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Array Set Addressing

Permanent Link: http://ufdc.ufl.edu/UFE0042319/00001

Material Information

Title: Array Set Addressing Enabling Efficient Hexagonally Sampled Image Processing
Physical Description: 1 online resource (113 p.)
Language: english
Creator: Rummelt, Nicholas
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: addressing, grid, hexagonal, image, lattice, optimal, processing, sampling
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: It has long been known that there are numerous advantages to sampling images hexagonally rather than rectangularly. However, due to various shortcomings of the addressing schemes, hexagonal sampling for digital images has not been embraced by the mainstream digital imaging community. The idea of using hexagonal sampling for digital imaging applications has been around since the early 1960s, yet no efficient addressing method for hexagonal grids has been developed in that time. This dissertation introduces a new hexagonal addressing technique called array set addressing (ASA), that solves the problems exhibited by other methods. This new approach uses three coordinates to represent the hexagonal grid as a pair of rectangular arrays. This representation supports efficient linear algebra and image processing manipulation. ASA-based implementations of several basic image processing operations are developed and shown to be efficient. A hexagonal fast Fourier transform (HFFT), based on the fact that the Fourier kernel becomes separable when using ASA coordinates, is also developed. An ASA implementation of the hexagonal discrete wavelet transform (DWT), which enjoys several advantages over the rectangular DWT due to the superior angular resolution and symmetry of the hexagonal grid, is demonstrated and shown to be efficient. The results of an experiment that quantifies and demonstrates the superior bandwidth of hexagonal sampling using real sensor hardware are presented.
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 Nicholas Rummelt.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Wilson, Joseph N.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042319:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042319/00001

Material Information

Title: Array Set Addressing Enabling Efficient Hexagonally Sampled Image Processing
Physical Description: 1 online resource (113 p.)
Language: english
Creator: Rummelt, Nicholas
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: addressing, grid, hexagonal, image, lattice, optimal, processing, sampling
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: It has long been known that there are numerous advantages to sampling images hexagonally rather than rectangularly. However, due to various shortcomings of the addressing schemes, hexagonal sampling for digital images has not been embraced by the mainstream digital imaging community. The idea of using hexagonal sampling for digital imaging applications has been around since the early 1960s, yet no efficient addressing method for hexagonal grids has been developed in that time. This dissertation introduces a new hexagonal addressing technique called array set addressing (ASA), that solves the problems exhibited by other methods. This new approach uses three coordinates to represent the hexagonal grid as a pair of rectangular arrays. This representation supports efficient linear algebra and image processing manipulation. ASA-based implementations of several basic image processing operations are developed and shown to be efficient. A hexagonal fast Fourier transform (HFFT), based on the fact that the Fourier kernel becomes separable when using ASA coordinates, is also developed. An ASA implementation of the hexagonal discrete wavelet transform (DWT), which enjoys several advantages over the rectangular DWT due to the superior angular resolution and symmetry of the hexagonal grid, is demonstrated and shown to be efficient. The results of an experiment that quantifies and demonstrates the superior bandwidth of hexagonal sampling using real sensor hardware are presented.
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 Nicholas Rummelt.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Wilson, Joseph N.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042319:00001


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Thanksgoouttomyfamilyfortheirsupport,understanding,andencouragement.Iespeciallywanttothankmyadvisorandcommitteechair,JosephN.Wilson,forhiskeeninsight,encouragement,andexcellentguidance.Iwouldalsoliketothanktheothermembersofmycommittee:PaulGader,ArunavaBanerjee,JefferyHo,andWarrenDixon.IwouldliketothanktheAirForceResearchLaboratory(AFRL)forgenerouslyprovidingtheopportunity,time,andfunding.TherearemanypeopleatAFRLthatplayedsomeroleinmysuccessinthisendeavortowhomIoweadebtofgratitude.IwouldliketospecicallythankT.J.Klausutis,RicWehling,JamesMoore,BuddyGoldsmith,ClarkFurlong,DavidGray,PaulMcCarley,JimmyTouma,TonyThompson,MartaFackler,MikeMiller,RobMurphy,JohnPletcher,andBobSierakowski. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTIONANDBACKGROUND ...................... 11 1.1Introduction ................................... 11 1.2Background ................................... 11 1.3RecentRelatedResearch ........................... 15 1.4RecentRelatedAcademicResearch ..................... 16 1.5HexagonalImageFormationandDisplayConsiderations ......... 17 1.5.1ConvertingfromRectangularlySampledImages .......... 17 1.5.2HexagonalImagers ........................... 19 1.5.3DisplayingHexagonalImages ..................... 21 2ARRAYSETADDRESSING(ASA) ......................... 24 2.1ForminganArraySetfromaHexagonalGrid ................ 24 2.2Neighborhoods ................................. 26 2.3FormsofArraySets .............................. 26 2.4EfcientandCompactStorageinDigitalMemory .............. 27 2.5ConvertingBetweenCartesianCoordinatesandASA ........... 28 2.6DistanceMeasuresUsingASA ........................ 34 2.6.1EuclideanDistance ........................... 34 2.6.2City-BlockDistance ........................... 34 2.7VectorArithmetic ................................ 36 2.7.1ProofoftheAssociativityofAddition ................. 37 2.7.2ProofofDistributivityofScalarMultiplication(w.r.t.ScalarAddition) 39 2.7.3ProofofCompatibilityofScalarMultiplication(withMultiplicationofScalars) ................................ 41 2.8ComparisonofASAtoSpiralAddressing .................. 42 3BASICIMAGEPROCESSINGOPERATIONS ................... 44 3.1Convolution ................................... 44 3.2GradientEstimation .............................. 46 3.3EdgeDetection ................................. 51 3.3.1Gradient-BasedEdgeDetector .................... 51 3.3.2LaplacianofGaussian(LoG)EdgeDetector ............. 52 5

