsineachlattic estructure.Inthisway,the box-splinecanberelatedtooneontheCartesianlatticevia change-of-variableskeeping mostpropertiessuchascontinuity,approximationorderan dBB-coecients.Thisis 13

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usefulsinceamaturetheoryofbox-splinesexistsfortheCa rtesianlattice[ 36 ].Moreover, byleveragingthesymmetryofrootlattices,wecanobtainsy mmetricbox-splinesthat cannotbeconstructedontheCartesianlattice.Foreachbox -spline,weinvestigateits degree,continuityandlinearindependenceofthesequence ofitsshiftsonthelattice. Theirquasi-interpolants,whichallowustoquicklyapprox imatetheinputwithgiven approximationorder(Section 3.6 ),arealsoinvestigatedanddocumentedforsomeof thebox-splines.Additionallythesplinescanbeevaluated inafastandstablewayvia theirBB-forms.Atechniquetocomputetheexact(rational) BB-coecientsoftheir polynomialpiecesisalsoproposed(Chapter 9 ).Asanapplication,a C 1 reconstruction ontheFCC(Face-CenteredCubic)latticewithasuitablebox -splineisimplementedand comparedtoa C 1 reconstructionontheCartesianlatticewiththetensor-pr oductB-spline (Section 5.3 ). 1.2Overview Wenowbrieyintroducethelatticeandbox-splinepairsde nedinarbitrarynumber ofvariables n .AsuccinctsummaryisgivenbyTable 1-1 The ( n +2 n 1 ) -directionbox-splineontheCartesianlattice, M Z n Tensor-productB-splinesarethemostpopularreconstruct ionltersontheCartesian lattice.Howeverwecanconstructotherbox-splinesthatar emoresymmetricandmostly havehigherapproximationorderwithgivendegreebylevera gingmoredirectionsofthe Cartesianlattice.Onewayistoincludethe 2 n 1 diagonaldirectionsinadditiontothe n mainaxisdirectionswhenconstructingabox-spline.Indim ensiontwo,thisresultsinthe well-knownZwart-Powellelement[ 99 ](Table 1-1 ,Figure 1-1 andFigure 4-1 ).Itsextension todimensionthree,a 7 -directiontrivariatebox-spline M Z 3 (seeSection 4.3 ),wasshown tobeusefulbyPeters[ 80 ]anditsdirectionmatrixcanbefoundinTable 1-1 (seealso Figure 4-2 andFigure 4-5 ). M Z 3 hasbeenusedrepeatedly[ 39 80 81 ],butevaluatingit exactlyviaitsBB-formhasnotbeenconsidered.Wederiveth estructureanddocument 14

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Table1-1.SymmetricBox-splinesontherootlattices.SeeS ection 3.6 for dim.box-splinelatticedirectionmatrixgeneratormatrix continuitybasis? ( f ( + j )) nM Z n Cartesian I n [f e n + n 1 X j =1 j e j : j 2f 1 gg I n C 2 n 2 nonotknown 2 M Z 2 = ZP-element 101 1 0111 C 1 no ( f 1 24 X 2 ZP D 2 f )( j ) 3 M Z 3 24 1001 11 1 01011 1 1 0011111 35 C 2 no ( f 1 24 X 2 Z 3 D 2 f )( j ) nM A n A n [ 1 i 3) ( yes ( n =3) no ( n> 3) notknown 3 M D 3 = M A 3 FCC 24 1111001 10011 001 11 1 35 24 10001 1 1 1 1 35 C 1 yes ( f 1 24 X 2 fcc D 2 f )( j ) nM D n D n I n [ 1 2 f e n + n 1 X j =1 j e j : j 2f 1 gg I n 1 j = 2 0 t 1 = 2 C 2 n 2 nonotknown 3 M D 3 BCC 1 2 24 2001 11 1 02011 1 1 0021111 35 24 101 = 2 011 = 2 001 = 2 35 C 2 no ( f 1 24 X 2 bcc D 2 f )( j ) 15

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thecoecientsofitspolynomialpiecesinBB-formusingthe techniquedescribedin Chapter 9 x y A x y B x y C Figure1-1.ConstructionofZP-elementviaconsecutivedir ectionalconvolutions: A ) [ 1001 ] B ) [ 101011 ] C ) [ 101 1 0111 ] The n ( n +1) = 2 -directionbox-splineonthe A n lattice, M A n .The A n lattice isgeneratedbytherootsystem A n (Section 5.1 )[ 23 60 ];itiscomposedoftheinteger linearcombinationsoftherootvectorsof A n .Althoughtherootvectorsof A n areusually expressedas ( n +1) -dimensionalvectorsembeddedinthe n -dimensionalhyperplane H n j (Section 3.1 )in R n +1 ,hereweembedthemdirectlyin R n byapplying n ( n +1) orthogonalmatrices.Geometrically,aswillbeshowninLem ma 5 ,thebasisforthe A n latticecanalsobeconstructedbytakingthe n edgessharingavertexofan n -dimensional equilateralsimplex.Thebox-spline M A n isconstructedbythedirectionstothe n ( n +1) numberofnearestlatticepointsofthe A n lattice. Indimensiontwo,thisisthewell-known 3 -directionlinearbox-splineonthe hexagonallattice.Indimensionthree,thisisthe 6 -directionbox-splineontheFCClattice (Section 5.3 )proposedbyEntezari[ 38 ].Asanapplication,volumetricdatareconstruction ontheFCClatticewiththe 6 -directionbox-splineisimplementedandcomparedwith reconstructionontheCartesianlatticewiththetensor-pr oducttri-quadraticB-spline. The ( n +1) -directionbox-splineonthe A n lattice, M A n .Afamily M r of n -variatebox-splinesisjustlypopularduetotheirlineari ndependenceandapproximation properties(Section 3.6.1 ).Membersof M r aredenedby r -foldconvolution,inthe n directionsoftheCartesianlatticeplusadiagonal,sothat thefootprintsofthese 16

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box-splinesareasymmetricallydistortedinthediagonald irection(Figure 1-1B ,Figure 6-3 andFigure 6-4 ).Tomakereconstructionofeldslessbiased,convolution andshiftson 2 -and 3 -dimensionalnon-Cartesianlattices,thehexagonallatti ceandtheBCClattice respectively,haverecentlybeenadvocated[ 38 40 42 92 94 ]. A n =1 B n =2 Figure1-2.Orthogonalprojectionofaslabofunitcubesalo ngthediagonaldirectionfor A ) n =1 and B ) n =2 Wegeneralizethebivariatebox-splinesonthehexagonalla tticeandthetrivariate box-splinesontheBCClatticetosymmetric n -variatebox-splines M r (Section 6.2 ) denedviathedirectionstothe 2( n +1) numberofnearestneighborpointsofthe A n lattice(seee.g.Figure 6-2A for n =3 ).The A n lattice,thedualofthe A n lattice,is well-knownincrystallographyanddiscretegeometrywhere itappearsembeddedinan n -dimensionalhyperplane H n j (R n +1 asinthecaseofthe A n lattice.Wedenethe A n latticedirectlyin R n .Thenthegeometricconstructionoftheshiftsofthesymmet ric linearbox-spline M 1 onthe A n latticesimpliestotheclassicalconstructionof n -variate box-splinesbyprojection:Theshiftsofthesymmetricline arbox-splineon A n arethe orthogonal projectionofaslabofthickness 1 decomposedintounitcubesalongthe diagonalofthecubes(Figure 1-2 ).Bycomparison, M 1 hasthesamepreimage,butits supportisdistortedbyitsanisotropicdirectionmatrix(F igure 6-4 ).Wedocumentthe support,itspartition,thedesirablepropertiessharedwi th M r and,fortheimportantcase 17

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r =2 ,thequasi-interpolantconstructionassociatedwith M 2 in any numberofvariables n The n ( n 1) -directionbox-splineonthe D n lattice, M D n .Therootlattice D n ,denedfor n 3 ,isgeneratedbytherootsoftherootsystem D n (Section 7.1 ).If consideredasasub-latticeoftheintegergrid(Cartesianl attice), D n iscomposedofthe integerpointswhosesumisalwayseven[ 23 ].Thebox-spline M D n isconstructedbythe directionstothe 2 n ( n 1) nearestlatticepointsofthe D n lattice.Indimensionthree,this yieldsthe 6 -directionbox-splineontheFCClattice[ 38 ]. The ( n +2 n 1 ) -directionbox-splineonthe D n lattice, M D n .Whenconsidered asasuper-latticeoftheintegergrid(Cartesianlattice), therootlattice D n ,thedualof D n ,canbebuiltbyinsertingadditionalpointsatthecenterof eachhypercube,hencecan beconsideredasageneralizationoftheBCClattice,altern ativeto A n .Thebox-spline M D n isconstructedbythe n mainaxisdirectionsand 2 n 1 directionstothecentersofthe 2 n hypercubesaroundtheorigin(seeFigure 8-1A for n =3 ).Notethatthesedirections arepairwiseparalleltothoseofthe ( n +2 n 1 ) -directionbox-splineontheCartesianlattice, M Z n ,buttheirlengthsaredierent. 1.3FastandStableEvaluationofBox-SplinesviatheBB-For m Applicationofbox-splinesrequiresexactvaluesofthem,b utevaluationonlattice edgesandfacesrequirescare.AlreadydeBoor[ 31 ]andlaterKobbelt[ 62 ]observeda fundamentalcombinatorialchallengeduetotheinclusiono rexclusionofcertain knot planes (Section 3.6 )anddealtwithitintwodierentwaysintheirrespectivere cursive box-splineevaluationalgorithms.Ourinterestwaspiqued bytheexampleofFigure 1-3A wheretheotherwisecarefullyconstructedalgorithmof[ 62 ]failsduetosubtlenumerical round-ointheunderlying MATLAB routine. Analternativeapproachtodirectrecursionistoevaluatea fterconversiontothe BB-form.ThisapproachwaspioneeredbyChuiandLai[ 18 ]andLai[ 65 ]intwovariables, andrecentlyextendedtoaclassoftrivariatebox-splinesb yCasciolaetal.[ 12 ].Sincethe 18

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A B Figure1-3.Isosurfacesfor 10 1 (blue), 10 2 (green), 10 3 (red)and 10 10 (purple)ofthe (non-centered) 6 -directionbox-spline(Section 9.4.1 ), A )evaluatedby[ 62 ]and B )thecorrectisosurface. splineswewanttoevaluatearelinearcombinationsofshift sofoneormorebox-splines, wefocusattentiononthesebox-spline(basisorgenerator) functions.Thepointof theconversionisthattheBB-formofthepolynomialpieces( Section 3.5 )hasastable evaluationalgorithmalsoontheknotplaneswheretherecur sivealgorithmsencounter diculties.Infact,alongknotplanes,thestandarddeCast eljau'salgorithmforevaluating polynomialsintheBB-formisofevenlowercomplexity.Thek eychallengesforthis approacharethederivationandexactrepresentationofthe changeofbasis.Ourrst contribution,generalizing[ 12 65 ],isTheorem 1 : (1)theBB-coecients,expressingthepolynomialpiecesof abox-splinewithaninteger directionmatrix,are rational Thisallowsustouseasimpleinterpolation-basedapproach forderivingachange-of-basis matrixwithexactintegerentries,scaledbyarationalnumb er. Whileboththerecursiveandtheconversionapproachbenet fromlocalization,i.e. fromdeterminingwhichbox-splinesinuencetheevaluatio n,theconversionapproach must ecientlydeterminethepolynomialpiecetobeevaluated.T hisforcesanunderstandingof thedecompositionofthedomainbytheknotplanesimpliedby thebox-splinedirections. Oursecondcontributionis (2)anindexingstrategy,basedonthebox-splinedirection s,forndingthedomain simplexofthepolynomialpieceforagivenparameter(Secti on 9.2.1 ). 19

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Theone-timedeterminationofthecombinatoricsoftheboxsplinesrequiredbythe indexingstrategyisatthecoreoftheimprovedspeed:compa redtorecursiveevaluators thatresolvethecombinatoricsatruntime,evaluationbase dontheBB-formisfasterby ordersofmagnitude(Table 9-2 ).Conversionplusindexing,bothpre-computed,storedand quicklyaccessed,yield (3)analgorithmforfastevaluationofsplinesgeneratedby box-splinesthatisstable,in particularalongknotplanes(Algorithm 9.3.1 ). 20

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CHAPTER2 LITERATUREREVIEW 2.1Multi-dimensionalSignalProcessingandSamplingLatt ices PetersenandMiddleton[ 82 ]extendedShannon'ssamplingtheorem[ 86 ]tohigher dimensionsandobservedarelationshipbetweenoptimalsam plinglatticeandthesphere packinglatticeproblem; theoptimalsamplinglatticeistheduallatticeofthedense st spherepackinglattice (Section 3.4 ).Theyderivedthe`canonicalreconstructionlter'in dimensionstwoandthree.Thiscanonicalreconstructionl terisdenedinthefrequency domainbythecharacteristicfunctionovertheVoronoicell ofthedualofthesampling lattice.Thismotivatedandjustiedseveralinvestigatio nsofnon-Cartesianlatticesas alternativesamplinglattices. Hexagonallattice .Basedon[ 82 ],Mersereau[ 75 ]investigatedthesignalprocessing onthehexagonallatticewhichistwo-dimensionaloptimals amplinglattice.Frederickson [ 44 45 46 ]wasrsttodiscussthe(symmetric)bivariatesplinesonth ehexagonallattice. LaterVanDeVilleetal.[ 92 93 94 ]proposed hex-splines onthehexagonallattice whichsharemanypropertieswiththe 3 -directionbox-splinesonthehexagonallattice (Section 5.2 andSection 6.2 ). BCC(Body-CenteredCubic)lattice .SincetheFCC(Face-CenteredCubic) latticeisthedensestspherepackinglattice,itsdual,the BCClattice,istheoptimal samplinglatticeindimensionthree[ 23 82 ].MatejandLewitt[ 70 ]comparedthree volumetricsamplinglatticesusingradialbasisfunctions asreconstructionlters.Ibanez etal.[ 56 ]proposedaray-castingalgorithmforthevolumetricdatas etontheBCC lattice.Theu letal.[ 90 91 ]investigatedvolumereconstructionontheBCClattice.In theirwork,sphericallterandvariousinterpolationsche mesareusedasreconstruction lters.XuandMueller[ 98 ]employedboth2Dand3Doptimalsamplinglatticesforthe acquisitionandreconstructionofCTdatasets,speedingup theprocessbyleveraging programmablegraphicshardware.Theyjustiedtheuniform ityoftheBCClatticeby 21

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measuringtheuniformityofalatticeasthesurfaceareaofi tsVoronoicellwithunit volume. D 4 lattice .Theself-dual D 4 lattice,theoptimal4Dsamplinglattice,wasusedby NeophytouandMueller[ 77 ]forthevisualizationoftime-varyingvolumetricdataset .The optimalityofthelatticeallowsthemtosavedatastoragewi thoutdegradingthequality andtospeeduptherenderingtime. FCC(Face-CenteredCubic)lattice .Qiuetal.[ 83 ]employedtheFCClatticefor globalilluminationacceleratedbythegraphicshardware. Theyleveragedthepropertyof theFCClatticethatithasthemaximumnumberofclosestneig hborlatticepoints[ 23 ]. Thisprovidestheoptimal3Dangulardiscretizationhencea llowsfastercomputationof multiplescattering.StelldingerandStrand[ 89 ]investigatedbothFCCandBCClattices fortopology-preservingdigitization. A n and A n lattices .Inarbitrarydimensions,therootlattices A n and A n (Section 5.1 andSection 6.1 )arewell-knownincrystallography,discretegeometry andrelatedareasandshowmuchbettersamplingeciencytha ntheCartesianlattice (Figure 3-8 ),althoughnotalwaysoptimal.Inthecrystallographylite rature,the A n and A n latticesareembeddedin R n +1 .Hamitoucheetal.[ 51 ]recognizedtheneedforsquare generatormatricestoembedthe A n and A n latticesin R n .Theirdenition,initerative bottom-upfashion,ismorecomplexandtheresultingmatric esaremorecomplicatedthan theonestobepresentedhere. 2.2SymmetricBox-SplinesontheRootLattices Cartesianlattice .Whilenotdevelopedinthecontextofbox-splines,the`Zwa rt-Powell' element[ 99 ](Section 4.2 )canbederivedasabox-splineandhasbeenwidelyusedto approximatebivariatefunctions[ 50 54 84 95 97 ].The 7 -directiontrivariatebox-spline, whichcanbeviewedasthegeneralizationoftheZwart-Powel lelementtothreevariables [ 39 ],wasshownusefulbyPeters[ 80 ].PetersandWittman[ 81 ]appliedittoCSG 22

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(ConstructiveSolidGeometry)modelingandEntezariandM oller[ 39 ]comparedits superioritywiththetensor-producttri-cubicB-spline. A n lattice .Entezari[ 38 ]proposedthe 6 -directiontrivariatebox-splineontheFCC lattice,constructedbythedirectionsto 12 nearestneighborlatticepointsoftheFCC lattice,whichisequivalenttothe A 3 and D 3 lattices(Section 5.3 ). A n lattice .Piecewiselinear hatfunctions ,inparticulartheshiftsofthebivariate 3 -directionlinearbox-splineandofthetrivariate 4 -directionlinearbox-splinesarepopular basisfunctionsforthe2Dand3DFiniteElementMethod,resp ectively.Linearhat functionsapplytoarbitrarytriangular,respectivelytet rahedralmeshes,buthigher-degree box-splines,obtainedbyconvolutionalongthemeshdirect ions,requirestructuredmeshes. Forasmallsampleoftheliteratureonthebivariate 3 -directionbox-splinesee[ 5 6 10 11 17 19 20 22 27 29 32 35 53 55 57 59 85 ].ChuiandLai[ 18 ]andLai[ 65 ]gave anecientevaluationofconvolutionsofhatfunctionsviat heBB-form.Theanalogous trivariate 4 -directionbox-splineevaluationhas,forexample,beendi scussedbyHeandLai [ 52 ]andCasciolaetal.[ 12 ]extendedtheapproachofChuiandLai[ 18 ]tothreevariables. Changetal.[ 15 16 ]andChangandQin[ 14 ]proposedavolumetricsubdivisionscheme basedonthetrivariate 8 -directionbox-spline, M 2 (Section 3.6.1 ). Interpolationonthe n -dimensionalCartesiangridby M r wasdiscussedbyArge andD hlen[ 3 ],andShiandWang[ 87 ]discussedtheassociatedsplinespace.Neuman [ 78 ]proposedaclosedbutintricateformulaforevaluationof M r .Theliteraturerefers tothespacedecompositioncorrespondingtothepolynomial piecesof M r as ( n +1) directionalmesh .Entezarietal.[ 40 41 42 ]andEntezari[ 38 ]werethersttoinvestigate the(symmetric)trivariate 4 -and 8 -directionbox-splinesontheBCClattice. 2.3ReviewofExistingEvaluationTechniquesofBox-Spline s Twodierent MATLAB packagesforevaluatingbox-splines,[ 31 ]and[ 62 ],arebasedon therecursiveformula( 318 ).Thesepackages,whicharefreelyavailablefromtheInter net, areimmenselyusefulbecausetheyaccommodatearbitrarydi rectionmatrices(Section 3.6 ), 23