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......................... 57 3.4Downsampling ................................. 66 4TRANSFORMS .................................... 68 4.1FourierTransforms ............................... 68 4.1.1TheHexagonalDiscreteFourierTransform(HDFT) ......... 68 4.1.2TheHexagonalFastFourierTransform(HFFT) ........... 69 4.1.3TheHexagonalDiscrete-SpaceFourierTransform(HDSFT) .... 79 4.2TheZTransform ................................ 80 4.3WaveletTransforms .............................. 81 5QUANTITATIVECOMPARISONUSINGREALIMAGES ............. 89 6CONCLUSION .................................... 92 6.1SummaryofAccomplishments ........................ 92 6.2SignicanceofAccomplishments ....................... 94 6.3FinalThought-RectangularComponentConjecture ............ 94 APPENDIX APROOFSOFASAZ-MODULEPROPERTIES .................. 96 A.1Preliminaries .................................. 96 A.1.1Domains ................................. 96 A.1.2Identities ................................. 96 A.2CommutativityofAddition ........................... 96 A.3IdentityElementofAddition .......................... 97 A.4InverseElementsofAddition ......................... 97 A.5DistributivityofScalarMultiplication(w.r.t.VectorAddition) ......... 97 A.6IdentityElementofScalarMultiplication ................... 99 BPYTHONSOURCECODEFORASATOHIPRUNTIMECOMPARISON .... 100 REFERENCES ....................................... 108 BIOGRAPHICALSKETCH ................................ 113 6

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Table page 2-1FormsofArraySets ................................. 27 2-2AverageNumberofOperationsExecuted(Yangvs.Hybrid) ........... 29 2-3PropertiesofArraySetAddressing(ASA)Vectors ................ 37 2-4RunTimeComparisonBetweenASAandHIP(s,mean(std)) ......... 43 2-5ComplexityComparisonBetweenASAandHIP .................. 43 4-1ComputationsRequiredforHexagonalFastFourierTransformArrays ..... 74 7

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Figure page 1-1RetesselatingaSourceImage ........................... 18 1-2ExampleofConvertingaSourceImage ...................... 19 1-3TrapezoidalImageSensorDesign ......................... 20 1-4DepictionofBrickWallDisplayTechnique .................... 21 1-5FourCommonDisplaysandTwoPotentialHexagonalRed-Green-Blue(RGB)Displays ........................................ 23 2-1HexagonalGridSpacing ............................... 25 2-2ExampleofSeparatingtheHexagonalGridintoanArraySet .......... 25 2-3DeterminingtheNeighborsofaGivenPixel .................... 26 2-4StorageofArraySetAddressing(ASA)Addresses ................ 28 2-5Representationofxs-ysSpace ........................... 30 2-6ResolvingtheBoundaryCase ............................ 31 3-1ASAConvolutionExample .............................. 45 3-2ComparisonofNearestNeighborNeighborhoodSizes .............. 45 3-3ExampleofFittingaPlanetoaHexagonalNeighborhood ............ 47 3-4HexagonalNeighborhoodIndexing ......................... 47 3-5ConvolutionMasksforLocalGradientEstimation ................. 50 3-6Example1ofGradient-BasedEdgeDetection ................... 53 3-7Example2ofGradient-BasedEdgeDetection ................... 54 3-8Example3ofGradient-BasedEdgeDetection ................... 55 3-9Example1ofLaplacianofGaussian(LoG)-BasedEdgeDetection ....... 58 3-10Example2ofLoG-BasedEdgeDetection ..................... 59 3-11Example3ofLoG-BasedEdgeDetection ..................... 60 3-12QuantizationDirectionsofCannyEdgeDetector ................. 62 3-13Example1ofCannyEdgeDetection ........................ 63 3-14Example2ofCannyEdgeDetection ........................ 64 8

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........................ 65 3-16DownsamplingwithASA ............................... 67 4-1StandardFastFourierTransform(FFT)ButteryDiagram-FullComputation 75 4-2StandardFFTButteryDiagram-ReducedComputation ............ 75 4-3Non-StandardTransform1(NST1)ButteryDiagram-FullComputation .... 76 4-4NST1ButteryDiagram-ReducedComputation ................. 76 4-5Non-StandardTransform2(NST2)ButteryDiagram-FullComputation .... 77 4-6NST2ButteryDiagram-ReducedComputation ................. 77 4-7Non-StandardTransform3(NST3)ButteryDiagram-FullComputation .... 78 4-8NST3ButteryDiagram-ReducedComputation ................. 78 4-9IdealizedWaveletTransformRegionsofSupport(Hexagonalvs.Rectangular) 83 4-10WaveletTransformAnalysisFilterBank ...................... 84 4-11WaveletTransformSynthesisFilterBank ...................... 85 4-12LiliesImage(HexagonallySampled) ........................ 86 4-13HexagonalDiscreteWaveletTransformofLiliesImage .............. 87 5-1TrapezoidalImager .................................. 89 5-2ExampleofaRadialSineWave(Left)anditsFourierTransform(Right) .... 90 5-3ExperimentalResults ................................ 91 9

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Ithaslongbeenknownthattherearenumerousadvantagestosamplingimageshexagonallyratherthanrectangularly.However,duetovariousshortcomingsoftheaddressingschemes,hexagonalsamplingfordigitalimageshasnotbeenembracedbythemainstreamdigitalimagingcommunity.Theideaofusinghexagonalsamplingfordigitalimagingapplicationshasbeenaroundsincetheearly1960s,yetnoefcientaddressingmethodforhexagonalgridshasbeendevelopedinthattime.Thisdisser-tationintroducesanewhexagonaladdressingtechniquecalledarraysetaddressing(ASA),thatsolvestheproblemsexhibitedbyothermethods.Thisnewapproachusesthreecoordinatestorepresentthehexagonalgridasapairofrectangulararrays.Thisrepresentationsupportsefcientlinearalgebraandimageprocessingmanipulation.ASA-basedimplementationsofseveralbasicimageprocessingoperationsaredevelopedandshowntobeefcient.AhexagonalfastFouriertransform(HFFT),basedonthefactthattheFourierkernelbecomesseparablewhenusingASAcoordinates,isalsodeveloped.AnASAimplementationofthehexagonaldiscretewavelettransform(DWT),whichenjoysseveraladvantagesovertherectangularDWTduetothesuperiorangularresolutionandsymmetryofthehexagonalgrid,isdemonstratedandshowntobeefcient.Theresultsofanexperimentthatquantiesanddemonstratesthesuperiorbandwidthofhexagonalsamplingusingrealsensorhardwarearepresented. 10

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1 ],[ 2 ].Curcio[ 2 ]claimsthatthepackinggeometryoftheconescontributestothefunctionalgrainoftheprimateretina. Researchintothesamplingoftwo-dimensionalsignalsbeganappearingintheliteratureduringthelate1950s.In1962,PetersenandMiddleton[ 3 ]developedamulti-dimensionalsamplingtheoryandshowedthataregularhexagonalgridistheoptimalsamplinglatticefortwo-dimensionalisotropicallyband-limitedsignals(anisotropicallyband-limitedsignalisonewhoseFouriertransformiszerooutsideofacircularbandregion).Aroundthatsametime,theILLIACIIIcomputerwasbeingbuiltattheUniversityofIllinoistoscanandprocessimages.Thedesignallowedtheprogrammertochoosebetweenrectangularandhexagonalsampling[ 4 ].Inthelate1960s,Golay[ 5 ]showedthelogicaldifcultythatarisesfromtheconnectivityambiguityinsquaresampledimageswhenperformingmorphologicaloperationsandshowedthegreatersimplicityofperformingtheseoperationsonhexagonallysampledimages.Theconnectivityambiguityarisesfromthefactthatfourofthenearestneighborpixelsshareasideandtheotherfourshareacorner,resultingintheneedtospecify4-wayor8-wayconnectedness.Thehexagonalgridhasnosuchambiguitysinceallnearestneighborpixelsshareaside.Golayalsopointedoutthegreaterangularresolutionthehexagonalgridhasoverrectangularones.Shortlyafterthat,Deutsch[ 6 ]experimentallycomparedtheperformanceofthinningalgorithmsonrectangular,hexagonal,andtriangulararrays.Hedeterminedthatthehexagonalarrayofferedthebestoverallperformance,duelargelytoitsrequiringsubstantiallylessprocessingtime.Despiteallthoseresultsandthebiologicalexamples,virtuallyalltwo-dimensionalsignalprocessingsystemsofthetimewereusingrectangularsampling.Thereasonsforthisarevaried:theprocessingalgorithmscouldbegeneralizedfromtheone-dimensionalcasestraightforwardly;thosealgorithmswereeasilyunderstoodandimplemented;anditwaseasiertobuildhardwarethatsampledrectangularly[ 7 ].In1979,Mersereauauthoredaseminalpaperprovidinganintroductorytheoryto 12