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andarewell-explainedintheircompanionpapers.Asthepap erspointout,evaluators basedonrecursionfaceakeydicultywhenevaluatingacomb inationofshiftsofthe characteristicfunction.Unlessthecombinatoricsofincl usionandexclusionofknotplanes arecorrectlyand consistently addressed,evaluationalongknotplanesyieldsincorrect results.Iftheevaluationisdonecorrectly,itiscalled`s table'. DeBoor[ 31 ]addressesthestabilityproblembyperturbingevaluation pointsthat aredeemedtooclosetoknotplanes.Kobbelt[ 62 ]untanglesthecombinatoricsexplicitly toavoidround-obydeferringtranslationofevaluationpo intsuntilthebaselevelofthe recursionwherepiecewiseconstantfunctions,thecharact eristicfunctions,areevaluated. Thisalgorithmalsopre-computesthenormalsoftheknotpla nesinadeterministicway toavoidthataknotplaneisdoublyincludedorcompletelyex cludedbytwoadjacent characteristicfunctions.Wefoundthat[ 62 ]workswellforbivariatebox-splines,butthe releasedcodefailsinhigherdimensionsastheexampleinFi gure 1-3A illustrates.After analyzingtheprobleminmoredetailthanwehadintended,we foundtheawinthe applicationofthe MATLAB null functioncall.Generically,thenullspaceisdetermined bySingularVectorDecomposition[ 71 ].ButevenminuteSVDround-oerrorscreate instability.Tellingly,wewereoftenabletoremovetheins tability,inthetrivariatecases wetested,byaddingthe 'r' parametertothe null functioncall,i.e.byenforcing close-to-rationalrepresentationviaGaussianeliminati on. ThealgorithmsofJetterandStockler[ 58 ]andMcCool[ 73 ]evaluatebox-splines approximately bysamplingintheFourierdomainfollowedbytheinverseFFT (Fast FourierTransform).Thisway,theyleveragetheclosedform ofbox-splinesintheFourier domain. Explicitformulasfortheconversionofbox-splinestothep olynomialsinBB-form havebeenderivedbyChuiandLai[ 18 ]andLai[ 65 ]intwovariablesandbyCasciolaetal. [ 12 ]foraclassoftrivariatebox-splines.Theapproachof[ 18 ]generatestheBB-formof bivariatebox-splinesbycomparingdirectionalderivativ esofbox-splineswiththoseofthe 24

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BB-form.Applyingthisapproachto3-and4-directionbivar iatebox-splines,[ 65 ]provides explicitFortrancodesandshowsthattheBB-coecientsare rational.Similarly,[ 12 ] convertstheimportantclassoftrivariatebox-splinesspa nnedbyfourdirections. CondatandVanDeVille[ 22 ]basedtheevaluationof3-directionbivariatebox-spline s onreductionofthebox-splinestoconesplines,i.e.trunca tedpowers.D hlen[ 28 ]went onestepfurtherbyconvertingtheconesplinestosimplexsp linesofonedimensionlower. Theapproachisshowntobeecientforbivariatebox-spline sand,withexplicitguidance alongknotlines,forthe4-directiontrivariatebox-splin e. Cavarettaetal.[ 13 ],page18(seealsodeBoor[ 31 ],page11)showthat,forfunctions satisfyingrenementrelationsandforbox-splinesinpart icular,theexactvaluesona latticecanbecomputedbysolvinganeigenvalueproblem(li stedasEquation( 97 )in Section 9.2.2 ).Valuesonarenedlatticecanthenbecomputedbytherene mentrelation (Equation( 331 )inSection 3.6 ).WeusethisfactinSection 9.2.2 toindependentlyverify exactnessoftheBB-coecientswecompute. Exceptfor[ 58 ],wherethegoalisinterpolation,alltheaboveaimatevalu ating individualbox-splines.Splinesinbox-splineformwouldo handbeevaluatedby evaluatingshiftsoftheunderlyingbox-splinesindividua lly,andthenaddingtheir contributionsweightedbythecoecients. 25

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CHAPTER3 BACKGROUND 3.1Notations ˆMatrices,includingthebox-splinedirectionmatrices(e .g., and T r )andthelattice generatormatrices(e.g., M A P and A n ),aretypesetinbolduppercase. ˆVectorsaretypesetinboldlowercase;e.g., e j and j ˆLatticesaretypesetincalligraphicuppercase;e.g., A n and D n ˆRootsystemsaretypesetas,e.g., A n and D n Thedimensionofvectorsandmatriceswillbedeterminedbyc ontext.Somenotationsfor thevectorsandmatricesare: ˆ x ( j ) 1 j n ,isthe j -thelementofthevector x 2 R n ˆ e j the j -thunitvector, ˆ I n the n n identitymatrix, ˆ 0 :=[0 0] t thezerovector, ˆ j :=[1 1] t the`diagonalvector', ˆ J n := jj t the n n matrixcomposedof 1 'sonlyand ˆ P n := I n J n =n : R n H n 1 j istheorthogonalprojectionalong j ontotheplane H n 1 j (seebelow) andthedotproductisdenedas x y := x t y 2 R : ˆFollowingtheconventionin[ 36 ],an n m matrixwillbeinterpretedas asetofcolumnvectorsallowingmultiplicityoralineartransformation R m R n Wheninterpretedasasetofcolumnvectors,repeatedcolumn sareconsideredas dierentelements. ˆ # A denotesthecardinalityoftheset A ˆColumnvectorsareusedaseithervectorsorpointsdependi ngonthecontext. 26

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ˆ H n j := f x 2 R n +1 : x j =0 g isthe n -dimensionalhyperplaneembeddedin R n +1 intersecting 0 withnormal j ˆ conv( P ) isthe convexhull ofthepointsin P Witha multi-index 2 Z n+ ˆ j j := P n =1 ( ) ˆ u := Q n =1 u ( ) ( ) for u 2 R n ˆ d := d = Q n =1 ( )! and ˆthe normalized -powerfunction [[]] isdenedas[ 36 ] [[ x ]] := x := n Y =1 x ( ) ( ) ( )! : Amatrix U 2 Z n m is unimodular [ 36 ,(II.57)]if det Z = 1 ; 8 Z U : Z issquareand rank( Z )= n: If m = n and U 2 Z n n isunimodularthen U 1 2 Z n n Lastly, ˆ D Z := 2 Z D isacompositionofdierentialoperators D := P n =1 ( ) D and ˆ r Z := 2 Z r isacompositionofbackwarddierenceoperatorssuchthat r := ( ) 3.2RootLattices Lattices .An n -dimensional lattice isadiscretesubgroupofrank n inaEuclidean vectorspace[ 69 ].Givenan l n matrix M with l n and rank( M )= n ,allintegerlinear combinationsofitscolumns, M Z n ,dene(thepointsof)an n -dimensionallattice,say L n embeddedin R l : L n := f M j 2 R l : j 2 Z n g : M iscalleda generatormatrix of L n ,andwecalleachcolumnof M a basis of L n .The choiceofageneratormatrixforalatticeisnotunique[ 69 ]. 27

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Lemma1. M and MU generatethesamelatticepoints, M Z n = MU Z n ,ifandonlyif U 2 Z n n is unimodular Proof. Since U Z n Z n M ( U Z n ) M Z n .Conversely, M Z n =( MU )( U 1 Z n ) MU Z n since U 1 Z n Z n Nextweshowthat U isunimodularif M Z n = MU Z n .Since rank( M )= n ( M t M ) 1 M t iswell-denedand Z n = U Z n hence U isinvertible.Since U 1 Z n = Z n U 1 2 Z n n hence U isunimodular. Ifalatticecanbeobtainedfromanotherbyarotation,reec tionandchangeofscale wesaytheyare equivalent ,written = [ 23 ]. Any n -dimensionallattice L n hasa dual latticegivenby L n := x 2 R l : x u 2 Z ; 8 u 2L n : (31) If M isasquaregeneratormatrixof L n ,then M t isasquaregeneratormatrixof L n [ 23 ]. Let Aut( L n ) bethe symmetrygroup (or automorphismgroup ),i.e.thesetof isometries (orthogonaltransformations)withoneinvariantlatticep ointthattransform L n to itself.If M isageneratormatrixof L n ,anorthogonalmatrix O 2 R l l isin Aut( L n ) if andonlyifthereisaunimodularmatrix U 2 Z n n suchthat[ 23 ](seealsoFigure 3-1 ) MU = OM (32) since,byLemma 1 M Z n =( MU ) Z n = O ( M Z n ) : Therefore,theorderof Aut( L n ) isthenumberoforthogonaltransformationsthatmap L n toitself,i.e.,itmeasureshow symmetric alatticeis.Notethat L n and L n havethesame symmetrygroup. FiniteEuclideanreectiongroups .Amongthemanylatticesinanydimension, aclassoflattices,called rootlattices ,areparticularlyinterestingduetotheirhighly 28

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b C b C b C b C b C b C b C A b C b C b C b C b C b C b C B Figure3-1.Atransformation O :=[ 10 01 ] whichbelongstothesymmetrygroupofthe hexagonallattice A ) M 1 := 1 2 210 p 3 and B ) M 2 := 1 2 h 2 1 0 p 3 i where M 2 = OM 1 = M 1 [ 1 1 01 ] (see( 32 )). symmetricstructure.Todenerootlattices,weneedtoexpl ain niteEuclideanreection groups In n -dimensionalEuclideanspace,wecanarrange n numberofnon-parallel one-sidedmirrorshavingoneintersectionpoint.Thenther eisaunique`innitecone' regionenclosedbythemirrors,called fundamentalchamber or Weylchamber (e.g.see Figure 3-3B andFigure 5-1B ),and virtualchambers generatedbythereectionsofthe fundamentalchamberbytherealandvirtualmirrorsobtaine dbyreections.Sincethe reectionoperationsbythemirrorssatisfy group axioms,theyforma reectiongroup Dependingontheconguration,thenumberofvirtualchambe rsiseitherinniteornite. Hereweconsideronlythe nitereectiongroups whereforeachpointinthefundamental chamberdoesnotmaptoanotherpointinthefundamentalcham ber(Figure 3-3B ), hence,e.g.,excludesthecasesinFigure 3-2 .Then,indimensiontwo,theanglesbetween twomirrorsarelimitedto =k k 2 Z toformanitereectiongroup(e.g.,theangle betweentwomirrorsis = 3 inFigure 3-3B )andthewholespace R 2 isdecomposedbythe mirrorsintonitenumberofnon-overlappingchambers(e.g ., 6 chambersinFigure 3-3B ). Therefore,thecongurationisinvariantunderalltheorth ogonaloperations,composedof reectionsandrotations,inthenitereectiongroup. Nowthequestionis,Whichcongurationscangeneratenitereectiongroupsi ndimension n ? 29

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b b c b c b c x R 1 x R 2 R 1 x R 1 R 2 R 1 x M 1 M 2 A b b c b c b c x R 1 x R 2 R 1 x R 1 R 2 R 1 x M 1 M 2 B Figure3-2.Reectiongroupswhereapoint( x )inthefundamentalchambermapstoa dierentpoint( R 1 R 2 R 1 x )inthefundamentalchamber.Theshadedarea denotesthefundamentalchamberand R 1 and R 2 denotethereections associatedwiththemirrors M 1 and M 2 ,respectively. andCoxeter[ 24 ]foundthecompletelistofsuchcongurationsinalldimens ions. (Thenitereectiongroupsarealsocalled niteCoxetergroups .)Interestingly,except theobviouscaseofdimensiontwo,thenumberofsuchcongur ationsisniteinany dimension.Correspondinglyonlylimitednumberofanglesa reallowed(again,except dimensiontwo).Suchcongurationscanbeeasilydescribed by Coxeterdiagrams (Figure 3-3A ).ACoxeterdiagramisa fully-connected graphcomposedof n nodeswhere eachnodedenotesarealmirror(or,equivalently,itsnorma lvector)ofthefundamental chamberandeachedgedenotestheanglebetweentwomirrors. Sincealltheangles allowedareoftheform =k k 2 Z ,onlythenumber k islabeledateachedge.Although thegraphisquitecomplicatedbecauseitisfully-connecte d,itappearsmuchsimplerafter applying`labelingconventions': ˆtheedgewithangle = 2 isomitted,whichisrelatedtothe reduciblereectiongroup tobeexplainedlater,and ˆtheedgewithangle = 3 isnotlabeledduetoitsimportance. SeeFigure 3-3A ,Figure 3-4 andFigure 5-1A forexamples. Rootsystems .ACoxetergroupcanbestudiedwiththehelpoflinearalgebr a intheformof rootsystems [ 60 ].Arootsystem R intheEuclideanspace E isanite 30

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A R 2 R 1 R 2 x = R 1 R 2 R 1 x M 1 M 2 b b c b c b c b c b c x R 1 x R 1 R 2 x R 2 R 1 x R 2 x B 1 + 2 2 1 1 2 1 2 + C Figure3-3.Threerepresentationsofthetwo-dimensionalr eectiongroupwithdihedral angle = 3 : A )Coxeterdiagram, B )real(black, M 1 and M 2 )andvirtual(gray) mirrorswiththefundamentalchamber(grayregion)and C )rootsystem (simpleroots( 1 and 2 )associatedwiththerealmirrors( M 1 and M 2 )in B ) aredenotedbythickarrows).In C ),therootvectorsaredecomposedintotwo disjointsubsetspositive( + )andnegative( )rootswithrespecttothesimple roots 1 and 2 3 2 5 = 5 Figure3-4.TwoequivalentCoxeterdiagramsaccordingtoth elabelingconventions[ 23 ]. setofnonzerovectors,eachrepresenting(thenormalof)am irroroftheCoxetergroup, satisfyingthefollowingtwoaxioms[ 60 ]. Axiom1. If 2 R ,then 2 R i = 1 Inotherwords,foreachmirror,twovectorswithsamelength andoppositedirections areassociated.Axiom2. If ; 2 R ,then R 2 R where R isthereectionbythemirrorwithnormal .Inotherwords, R isclosed underthereectionoperatorsassociatedwiththevectorsi n R .Moreover,therootsystem R isinvariantunderthereectionoperatorsassociatedwith thevectorsin R .Notethat 31

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thelengthsofthevectorsshouldbechosencarefullytomeet theaxioms,althoughthey don'tnecessarilyneedtobeallthesame. Givenarootsystem R R isa fundamentalsystem of R if[ 60 ] ˆ islinearlyindependentand ˆeveryrootof R isalinearcombinationoftherootsof ,wherethecoecientsare allnon-negativeorallnon-positive. Therootsin R arecalled fundamental (or simple )roots( 1 and 2 inFigure 3-3C ). Asinthecaseofthereectiongroups,arootsystemalsocanb edescribedbya Coxeterdiagram(Figure 5-2 andFigure 7-1 ).Whileonlythedihedralanglesbetweenthe mirrorsareconsideredforthereectiongroups,thelength saretakenintoaccountforthe rootvectors.Thereforeadditionalnotationisrequiredfo rtheirCoxeterdiagrams[ 23 60 ]. Butwedonotdiscussitherebecausealltherootsystems( A n and D n )consideredinthis dissertationhaverootvectorsofthesamelengths. Onesimplewaytoconstructan ( n +1) -dimensionalrootsystemfroman n -dimensional rootsystemistoaddtwovectors v and v thatareorthogonaltoalltheotherroot vectorssothatAxiom 1 andAxiom 2 .Sucharootsystemiscalled reducible .Areducible rootsystemis,duetoorthogonality,thedirectsumoftwo(o rmore)lowerdimensional rootsystemshencecanbestudiedthrougheachlowerdimensi onalones.Thereforewe focusonthe irreducible (or indecomposable [ 23 ])rootsystemswherenorootvectoris orthogonaltoalltheotherroots.Duetothelabelingconven tiondiscussedearlier,a Coxeterdiagramassociatedwithareduciblerootsystemcon tains(atleast)onenode disconnectedfromtheothernodes.ThereforetheCoxeterdi agramassociatedwithan irreduciblerootsystemisaconnectedgraphhencejusties therstlabelingconvention. Rootlattices .Althoughtherootsystems,hencetheniteCoxetergroups, are veryimportanttoinvestigatethe regularpolytopes [ 25 ],notallrootsystemscangenerate lattices.By`generatingalattice'wemeanthatallofthein tegerlinearcombinationsof therootvectorsinarootsystemarelocatedatthelatticepo ints.Sincetherootsystem 32

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isinvariantundertheorthogonaltransformationsofthere ectiongroup,thismeansthat alltheorthogonaltransformationsmustbeinthesymmetryg roupofthelattice.For example,fortherootsysteminFigure 3-5 notalloftherootvectorsarelocatedatthe latticepointsgeneratedbythesimpleroots.Therefore,fo rarootsystemtogenerate alattice,called rootlattice ,anadditionalconditionisrequired,the crystallographic restriction [ 23 25 26 ]: Axiom3. 2( ; ) ( ; ) 2 Z forany ; 2 R where ( ; ) istheinnerproductof and .Thenitereectiongroupssatisfying b C b C b C b C b C b C b C b C b C b C b C b C b C Figure3-5.Anon-crystallographicrootsystem.Thelattic epointsaretheintegerlinear combinationsofthesimpleroots(thickarrows). allthreeaxiomsarealsocalled Weylgroups [ 23 60 ]andtherootsystem R satisfying allthreeaxiomsiscalleda crystallographicrootsystem .Crystallographicsrootsystems generaterootlattices.Duetothecrystallographicrestri ction,onlythedihedralangles =k k 2f 2 ; 3 ; 4 ; 6 g ,areallowedbetweenanypairofmirrors(rootvectors)[ 23 26 ]. Also,with GL denotingthegenerallineargroup[ 60 ],thecrystallographicreection groupbecomesasubgroupof GL n ( Z ) notjustof GL n ( R ) .Ageneratormatrixcanbe constructedbyanysetofthefundamentalroots.Sincethero otsystem R isinvariant undertheorthogonaltransformationsassociatedwithther eectiongroup,theroot latticegeneratedby R isalsoinvariantunderthoseoperations.Moreover,sincel attices arealreadyinvariantundershiftstolatticepoints,rootl atticesareinvariantunderthe 33

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orthogonaltransformationsateverylatticepoint.Thisma kesrootlatticesgoodsampling candidatesforisotropicsignals. 3.3Multi-DimensionalSignalProcessing A Diraccomb or Shah functiononthelattice M Z n M 2 R n n ,isdenedas M ( x ):= X k 2 Z n ( x M k ) (33) where isthe Diracdelta function.ItsFouriertransformis(see[ 37 ]) c M ( ):= Ff M g ( )= 2 j det M j X k 2 Z n ( 2 M t k ) (34) where Ffg istheFouriertransformoperator.Samplingafunction f onthelattice M Z n is equivalenttomultiplying f with M ( f M )( x )= X k 2 Z n f ( M k ) ( x M k ) : Therefore Ff f M g ( )= b f ( ) c M ( ) = 2 j det M j X k 2 Z n b f ( 2 M t k ) (35) where b f ( ):= Ff f g ( ) .Inotherwords,the(scaled)Fouriertransformof f 2 b f ( ) = j det M j ,isreplicatedon 2 M t Z n ,the(scaled)duallatticeof M Z n [ 37 ]. 3.4DensestSpherePackingLatticeandOptimalSamplingLat tice The spherepacking problem, howdenselycanwepackidenticalspheresin R n ? ,is oneoftheoldestproblemsingeometry[ 23 ].Whilethespherepackingdoesnotrequire latticestructureforthearrangementofthespheres,the latticepacking problemisto ndthelatticethatinducesthedensestspherepackingwhen thespheresarelocated atthelatticepoints.Surprisingly,eventheanswerforthe latticepackingproblemhas beenfoundonlyforlimitednumberofdimensions[ 23 ];amongthemaredimensionstwo 34

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andthree,wheretheanswersarethe hexagonallattice (Figure 3-7 )andthe FCClattice respectively[ 23 ]. A Ff f M 1 g ( ) b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b B f M 1 ( x ) C Ff f M 2 g ( ) b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b D f M 2 ( x ) Figure3-6.Illustrationoftherelationbetweenlatticepa ckinginthefrequencydomain andecientsamplingintheprimaldomain.Attheircenters, in orange Figures A )and C )eachshowstheband-limited,star-shapedFouriertransfo rm Ff f g ofagivenfunction f .Fouriertransforms Ff f M 1 g and Ff f M 2 g ( gray )ofthefunctionsamples f M 1 and f M 2 consistof(scaled)replicasof Ff f g onthetwodierently-spacedlatticeswithgeneratormatri ces, M 1 and M 2 respectively.Frombothtransforms,theoriginalsignalca nbe reconstructedwithoutaliasingbyremovingthereplicaswi thalow-passlter (thickcircle)andapplyingtheinverseFouriertransform. Butthedenser packingofreplicasinFigure A )ismoreecientsinceitcorrespondstoa sparsersamplinglatticeintheprimalspace,Figure B ). Therelationbetweentheoptimalsamplinglattice,thelatt iceallowinglowest densitytoreconstructasignalwithoutaliasing,andthede nsestspherepackinglattice problemwasrstinvestigatedbyPetersenandMiddleton[ 82 ].Whilethereisonlyone latticeindimensionone,henceitsscalingistheonlyvaria bleforthelattice,thereare innitenumberofpossiblelatticesinhigherdimensionswh ichaectthequalityofthe reconstruction.Assumingthefrequencyoftheinputsignal isisotropicandband-limited, wecanreconstructtheoriginalsignalwithoutaliasingusi ngasphere-shapedlterinthe frequencydomain(Figure 3-6A and 3-6C ).Sincethelatticeinthefrequencydomainis thedualofthesamplinglattice(Section 3.3 ),themoredenselywecanpackthespheres (reconstructionlters)inthefrequencydomain,thespars erasamplinglatticewecan chooseinthespacedomaintoreconstructtheoriginalsigna l(Figure 3-6 ).Therefore,for inputsignalswithisotropicband-limitedfrequencies,th eoptimal n -dimensionalsampling 35

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latticeisthe dual oftheoptimalspherepackinglattice[ 38 40 64 ].Figure 3-7 shows howthesamplingeciencydiersaccordingtothelatticest ructure.Ifthedensityofthe latticeislow(hencesamplingdensityishigh)enoughasint heleftmostcase,theoriginal signalcanbereconstructedwithoutaliasingregardlessof thesamplinglattice.Butascan beseenfromtherightmostcase,aliasingbetweentheprimar yspectrumanditsreplicas mayormaynotappeardependingonthesamplinglattice. bb b b b b b bb b b b b b bb b b bb b bb b bb bb bb bb b bb bb bb bb b bb bb bb =4 = 25 =1 = 4 =1 = 2 p 3 Figure3-7.Unitsphere(primaryspectrumanditsreplicas) packingwithrespecttothe density for(top)hexagonallatticeand(bottom)Cartesianlattice .(left) samplingdensityishighenoughsothatnopairofspectraove rlap(bottom center)theprimaryspectrum`touches'itsreplicaontheCa rtesianlatticebut (topcenter)thespectraarestillawayfromeachotheronthe hexagonal lattice.(topright)thespectratoucheseachotheronthehe xagonallattice, hencetheoriginalsignalstillcanbereconstructedwithou taliasingbut (bottomright)aliasingappearsontheCartesianlattice. The density ofalatticepackingistheproportionofthespaceoccupiedb ythespheres whenpacked.The centerdensity ofalatticeisthenumberofthelatticepointsperunit volume,whichcanbeobtainedbydividingitsdensitybythev olumeoftheunitsphere [ 23 ].Therefore,larger(center)densityimpliesthat itsdualisamoreecientsampling 36

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lattice .Table 3-1 andFigure 3-8 respectivelyshowthecenterdensityandthedensityof therootlatticestobeconsideredinthiswork.Bothimplypo orersamplingeciencyof theCartesianlattice Z n comparedtootherrootlattices. Table3-1.Centerdensityofseveralrootlattices[ 23 ].SeeSections 5.1 6.1 7.1 and 8.1 for thedenitionsof A n A n D n and D n ,respectively. Z n A n A n D n D n 2 n 2 n= 2 ( n +1) 1 = 2 n n= 2 2 n ( n +1) ( n 1) = 2 2 ( n +2) = 2 ( 3 1 : 5 2 5 ( n =3) 2 ( n 1) ( n> 3) u t u t u t u t u t u t u t u t u t u t r s r s r s r s r s r s r s r s r s r s l d l d l d l d l d l d l d l d l d l d b C b C b C b C b C b C b C b C b b b b b b b b b hexagonal b FCC b BCC u t r s l d b C b Z n A n A n D n D n dimension 0 1 12345678910 Figure3-8.Densityofseveralrootlatticesuptodimension 10 3.5TheBB-FormofaMultivariatePolynomial Let f v j 2 R n : j 2f 1 ;:::;n +1 gg betheverticesofanon-degeneratesimplex .The map : R n R n +1 : x 7! 264 v 1 v n +1 1 1 375 1 264 x 1 375 (36) iscalledthe barycentriccoordinatefunction withrespectto [ 30 ].TheBB-formofan n -variatepolynomialoftotaldegree d andcoecients f c 2 R : j j = d; 2 Z n +1 + g 37

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denedon ,is P ( u ):= X j j = d c b ( u ) (37) where ˆ u isinbarycentriccoordinatesw.r.t. and ˆ f b ( u ):= d u : j j = d; 2 Z n +1 + g aretheBernsteinbasispolynomialsofdegree d Denotethe j -thunitvectorby e j .Then b ( e j )=1 if = d e j andzerootherwise. Therefore, P j v j = P ( e j )= c d e j .Inotherwords,theBB-formofamultivariatepolynomial interpolates its vertexcoecients f c : = d e j ;j 2f 1 ;:::;n +1 gg Thedirectionalderivativeof P alongoneoftheedgesof ,say v i v j ,is D v i v j P = D v i v j X j j = d c b = d X j j = d 1 ( c + e i c + e j ) b : (38) 3.6Box-Splines Weusethenotationanddenitionsmadestandardby[ 36 ].Inparticular,abox-spline isasmoothpiecewisepolynomialofnitesupportanda spline inbox-splineformisa linearcombinationoftheshiftsofabox-spline.Ifthesequ enceoftheshiftsofabox-spline arelinearlyindependent,thebox-splineisa basis function. Denition .Geometrically,thevalueofabox-splinewithdirectionma trix 2 R n m b c b c b c 0123 A B Figure3-9.Geometricdenitionofthebox-splinewiththed irectionmatrix := 111 and 1 f x g = 1 3 111 t f x g + H 2 j ; thetranslatesofthe hyperplanesorthogonalto j := 111 t 38

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at x 2 ran R n istheshadow-density[ 36 ,(I.3)](see,e.g.,Figure 1-2 andFigure 3-9 ) M ( x ):=vol m rank( ) 1 f x g\ = j det j ; i.e.thenormalizedvolumeoftheintersectionofacube R m m n ,withthepreimage 1 f x g of x ,an ( m dimran ) -dimensionalanesubspacein R m .Thecubeorboxgives thebox-splineitsname.Inmoredetail, ˆ :=[0 :: 1) m isan m -dimensionalhalf-openunitcube, ˆ isthe n m directionmatrix,possiblywithrepeatedcolumns,ofthebo x-spline, ˆ ran isthesubspacespannedbythecolumnvectors f : 2 g ˆ 1 f x g := f y : y = x g isthepreimageof x thatcanbeexpressedas 1 f x g = t t 1 f x g +ker ; (39) whenviewing asalineartransformation : R m R n and ˆ vol d ( ) isthe d -dimensionalvolumeofitsargument. Inthefollowing,unlessmentionedspecically,weassume rank( )= n hence ran = R n : Alternatively,wecanconstruct M viaconsecutivedirectionalconvolutionsalongthe directionsin asinFigure 1-1 [ 36 ,(I.8)]: M [ = Z 1 0 M ( t ) dt: DegreeandContinuity .Abox-spline M isapiecewisepolynomialon ran .Its degreeislessthanorequalto # n where # denotesthenumberofcolumnsof Thepolynomialpiecesjointoformafunctionin C m 1 (ran ) where[ 36 ,page9] m := m ( ):=min f # Z : Z 2A ( ) g 1 (310) and[ 36 ,page8] A ( ):= f Z : n Z doesnotspan R n g 39

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CenteredBox-Spline .Thecenteredbox-spline M c of M is[ 36 ,(I.21)] M c := M ( + X 2 = 2) : (311) FourierTransform .TheFouriertransformof M is[ 36 ,(I.17)] c M ( ):= Ff M g ( )= Y 2 1 exp( i ) i ;i := p 1 ; (312) where Ffg istheFouriertransformoperator.Forthecenteredbox-spl ine M c ,[ 36 page11] Ff M c g ( )= Y 2 sinc( ) ; sinc( ):= sin( != 2) != 2 : (313) Support .By[ 36 ,page9],the(closed)supportof M consistsoftheset(see Figure 4-1 ,Figure 4-2B ,Figure 5-4B ,Figure 6-2B andFigure 8-1B ) supp M = = f X 2 t :0 t 1 g where :=[0 :: 1] m istheclosedunitcubeand t istheelementof t associatedwiththe column 2 by t .Assuming ran = R n ,thesetofall bases of isdenoted[ 36 page8] B ( ):= f Z :# Z =rank( Z )= n g : (314) Thesupportof M iscomposedoftheparallelepipedsspannedby Z 2B ( ) :For ran = R n thereexistspoints Z 2 f 0 ; 1 g m Z 2B ( ) ,sothat istheessentially disjointunionofthesets[ 36 ,I.53](Figure 6-6 ) Z + Z ; Z 2B ( ) : (315) Dierentiation .Aderivativeof M inthedirections Z equalsthebackward dierencesof M n Z alongthem[ 36 ,(I.30)]: D Z M = r Z M n Z : (316) 40

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KnotPlanes .Accordingto[ 36 ,(I.37)],abox-splineisapiecewisepolynomialwith piecesdelineatedbyashift-invariantmeshon Z n generatedbythecollectionofknot planes(hyperplanesspannedbycolumnsof ) H ( ) [ 36 ,page16](Figure 4-3 ): ( ):= [ H 2 H ( ) H + Z n : (317) Themesh ( ) decomposes R n intoconvexpolytopes(seeFigure 9-1 ). TheRecurrenceRelation .Aslongas M n for 2 iscontinuousat x = t := P 2 t 2 R n ; t 2 R m ,thebox-spline M canbeevaluatedrecursivelywiththe recurrence[ 36 ,(I.43)] ( m n ) M ( x )= X 2 t M n ( x )+(1 t ) M n ( x ) : (318) CardinalSplineSpace .Thecardinalsplinespace[ 36 ,(II.1)] S :=span( M ( j )) j 2 Z n (319) isthesplinespacespannedbytheshiftsof M on Z n andeachspline s 2 S hastheform s := X j 2 Z n M ( j ) a ( j ) withameshfunction(splinecoecients) a : Z n R LinearIndependence .Thesequence ( M ( j )) j 2 Z n islinearlyindependentifand onlyif is unimodular [ 36 ,page41]. PolynomialsinSplineSpace .Let bethesetofallthepolynomialson R n [ 36 (II.21)]and bethesetofpolynomialsof(total)degreeupto .Themap M 0 : f 7! X j 2 Z n M ( j ) f ( j ) (320) mapsthepolynomialspace M := \ S ontoitself,i.e., M 0 isone-to-oneon M Specically, m ( ) M 41

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Quasi-Interpolation .Whilethemap M 0 isone-to-oneonthepolynomialspace M ,itistheidentityonlyon 0 [ 36 ,(II.10)].Thefollowingquasi-interpolant Q M for thesplinespace S providesafastwayofapproximatingafunction f byaspline Q M f 2 S [ 36 ].Wefocusonthequasi-interpolantsthatprovidetheoptim alapproximationorder m ( )+1 by reproducing allthepolynomialtermsuptodegree m ( ) of(theTaylor expansionof)acontinuousfunction,leavinghighertermsa sapproximationerror:[ 36 page72] ( Q M f )( x ):= X j 2 Z n M ( x j ) M ( f ( + j )) (321) where M isthelinearfunctional[ 36 ,(III.22)] M f := X j j m ( ) g ( 0 )( D f )( 0 ) (322) and 2 Z n+ isamulti-index.The Appellsequence f g g in( 322 )canbecomputedeither recursivelyas 8>><>>: g 0 = [[]] 0 g = [[]] P 6 = ( [[]] ) g ([ 36 ,(III.19)]) where f := X j M ( j ) f ( j ) ; (323) orfromtheFouriertransform c M [ 36 ,(III.34)]: g ( 0 )= [[ iD ]] 1 = c M ( 0 ) : (324) Box-splinesonNon-CartesianLattices .Givenaninvertiblelinearmap L on R n [ 36 ,(I.23)] M = j det L j M L L : (325) Hence,givenasquaregeneratormatrix M ,anyweightedsumoftheshiftsofthe(scaled) box-spline f M := j det M j M M (326) 42

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onthe(possiblynon-Cartesian)lattice M Z n canbeexpressedasaweightedsumofthe shiftsof M ontheCartesianlattice Z n by changeofvariables : X j 2 M Z n f M ( j ) a ( j )= X k 2 Z n M ( M 1 k ) a ( M k ) (327) where a : M Z n R isthemeshfunction(splinecoecients)on M Z n .DeBoorandHollig [ 32 ,page650]alreadypointedtothisrelationshipinthebivar iatesetting. Wedenotethesplinespacespannedbytheshiftsof f M on M Z n by S M :=span f M ( j ) j 2 M Z n : Thisnotationbecomesconsistentwith( 319 )byomitting M = I n anddening S := S I n : Bythechangeofvariablerelation,thesequence f M ( j ) j 2 M Z n islinearly independentifandonlyif isunimodular. Lemma2 (Quasi-interpolant) Let D M := Q v 2 M D v v bethecompositionofdirectional derivatives D v := P nj =1 v ( j ) D j alongthecolumnsof M and f g g theAppellsequenceof M ( 322 ) .Thequasi-interpolant Q MM for S M denedbythefunctional MM ( f ( + j )):= M ( f M ) + M 1 j (328) = X j j m ( ) g ( 0 )( D M f )( j ) ; j 2 M Z n ; (329) providesthesamemaximalapproximationorderasdoes Q M denedby M for S Proof. Ifwedene Q MM f ( x ):= Q M ( f M ) ( M 1 x ) then,since f = f M M 1 f Q MM f ( x )= ( f M ) Q M ( f M ) ( M 1 x )= e f Q M e f ( e x ) ; 43

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for e f := f M and e x := M 1 x ,i.e., Q MM hasthesameapproximationorderas Q M .Since Q M ( f M ) ( M 1 x )= X k 2 Z n M ( M 1 x k ) M (( f M )( + k )) = X j 2 M Z n j det M j M M ( x j ) M ( f M )( + M 1 j ) ; and D k ( f M )=lim h 0 f ( M + h Me k ) f ( M ) h =( D Me k f ) M ; thecorrespondingfunctional MM for j 2 M Z n is MM ( f ( + j ))= M ( f M ) + M 1 j = X j j m ( ) g ( 0 )( D ( f M )) M 1 j (by( 322 )) = X j j m ( ) g ( 0 )( D M f )( j ) : DiscreteBox-splines .Adiscretebox-spline b h associatedwiththedirectionmatrix 2 Z n m and h 2 1 = N canbeconstructedbytherecurrencerelation[ 36 ,(VI.5)] b h = h 1 =h 1 X j =0 b h n ( jh ) (330) withthebasecaseof b h[] = ,theKroneckerdeltafunction. TheRenementequation .Abox-spline M with 2 Z n m hastherenement equation[ 36 ,(VI.10)] M = X k 2 h Z n M (( k ) =h ) m h ( k ) (331) wherethe renementmask m h := b h =h n [ 36 ,(VII.7)]. 3.6.1ExamplesofBox-Splines TheBox-Spline M r .Box-splinesdenedbypossiblyrepeated ( n +1) distinct convolutiondirectionsarealsocalled box-splinesonthe ( n +1) -directionalmesh [ 3 ].Given 44

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the n ( n +1) directionmatrix T 1 := I n j = e 1 e n j 2 Z n ( n +1) ; (332) thebox-spline withmultiplicity r ineachdirectionisdenedbythe n r ( n +1) direction matrix[ 36 ,page80] T r := r [ j =1 T 1 andweabbreviate M r := M T r : AspointedoutinSection 2.2 ,thisfamilyofbox-splineshasbeenwidelyused.Since T 1 =[1 1] intheunivariatecase, M r canbeviewedasageneralizationoftheuniform B-splinesofodddegreetoarbitrarydimensions. Tri-quadraticTensor-ProductB-Spline .With I 3 the 3 3 identitymatrix,the tri-quadratictensor-productB-splinehasthecenteredbo x-splinerepresentation M 333 := M 333 ( +( 3 2 ; 3 2 ; 3 2 )) ; 333 := I 3 I 3 I 3 : Sinceuptosixdirectionsof 333 lieinahyperplane, m ( 333 )=(9 6) 1=2 .Therefore allquadraticsarecontainedin S 333 :=span( M 333 ( j )) j 2 Z 3 (333) andallthesplinesin S 333 are C 1 .Thetotaldegreeofthepolynomialpiecesis 9 3=6 Thequasi-interpolant Q 333 withmaximalapproximationorderfor S 333 isdenedbythe functional 333 f := f ( 0 ) 1 8 ( D 2 1 + D 2 2 + D 2 3 ) f ( 0 ) : Foradiscreteinput f : Z 3 R 333 f 7 4 f ( 0 ) 1 8 3 X k =1 ( f ( e k )+ f ( e k )) : (334) DeBoor'salgorithmisthethestandardstableevaluationme thodforsplinesin S 333 45

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CHAPTER4 SYMMETRICBOX-SPLINEONTHECARTESIANLATTICE 4.1DenitionandProperties TheCartesianlatticehasbeenusedasasamplinglatticefor alongtime,sinceit naturallymatchestheCartesiancoordinates.Asoneofther ootlatticesgeneratedby therootsystem B n [ 23 ],itssymmetrygroupconsistsofall n permutationsand 2 n sign changesofthecoordinates.Hencetheorderis 2 n n [ 23 ]. Tensor-productB-splinesarethemostpopularreconstruct ionltersontheCartesian lattice.Theirtensoredstructuremakesthecomputationsh andy,andtheshiftsofthem ontheCartesianlatticearelinearlyindependent,henceth eyarebasisfunctions.Butthey possessrelativelylowcontinuitywithhighdegree,andhav elowapproximationorder.This canbeovercomebyconsideringbox-splinesconstructedbym oredirectionsinadditionto the n mainaxisdirectionsfor n 2 .Onesuchextensionistoincludethe 2 n 1 `diagonal' directions,i.e.,thedirectionmatrixisdenedas Z n := I n [f e n + n 1 X j =1 j e j : j 2f 1 gg ;n 2 : Wedenethecenteredbox-spline M Z n asfollows: M Z n := M c Z n = M Z n ( + X 2 Z n = 2) : (41) Thedegreeof M Z n is n +2 n 1 n =2 n 1 .Thecontinuityof M Z n canbedetermined withthehelpofthefollowinglemma.Lemma3. Let ? v := f 2 : v =0 g (42) bethevectorsin orthogonalto v 2 R n .Then max v 2 R n # ? v Z n =# ? ( e n + e n 1 ) Z n =2 n 2 + n 2 : 46