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7 ].Heconrmedthattheregularhexagonalgridistheoptimalsamplinglatticefortwo-dimensionalisotropicallyband-limitedsignals.HealsodevelopedthehexagonaldiscreteFouriertransform(HDFT)butfoundtheFourierkerneltobenon-separableinhisrepresentation.Lastly,heshowedthatrecursivesystemsforhexagonalarrayscanbedevelopedandthatFIRlterscanbeeasilydesignedandimplemented.Heconcludedwithaclaimthathexagonalsamplingprovidesasavingsrangingfrom13to50percentinbothcomputationandstorageoverrectangularsamplingformanytwo-dimensionalsignalprocessingapplications. Oneofthenicethingsaboutrectangularsamplingisthatitproducesasimplearraythatiseasytostoreandprocess.Thisfactalmostcertainlycontributedtorectangularsamplingbeingusedoverhexagonalsamplingdespitethetheoreticalworkthatshowedthesuperiorityofhexagonalsampling.Thehexagonalgridcannotbeindexedasasimplearrayandrequiresaspecialaddressingmethod.In1980,Burtdevelopedsept-treeandpyramidaldatastructuresforprocessingbinaryhexagonallysampledimages[ 8 ].AccordingtovanRoessel[ 9 ],LucasandGibsonintroducedtheGeneralBalancedTernary(GBT)addressingapproachinanunpublisheddocumentin1970.However,thatdateisinquestion.Thesamedocument[ 10 ]iscitedbyGibsonherselfinanalreportforanAirForceOfceofScienticResearch(AFOSR)contractedeffort[ 11 ]andisdated1980inthatcitation.Regardless,GBTisahierarchicaladdressingmethodthatcanbeusedforn-dimensionaltessellationsofEuclideanspaces.Intwodimensions,itcanbeusedtoaddressthehexagonalgrid.Thetwo-dimensionalformofGBTistheonlyonebeingconsideredhere.GBTisbasedonahierarchyofcells,whereateachlevelthecellsareaggregatesofsevencellsfromthepreviouslevel,withthelowestlevelbeingthehexagonstilingtheplane[ 11 ].Eachhexagonisaddressedwithastringofbase-7digitswhereeachdigitindicatesthehexagon'spositionwithinthatlevelofthehierarchy.TherearealsoslightlymodiedversionsofGBTintheliteraturesuchasHIP,describedinMiddleton'sHexagonal

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12 ]andspiralarchitecture,describedinSheridan'sPh.D.dissertation[ 13 ].Crettezintroducedthesameaddressingschemeforahexagonalgridin1980[ 1 ]andcalledthearrangementofheptupletsaheptarchy.NotethatalloftheseGBT-basedapproacheswillbecollectivelyreferredtoasspiraladdressing.Alternatively,non-orthogonalaxescanbeusedtoaddressahexagonalgrid.Thereareseveralvariationsofthisapproachduetothedifferentpossibleorientationsoftheobliqueaxes.AdescriptionofthevariationsandcitationsoftheresearchersthathaveusedeachcanbefoundinMiddleton'sbook[ 12 ].Anotheraddressingapproachwasintroducedintheearly1990sbyHer[ 14 ].Her'sapproachisaninterestingthreecoordinatesystemthatplacestheimageonanobliqueplaneinthreedimensions,therebypreservingthenaturalsymmetryofthehexagonalgrid. Acoordinatesystemtorepresentthepointsofanimageshouldhaveacompactandefcientmachinerepresentation,beabletousethetechniquesoflinearalgebra,andbeabletotakeadvantageofthetoolsthathavebeendevelopedforprocessingrectangularlysampledimages.Noneoftheapproachesdescribedintheliteraturetodate,namelyspiraladdressingapproaches,variantsofobliqueaddressing,andHer'sapproach,possessalloftheseproperties.Snyderetal.[ 15 ]describeandanalyzeatypicalobliqueaddressingapproach,whichisreferredtoastheh2system.Suchasystemofobliquecoordinatesusuallyintroducesnegativeindicesinordertohaveanon-oblique(rectangular)image.Theh2systemavoidsnegativeindicesbykeepingtheimageanappropriatedistanceawayfromthecoordinateorigin,howeverthatintroducestheneedtokeeptrackoftheoffset.Inanattempttodemonstrateefcientmemoryallocation,theh2systemintroducesanadditionalsetofrowandcolumncoordinatestoaddressthepixelsthroughanarrayofpointerstoeachrow.However,therowandcolumncoordinatesdonotdirectlymaptotheh2coordinates,makingtheoverallsystemcumbersomeforimageprocessing.TheapproachintroducedbyHer[ 14 ]placestheimageonanobliqueplaneinthreedimensions.Theconstraintthat 14