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Proof. Let ( v ):= f j 2 Z : v 2 R n ; v ( j ) 6 =0 g (43) bethesetofnonzeroelementindicesofavector v 2 R n and k := k ( v ):=# ( v ) .Also,let Z := Z n n I n sothat Z n = Z t I n andthematrix F j :=[ e 1 e j e n ] t 2 R n n bethetransformationwhichinversesthesignof j -thelement.For v 2 R n 2 Z and F j with j 2 ( v ) ,if v =0 then ( F j ) v 6 =0 Let Z 0 := f P nj =1 e j g .If j 2 ( v ) ,foranyvector 2 Z 0 wehaveanothervector F j 2 Z 0 .Therefore,atmost (2 k = 2)2 n k =2 n 1 vectorsin Z 0 areorthogonalto v Since Z arethevectorsin Z 0 withthelastelement 1 # Z ? v 2 n 2 .Ontheotherhand, # I ? v n = n k .Therefore, # ? v Z n =# Z ? v +# I ? v n 2 n 2 + n k: Since Z ? v = ; if k =# ( v )=1 # ? v Z n canbemaximizedif k =2 andtheexistenceof such v isveriedby ? ( e n + e n 1 ) Z n = f e n e n 1 + n 2 X j =1 e j g[ ( I n nf e n 1 ; e n g ) : ByLemma 3 ,atmost (2 n 2 + n 2) directionsspanahyperplane,therefore( 310 ) m ( Z n )=(( n +2 n 1 ) (2 n 2 + n 2)) 1=2 n 2 +1 hence M Z n 2 C 2 n 2 Lemma4. Thesequence ( M Z n ( j )) j 2 Z n n 2 ,islinearlydependent. 47

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Proof. Forthefull-rankmatrix Z := n [ i =1 ( i 1 X j =1 e j + n X j = i e j ) = 266666666664 1 1 1 1 11 1 1 ... ... ... ... 11 1 1 11 11 377777777775 Z n det Z canbecomputedrecursively,e.g.,bytheformulain[ 88 ],as det Z =2 n 1 6 = 1 ; 8 n 2 : 4.2TheZP-Element A 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B Figure4-1. A )DirectionsandsupportoftheZP-elementand B )itsBB-coecients multipliedby 8 [ 18 ]. Indimensiontwo, Z 2 isthedirectionmatrixofthefamousZP-element[ 99 ] (Figure 4-1 ): ZP := 264 101 1 0111 375 =: Z 2 : (44) Figure 1-1 showsconstructingtheZP-elementviaconsecutivedirecti onalconvolutions alongthedirectionsin ZP .NotethattheZP-elementisnotcentered(Figure 4-1A ),while M Z 2 isdenedtobecenteredby( 41 ). 48

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The(total)degreeof M Z 2 is 2 2 1 =2 and M Z 2 2 C 2 2 2 = C 1 .TheBB-coecients (BB-net)of M Z 2 canbefoundin[ 18 ]andarerepeatedinFigure 4-1B .Followingthe procedureinSection 3.6 ,thequasi-interpolantwithoptimalapproximationorder 3 for S Z 2 isdenedbythefunctional Z 2 f := M Z 2 f =( f 1 8 ( D 2 1 + D 2 2 ) f )( 0 ) =( f 1 24 ( D 2 1 + D 2 2 +( D 1 + D 2 ) 2 +( D 1 + D 2 ) 2 ) f )( 0 ) =( f 1 24 X 2 ZP D 2 f )( 0 ) : (45) 4.3The 7 -DirectionTrivariateBox-Spline 4.3.1Denition A B Figure4-2. A )Directionsand B )support(truncatedrhombicdodecahedron)of M Z 3 Indimensionthree, Z 3 isthedirectionmatrixofthe 7 -directiontrivariatebox-spline whichwasshownusefulbyPeters[ 80 ](Figure 4-2 ): Z 3 := 266664 1001 11 1 01011 1 1 0011111 377775 : (46) 49

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The(total)degreeof M Z 3 is 2 3 1 =4 and M Z 3 2 C 2 3 2 = C 2 .Thesamecontinuitycanbe achievedbythetri-cubicB-splineofwhichtotaldegreeis 12 3=9 4.3.2PolynomialStructure Figure4-3.Sixnon-axis-alignedknotplanesof M Z 3 spannedbythedirectionsof Z 3 whichdecomposetheVoronoicelloftheCartesianlattice. Thesetofknotplanes H ( Z 3 ) spannedbythedirectionsin Z 3 iscomposedofthree axis-alignedplanesandsixplanesofwhichnormalvectorsa rethecolumnsof(Figure 4-3 ) N Z 3 := 266664 0 1 1110 101101110011 377775 : (47) Theshiftsofthreeaxis-alignedplanesdecompose R 3 intotheVoronoicells j +( :: ) 3 at eachlatticepoint j 2 Z 3 .AndeachVoronoicellisfurtherdecomposedinto 24 tetrahedra bytheremainingsixnon-axis-alignedplanesof H ( Z 3 ) (Table 4-4 ). 000111100110100111000110 011001111000111001011000 001111011101011111001101 100010110000110010100000 110111111011111111110011 000100001000001100000000 Figure4-4. 24 tetrahedradecomposedbythesixknotplanesinFigure 4-3 .Theindices ( i 4 )areassignedasdescribedinSection 9.2.1 50

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Table4-1.Orderingofthevertices(scaledby 2 )ofthedomaintetrahedraintheVoronoi cellof (0 ; 0 ; 0) (seeFigure 4-4 ). i 4 v 1 v 2 v 3 v 4 i 4 v 1 v 2 v 3 v 4 000111 0 ,( 1 0 0 ),( 1 1 0 ),( 1 1 1 ) 011001 0 ,( 1 0 0 ),( 1 1 1 ),( 1 1 1 ) 100110 0 ,( 1 0 0 ),( 1 1 1 ),( 1 1 1 ) 111000 0 ,( 1 0 0 ),( 1 1 1 ),( 1 1 1 ) 100111 0 ,( 1 0 0 ),( 1 1 1 ),( 1 1 1 ) 111001 0 ,( 1 0 0 ),( 1 1 1 ),( 1 1 1 ) 000110 0 ,( 1 0 0 ),( 1 1 1 ),( 1 1 1 ) 011000 0 ,( 1 0 0 ),( 1 1 1 ),( 1 1 1 ) 001111 0 ,( 0 1 0 ),( 1 1 1 ),( 1 1 1 ) 100010 0 ,( 0 1 0 ),( 1 1 1 ),( 1 1 1 ) 011101 0 ,( 0 1 0 ),( 1 1 1 ),( 1 1 1 ) 110000 0 ,( 0 1 0 ),( 1 1 1 ),( 1 1 1 ) 011111 0 ,( 0 1 0 ),( 1 1 1 ),( 1 1 1 ) 110010 0 ,( 0 1 0 ),( 1 1 1 ),( 1 1 1 ) 001101 0 ,( 0 1 0 ),( 1 1 1 ),( 1 1 1 ) 100000 0 ,( 0 1 0 ),( 1 1 1 ),( 1 1 1 ) 110111 0 ,( 0 0 1 ),( 1 1 1 ),( 1 1 1 ) 000100 0 ,( 0 0 1 ),( 1 1 1 ),( 1 1 1 ) 111011 0 ,( 0 0 1 ),( 1 1 1 ),( 1 1 1 ) 001000 0 ,( 0 0 1 ),( 1 1 1 ),( 1 1 1 ) 111111 0 ,( 0 0 1 ),( 1 1 1 ),( 1 1 1 ) 001100 0 ,( 0 0 1 ),( 1 1 1 ),( 1 1 1 ) 110011 0 ,( 0 0 1 ),( 1 1 1 ),( 1 1 1 ) 000000 0 ,( 0 0 1 ),( 1 1 1 ),( 1 1 1 ) TheBB-coecientsof M Z 3 ,obtainedbythetechniqueproposedinChapter 9 ,are tabulatedinTable 4-2 where J Z 3 ,called`stencil',isthesetofshiftsof M Z 3 thathave nonzerovalueatthepointinsidethe`standard'domaintetr ahedron(Figure 4-6 ).There are # J Z 3 =53 suchshiftsatanypointinthesplinedomain,i.e.,eachspli nevalueis determinedbyalinearcombinationof 53 neighborcoecientsweightedbythevalue of M Z 2 shifted.Sincethereare 24 typesofdomaintetrahedra, M Z 3 iscomposedof 53 24=1 ; 272 polynomialpieces.Figure 4-5 showsseverallevelsetimagesof M Z 3 Figure4-5.Ray-tracedimagesofseverallevelsetsofthe M Z 3 .Inthebottomimages,a randomcolorisassignedtoeachpolynomialpiece.Theimage sarerenderedby POV-Ray[ 1 ]. 51

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Table4-2.BB-coecients(multipliedby 2 9 3=1 ; 536 )ofthepolynomialpiecewith domainof i 4 =000111 (seeChapter 9 ,Figure 4-4 andFigure 4-6 ). 43210321021010032102101002101001000 0123401230120100123012010012010010000000111122233400001112230001120010 J Z 3 00000000000000011111111112222223334 ( 2 1 1 ) 11 22448816 ( 2 1 0 ) 1427412816162741282024412820328162416 ( 2 1 1 ) 1248161248 ( 2 0 1 ) 141 2724124816816 ( 2 0 0 ) 481628448162844162436162416816284416284424362416243 624363216242416 ( 2 0 1 ) 1427412816161248 ( 2 1 1 ) 1 1 ( 2 1 0 ) 141 1 ( 2 1 1 ) 11 ( 1 2 1 ) 4816 ( 1 2 0 ) 1224488121624488162448816328122416 ( 1 2 1 ) 416 8 ( 1 1 2 ) 416 ( 1 1 1 ) 121620242716202427162024162016243240463240463240324 8647664766496112112144 ( 1 1 0 ) 324460779256769611396116132128144144567696113961201 4014416417696116132144164192128144176144 ( 1 1 1 ) 121620242724324046486476961121441620242732404664761 1216202432406416203216 ( 1 1 2 ) 416 ( 1 0 2 ) 12 24 4881216 ( 1 0 1 ) 324460779232445871324048242816567696113567286485632 961161328810064128144112144 ( 1 0 0 ) 128152176200220152176200220160184204160176144152176 200220176200220184204176160184204184204192160176176 144 ( 1 0 1 ) 324460779256769611396116132128144144324458715672868 810011232404848566424283216 ( 1 0 2 ) 12244881216 ( 1 1 1 ) 12162024278888 16202427888162024881620816 ( 1 1 0 ) 324460779232445871324048242816324458713244563240243 2404832403224282416 ( 1 1 1 ) 1216202427162024271620241620168888888888 ( 1 2 0 ) 12 ( 0 2 1 ) 84816 ( 0 2 0 ) 444328864161281612168864161282416241612824163216122 416 ( 0 2 1 ) 8416 8 ( 0 1 2 ) 8416 ( 0 1 1 ) 323232302732323027322824242016565652465652464840329 68876887664128112112144 ( 0 1 0 ) 128128120107921521441301131601481321601441441521441 301131761601401841641761601481321841641921601441761 44 ( 0 1 1 ) 323232302756565246968876128112144323230275652468876 11232282448406424203216 ( 0 1 2 ) 8416 ( 0 0 2 ) 44432 886416128161216 ( 0 0 1 ) 128128120107921049684716456483228161521441301131121 008664563216014813211210064160144112144 ( 0 0 0 ) 264264256240220264256240220240224204192176144264256 240220256240220224204176240224204224204192192176176 144 ( 0 0 1 ) 128128120107921521441301131601481321601441441049684 711121008611210011264564864566432283216 ( 0 0 2 ) 44432886416128161216 ( 0 1 1 ) 32323230278888 32323027888322824882420816 ( 0 1 0 ) 128128120107921049684716456483228161049684718068564 8402464564848403232282416 ( 0 1 1 ) 3232323027323230273228242420168888888888 ( 0 2 0 ) 44432 ( 1 1 1 ) 1284218421 16842842168484168816 ( 1 1 0 ) 322012743220127322012241616322012732201232202432201 232203224162416 ( 1 1 1 ) 128421168421684168168421842848 ( 1 0 1 ) 322012748421 3220127842322012842416816 ( 1 0 0 ) 128104806044104806044644836322416104806044806044483 62464483648363232242416 ( 1 0 1 ) 3220127432201273220122416168421842848 ( 1 1 1 ) 128421 8421 ( 1 1 0 ) 322012748421 8421 ( 1 1 1 ) 1284218421 ( 2 0 0 ) 4 v 0 =(0 ; 0 ; 0) v 1 =( ; 0 ; 0) v 2 =( ; ; ) v 3 =( ; ; ) Figure4-6.Domaintetrahedronoftype i 4 =000111 (seeFigure 4-4 ). 52

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4.3.3Quasi-Interpolation Thequasi-interpolant Q Z 3 := Q M Z 3 withoptimalapproximationorder 4 for S Z 3 is denedbythefunctional Z 3 f :=( f 5 24 ( D 2 1 + D 2 2 + D 2 3 ) f )( 0 ) (48) =( f 1 24 ( D 2 1 + D 2 2 + D 2 3 +( D 1 + D 2 + D 3 ) 2 +( D 1 + D 2 + D 3 ) 2 +( D 1 D 2 + D 3 ) 2 +( D 1 D 2 + D 3 ) 2 ))( 0 ) =( f 1 24 X 2 Z 3 D 2 f )( 0 ) : (49) Fordiscreteinput f : Z 3 R ,approximatingthedirectionalderivativesbynite dierences, Z 3 ( f ( + j )) 19 12 f ( j ) 1 24 X 2 Z 3 f ( j + )+ f ( j ) ; j 2 Z 3 : Alternatively,wecanapproximateasfollowsleveragingth eform( 48 ): Z 3 ( f ( + j )) 9 4 f ( j ) 5 24 3 X i =1 ( f ( j + e i )+ f ( j e i )) ; j 2 Z 3 : 4.3.4SplineEvaluationAlgorithm4.3.1: EvaluateSplineOf M Z 3 ( a; x ) x NearestCartesianPoint ( x ) i 4 U ( N tZ 3 ( x x )) u ComputeBarycentric ( i 4 ; x ) P P j 2 J Z 3 a ( x + V i 4 j ) B j ( u ) return EvaluateBB ( P; u ) Algorithm 4.3.1 showsthepseudocodetoevaluateasplineinthesplinespace S Z 3 :=span( M Z 3 ( j )) j 2 Z 3 withcoecients a : Z 3 R where 53

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Table4-3.Orthonormaltransformationsfromthedomaintet rahedron i 4 =000111 to eachdomaintetrahedra. i 4 V i 4 i 4 V i 4 000111 I 3 011001[ e 1 e 2 e 3 ] t 100110[ e 1 e 2 e 3 ] t 111000[ e 1 e 2 e 3 ] t 100111[ e 1 e 3 e 2 ] t 111001[ e 1 e 3 e 2 ] t 000110[ e 1 e 3 e 2 ] t 011000[ e 1 e 3 e 2 ] t 001111[ e 2 e 1 e 3 ] t 100010[ e 2 e 1 e 3 ] t 011101[ e 2 e 1 e 3 ] t 110000[ e 2 e 1 e 3 ] t 011111[ e 3 e 1 e 2 ] t 110010[ e 3 e 1 e 2 ] t 001101[ e 3 e 1 e 2 ] t 100000[ e 3 e 1 e 2 ] t 110111[ e 2 e 3 e 1 ] t 000100[ e 2 e 3 e 1 ] t 111011[ e 2 e 3 e 1 ] t 001000[ e 2 e 3 e 1 ] t 111111[ e 3 e 2 e 1 ] t 001100[ e 3 e 2 e 1 ] t 110011[ e 3 e 2 e 1 ] t 000000[ e 3 e 2 e 1 ] t ˆ U ( x ) isthestepfunction(acceptingvectorinput), U ( x ):= ( 1 x 0 0 otherwise ˆ N Z 3 ( 47 )isthesetofnormalvectorsofthenon-axis-alignedknotpl anesinthe Voronoicell V Z 3 :=( :: ) 3 of Z 3 and ˆ B j ( u ) isthepolynomialinBB-formconstructedbytheBB-coecien tsassociated with j 2 J Z 3 AsdescribedinChapter 9 i 4 2f 0 ; 1 g 6 (Section 9.2.1 )uniquelydeterminesoneofthe 24 polynomialpiecesintheVoronoicell(Figure 4-4 ).Duetothesymmetry,theoset valuesin J Z 3 (Table 4-2 )canbetransformedbythecorrespondingorthonormalmatri x inTable 4-3 toproperosets.Foreachinputpoint x ,onepolynomialinBB-formis constructedasalinearcombinationofthepolynomialpiece sweightedbytheinput coecients a .Notethat,whileAlgorithm 4.3.1 iscustomizedtothesplinesof M Z 3 leveragingitssymmetry,Algorithm 9.3.1 canbeappliedtogeneralbox-splinecases. 54

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CHAPTER5 SYMMETRICBOX-SPLINEONTHE A n LATTICE 5.1DenitionandProperties A B C D Figure5-1. A )Coxeterdiagramoftherootsystem(reectiongroup) A 3 B ) decompositionofasphereby 6 hyperplanes(mirrors), C ) 12 rootsassociated withthehyperplanesin B )withcoordinates f e i e j :1 i 6 = j n g and D ) theVoronoicell(rhombicdodecahedron)ofthe A 3 (FCC)lattice.Thethick sphericaltrianglein B )denotestheintersectionofthesphereandthe fundamentalchamberconstructedbyplanesassociatedwith thethicklines. Thethreethickarrowsin C )areassociatedwiththeplanesboundingthe sphericaltriangleemphasizedin B ).Thedihedralanglesofthefundamental chamberin B )are f = 2 ;= 3 ;= 3 g asdenotedbytheCoxeterdiagram A ). Notethatthe 12 rootsof C ),whoselengthsareallthesame,areassociated withthe 12 nearestFCClatticepointsandthe 12 facesoftheVoronoicellin D )intersectthemiddleofeachrootvectorin C )orthogonally.Eachfaceof therhombicdodecahedronisassociatedwithapairofadjace ntspherical trianglesin B ). 55

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e 1 e 2 e 2 e 3 e n 1 e n e n e n +1 Figure5-2.Coxeterdiagramoftherootsystem A n withtheir ( n +1) -dimensional Cartesiancoordinates. TheRootsystem A n .Thenitereectiongroup A n iscomposedof n hyperplanes withtheirdihedralanglesdescribedbytheCoxeterdiagram inFigure 5-2 .Itcanbe formulatedbytherootsystemwithCartesiancoordinatesas follows[ 23 60 ]: A n = e i e j 2 R n +1 :1 i 6 = j n +1 : (51) Figure 5-1 illustrates A 3 .Notethat,whiletherootsin( 51 )are ( n +1) -dimensional, therootsinFigure 5-1 aretransformedtodimensionthree.Ingeneral,Wecanobtai n n -dimensionalrootsof A n byanyorthogonaltransformationthatmaps H n j to R n .Apair ofsuchmatricesare X n :=( A P ( A n ) 1 ) t = A n A P t = A n I n j : H n j R n (52) where A P := 1 n +1 264 ( n +1) I n J n j t 375 ; (seeLemma 10 ) A n := I n + 1 n 1 p n +1 J n (seeLemma 6 ) and A n := I n + 1 n 1 1 p n +1 J n : (seeLemma 11 ) Notethat,while ( X n ) t X n = I n +1 1 n +1 J n +1 6 = I n +1 ; (53) 56

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X n isanorthogonaltransformationthatpreserveslengthsand angles,since J n x =0 for x 2 H n j andhence ( X n x ) ( X n x )= x x : TheRootlattice A n .Therootlattice A n isgeneratedbyalltheintegerlinear combinationsoftherootsof A n A n canbeeitherembeddedin H n j (R n +1 denedas H n j \ R n +1 withthegeneratormatrix,e.g.[ 23 ,page109] A C := 266666666664 11 11 377777777775 2 Z ( n +1) n ; (54) orbeembeddeddirectlyin R n bythegeneratormatrix A n constructedinLemma 6 .In lowdimensions, ˆ A 2 = A 2 isequivalenttothehexagonallatticeand ˆ A 3 = D 3 isequivalenttotheFCClattice. Thebasisof A n canbetakenfroman n -dimensionalequilateralsimplex. Lemma5 (Geometricconstructionof A n in R n +1 ) Let n bean equilateral n dimensionalsimplexoneofwhoseverticesislocatedattheo rigin.Thenthe n edgesof n emanatingfromtheoriginformabasisofalatticeequivalen tto A n Proof. Let U := 266666666664 111 ... ... 11 1 11 11 377777777775 2 Z n n ; hence U 1 = 266666666664 1 11 . . 1 11 377777777775 : 57

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ByLemma 1 A C U = 264 I n j t 375 2 Z ( n +1) n ; (55) alsogenerates A n .Since 8>><>>: k v k 2 = p 2 8 v 2 A C U k v j v k k 2 = p 2 8 v j ; v k 2 A C U ; v j 6 = v k ; thesimplex conv( f 0 g[ S v 2 A C U f v g ) isequilateralhenceequivalenttoany n BasedonLemma 5 ,weconstructasquaregeneratormatrixwhichembeds A n directly in R n .Consideralinearmapthatscalesalongthediagonal j bytransformingapoint x 2 R n accordingto x 7! x + c n ( j x ) j ; where c isthescalingfactor. b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C A Z 2 b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C b C B A +2 Z 2 = A 2 Figure5-3.Geometricconstructionof A n in R n Lemma6 (Geometricconstructionof A n in R n ) A n canbegeneratedby A n := I n + c n n J n 2 R n n with c n := 1 p n +1 : (56) Proof. Anyvector e j e k for j 6 = k isparallelto H n j andhenceitslengthremains p 2 unchangedby A n andregardlessofthedimension n .Toshowthatthe n -dimensional 58

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simplex conv( f A n e j :1 j n g[f 0 g ) is equilateral ,weverifythatthevectors e j satisfy k A n e j k 2 = p A n e j A n e j = s c n n +1 2 +( n 1) c n 2 n 2 = p 2 : (57) TheclaimfollowsbyLemma 5 Thetwodierentchoicesof c n produceequivalentresultswithrespectto H n j because I n J n =n projects e j on H n 1 j Thesymmetrygroupof A n consistsofthesymmetricgroupofall ( n +1)! permutations ofitscoordinatesandthegroupofchangingthesignofallth ecoordinates,henceitsorder is ( n +1)!2 [ 23 ].Indimensiontwo,thiscanbeobservedbythe 12 -foldsymmetryofthe hexagon.Indimensionthree,thiscanbeobservedbyconside ringthehalfofthespherical triangleinFigure 5-1B where 48 ofitscopies(orthogonaltransformations)llthewhole sphere. TheBox-spline M A n .The n ( n +1) = 2 -directionbox-splineonthe A n latticeis denedbythedirectionstothe n ( n +1) nearestlatticepointsof A n transformedby X n : A n := [ 1 i
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arearrangedsothatrst k elementsareidenticaland k isthemaximumnumberofthe identicalelementsof v .Notethat k
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for 1 i><>>: e i ;j = n +1 e i e j ;j
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andthegeneratormatrix(Lemma 6 ) A 2 := 1 2 264 1 p 3 1 p 3 1 p 31 p 3 375 : Thebox-spline M A 2 denedby A 2 and A 2 isequivalenttothe 3 -directionlinear box-splineonthehexagonallattice. 5.3 C 1 ReconstructionontheFCCLattice Weintroduce C 1 reconstructionontheFCClatticebythesplinedenedbya weightedsumoftheshiftsofthebox-spline M fcc := M A 3 .Werstanalyze M fcc then deriveitspolynomialpiecesexplicitlytoallowforecien tuse. 5.3.1The 6 -DirectionTrivariateBox-SplineontheFCCLattice A B Figure5-4. A )Directionsand B )support(truncatedoctahedron)of M fcc Indimensionthree,withtheorthogonalmatrix( 52 ) X 3 := 1 2 266664 1 1 11 11 11 1 111 377775 ; 62

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wegetthedirectionmatrix( 58 ) fcc := A 3 = [ 1 i
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eventhen j isontheFCClattice(Figure 5-5A ).Otherwise,wedeterminewhichof 6 Voronoicell j belongsto(Figure 5-5B ). b b A j 2 M fcc Z 3 b b B j = 2 M fcc Z 3 Figure5-5. A )RegionwheremappingtothenearestCartesianlatticepoin t(green)yields thenearestFCClatticepoint(red); B )Regionrequiringfurtherwork (Algorithm 5.3.1 ). Algorithm5.3.1: FindNearestFCC ( x 2 R 3 ) j NearestCartesianlatticepointfrom x if j isnotanFCClatticepoint then 8>>>><>>>>: e x x j k argmax j j e x ( j ) j j ( k ) j ( k )+sign( e x ( k )) return ( j ) 5.3.3Quasi-Interpolation Theoptimalapproximationorderofasplinein S fcc is m ( fcc )+1=3 (seeSection 3.6 andSection 5.1 )anditcanbeachievedbythefollowingquasi-interpolant. Lemma9 (quasi-interpolant) Aquasi-interpolant Q fcc for S fcc reproducingquadratic polynomialsisdenedbythefunctional fcc f := f ( 0 ) 1 24 X 2 fcc ( D 2 f )( 0 ) : (511) 64

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Proof. Let e fcc := M 1 fcc fcc = 1 2 266664 1111 11 11 1 377775 266664 1100 1 1 10 110 1 0 1 1 1 10 377775 = 266664 1 1 1000 100 1 10 01010 1 377775 : (512) Thequasi-interpolantforthesplinespace span M c e fcc ( j ) j 2 Z 3 isdenedbythefunctional e fcc f := f ( 0 ) 1 8 (( D 2 1 + D 2 2 + D 2 3 ) f )( 0 ) + 1 12 (( D 1 D 2 + D 2 D 3 + D 3 D 1 ) f )( 0 ) = f ( 0 ) 1 24 ((( D 1 ) 2 +( D 2 ) 2 +( D 3 ) 2 +( D 1 + D 2 ) 2 +( D 2 + D 3 ) 2 +( D 3 D 1 ) 2 ) f )( 0 ) = f ( 0 ) 1 24 X e 2 e fcc ( D 2 e f )( 0 ) : ByLemma 2 ,thequasi-interpolant Q fcc for S fcc canbeobtainedfrom e fcc byreplacing D e with D M fcc e andthisyieldstheclaim. Foradiscreteinput f : M fcc Z 3 R fcc f 3 2 f ( 0 ) 1 24 X 2 fcc ( f ( )+ f ( )) (513) whichcorrespondsnicelytoaveragingwithintheFCClattic e. 65

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5.3.4ThePolynomialPiecesandEvaluationoftheSpline Themeshgeneratedby M fcc is( 317 ) fcc := [ H 2 H fcc + M fcc Z 3 ; where H fcc istheplanesspannedbythecolumnsof fcc whoseplaneequationsare 266664 10011 1 1 0101 11 1 0011111 377775 x = 0001111 : Amongthe 7 knotplanesin H fcc ,theshiftsof 4 non-axis-alignedplanesdecompose R 3 into octet-truss [ 49 ]structures(Figure 5-6 ),andtheshiftsof 3 remainingaxis-aligned knotplanesdecomposeeachoctahedronfurtherintoeightco ngruenttetrahedrasuchas theoneshowninFigure 5-8A .Notethattheequilateraltetrahedraoftheoctet-truss Figure5-6.Theoctet-trussstructure:layersofoctahedra inaneggcartonpatternwith twotypesoftetrahedrallinginthevoids. (equilateraltetrahedronwithblueedgesinFigure 5-8B )arenotsplitbyknotplanes andhenceareeachthedomainofasinglepolynomialpiece,no toffour.Shiftingthe truncatedoctahedron(Figure 5-4B )thatrepresentsthesupportof M fcc ,wendthat twodierenttypes( J and J + inTable 5-2 )of 16 dierentshiftsof M fcc overlapatany genericlocation.Sincethereare 10 typesoftetrahedra( 8 composinganoctahedronand 2 66

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equilateraltetrahedrawithdierentorientation(Figure 5-6 )), M fcc iscomposedofoverall 10 16=160 polynomialpieces.Figure 5-7 showsseverallevelsetimagesof M fcc Figure5-7.Ray-tracedimagesofseverallevelsetsof M fcc .Inthebottomimages,a randomcolorisassignedtoeachpolynomialpiece.Theimage sarerenderedby POV-Ray[ 1 ]. Toevaluateaspline s in S fcc at x 2 R 3 viathepolynomialform,let x bethenearest FCClatticepointfrom x .Sincethesetupissymmetricwithrespecttothethreeaxes, we needonlyconsideroneoctant,e.g., (+ ; + ; +) .Therearetwotypesoftetrahedra, and + ,intheVoronoicellof x (Figure 5-8 ).Correspondingly,for e x := x x inthe (+ ; + ; +) Table5-1.Twotypesofdomaintetrahedraforthepolynomial piecesoftheFCCsplines. RefertotheorangetetrahedroninFigure 5-8A andtheequilateraltetrahedron withblueedgeinFigure 5-8B foreachtype.Notethattheorderofvertices shouldbeconsistentwiththemulti-index inTable 5-2 + v 0 0,0,01,1,1 v 1 1,0,00,1,0 v 2 0,1,01,0,0 v 3 0,0,10,0,1 67

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octant, s ( x ):= X k 2 M fcc Z 3 a ( k ) M fcc ( x k ) (514) = 8>><>>: P j 2 J a ( x + j ) B j ( ( e x )) if k e x k 1 1( type ) P j 2 J + a ( x + j ) B j + ( ( e x )) if k e x k 1 > 1( type +) where,for denotingeither or + ˆ J := f j 2 Z 3 : M fcc ( e x j ) 6 =0 ; e x interiorof g ,thesetof16FCCshiftsofthe box-spline,alsocalled`stencil'(Figure 5-8 androwindicesinTable 5-2 ), ˆ B j ( u ):= P j j =3 c j ( ) b ( u ) isthepolynomialinBernstein-Bezier(BB)form associatedwiththeshift j oftype viaitsBB-coecients f c j ( ) g listedin Table 5-2 and ˆ ( e x ) thebarycentriccoordinatesof e x withrespecttothedomaintetrahedron (Figure 5-8 and( 5-1 )). TheBB-coecientsinTable 5-2 havebeencomputedwiththetechniquein Chapter 9 b ( 1 ; 1 ; 0) b (0 ; 0 ; 0) b (1 ; 0 ; 1) b (0 ; 1 ; 1) b ( 1 ; 0 ; 1) b (0 ; 1 ; 1) b ( 1 ; 1 ; 0) b (1 ; 1 ; 0) b (1 ; 1 ; 0) b (1 ; 0 ; 1) b (0 ; 1 ; 1) b ( 1 ; 0 ; 1) b (0 ; 1 ; 1) b (2 ; 0 ; 0) b (0 ; 2 ; 0) b (0 ; 0 ; 2) A : k e x k 1 1 b ( 1 ; 1 ; 0) b (0 ; 0 ; 0) b (0 ; 1 ; 1) b (1 ; 0 ; 1) b ( 1 ; 0 ; 1) b (0 ; 1 ; 1) b (1 ; 1 ; 0) b (1 ; 0 ; 1) b (0 ; 1 ; 1) b (1 ; 1 ; 0) b (1 ; 2 ; 1) b (1 ; 1 ; 2) b (2 ; 1 ; 1) b (2 ; 0 ; 0) b (0 ; 2 ; 0) b (0 ; 0 ; 2) B + : k e x k 1 > 1 Figure5-8.Twotypesoftetrahedra: A ) and B ) + .16latticepoints,ofwhich13are sharedbetweenthetwotypes,determinethepolynomialpiec eon 2f ; + g .Allotherpolynomialpiecescanbeobtainedfrom and + by symmetry.Theblacklinesoftherhombicdodecahedrondelin eatetheVoronoi celloftheFCClattice.The16FCCpointsmarkedineachgure ,areusedas rowindicestoselectBB-formcoecientsoftheresultingsp lineinTable 5-2 68

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Table5-2.BB-coecients(scaledupby24)ofthepolynomial piecesoftype J and J + ThecolumnsindextheBB-coecientsbytheirbarycentricco ordinates(scaled upby3)withrespecttotheverticesin( 5-1 )andtherowindicesaretheshifts ofthebox-spline M fcc tothepositionsmarkedinFigure 5-8 32102101002101001000 012301201001201001000000111223000112001000000000001111112223 J 0,0,012128412128884121281212888840,1,111224442244844841,0,112441242482448441,1,01244248484124244 -1,0,11122444-1,1,01244124 0,-1,111224440,1,-111224441,-1,012441241,0,-112441240,-1,-111 -1,0,-111-1,-1,01 1 2,0,040,2,0 4 0,0,2 4 J + 0,0,0444848448412848840,1,1488444481284488441,0,1448484484124884841,1,0488481288844444441,1,24 4 1,2,1442,1,1440,2,044 -1,1,044 0,1,-1442,0,0441,-1,0 44 1,0,-1440,0,2 44 -1,0,1 44 0,-1,1 44 69

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Algorithm 5.3.2 showsthepseudocodeforevaluatingasplinein S fcc at x .Werst transform x tothelocalcoordinate e x Dependingonthetype, or + ,andleveraging symmetry,wesumthe 16 polynomials f B j : j 2 J g denedinTable 5-2 weightedbytheir coecients a ( j ) Algorithm5.3.2: EvaluateFCCspline ( x 2 R 3 ) x nearestFCClatticepointfrom x e x x x sign( e x ) if k e x k 1 1 then y P j 2 J a ( x + j ) B j ( ( e x )) else y P j 2 J + a ( x + j ) B j + ( ( e x )) return ( y ) 5.3.5Reconstruction Therearethreeapproachestoreconstructingafunctionint ermsof M fcc fromdataon theFCClattice: ˆ approximation :interpretdatapointsascontrolpointsoftheshiftsof M fcc ˆ quasi-interpolation :pre-processthedataaccordingto( 513 ),or ˆ interpolation :computethecoecientsbysolvingalinearsystem. Let X j 2 M fcc Z 3 M fcc ( j ) a ( j ) 2 S fcc bethereconstructedspline.Interpretingdatapoints f ( j ) ascontrolpoints,i.e.setting a ( j ):= f ( j ) ,isclearlythequickestoptionbutresultsinadditionalsm oothingandshifts ofthelevelsetsthatareusuallynotacceptable.Interpola tionontheotherextreme,while 70

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alwayspossiblebecausethelinearsystemsforinterpolati onareinvertible(Lemma 8 ), hasaprohibitivecostforlargedatasets.And,whilethedat aarenowexactlymatched, therationalefordoingsoisweaksincethedataaretypicall ytheresultoflteringand discretization;sopoint-wiseexactreplicationbyitself sayslittleaboutthedelityof thereconstructedfunction.Infact,splineinterpolation iswell-knowntosuerfrom Gibbsphenomenon[ 68 ]wheregradientschange.Ourmethodofchoiceistherefore quasi-interpolationsinceitbalancescostandapproximat ionquality:theapproximation powermatchesthatofinterpolation,butwithasmall,local footprint( 513 ). Choosingquasi-interpolation,thecoecientsinthedeni tionofthereconstructed functionare(see( 513 )) a ( j ):= fcc ( f ( + j ))= 3 2 f ( j ) 1 24 X 2 fcc ( f ( j + )+ f ( j )) : Thepolynomialpiecesofthereconstructedfunctioncanbed enedbyTable 5-2 .The functioncanthenbeevaluated,forexample,usingdeCastel jau'salgorithm[ 30 43 ]and exactlevelsetscanberenderedbyray-tracing(seeSection 5.3.6 ). 5.3.6Results Volumetricdataaretraditionallysampledandreconstruct edontheCartesian lattice.Asuitableanti-aliasinglterisappliedtobandlimitthespectrumofthesampled datawithintheNyquistregion,whichistheVoronoicelloft heduallattice.TheFCC latticecellisclearlydierentfromtheCartesiantherefo retestingFCCsamplingand reconstructiononreal-lifedatasetsisonlypossibleonce trueFCCsamplingdevicesare available.WeconstructedcomparableFCCandCartesiandat asetsbysub-samplinga high-densityCartesiandatasetintocoarserFCCandCartes iandatasetswithcomparable resolutions. WeexperimentedwiththeMarschner-Lobbbenchmarkdataset (Figure 5-9B ) forevaluatingtheaccuracyofreconstructionalgorithms[ 68 ].Figure 5-11 compares threereconstructionmethodsondierentlatticesandlte rsdescribedinTable 5-4 .It 71

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A B Figure5-9.Benchmarkdataset: A )Carpdatasetattheresolutionof 256 256 512 (density 1 )reconstructedbyquasi-interpolatingtri-quadraticB-s pline (Courtesyof[ 2 ]).Thisrelativelyhigh-resolutiondatasetisusedasthe standard`truth'functionforourexperimentsatlowresolu tions. B ) Marschner-Lobbtestfunction. demonstratesreducedaliasinginFCCmethodcomparedtothe commonly-usedCartesian method.Inparticular,thecircularrimsarealmostperfect lyreconstructedintheFCC methodwhiletheCartesianmethodshowssevereoscillation s.ComparedwiththeBCC method,FCCmethodcapturesthe`valley'oftheoriginalfun ctionbetterthantheBCC methodwherethevalleysaresmoothedoutduetothehigherde greelter.Thiscan bealsojustiedbythehigher smoothingmetric [ 68 ]ofthereconstructionlterofthe BCCmethod(Table 5-4 ).ButtheFCCmethodshowshigh-frequencyartifactsonthe isosurfacewhichdonotappearintheBCCmethod.Inanycase, itisobviousthatboth reconstructionmethodsshowsuperiorresultsthantheCart esianmethod.Figure 5-10 showsanotherreconstructionofMLdatasetandtheircorres pondingerrorimages.The 72

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angularerror,cappedat 0 : 2 radians,ismappedtowhiteand 0 errortoblack.Increased blacknessintheFCCmethodimpliesincreasedaccuracy. A B C D Figure5-10.Marschner-Lobbtestimages. A ) C ) C 1 reconstructiononCartesianwith 41 41 41=68 ; 921 samplesand B ) D ) C 1 reconstructiononFCCwith 25 25 25 4=62 ; 500 samples.(bottom)theangularerror,cappedat 0 : 2 radians,ismappedtowhiteand 0 errortoblack.Increasedblacknessfrom lefttorightimpliesincreasedaccuracy. Forourexperiments,weusedtheCarpshdatasetoforiginal resolutionof 256 256 512 (density 1 ,seeFigure 5-9A ).Thisdatasetwassub-sampledto 100 100 200 resolutionusingarationalsub-samplingscheme.Forthiss ub-sampling,aproper up-samplingoperationwasappliedbyzero-paddinginthefr equencydomainsothat adown-samplingoperationproducesthetargetresolutiono f 100 .Weperformedthe equivalentrationalsub-samplingtoaCartesianvolumeofr esolution 126 126 126 to obtainthematchingresolutionontheFCClattice.Thenthis Cartesianvolumeissimply down-sampledontoaFCClatticebyretainingCartesianpoin tswhosesumofcoordinates areeven,halvingthenumberofsamples.Hencetheresulting FCClatticeisofresolution 73