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16 ][ 19 ]ontopicssuchasimagetranslationandrotation,uniformpartitioning,samplinggridconversions,andimageanalysis.Thesewereallbasedonusinghexagonallysampledimagesaddressedwithspiraladdressing.Condatetal.publishedaseriesofpapers[ 20 ][ 22 ]onsplinefunctionsforhexagonallysampleddata.Thesplinefunctionsrepresentthediscretedatainacontinuousfunctionthatcanberesampledtoperforminterpolation,gridconversion,etc.Theypublishedanotherseriesofpapers[ 23 ][ 25 ]onreversibleoperationsthatcanbeperformedvia1-Dlteringtoapplyasequenceofshearingoperations.Theydemonstratereversiblegridconversionandreversibleimagerotation.Thisauthoralsopublishedapaper[ 26 ]thatintroducesafamilyofwaveletsonthehexagonalgrid.Jiangpublishedseveralpapers[ 27 ][ 29 ]aboutthedesignoflterbanksforwaveletsonhexagonalgrids.FailleandPetroupublishedapaper[ 30 ]aboutusingsplinestoreconstructimagesfromirregularlyplacedsamplesonhexagonalgrids.Gardineretal.publishedacoupleofpapers[ 31 ],[ 32 ]ondesigninggradientoperatorsforhexagonallysampledimages.PuschelandRottelerapplied 15

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33 ],[ 34 ].Andlastly,VinceandZhengshowedhowtocomputethediscreteFouriertransform(DFT)usinggrouptheoryandGBT[ 35 ]. 36 ].Thedissertationdoesnotdiscusstheaddressingscheme,butratherusesahexagonalgridtodevelopafastbinningalgorithmandsubsequentsmoothingoperations.Hegenerallyusesthethreeprincipallatticelinesandperformssmoothingalongthoselatticelines.ThebinningalgorithmconvertsarbitraryCartesiancoordinatestothenearesthexagonalgridpoint.ThisisalsothegoalofaCartesiantoASAconversionalgorithm.Therefore,theYangbinnningalgorithmwasimplementedandcomparedtotheoriginalalgorithmdevelopedforASAwithinconclusiveresults:oneranfasterinMatlabwhiletheotherranfasterwithPython.Beforeaformalanalysiswascompleted,itwasrealizedthatahybridversioncouldbefasterthanbothoftheoriginalalgorithms.Indeed,thatturnsouttobethecaseandthehybridversionhasbeenadoptedastheCartesiantoASAconversionalgorithm.RefertoSection 2.5 foradditionaldetails.Barbourusesahexagonalgridinpartofhisthesis[ 37 ]onpathnding.Headdressesthehexagonalgridusingstandardrowandcolumnaddressingwhereoneofthecoordinateaxesfollowsasawtoothpathratherthanastraightline.Thisisnotausefuladdressingschemefordigitalimageprocessing.Boydalsousesthesameschemeinhisthesis[ 38 ]butreferstoitasatexgrid.ThereissomeambiguitywiththatnamebecauseBarbouralsousedastructurethathereferredtoasatexgrid,butitwasdescribedasbeingsquaretilesarrangedinahexagonalpattern,similartothebrickwallstructuredescribedinSection 1.5.3 andshowninFigure 1-4 .Guenetteusesahierarchicalhexagonalgridstructureforglobalmappingtechniquesinhisthesis[ 39 ].Thisstructureisindexedsimilarlytothespiraladdressingtechniques, 16

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40 ]andZheng[ 41 ]bothusehexagonalgridsandobliqueaddressingintheirdissertations. 1. Thelackofanefcientaddressingscheme 2. Thelackofimagersthatsamplehexagonally 3. Thelackofdisplaysforhexagonallysampledimages Themainportionofthisdocumentintroducesanapproachthatsolvesitem1).Thefollowingsectionswilldiscusssolutionstoitems2)and3).Itisimportanttonotethatthesearenottheonlyhurdlestotheuseofhexagonalsampling.Thistechnologynowfacesacultureofrectangularsamplingthathasexistedfordecades.Evenifitcanbeshowntobesuperior,itmaybedifculttogainafootholdinthemarketplaceandbreakoutoftheexistingculture.Asanexample,considerthebattlebetweentheVHSandBetamaxvideotapeformats.Betamaxisconsideredbymanytobeasuperiorformat,butforvariousreasonstheVHSformatwonoutinthemarketplaceafteraroughlytenyearbattle.Forhexagonalsampling,thebattlemaybeevenmoredifcultsincerectangularsamplingisalreadysormlyestablishedindigitalimaging. 17

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Itisassumedthatthesourceimagewassampledwithunitareasquaresampling.Thesourceimageisthenresampledtoconvertunitsquarepixelsintoappropriaterectangles.Theimageisretesselatedbyconglomerating36pixelsintotheappropriateshape(hexagonorsquare)asshowninFigure 1-1 .AnexampleoftheresultsofthisprocessisshowninFigure 1-2 Figure1-1. Retesselatingasourceimagetoproduceroughlyequivalenthexagonallysampledandrectangularlysampledimages. 18

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Exampleofconvertingasourceimageintoequivalenthexagonallyandrectangularlysampledimages. 42 ]andtheprototypedescribedinHauschild'spaper[ 43 ],buttherearenocommerciallyavailablehexagonallysamplingimagers.Thereisnophysicalreasonwhyitcan'tbedone,ratheralackofmotivationtodoso,arguablyduetotheabsenceofanefcientaddressingscheme. Oneofthemaingoalsofthisresearcheffortistoconcretelyshowthattheadvantagesofhexagonalsamplingcanberealizedwithlittleornomoreprocessingcomplexitythanisneededforprocessingequivalentrectangularlysampledimages.Tothatend,atrapezoidalimagerhasbeendevelopedthatallowssimultaneoushexagonalandrectangularsamplingofascene.Byusingappropriatetrapezoidalunitcells,theunitscanbecombinedtoformeitherhexagonalorrectangularpixels.Thisproducestwoimagesforeachframe,wheretheonlydifferencebetweenthetwoisthesamplingmethod. Figure 1-3 showsthepixellayoutofaportionofthetrapezoidalimagesensor.Thebrightgreenareasarethephotosensitiveareas,approximatingtrapezoids.Thesizeof 19

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Figure1-3. Partofthetrapezoidalimagesensordesign. 20

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12 ],[ 44 ],whichusesfoursquarepixelsperhexagonalpixeltodisplaythehexagonallysampledimage.Thisresultsinafactoroffourreductionintheresolutionanddisplaysthehexagonsasrectangles,butproducesqualitativelygoodresults.Thebrickwalltechniqueworksbyusinga22blockofsquarepixelstorepresenteachhexagonwitheveryotherrowshiftedbyonepixelasdepictedinFigure 1-4 Figure1-4. Depictionofbrickwalldisplaytechnique. TheleftsideofFigure 1-5 showsanimage[ 45 ]ofsomecommoncolor(RGB)displaygeometries.Thetwoupperimagesareofcathoderaytubes(CRT),adisplaytechnologythatisseeminglybecomingobsolete.Itisinteresting,however,tonotethehexagonalarrangementofthepixelsintheseimages.Thetwolowerimagesareofliquidcrystaldisplays(LCDs).TheXO-1LCDcouldemploythebrickwalltechniqueonthesub-pixelscreatingapseudo-hexagonalRGBdisplay,butthecolordistributionperpixelwouldbemismatched,andwouldneedtobecorrected.ThestandardLCDdesign 21

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1-5 tocreateahexagonalRGBdisplay.Notethatthescopeofthiseffortislimitedtograyscaleimages,thoughitwouldbestraightforwardtoextendASAaddressingtocolorimages. 22

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Fourcommondisplaygeometries(left)[ 45 ]andtwopotentialhexagonalRGBdisplaygeometries(right).

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2-1 .Anintegerindexingcouldbeachievediftherequirementthatbothaxesusethesameunitofmeasurewereremoved,however,theresultingarraywouldresembleacheckerboardsincesomeindiceswouldnotcorrespondtoimagelocations. TheASAapproachisbasedonrepresentingthehexagonalgridastworectangulararrayswhichcanbeindividuallyindexedbyinteger-valuedrowandcolumnindices.Thehexagonalgridissimplyseparatedintorectangulararraysbytakingeveryotherrowasonearrayandtheremainingrowsastheotherarray,astheexampleinFigure 2-2 demonstrates.Thearraysaredistinguishedusingasinglebinarycoordinatesothatafulladdressforanypointonthehexagonalgridisuniquelyrepresentedbythreecoordinates wherethecoordinatesrepresentthearray,row,andcolumnrespectively.Notethatthisdomainisvalidforahexagonaltilingoftheentireinniteplane.Hexagonalimages,however,willtypicallyberestrictedtohavingnon-negativerowandcolumnindices. 24

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Thespacingisimportanttomaintainingthenaturalsymmetryofthehexagonalgrid. Figure2-2. ExampleofhowthehexagonalgridisseparatedintotwoarraysusingASA. 25