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Figure5-11.Comparisonofthreereconstructionmethodswi thsamplingdensities:top row 0 : 05 3 ,middlerow 0 : 06 3 ,andbottomrow 0 : 07 3 .(left) C 1 onFCC (copperwithredcrosssection)and C 2 onBCC(silverwithgreencross section)juxtaposedtogether.(middle)theviewpointands licingplanrotated by45degrees.(right)showsthecorresponding C 1 ontheCartesianlattice. 63 63 126 4 whichisveryclosetotheCartesianvolumeTheimagesareren deredwith (customized)POV-Ray[ 1 ]. Figure 5-12 compares C 1 reconstructionontheFCCandCartesianlattices(see Table 5-4 ).Ascanbeseen,theFCCmethodcapturesthedetailedfeatur esofthe datasetbetterthantheCartesianmethod.Mostoftheribsin theCartesiandataset aredisconnectedwhileontheFCCdatasetaremostlyconnect ed.Alsostaircaseartifacts ontheCartesianmethodismoresevere(see,e.g.,theribsar eaofFigure 5-12 ).Itis clearthattheCartesianmethodhasdicultytocapturethef eatureswhicharenot axis-aligned.5.3.7ComputationalCost AnotheradvantageofFCCmethodistheecientevaluation.S incethenumberof neighborsamples(stencilsize)requiredtoevaluateapoin tissmall( 16 comparedto 27 fortheCartesianmethod),ittakeslesstimetoevaluatethe reconstructedeldifthe 74

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A B Figure5-12.Carpdatasetatroughly 6% oforiginalresolution.Reconstructedby A ) C 1 onCartesian( 100 100 200=2 ; 000 ; 000 resolution)and B ) C 1 onFCC ( 63 63 126 4=2 ; 000 ; 376 resolution). memoryaccessintroducesrelativelybigoverhead.Thisove rheadbecomesdominant, duetocachemiss,asthedatasetsizegrows.Forexample,whi leittakes 125 secondsto generateFigure 5-10A ,ittakesonly 98 seconds( 72% )togenerateFigure 5-10B .These imageswererenderedattheresolutionof 500 500 pixels.Table 5-3 comparesthetime forray-castingtheimagesinthispaper.Theseexperiments wereperformedonadualcore Intel 2 : 13 GHzmachinewith 2 GBmemory. Table5-3.Renderingtime(inseconds)togenerateray-cast edimagesinFigure 5-10 DatasetCartesianFCCRatio Marschner-Lobb1359872% Carp-Fish51535869% 75

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5.3.8ReconstructionProperties OurproposedboxsplinereconstructionontheFCClatticeha sthesameapproximation orderandcontinuityasthatofthetri-quadraticB-splineo ntheCartesianlattice. However,asimilarreconstructionontheBCClatticeisnoty etknown.Theonlyavailable symmetricbox-splinereconstructionlterontheBCClatti ceisa 8 -directionbox-spline [ 42 ].Table 5-4 summarizesthereconstructionpropertiesoftheseschemes Smoothingandpost-aliasingmetricshavebeenproposedtoc omparereconstruction schemes[ 68 ].Wehavealsonumericallyevaluatedthesemetricsforourr econstruction schemewhicharetabulatedinTable 5-4 .Whilebothsmoothingandpost-aliasing measuresarerelativelyclose,theFCCboxsplinelterhasl esssmoothingartifactsbut morepost-aliasingartifacts,whencomparedtotri-quadra ticB-spline. Table5-4.ReconstructionpropertiesofFCCboxspline,Car tesiantri-quadraticB-spline, andtheBCCboxspline. C 1 onFCC C 1 on Z 3 C 2 onBCC lter M fcc M 333 8-dir.box-spline degree365 order334 stencilsize162732 smoothing0.8043810.8418650.852867 post-aliasing0.01148940.008257020.00401669 5.3.9DiscussionandExtensions Despitetheirtheoreticaladvantage,non-Cartesianlatti ceshaveonlyhadalimited impactonpracticalapplications.Ononehand,therearenoa cquisitionstrategiesthat takeadvantageofthesuperiorsamplingonoptimalnon-Cart esianlattices;ontheother hand,majorsignalprocessingtools,suchasreconstructio n,toprocessandanalyzethe resultingdatahavebeenmissing.Hereweaddressthesecond issue,andtherebymotivate engineeringresearchintotherst,byprovidinganecient reconstructionalgorithm fordataontheFCClattice.Tothisend,weintroducedthe6-d irectionbox-spline M fcc 76

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deriveditssimplequasi-interpolantformulaandshowedfa standexactevaluationviain piecewisepolynomialform. Inthiswork,wehavenotyettakenfulladvantageofthelowal gebraicdegreeofthe levelsetsof M fcc .Inprinciple,wecanparametrizetheisosurfacesofthetri variatecubic polynomialpiecesofthesplinebasedon M fcc [ 4 ]andthenexplicitlywriteoutallray intersections.(Bycomparison,theray-isosurfaceinters ectionfor M 333 cangenericallyonly becomputednumerically.)However,POV-Rayallowsusonlyt oprovidetheevaluatorof animplicitfunctionandnottheexactlocationoftheray-le velintersection.Therefore, weplantoexploitthislowpolynomial-degreepropertytoex peditetheray-intersection operations. 77

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CHAPTER6 SYMMETRICBOX-SPLINEONTHE A n LATTICE 6.1DenitionandProperties TheRootlattice A n .The A n lattice,thedualof A n lattice,canbeembeddedin thehyperplane H n j bythenon-squaregeneratormatrix[ 23 ] A C := 266666666664 1 1 n= ( n +1) 11 = ( n +1) ... 11 = ( n +1) 1 = ( n +1) 377777777775 = 266664 j t n= ( n +1) I n 1 j = ( n +1) 0 t 1 = ( n +1) 377775 2 R ( n +1) n : (61) Thesamelatticecanbealternativelyconstructedbyorthog onallyprojectingthe ( n + 1) -dimensionalCartesianlatticealongthediagonaldirecti on. Lemma10 (Geometricconstructionof A n in R n +1 ) A n canbegeneratedbythenoninvertibleelementarymatrix P n +1 := I n +1 1 n +1 J n +1 ; (62) theorthogonalprojectionofthe ( n +1) -dimensionalCartesianlattice Z n +1 alongthe diagonaldirection j Proof. For U := 264 0 I n 1 j t 375 2 Z ( n +1) ( n +1) hence U 1 = 264 j t 1 I n 0 375 ; weverifythat A C U = A P = 1 n +1 264 ( n +1) I n J n j t 375 2 R ( n +1) n ; (63) where A P isthematrixoftherst n columnsof P n +1 .Thelastcolumnof P n +1 isan integerlinearcombinationoftherst n columns, A P .ByLemma 1 ,theclaimfollows. 78

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Asinthecaseofthe A n lattice,wecanconstructasquaregeneratormatrixfor A n by diagonalscaling. b C b C b C b C b C b C b C b C b C A Z 2 b C b C b C b C b C b C b C b C b C B A + 2 Z 2 = A 2 Figure6-1.Geometricconstructionof A n in R n Lemma11 (Geometricconstructionof A n in R n ) A n canbegeneratedby A n := I n + c n n J n 2 R n n with c n = 1 1 p n +1 : (64) Proof. Since 1= p n +1 1 p n +1 = c n +1 c n +1 = c n c n + c n + c n +1 ; c n c n + c n + c n =0 andhence ( A n ) t A n = I n +( c n c n + c n + c n ) J n = I n : As A n c n resultinequivalenttransformationswithrespectto H n j Underthediagonalscaling A n ,thelengthof j becomesthesameasthoseofthe transformedunitvectors(Figure 6-1 ): j A n j j = j A n e j j ; 8 1 j n: (65) Examplesinlowdimensionsare ˆ A 2 = A 2 isequivalenttothehexagonallatticeand ˆ A 3 = D 3 isequivalenttotheBCClattice. 79

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Forexample,for n =2 A 2 := 1 2 264 1 1 = p 3 1 1 = p 3 1 1 = p 31 1 = p 3 375 andfor n =3 ,thetwochoicesare A + 3 := 1 6 266664 5 1 1 15 1 1 15 377775 and A 3 := 1 2 266664 1 1 1 11 1 1 11 377775 : A n istheoptimalsamplinglatticeindimensionstwoandthree[ 38 40 42 64 74 75 90 ]. Indimensionshigherthanthree,Figure 3-8 showsthat A n packsspheresbetterthanthe Cartesianlattice,making A n abettersamplinglatticethan Z n TheBox-spline M A n .Onthelattice A C Z n = A P Z n = A n embeddedin R n +1 ,there A B Figure6-2. A )Directionsofthe 4 -directiontrivariatebox-splineand B )thesupport (rhombicdodecahedron). are 2( n +1) nearestneighborlatticepointswiththeirCartesiancoord inates f ( e j 1 n +1 j ) 2 H n j :1 j n +1 g : (66) The ( n +1) -directionbox-spline M A n onthe A n latticeisconstructedbythedirectionsto the 2( n +1) nearestlatticepoints.AsinSection 5.1 ,wetransformthedirectionsof( 66 ) 80

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to R n by X n :( X n 2 R n ( n +1) ,and I n ; J n ; A n 2 R n n ) T 1 := X n I n +1 1 n +1 J n +1 = A n I n j 264 I n J n = ( n +1) j = ( n +1) j t = ( n +1) n= ( n +1) 375 = A n I n j : Thenotation T 1 willbecomeclearinthenextsectionwhereitwillbederived froma dierentapproach. Withthesquaregeneratormatrix A n ,the ( n +1) -direction(linear)box-splineonthe A n latticeisdenedas M A n := M 1 := j det A n j M T 1 ( + X 2 T 1 = 2)= j det A n j M T 1 : Inthenextsection,weinvestigatethefamilyofbox-spline s M r constructedby T r := r [ j =1 T 1 ; an r -foldrepetitionof T 1 6.2TheSymmetricBox-Splineonthe A n Lattice Thebox-splinefamily M r (Section 3.6.1 )andthe A n latticehaveacloserelationship thatbecomesapparentwhenwecomparethesplinespaces S A P P n +1 :=span( j det A P j M P n +1 ( j )) j 2 A P Z n and S T 1 :=span( M 1 ( j )) j 2 Z n where j det A P j = p A P t A P =1 = p n +1 .Since P n +1 = I n +1 J n +1 = ( n +1)= A P T 1 ,the twospacesarerelatedby X j 2 A P Z n j det A P j M P n +1 ( j ) a ( j )= X k 2 Z n M 1 ( A P 1 k ) a ( A P k ) 81

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where A P 1 isdenedas( 39 ).(Thisissimilarto( 327 )but A P isnotasquarematrix.) Thesplinespace S T 1 ,thoughwidelyused,correspondstotheCartesiandomainla ttice thathaspoorersamplingeciencycomparedtootherrootlat tices(Section 3.4 ). Moreover,while M P n +1 issymmetricaswillbeshownlater, M 1 isnot(Figure 6-3 and Figure 6-4 ),since A P : R n R n +1 isnotanorthonormaltransformation: A P t A P = I n 1 n +1 J n 6 = I n : Therefore M 1 isabiased(non-symmetric)reconstructionlter. Table6-1.Box-splinespacesrelatedbychangeofvariables splinespace S T 1 S A P P n +1 S A n T 1 M 1 on Z n j det A P j M P n +1 on A P Z n M 1 on A n Z n symmetricbox-spline XX domainlatticeis A n XX domainis R n X X Bycontrast,thedomainlatticeofthebox-spline M P n +1 is A n ,anecientsampling lattice,and M P n +1 issymmetricsincethedirections,i.e.thecolumnsof P n +1 ,are ˆisometric:theyhavethesamelengthsandˆisotropic:theinnerproduct(hencetheangle)betweenany twodirectionsisthe same. Thesupportof M P n +1 inheritsthesymmetryof A n ,or A n ,sincethedirectionsin P n +1 are takenfromthedirectionsfromtheorigintothenearestlatt icepoints.Thereare 2( n +1) nearestlatticepointsfromeach,equaltothe kissingnumber of A n [ 23 ]. Theshiftsof M P n +1 arethebox-splinesobtainedbyprojectingaslabasshown inFigure 1-2 .Byinterposingthelattice A n onthehyperplane H n j (R n +1 ,weseeit partitiontheslabthatservesasanalternativepreimageof (theshiftsof) supp M 1 besides thecube 2 R n +1 thatdenesthebox-spline. 82

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Lemma12. Let :=[0 :: 1] n +1 and P n +1 := I n +1 J n +1 = ( n +1) .Thepreimageof supp M 1 withrespecttothemap T 1 decomposesinto P n +1 ( H n j (R n +1 and ker T 1 =span( j ) : T 1 1 ( supp M 1 )= P n +1 span( j ) : Therefore T 1 = supp M 1 = T 1 ( P n +1 ) Proof. Recallthat A P iscomposedoftherst n columnsof P n +1 (Lemma 10 ).By( 332 ) and( 39 ), T t1 ( T 1 T t1 ) 1 = 1 n +1 264 ( n +1) I n J n j t 375 = A P 2 R ( n +1) n ; andtherefore T 1 1 f x g = A P x +span( j ) ; x 2 R n ; j 2 R n +1 : (67) Bythedenitionofbox-splines, supp M 1 = T 1 (R n hence T 1 1 ( T 1 )= A P T 1 span( j )= P n +1 span( j ) : P n +1 issymmetricsincethedirections,i.e.thecolumnsof P n +1 ,areisometricand isotropic.However,thedomainembeddedinthehyperplane H n j makes M P n +1 dicult touseinapplications.Thiscanberesolvedbytransforming thedirections T r withthe squaregeneratormatrix A n andinterpretingthecolumnsofthematrix T r := A n T r as directionvectorsin R n Lemma13. T 1 isisometricandisotropic. Proof. Since( 65 )impliesisometry,weneedonlyverifyisotropy, A n ( j ) A n e j = 1 n +1 ; 8 e j and A n e k A n e j = 1 n +1 ; 8 e j 6 = e k : 83

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=[0 :: 1] n +1 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b P n +1 ( H n j (R n +1 b b b b b b b b b b b b b b b b b b b b b b b b b supp M 1 = T 1 (R n T 1 = A P 1 P n +1 P n +1 A P Figure6-3.Symmetryofthesupportof M P n +1 andasymmetryofthesupportof M 1 A M 1 on Z R 2 H 1 j B j det A P j M P 2 on H 1 j (R 2 C M 1 on A 1 Z(R b b b b b b b b b b b b b b b b b b b b b b b b b D M 1 on Z 2 b b b b b b b b b b b b b b b b b b b b b b b b b E j det A P j M P 3 on H 2 j (R 3 b b b b b b b b b b b b b b b b b b b b b b b b b F M 1 on A 2 Z 2 (R 2 Figure6-4.(top)Shiftsofunivariatelinearbox-splinesa nd(bottom)shiftsof(the support)ofbivariatelinearbox-splines. Therefore M T r issymmetricand A n Z n = A n canserveasadomainlatticeforthe box-splinefamily(Figure 6-4C and 6-4F ) M r := j det A n j M T r = M r ( A n ) 1 : 84

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Since,incontrastto( 67 )for M 1 ( A P ( A n ) 1 ) t ( A P ( A n ) 1 )= I n ; (68) hencethesymmetryof P n +1 ispreservedwhencomputingthepreimage, ( T 1 ) 1 f x g = A P ( A n ) 1 f x g +ker T 1 : (69) By( 327 ),theweightedsumoftheshiftsof M r on A n Z n = A n canbeexpressedas X j 2 A Z n M r ( j ) a ( j )= X j 2 Z n M r ( A n 1 j ) a ( A n j ) : (610) Therefore M r inheritsmostpropertiesof M r Lemma14. M r iscentered. Proof. By( 311 ), M r c := M r 0@ + 1 2 X 2 T r 1A = M r (611) since P 2 T r = 0 Lemma15. M T r = M T r Proof. By( 313 ), F M T r ( )= Y 2 T r sinc( ) and F M T r ( )= Y 2 T r sinc( )= Y 2 T r sinc( )= Y 2 T r sinc( ) because sinc isanevenfunction.TheclaimholdssincetheFouriertransf ormisinvertible. Lemma16 (evenfunction) M r = M r ( ) 85

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Proof. By( 325 )andLemma 15 M T r = j det( I n ) j M T r ( I n )= M T r ( ) : (612) Lemma17 (basisfunctions) Thesequence ( M r ( j )) j 2 A n Z n islinearlyindependent. Proof. Sinceany n directionsin T 1 span R n det Z = 1 ; 8 Z 2 [ 2 T 1 T 1 n = B ( T 1 )= B ( T r ) andthesequence ( M r ( j )) j 2 Z n islinearlyindependent.Theclaimfollowssinceby ( 610 ),theshiftsof M r ontheintegergridandtheshiftsof M r on A n Z n arerelatedby aninvertibleanechangeofvariables. M r ,hence M r ,isapiecewisepolynomialof(total)degreelessthanorequ alto ( n +1) r n Lemma18 (polynomialreproduction) Themap M r 0 ( 320 ) mapsthepolynomialspace m ( T r ) ( 310 ) ontoitselfwhere m ( T r )=2 r 1 : Proof. Dueto( 610 ), m ( T r )= m ( T r ) .For M 1 ,wehavetoremoveatleast 2 directions sothattheremainingdirectionsin T 1 nolongerspan R n ,hence m ( T 1 )=(( n +1) ( n 1)) 1=2 1=1 : Inthesameway,atmost r ( n 1) directionsin T r spanahyperplane,therefore m ( T r )=( r ( n +1) r ( n 1)) 1=2 r 1 : Notethat m ( T r ) doesnotdependonthedimension n 86

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Also,sincetheknotplanesgeneratedby T r arethoseof ( T r ) underinvertible lineartransformation,themeshinheritsthetopologyofth e ( n +1) -directionalmesh. Lemma19. Thereare n ( n +1) = 2 non-parallelplanesin H ( T r ) Proof. Thereare n planesgeneratedbythe n unitvectorsin I n and n n 2 additional non-parallelplanesarespannedbythediagonaldirection j and n 2 additionalunit vectorsyieldingatotalof n + n n 2 = n + 1 2 n ( n 1)= 1 2 n ( n +1) non-parallelplanesin H ( T r ) Next,wecharacterizethepartitionof R n intosimplicesbytheknotplanesin ( T r ) Figure6-5.Kuhntriangulationfor n =3 Lemma20 (Partitionof ( n +1) -directionalmesh=Kuhntriangulation) Let S n bethe setofallthepermutationsof f 1 ; ;n g .Theknotplanesin H ( T r ) partitiontheunitcube into n simplices(Figure 6-5 ) :=conv( V ) ;V := f 0 g[f n [ i =1 i X j =1 e ( j ) g ; 2 S n : (613) Thepartition f g 2 S n iscalled Freudenthaltriangulation [ 47 ]or Kuhntriangulation [ 63 ,Lemma1],[ 66 ,page140]and[ 3 8 9 ]. Proof. Recallthat T 1 = I n j .Allplaneswithnormaldirection e j e k j 6 = k ,intersect theinteriorof andaregeneratedby T 1 nf e j ; e k g i.e.,asknotplanesof M 1 generatedby n 1 vectorsincluding j .Unlesstwovertices v j ; v k arebothin V forsomepermutation 87