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1.2 .Itmustbespeciedwhethertheimmediateneighborsarethefourthatshareasidewiththegivenpixeloralleightthatareadjacent.Regardless,theiraddressescaneasilybecalculatedbyasimple1offsetfromthegivenpixel'scoordinates.Foranyofthespiraladdressingapproaches,thistaskseemstorequireasimpleadditionforeachofthesixneighbors,however,thespecialmathematicsdenedfortheseapproachesrequiresalook-uptableoperationforeachpairofdigitsbeingaddedwiththepossibilityofdouble-digitcarries,leadingtoacomputationalcomplexityofO((logN)2),whereNisthenumberofpixelsintheimage.UsingASA,thereisnoambiguityandnospecialmathematicsarerequired.TheneighborsofagivenpixelcanbecalculatedbysimpleoffsetsasshowninFigure 2-3 Figure2-3. Theaddressesofthenearestneighborsofpixel(a,r,c)areeasilydeterminedusingtheseexpressions. 2-1 liststheninepossibleformsofanarraysetgiventhattheorigin(inCartesiancoordinates)isalwaysatASAaddress(0,0,0).RefertoSection 2.5 foranexplanationoftheB1designationinTable 2-1 .Workingwithasingleformforimagessimplies 26

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Table2-1. FormsofArraySets CartesianCartesianCoordinates(B1)Coordinates(B1)Form0-ArraySizeof(0,0,0)1-ArraySizeof(1,0,0) 1RxC(0,0)RxC1 2,p 22RxC(0,0)R-1xC1 2,p 23RxC(0,0)R-1xC-11 2,p 24RxC(0,0)RxC-11 2,p 25RxC(0,0)RxC+11 2,p 26RxC(0,0)R-1xC+11 2,p 27RxC(0,0)R+1xC1 2,p 28RxC(0,0)R+1xC-11 2,p 29RxC(0,0)R+1xC+11 2,p 2 2-4 .TheimageaspectratioinCartesianspaceis2j:(p 2)N.Forexample,a443512formatimagewouldhaveanapproximately4:3aspectratio(inCartesianspace)andthememoryaddresseswouldbeoftheformXXXXXXXXXXXXXXRRRRRRRRACCCCCCCCC,witheachmemorylocationholdingthecorrespondingpixel'svalue.Readingthedatainasequentialmannerprovidesaprogressivescansweepofthedata. 27

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Illustrationofrow-majororderstorage. 1. 2. 3. theanglebetweenr1andr2isnot0or180. Letd=kr1k=kr2kandA=ff0,1gZZgandC=fRRg.StartingwiththeconversionfromASAtoCartesian,thefunctionfd:A7!Cisasimplematrixmultiplication.Let 2(d)(p

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ConvertingfromCartesiantoASAisalittlemoreinvolvedbutstillveryefcient.Thefunctiongd:C7!Aisdenedbyaprocess.ThisprocessispartiallyinspiredfromYang'sfastbinningalgorithm[ 36 ].ItwasdeterminedthatahybridalgorithmusingYang'sfastbinningalgorithmandtheoriginalCartesiantoASAconversionalgorithmcouldbefasterthaneitheroriginalalgorithm.AlthoughallhaveO(N)complexity,ananalysisoftheaveragenumberofoperationsperformedbythealgorithmswascompletedonYang'sbinningalgorithmandthehybridversion.Theportionsofthealgorithmsthatareequivalent(scalingthecoordinates,round/ooroperations,andconvertingcoordinatestoASA)werenotincludedintheanalysis.Notethattheoor()andround()functionsareassumedtobeequivalentintermsofruntime.TheresultsareshowninTable 2-2 .Themeanruntimes(10,000runs)forYang'sfastbinningalgorithm,theASAoriginalCartesiantoASAalgorithm,andthehybridCartesiantoASAalgorithmusingthecodeprovidedinAppendix B were7.6s,5.6s,and4.5s,respectively. Table2-2. AverageNumberofOperationsExecuted(Yangvs.Hybrid) AlgorithmComparisonsAdd/subtractMultiply Yang261Hybrid2.52.50.5 Thehybridalgorithmproceedsasfollows.First,thexandydimensionsoftheCartesianspacearescaledasxs=2

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2-5 showsarepresentationofthisspacewhere(xs,ys)coordinatesthatfallwithinthegreenboxeshavebeenroundedtothehexagoncentersandtheonesthatfallwithintheredboxeshavebeenroundedtotheboundarybetweentwohexagons.Notice Figure2-5. Representationofxs-ysspace. thatitiseasytodistinguishbetweenthesetwocasesbycheckingifxrandyrhavethesameevennessoroddness.Iftheyarethesame,then(xr,yr)isahexagoncenterand(xs,ys)isinthathexagon,otherwiseitisaboundary.Anadditionalstepisnecessaryfortheboundarycasetodeterminewhichofthefoursurroundinghexagonscontainstheoriginalpoint.ReferringtoFigure 2-6 ,itiseasytodeterminewhichquadrant(I,II,III,orIV)(xs,ys)fallsinbycomparingxstoxrandystoyr.Oncethequadrantisknown,itisamatterofdeterminingwhether(xs,ys)liesaboveorbelowthelinedividingthatquadrant.Duetothegeometryofthespace,theslopeofthedividinglineisalwaysknownandis1 3,dependingonthequadrant.Apointonthelineisalsoalwaysknownandis(xr,yr1 3),againdependingonthequadrant.Usingtheslopeandthepointtocalculatetheequationoftheline,itiseasytodeterminewhichhexagoncontainstheoriginalpointandoffset(xr,yr)appropriately.Lastly,convert(xr,yr)toASAcoordinates. 30

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Illustrationofhowtheboundarycaseisresolved. ThismethodforconvertingasetofCartesiancoordinates,(x,y),toASAcoordinates,(a,r,c),givend,isprovidedinpseudo-codebelow.