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,thereexistindices and sothat v j ( )=1 ; v k ( )=0 and v j ( )=0 ; v k ( )=1 andhencetheknotplanewithnormal e e separatesthem, ( e e ) v j =1 > 0 and ( e e ) v k = 1 < 0 : (614) Conversely,sinceknotplanesexcluding j areaxis-aligned,neithertheynortheirshifts on Z n intersecttheinterioroftheunitcube .Itremainstoshowthatnoshiftsofthe knotplaneswithnormal e e separateverticesofasimplex forthesamexed permutation .Sinceshiftsby j 2 Z n withintheknotplane,i.e. j ( e e )= j ( ) j ( )=0 ,resultin ( e e ) ( x j )=0 ; wecanassumethat j ( e e )= j ( ) j ( ) > 0 Then,forall v 2f 0 ; 1 g n ( e e ) ( v j )= 8>>>>>><>>>>>>: j ( )+ j ( )+1 < =0 v ( )=1 ; v ( )=0 j ( )+ j ( ) 1 < 1 v ( )=0 ; v ( )=1 j ( )+ j ( ) < 0 v ( )= v ( ) 0 : Thecase j ( e e ) < 0 correspondstoaippednormalandyields ( e e ) ( v j ) 0 Wecaninvestigatethestructureof supp M r byrstdecomposingitintoparallelepipeds. Therearetwodecompositions.Lemma21. The(closed)supportof M 1 istheessentiallydisjointunionofthe ( n +1) parallelepipeds Z : Z 2B ( T 1 ) (615) or,alternatively, Z + Z : Z 2B ( T 1 ) (616) 88

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A B C Figure6-6. A )Rhombicdodecahedron:supportof M 1 := 1 2 h 1 1 11 11 11 1 111 i and A 3 = 1 2 h 1 1 1 11 1 1 11 i B ) C )twodecompositionofthesupportinto parallelepipeds: B )by( 615 )and C )by( 616 ). where Z := T 1 n Z .Ineitherdecomposition,alltheparallelepipedsarecong ruent. Proof. Duetotherelation( 610 ),weneedonlyconsider M r .Let Z j 2B ( T 1 ) beabasisof T 1 and j := T 1 n Z j = X 2 Z j : For j := Z j in( 315 ),thereareonlytwochoices, j 2 0 ; j sinceno 2 Z j doubled tsintothesupport Z j + 3 + = 2 T 1 ; 8 2 Z j : Nowassume j =0 and k = k for Z j ; Z k 2B ( T 1 ) ; Z j 6 = Z k : Thisleadstoacontradictionasweprovethatthetwoparalle lepipeds Z j + j and Z k + k arenotessentiallydisjointbutshareapoint p interiortoboth.Let p := 1 2 k + 1 4 X 2 Z j \ Z k : 89

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Then p 2 Z j (0 ; 1) n + j since p = 0@ 1 2 X 2 Z j n Z k + 1 4 X 2 Z j \ Z k 1A + j where j = 0 Z j =( Z j n Z k )+( Z j \ Z k ) and + denotesdisjointunion.Also p 2 Z 2 (0 ; 1) n + k since p = 1 2 k + 1 4 X 2 Z j \ Z k + 1 2 X 2 T 1 ( P 2 T 1 = 0 ) = 1 2 k + 1 4 X 2 Z j \ Z k + 1 2 X 2 Z k + 1 2 k = 0@ 3 4 X 2 Z j \ Z k + 1 2 X 2 Z k n Z j 1A + k : Thisestablishesthatthereareonlytwoalternatives.Next ,weprovethatallparallelepipeds arecongruent.Let'sconsiderthedecomposition Z : Z 2B ( T 1 ) : (617) Thefollowinglemmasimpliestheproof.Lemma22. Letthematrices A ; B 2 R n n bedenedasfollows: A ( j;k )= 8>><>>: 1 k = 0 otherwise and B ( j;k )= 8>><>>: 1 j = k = 0 otherwise : Thenfor Z j := I n A j B j Z j ( I n + J n ) Z tj = I n + J n and Z 2j = I n : SeeFigure 6-6 for n =3 Proof. Bothcanbeveriedusingthefollowingrelations. A j J n = J n ; B j J n = A tj ; A j A tj = J n ; A j B j = A ; B j A j = B j ; A 2j = A j : 90

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Firstnotethat ( A n ) 2 = I n + J n .For Z j := A n Z j and Z k := A n Z k Z j =( A n Z j Z k ( A n ) 1 ) Z k wherewecanverifythat ( A n Z j Z k ( A n ) 1 )( A n Z j Z k ( A n ) 1 ) t = A n Z j Z k ( A n ) 2 Z tk Z tj A n ( ( A n ) 1 = A n ) = A n ( I n + J n ) A n (Lemma 22 ) = I n : Therefore A n Z 1 Z 2 ( A n ) 1 isanorthonormal(rigid)transformationhenceallthe parallelepipeds Z Z 2B ( T 1 ) arecongruent.It'squitestraightforwardwhenoneofthe basesis A n I n .Notethatanypermutationofthecolumnsof Z j 2B ( T 1 ) canbedone withapermutationmatrix,whichisalsoanorthonormaltran sformation. Theotherdecompositioncanbeveriedinthesameway. Lemma 21 iseasilyextendedto T r since T r = f X 2 T r t :0 t 1 g = f X 2 T 1 t :0 t r g = T 1 ( r ) : For Z 2B ( T 1 ) ,thepair ( Z ; Z ) isalineartransformationofthepair ( I n ; j ) .Therefore Z isdecomposedinthesamewayastheunitcube isdecomposedbytheKuhn triangulationand supp M 1 consistsof ( n +1)! simplices.Thiscountalsoagreeswiththe numberof modularcells intherstneighborpolytopeof A n [ 51 ].Thetwotypesofthe decompositionof supp M 1 inLemma 21 canbeviewedascubicalmeshessuchthatone isthe ip oftheother[ 7 ]sinceeachcubicalmeshcanbeviewedastheprojectionofth e ( n +1) -dimensionalcubealongonexeddiagonalintwooppositedi rections. 91

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Next,weexpandonLemma 18 andshowamap(quasi-interpolant)canreproduceall cubicpolynomials.Lemma23 (Quasi-interpolantfor M 2 ) Thequasi-interpolantof M 2 denedbythe functional 2 ( f ( + j )):= A n T 2 ( f ( + j )):= 0@ f 1 12 X 2 T 1 D 2 f 1A ( j ) ; j 2 A n Z n (618) providestheoptimalapproximationorder m ( T 2 )+1=4 Proof. Followinglemmawillmakethecomputationeasier. Lemma24. Foranoddfunction f T 2 f =0 ( 323 ) Proof. T 2 f = X j M 2 ( j ) f ( j ) = X j M 2 ( j ) f ( j ) (by( 16 )) = X j M 2 ( j ) f ( j ) ( f = f ( ) ) = X j M 2 ( j ) f ( j ) (changeofindex) therefore T 2 f =0 Wecompute g ( 0 ) foreachdegree j j 1. j j =0 g ( 0 )= g 0 ( 0 )=1 By[ 36 ,page68]. 2. j j =1 ByLemma 24 T 2 [[]] =0 and g = [[]] X 6 = T 2 [[]] g = [[]] ( T 2 [[]] ) g 0 = [[]] therefore g ( 0 )=0 92

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3. j j =2 By[ 36 ,page11], c M T 2 ( ):= Ff M T 2 g ( )= Y 2 T 2 sinc( )= Y 2 T 1 sinc 2 ( ) : Therefore,By( 324 ),for j 6 = k D j D k 1 c M T 2 ( ) = 0@ Y 2 T 1 nf j ; e j ; e k g 1 sinc 2 ( ) 1A D j D k 1 sinc 2 ( j )sinc 2 ( j )sinc 2 ( k ) Since sinc(0)=1 ,withthehelpof MAPLE ,wecancompute D j D k 1 c M T 2 (0)= D j D k 1 sinc 2 ( j )sinc 2 ( j )sinc 2 ( j + k ) (0)= 1 6 : Also, D 2 j 1 c M T 2 ( )= 0@ Y 2 T 1 nf j ; e j g 1 sinc 2 ( ) 1A D j 1 sinc 2 ( j )sinc 2 ( j ) : Again,withthehelpof MAPLE ,wecancompute D 2 j 1 c M T 2 (0)= D 2 j 1 sinc 4 ( j ) (0)= 1 3 : By( 324 ),for j 6 = k g e j + e k (0)= [[ iD ]] e j + e k 1 c M T 2 ( 0 )= D j D k 1 c M T 2 ( 0 )= 1 6 and g 2 e j ( 0 )= [[ iD ]] 2 e j 1 c M T 2 ( 0 )= 1 2 D 2 j 1 c M T 2 ( 0 )= 1 6 : 4. j j =3 93

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By[ 36 ,(III.19)], g = [[]] X 6 = T 2 [[]] g = [[]] 0@ ( T 2 [[]] ) g 0 + X j j =1 T 2 [[]] g + X j j =2 T 2 [[]] g 1A = [[]] hence g ( 0 )=0 because ˆ T 2 [[]] =0 byLemma 24 ˆ g =0 for j j =1 and ˆ T 2 [[]] =0 for j j =2 hence j j =1 byLemma 24 Summingup, T 2 f = X j j m ( T 2 ) g ( 0 )( D f )( 0 ) = f ( 0 ) 1 6 X j j =2 ( D f )( 0 ) = f ( 0 ) 1 12 n X k =1 ( D 2 k f )( 0 )+(( n X k =1 D k ) 2 f )( 0 ) = f ( 0 ) 1 12 X 2 T 1 ( D 2 f )( 0 ) : (619) Now,by( 329 ), 2 f = f ( 0 ) 1 12 X 2 T 1 ( D 2 f )( 0 ) : Fordiscreteinput f : A n Z n R ,weapproximatethedirectionalderivativealong 2 R n bynitedierences,e.g., D 2 f f ( + )+ f ( ) 2 f: (620) 94

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Therefore 2 f f ( 0 ) 1 12 X 2 T 1 ( f ( )+ f ( ) 2 f ( 0 )) = 1+ n +1 6 f ( 0 ) 1 12 X 2 T 1 ( f ( )+ f ( )) : (621) Whenspecializedtotwovariables,thisagreeswithLevin's formula[ 67 ]. 95

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CHAPTER7 SYMMETRICBOX-SPLINEONTHE D n LATTICE 7.1DenitionandProperties TheRootsystem D n .Therootsystem D n canbedescribedbytheCoxeter diagraminFigure 7-1 whereitiscomposedofalltheintegervectorsoflength p 2 ,i.e., [ 23 60 ] D n = f e i e j :1 i 6 = j n g (71) hence # D n =2 n ( n 1) .Notethat,bydenition, D n isonlydenedfor n 3 ,due tothethreerightmostmandatorynodesoftheCoxeterdiagra mfor D n (Figure 7-1 ).In dimensionthree, D 3 = A 3 (Figure 5-1 ). TheRootlattice D n .Therootlattice D n isgeneratedbyallintegerlinear combinationsoftherootsof D n .Alternatively, D n isdenedasallintegerpointsof Z n wherethesumoftheirelementsisalwaysevenwhen D n isformulatedas( 71 )[ 23 ]: D n := f i 2 Z n : i j iseven g : e 1 e 2 e 2 e 3 e n 2 e n 1 e n 1 e n e n 1 + e n Figure7-1.Coxeterdiagramoftherootsystem D n withCartesiancoordinates. 96

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AsetofsimplerootsassociatedwiththeCoxeterdiagraminF igure 7-1 ,[ 60 ] M D n := f e n 1 e n g[ n 1 [ j =1 f e i e n g = 266666666664 1 1 1 1 1 1 1 1 377777777775 = 264 I n 1 e n 1 j t 1 375 (72) servesasthegeneratormatrixforthe D n lattice. For n 6 =4 ,thesymmetrygroupof D n consistsof ˆall n permutationsofcoordinates, ˆ 2 n 1 signchangesofevenlymanycoordinatesand ˆ 2 signchangesofthelastcoordinates. Overall,theorderis n !2 n .For n =4 ,whichhasthesamesymmetrygroupasthe 24 -cell [ 23 25 ],thesymmetrygroupconsistsofthersttwooftheaboveand the 3! permutation ofthreerootsduetothesymmetryofitsCoxeterdiagram(Fig ure 7-2 )hencethetotal orderis 4! 2 3 3!=1152 Figure7-2.Coxeterdiagramoftherootsystem D 4 TheBox-spline M D n .The n ( n 1) -directionbox-splineonthe D n latticeis constructedbytakingthedirectionstothe 2 n ( n 1) nearestlatticepointsof D n .Inother 97

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words,thedirectionmatrixisoftheform D n := [ 1 i><>>: 3( n =3) ( n 1)( n 2)( n> 3) : Proof. Let ( v ) and ? v D n for v 2 R n v 6 = 0 ,bedenedas( 43 )and( 42 ),respectively. Weconsidertherelationbetween # ( v ) and # ? v D n Let k :=# ( v ) 0 ><>>: =0( k =1) #Z 1 = 2= k 2 otherwise : 98

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Alsothefollowingsarestraightforward. #Z ? v 2 =0 (Since ( e i e j ) v = v ( i ) 6 =0 .) #Z ? v 3 = 8>><>>: 0( k n 1) #Z 3 =2 n k 2 otherwise : Therefore, 1. k =1 # ? v D n =#Z ? v 2 +#Z ? v 3 =2 n 1 2 =( n 1)( n 2)=: f 1 ( n ) : 2. k n 1 # ? v D n =#Z ? v 1 +#Z ? v 2 k 2 = 8>><>>: 1 2 ( n 1)( n 2)=: f 2 ( n )( k = n 1) 1 2 n ( n 1)=: f 3 ( n )( k = n ) : 3. 2 k n 2 Keepinginmindthat n 3 # ? v D n =#Z ? v 1 +#Z ? v 2 +#Z ? v 3 k 2 +2 n k 2 = 1 2 (3 k 2 +2 k 4 nk +2 n 2 2 n ) = 1 2 3 k 2 n 1 3 2 + 2 3 n ( n 1) 1 3 := g ( k;n ) : 99

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Since (2 n 1) = 3 >n= 2 for n 3 g ( k;n ) g (2 ;n )= 22 +2 n 2 2 =1+( n 2)( n 3) < ( n 1)( n 2)= f 1 ( n ) : Therefore # ? v D n ismaximizedbyoneofthequadraticfunctions f 1 ( n ) f 2 ( n ) and f 3 ( n ) Comparingthethree,weget 8>>>>>><>>>>>>: f 3 (3)=3 >f 1 (3)=2 >f 2 (3)=1( n =3) f 1 (4)= f 3 (4)=6 >f 2 (4)=3( n =4) f 1 ( n ) >f 3 ( n ) >f 2 ( n )( n> 4) : Theclaimholdssince maxv 2 R n # ? v D 3 =# ? j D 3 =3 andfor n> 3 maxv 2 R n # ? v D n =# ? e j D n =( n 1)( n 2) where 1 j n ByLemma 25 m ( D n )= 8>><>>: (6 3) 1=2( n =3) ( n ( n 1) ( n 1)( n 2)) 1=2 n 3( n> 3) therefore M D n 2 8>><>>: C 1 ( n =3) C 2 n 4 ( n> 3) : Lemma26. Thesequence ( M D n ( j )) j 2 M D n Z n islinearlydependentexceptfor n =3 100

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Proof. Since[ 88 ] det M D n =det 264 I n 1 e n 1 j t 1 375 =det I n 1 det( 1 j t I 1 n 1 e n 1 )= 2 ; thesequenceislinearlyindependentifandonlyif detZ 2f 0 ; 2 g ; 8 Z D n : Theclaimholdsduetothecounterexample := 8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>: 26666664 B B 37777775 =: e ( n even ) 266666666664 e 111 11 1 377777777775 ( n odd ) ; where D n and B := 264 111 1 375 ; since det=( 2) b n= 2 c by[ 88 ]. 7.2The 6 -DirectionBox-SplineontheFCCLattice Indimensionthree, D 3 := 266664 1111001 10011 001 11 1 377775 101

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and M D 3 := 266664 100011 1 11 377775 therefore M D 3 = M fcc sinceitiscenteredandre-normalized(seeSection 5.3 ). 102

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CHAPTER8 SYMMETRICBOX-SPLINEONTHE D n LATTICE 8.1DenitionandProperties TheRootlattice D n .Oneofthegeneratormatricesforthe D n latticeis M t D n = 264 I n 1 e n 1 j t 1 375 t = 266664 I n 2 00 0 t 1 1 j t 1 1 377775 t = 266664 I n 2 00 j t = 21 = 2 1 = 2 j t = 2 1 = 2 1 = 2 377775 : Butinsteadwechoosethesimplergeneratormatrix[ 23 ] M D n := 1 2 j [ n 1 [ j =1 e j = 264 I n 1 j = 2 0 t 1 = 2 375 : (81) TheBox-spline M D n .The ( n +2 n 1 ) -directionbox-splineonthe D n latticeis constructedbythedirectionsimpliedbythelatticepoints D n := I n [ 1 2 f e n + n 1 X j =1 e j g (82) correspondingtothecentersofthe 2 n unitcubesadjacenttotheoriginandthe n main axisdirections. Thecenteredandre-normalizedbox-splineisdenedas M D n := j det M D n j M c D n = j det M D n j M D n ( + X 2 D n = 2) : (83) Thedegreeof M D n is n +2 n 1 n =2 n 1 .Sincethedirectionsin D n arepairwise paralleltothoseof Z n (Section 4.1 ), M D n 2 C 2 n 2 Lemma27. Thesequence M D n ( j ) j 2 M D n Z n islinearlydependent. 103

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Proof. Theclaimholdssince I n D n while det M D n =1 = 2 8.2The 7 -DirectionBox-SplineontheBCCLattice A B Figure8-1. A )Directionsand B )supportof M D 3 Indimensionthree, D 3 = A 3 = BCClattice.Withthedirectionmatrix(Figure 8-1A ) bcc := D 3 := 1 2 266664 2001 11 1 02011 1 1 0021111 377775 andthegeneratormatrix(insteadof M D 3 ) M bcc := 1 2 266664 1111 11 11 1 377775 wedene M bcc := M D 3 := j det M bcc j M bcc ( + X 2 bcc = 2)= 1 2 M bcc ( +( 1 2 ; 1 2 ; 3 2 )) : Lemma28. Thequasi-interpolantwithoptimalapproximationorder 4 forthesplinespace S bcc :=span( M bcc ( j )) j 2 M bcc Z 3 104

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isdenedbythefunctional bcc f :=( f 1 24 X 2 bcc D 2 f )( 0 ) : Proof. With e bcc := M 1 bcc bcc = 266664 01111001011010110100 1 377775 ; thequasi-interpolantforthesplinespace span( M c e bcc ( j )) j 2 Z 3 isdenedbythefunctional e bcc f :=( f 1 6 ( D 2 1 + D 2 2 + D 2 3 + D 1 D 2 + D 2 D 3 + D 3 D 1 ))( 0 ) =( f 1 24 (( D 1 + D 2 + D 3 ) 2 +( D 1 + D 2 ) 2 +( D 2 + D 3 ) 2 +( D 3 + D 1 ) 2 + D 2 1 + D 2 2 + D 2 3 ))( 0 ) =( f 1 24 X 2 e bcc D 2 f )( 0 ) thereforethequasi-interpolantforthesplinespace S bcc isdenedbythefunctional bcc f :=( f 1 24 X 2 bcc D 2 f )( 0 ) : Foradiscreteinput f : M bcc Z 3 R bcc ( f ( + j )) 19 12 f ( j ) 1 24 X 2 bcc ( f ( j + )+ f ( j )) : 105

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CHAPTER9 FASTANDSTABLEEVALUATIONOFBOX-SPLINESVIATHEBB-FORM Beforediscussingtheevaluationofbox-splines,werstpr ovethattheBB-coecients ofthebox-splinesarerationalnumbers,whichallowsusexa ctevaluationofbox-splines. 9.1Box-SplinewithRationalBB-Coecients In[ 65 ], Lai provedfor 3 -and 4 -directionbivariatebox-splinesandin[ 12 ], Casciolaetal. provedfor 4 -directiontrivariatebox-splinesthattheBB-coecients ofthosebox-splines arerational.Inthissection,wegeneralizethisobservati on. Westartbyshowingthatrationalityispreservedbybox-spl ineswithrational directionmatrices.Lemma29. Let 2 Q n m and rank( )= n .If x 2 Q n then M ( x ) 2 Q Proof. Weuseinductionon # .Let 2 Q n n .Then 1 = j det j2 Q and M ( x )= = j det j2 Q where isthecharacteristicfunctionofparallelepiped ,alinear transformationoftheunitcube Nowassumetheclaimholdsfor M n .Thatis M n ( x ) 2 Q and M n ( x ) 2 Q since x 2 Q n .Since rank( )= n ,thereexistsaninvertiblesubmatrix 0 2 Q n n of Since 0 1 2 Q n n ,therealsoexists t 2 Q m sothat x = t .Therecursion( 318 ), ( m n ) M ( x )= X 2 t M n ( x )+(1 t ) M n ( x ) ; thenimpliesthat M ( x ) 2 Q Nowdenoteby ( ) thecollectionofallshiftsof ( n 1) -dimensionalhyperplanes spannedbycolumnsof ( 317 ).Each H i 2 ( ) isdenedbyaplaneequation n ti ( j i )=0 .Wedenoteby knotvertex theintersection x of n hyperplanes H 1 ; ;H n 2 ( ) 106

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whosenormalsspan R n : x = N 1 ; where N := 266664 n t1 ... n tn 377775 and := 266664 n t1 j 1 ... n tn j n 377775 : (91) Lemma30. Let 2 Q n m and rank( )= n .Thenthepolynomialpiecesof M canbe representedinBB-formwithvertexcoecientsin Q Proof. BySection 3.6 ,thepiecewisepolynomialsof M canbeexpressedoverconvex polytopesdelineatedbyknotplaneswithrationalnormals. Since 2 Q n m ,wehave n i 2 Q n andhence N 1 2 Q n n in( 91 ).Sincetheshiftsareontheintegergrid, j i 2 Z n andthereforeallknot-verticesarein Q n .Sinceany n -dimensionalconvexpolytopecanbe decomposedinto n -dimensionalsimpliceswithoutintroducinganynewvertex ,theclaim followsfromLemma 29 Lemma 30 canbeextendedtoyieldthemainconclusion.Toaccommodate shiftson theCartesianlattice, isrequiredtohaveintegerentries. Theorem1. Let 2 Z n m and rank( )= n .Thenthepolynomialpiecesof M canbe representedinBB-formwithcoecientsin Q Proof. Weuseinductionon # .If 2 Z n n ,thenthepolynomialisconstantandequals thevalueatthevertex.ByLemma 30 ,thisvalueisrational. Since rank( )= n ,forany w 2 Q n thereexistsan y 2 Q m sothat w = y .By linearityofdierentiationand( 316 ), D w M = X 2 y D M = X 2 y r M n : Bytheinductionhypothesis, M n ispiecewisepolynomialinBB-formwithcoecients in Q .Sincetheknotplanesareinvariantunderintegershiftsan d 2 Z n m r M n isadierenceofpolynomialsinBB-formwithcoecientsin Q .Therefore D w M isa polynomialinBB-formwithcoecientsin Q .Nowlet v i and v j beanytwoknotvertices 107

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ofthedomainsimplexofapolynomialpiece,possiblyobtain edbydecompositionofan n -dimensionalconvexpolytope.Then w := v i v j 2 Q n and D v i v j M hasrational coecients,i.e.,inthenotationof( 38 ), ( c + e i c + e j ) 2 Q .Thevertexcoecients arerationalbyLemma 30 .Sorationaldierencespropagatemembershipin Q toallthe BB-coecients c in( 38 ). Notethatallthebox-splinesproposedareconstructedbyth edirectionsassociatedto thelatticepoints.Therefore,afterchangeofvariabletot hebox-splinesontheCartesian lattice,allthedirectionmatricesbecomeintegermatrice shencetheirBB-coecientsare rational. 9.2Pre-processingBox-Splines Werstdiscusshowtoencode(orindex)thedomainsimplices ofthepolynomial piecesforaparticulardirectionmatrix andthendescribehowtondthechange-of-basis forconversionofbox-splinestotheBB-form.Wenotethatth ecombinatorialwork ofdeningthepartitionintodomainsimplicesisdoneonlyo nceeverperbox-spline generatororbasisfunctionsincetheresultistabulatedan dquicklyaccessedbythe followingindexingstrategy.9.2.1IndexingPolynomialPieces(Domains) Unlessthedirectionsformatensor-product,themostconve nientrepresentationofthe polynomialpiecesofabox-splineistheBB-formonasimplex (Section 3.5 ).Thechallenge istosmartly index eachdomainpiecein supp M toderive,storeandecientlyaccessthe BB-coecients.OurdecompositionisinspiredbyBSP(Binar ySpacePartitioning)trees [ 48 76 ].Let ( ) ( ) ( 317 )bethesetofknotplanesof M ,eachofwhichsplits intotwo n -dimensionalsubspaces.Theneach path ofthetreeisconvertedintoapairof indexvectors ( i 2 ; i 4 ) 2 Z n f 0 ; 1 g q where q := q ( ):=# ( ) isthenumberofknot planesin ( ) Specically(cf.Figure 9-1 ),werstcircumscribethesupportofthebox-splineas follows.Let I 2 Z n betheminimalsetofgridpoints(e.g.,blackpointsinFigur e 9-1A ) 108

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suchthat supp M I + ; i.e. I := f j 2 Z n :( j + \ supp M ) 6 = ;g .Eachcube j + for j 2 I isfurther partitionedintoconvexpolytopesbytheknotplanes(e.g., blacklinesinFigure 9-1B )in j + ( ) A i 2 B i 4 Figure9-1.IndexingofZP-element(Section 4.2 )where I ZP = f 1 ; 0 ; 1 gf 0 ; 1 ; 2 g Byshift-invariance,eachcubeispartitionedalike.Wenow indexadomainpieceby i 2 2 I and i 4 2f 0 ; 1 g q : Inotherwords, i 2 := bc identiesacubeintersecting supp M and i 4 identiesa polynomialpieceinsidethecube(Figure 9-1 ).Theindexvector i 4 iscomputedby membershiptestagainstalltheknotplanesin ( ) : i 4 ( x ):= U ( N ( x b x c ) ) ;U ( t ):= 8>><>>: 1 ;t 0 0 ;t< 0 ; (92) where N 2 Q q n and 2 Q q denetheknotplanesin ( ) 9.2.2ComputingtheChangeofBasis Toconvert M topolynomialpiecesinBB-forminthepre-processingstep, werst needtodetermineadomainsimplexforeachpiece.Thedomain ofthepolynomialpieceis aconvexpolytope delineatedbytheknotplanes.Typically,thepolytopeisas implex.If not,wehaveseveraloptions. 109

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ˆWecanchooseany n +1 verticesfrom todenethedomainsimplex.This,however, implieslossoftheconvexhullpropertyastheBB-formwilla lsobeevaluatedoutside thesimplex. ˆWecansplit intosimpliceswithoutintroducingnewvertices.Butthism akesthe evaluationprocessmorecomplex. ˆThemostpracticalstrategyistochooseanysimplexwithra tionalverticesthat encloses Inthelastcase,itdoesnotmatterthatthedomainsimplices ofdierentpolynomial piecesoverlap,becausethedomainisalreadydeterminedbe foreevaluatingtheBB-form andweonlyevaluatetheBB-formforpointsinthedomainrath erthanonthewhole enclosingsimplex. Oncethedomainsimplex isdetermined,thechange-of-basisequalityfor x inthe domainis M ( x )= X j j = m n c b ( ( x )) : (93) Wewanttodeterminethe m n BB-coecients c ofthepolynomial M ofdegree d := m n in n variables.Wesetupandsolvethe m n by m n linearsystemofconstraints d d c = M V e d (94) arisingfrom( 93 )whenwechooseuniformlydistributedrationalpoints x := V e =d ina simplex e ( 6 = )withvertices V e := e v 1 e v n +1 e v j 2 Q n for 1 j n +1 ; 2 Z n +1 + ; j j = j j = d: Forexactness,thevertices V e of e shouldlieatrationalpointsandforstabilitystrictly insidethedomainpolytope: e ( .Theexistenceanduniquenessofasolution c ofthislinearsystemisguaranteedby[ 79 ],aspecialcaseofChung-Yaointerpolation[ 21 ]. Notethatthematrix B := h d d i dependsonlyonthedimensions, n and m ,of .In particular,itisindependentofthesimplex e sothatitsinversecanbecomputedonceto beusedforallpolynomialpiecesofabox-spline. 110

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Whilethematrix B iseasilyexactlyinverted,thechallengeistoobtainanexa ct righthandside M ( V e =d ) .Reimplementing[ 31 ](or[ 62 ])inrational(integer-based) arithmeticdoesnotseempractical;andrationalarithmeti ccanbeslowforhigh m even whenwecantakeadvantageofsymmetriesinthebox-splinede nition. Wethereforeapplythefollowingalternativeapproach. 1.Choosethe n +1 verticesof V e strictlyinsidethepolynomialpiece'sdomain,away fromtheknotplanes. 2.Compute M ( V e =d ) withthehelpof[ 31 ](or[ 62 ])in MATLAB (dueto(a),the evaluationsarestable). 3.Solve( 94 )in MATLAB toobtainapproximatecoecients ~ c 4.Round ~ c to c usingthe MATLAB rat functioncalltoroundtorationalnumbers [ 72 ]. Eventhoughtheroundingisinourexperiencenomorethan 10 10 timesthecoecient size,formallyonehastocheckcorrectnessoftherounding. Therenementequation ( 331 )providestheappropriateandecienttool.Weusethefactt hat m h canbe computedusingonlyintegerarithmetic.Lemma31. On h Z n m h 2 Z n m ,canbecomputedexactlyusingintegercomputation. Proof. Let B h := 1 h # b h ( h )= 1 h # n m h ( h ) hence,for k = h i 2 h Z n i 2 Z n b h ( k )= h # B h ( i ) .Thentherecurrencerelation( 330 ) becomes B h = 1 =h 1 X j =0 B h n ( j ) : (95) Since 2 Z n m ,thisallows B h tobecomputedbyintegercomputationon Z n .Therefore b h andhence m h canbecomputedexactlyon h Z n andhasthedenominator 1 =h # n 111

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ToverifycorrectroundingoftheBB-coecientsof M ,weobservethatthe renementequation( 331 )canbeconvertedto M ( )= X 2 Z n M ( =h ) m h ( h ) ; 2 Z n : (96) Since supp M isnite,sois supp M \ Z n andwecanchooseanynitesuperset K Z n containing supp M \ Z n .Cavarettaetal.[ 13 ],page18provedthatanyconvergent subdivisionschemehasthesimple(orsingle)eigenvalue 1 sothattheeigenvalueproblem ( 97 ),augmentedwiththecondition P j 2 supp M \ Z n M ( j )=1 ; resultsinthesystem M ( )= X 2 Z n M ( =h ) m h ( h ) ; 2 K; X 2 K M ( )=1 (97) andthe unique eigenvector f M ( ): 2 K g ; (98) whoseentriesarethebox-splineevaluatedon Z n Wecannowconstructwhatwehopetobetheexacteigenvector( 98 )byevaluating theBB-formwithitsrationalcoecientsthatweobtainedby roundingtoarational representation.Sinceeachpolynomialpiecehasnitedegr ee,itsucestotestonarened grid h Z n sothateachdomaincontainsmoreevaluationpointsthanits BB-coecients.We emphasizethatwedonotsolvetheeigenvalueproblem( 97 )butonlyverify,bycomparing thevalues,thatequalityholds.Sinceweknowalltheexactd enominatorsinvolvedin theverication,wecanpre-scale( 331 )and( 97 )sothatonlyintegercomputationsare required. Wealsopre-computethematricesinEquation( 36 )thatcomputebarycentric coordinates u withrespecttoadomainsimplex forapoint x 2 R n inCartesian coordinates. 9.3TheSplineEvaluationAlgorithm Giventheindexing(hashtable),thetableofBB-coecients andthepre-computed inversematricesforcomputingbarycentriccoordinates,w ecanevaluatesplines,i.e.linear 112

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combinationsoftheshiftsofbox-splines,ecientlyandst ably.Thefollowingalgorithm evaluatesasplinewithbox-splinedirections andbox-splinecoecients a (treatedas ameshfunction)atapoint x .Thestepsareasfollows.First,wendthedomainindex i 4 ofthepoint x usingthemembershiptest( 92 ).Wethencomputethebarycentric coordinates u of x withrespecttothedomainsimplex.Shiftsoftheindexareus edto pickup,viathehashtable,allBB-coecientsthatstemfrom box-splineshiftswhose supportsoverlap x ;andtoformtheirlinearcombinationwithweightsfrom a .The coecientsoftheresultingpolynomialarestoredin P .Finally,thealgorithmevaluates thispolynomialwithcoecientvector P .Thefollowingisthepseudocodeforthespline evaluation.Algorithm9.3.1: EvaluateSpline ( a; x ) i 4 U ( N ( x b x c ) ) u ComputeBarycentric ( i 4 ; x b x c ) P P i 2 2 I a ( b x c i 2 ) C ( i 2 ; i 4 ) return EvaluateBB ( P; u ) Herethesubscript ,ratherthananargument ,emphasizesthatthealgorithm requiresthepre-processingwithrespectto accordingtoSection 9.2 ˆ a ( b x c i 2 ) 2 R isthebox-splinecoecientwithindex b x c i 2 2 Z n ˆ x istheinputpoint(inCartesiancoordinates)tobeevaluate d, ˆ N and denetheknotplanesin ( ) ,(Section 9.2.1 ) ˆ ComputeBarycentric computesthebarycentriccoordinateusing( 36 ), ˆ C ( i 2 ; i 4 ) isavectorofallcoecientsofthepolynomialpieceinBB-fo rmwithindex ( i 2 ; i 4 ) ,retrievedfromthehashtable,and ˆ EvaluateBB evaluatesthepolynomialinBB-formwithcoecients P at u 113

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Inmostofapplicationsbox-splinesaremoresymmetricther eforemoreecientevaluation algorithm(See,e.g.,Section 4.3.4 andSection 5.3.4 )ispossible. 9.4Example Weillustratetheinitialconversionandgenerationofthei ndexfunctionforthe 6 -directionbox-splineontheFCClattice(Section 5.3 ).Whilethesymmetricstructureof theFCClatticewastakenintoaccountforevaluationinSect ion 5.3 ,hereweconsiderit transformedtotheCartesianlatticetodemonstratetheove rallprocess. 9.4.1 6 -DirectionBox-SplineontheFCCLattice Denition .A 6 -directiontrivariatebox-splinecanbedenedbythefollo wing directionmatrix[ 38 ]( 59 ) fcc := M fcc e fcc := 266664 001 111 1 11100 11001 1 377775 where( 512 ) M fcc := 266664 011101110 377775 and e fcc := 266664 10010 1 010 110 0 11001 377775 : SplineSpace .The 6 -directionbox-splineisassociatedwiththeFCC(Face-Cen tered Cubic)lattice.By( 327 ),splinesontheFCClatticearegeneratedbytheshiftsofth e (scaled)box-spline j det M fcc j M fcc since M fcc isthegeneratormatrixfortheFCClattice [ 23 ]. DegreeandContinuity .Thebox-spline M fcc ispiecewisepolynomialoftotal degree k ( fcc )=6 3=3 .Atmost3directionsin fcc spanahyperplane.Therefore m ( fcc )=(6 3) 1=2 and M fcc 2 C 1 ( R 3 ) 114

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IndexingDomainTetrahedra .Wehave q =# ( e fcc )=5 planesdenedby (Figure 9-2 ) N e fcc := 266664 111101110111011 377775 t and e fcc := 12111 t : issplitinto10tetrahedraasspeciedinTable 9-1 andFigure 9-3 (1 ; 1 ; 1)(1 ; 1 ; 1)(1 ; 1 ; 0)(1 ; 0 ; 1)(0 ; 1 ; 1) Figure9-2.5knotplanesin ( e fcc ) Table9-1.The 6 -directionbox-spline M e fcc .Theknotplanesin ( e fcc ) split into10 tetrahedra( v c :=( ; ; ) ). i 4 vertices i 4 vertices 00000 000100010001 11111 111101011110 10010 v c 101001100 10011 v c 011001101 10001 v c 001011010 10000 v c 001010100 10110 v c 101100110 10111 v c 011101110 10100 v c 010110100 10101 v c 011110010 0000010010100011011010100 1111110011100001011110101 Figure9-3.10tetrahedrageneratedbythe5knotplanesin ( e fcc ) 115

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Table9-2.Comparisonofevaluationtimeofthreepackagesf or N 3 pointsdistributed over: [0 : 5 :: 3] 3 forthe 7 -directionbox-splineand [1 :: 3] 3 forthe 6 -direction box-spline. algorithmspline time( multipleofAlgorithm 9.3.1 ) 21 3 31 3 41 3 [ 31 ] 7 -dir.20.273( 144 )75.297( 154 )187.716( 153 ) 6 -dir.1.867( 34 )7.088( 39 )18.147( 41 ) [ 62 ] 7 -dir.52.728( 375 )207.841( 424 )550.423( 450 ) 6 -dir.3.645( 66 )14.035( 78 )37.232( 84 ) Alg. 9.3.1 7 -dir.0.1410.4901.223 6 -dir.0.0550.1810.445 9.5ComparisonandanApplication Table 9-2 illustratestherelativeeciencyof[ 31 ],[ 62 ]andAlgorithm 9.3.1 .The tableentriesaretheresultofdenselyevaluatinginoneoct antofthe 6 -directiontrivariate box-spline,respectivelyofthe 7 -directiontrivariatebox-spline.Nolinearcombination ofbox-splinesisevaluated.Allthree MATLAB implementationsaredesignedtohandle vectorinputandavoidfor-loops.Themeasurementsused MATLAB onaLinuxsystem withIntel Core — 2CPU6400@2.13GHz(2MBcache)and2GBmemory.Thecompariso n showsAlgorithm 9.3.1 tobefasterbyordersofmagnitude.Weexplainthedierence inspeedastheresultofpre-resolutionofbox-splinecombi natorics,asencodedinthe indexingandthetabulationofBB-coecientspriortorunni ngAlgorithm 9.3.1 .Theother twopackagesaremoregeneralandresolvethecombinatorics atruntime. Tocomputehigh-qualityray-tracingoflevelsetsofa3Del dreconstructedby theshiftsofatrivariatebox-spline,weimplementeda MATLAB scriptthatexportsthe BB-formofasplineformedbyabox-spline[ 61 ].Specically,weoutputPOV-Ray[ 1 ]script format.POV-Rayisapopularandfreelyavailableray-traci ngengine.Thesetuprequires onlyaddingoneinternalfunctiontoPOV-Raythatevaluates apolynomialinBB-form, e.g.usingdeCasteljau'salgorithm.Figures 4-5 and 5-7 showexamples. 116

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Suchhigh-qualityray-tracingishardtoobtainbysubdivis ion,unlesstheray-tracing algorithmiscarefullydesignedforrecursion.Useof[ 31 ]or[ 62 ]isprecludedbytheirlack ofspeed. 117

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BIOGRAPHICALSKETCH MinhoKimwasborninMarch3,1974,inSeoul,RepublicofKore a.Hereceivedhis BachelorofSciencedegreeinelectricalengineeringin199 7fromSeoulNationalUniversity, Seoul,RepublicofKorea.In2004,hereceivedhisMasterofS ciencedegreefromthe DepartmentofComputerandInformationScienceandEnginee ringattheUniversityof Florida.Hismajorresearchareaiscomputergraphics. 125