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2.6.1EuclideanDistance ThisformulaiseasilyconvertedforusewithASAcoordinatesbysubstitutingtheexpressionsforxandyfromEquation 2 .TheresultingformulafortheEuclideandistancebetweentwopointsrepresentedinASAcoordinates,p1=(a1,r1,c1)andp2=(a2,r2,c2),isde(p1,p2)=(d)q 2(a1a2)+(c1c2))2+(3)(1 2(a1a2)+(r1r2))2. ThismethodofsubstitutingtheexpressionsforxandyfromEquation 2 intoexistingformulasisaneasywaytoconvertexistingusefulmethodsintoASAcoordinaterepresentations. 46 ]developedtheequationsforthecity-block,ord6,distanceonahexagonalgridusingobliqueaddressing.Theyshowedthat 34

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20p 21CCA0B@pq1CA=0BB@p+q 21CCA.(2) SolvingEquation 2 forpandqintermsofxandyyieldsp=xy Now,substitutingtheASAexpressionsforxandy(withd=1)fromEquation 2 andsimplifyingresultsinp=crq=a+2r. Next,leti=c1r1h=c2r2j=a1+2r1k=a2+2r2 2 providesthecity-blockdistancebetweentwopointsrepresentedinASAcoordinates,p1=(a1,r1,c1)andp2=(a2,r2,c2), ThisisanexampleofhowonecaneasilyconvertequationsdevelopedusingobliqueaddressingintoASAcoordinaterepresentations. 35

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Anegationoperator,(ASA,),canbedenedasp0BBBB@araca1CCCCA Scalarmultiplicationcanbedenedfornon-negativeintegerscalarmultipliers,k2N,askp0BBBB@(ak)mod2kr+(a)k 36

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PropertiesofArraySetAddressing(ASA)Vectors PropertySignicance Commutativityofadditionp1+p2=p2+p1Associativityofadditionp1+(p2+p3)=(p1+p2)+p3Identityelementofaddition902S:p+0=p,8p2SInverseelementsofaddition9q2S:p+q=0,8p2SDistributivityofscalarmultiplicationk(p+q)=kp+kq(w.r.t.vectoraddition)Distributivityofscalarmultiplication(k+j)p=kp+jp(w.r.t.scalaraddition)Compatibilityofscalarmultiplicationk(jp)=(kj)p(withmultiplicationofscalars)Identityelementofscalarmultiplication1p=p 2-3 ,makingASAamoduleovertheringofintegers,oraZ-module.Notethatinanearlierpaper[ 47 ]itwaserroneouslystatedthatASAformedavectorspace.Thedifferencebetweenamoduleandavectorspaceisthatthescalarsofavectorspacearerequiredtolieinaeldratherthananarbitraryring.Forthefollowingsections,refertoSection A.1 ofAppendix A forthedomaindenitionsandidentitiesused.ProofsoftheremainingpropertiescanbefoundinAppendix A Lemma1. Proof.

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1 ),x^y+((xy)^z)=(x^y)_((xy)^z)=((x^y)_(xy))^(((x^y)_z)=((x^y)_((x_y)^:(x^y)))^((x^y)_z)=((x^y)_(x_y)^:(x^y))^((x^y)_z)=(x_y)^((x^y)_z)=(x_y)^(x_z)^(y_z). Proof. 38

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(2) TheequalityofEquations 2 and 2 isprovenbyconsideringthecoordinatesseparately.First,becauseexclusive-orisassociative,(a1a2)a3=a1a2a3=a1(a2a3). Second,byLemma 2 ,(a1^a2)+((a1a2)^a3)=(a1^a2)_(a1^a3)_(a2^a3)=(a2^a3)_(a2^a1)_(a3^a1)=(a2^a3)+((a2a3)^a1)=(a2^a3)+(a1^(a2a3)). Thusr1+r2+r3+(a1^a2)+((a1a2)^a3)=r1+r2+r3+(a2^a3)+(a1^(a2a3)). Finally,c1,c2,c3canbesubstitutedforr1,r2,r3yieldingc1+c2+c3+(a1^a2)+((a1a2)^a3)=c1+c2+c3+(a2^a3)+(a1^(a2a3)). Theorem2.2.

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andkp+jp=0BBBB@(ak)mod2kr+(a)k (2) TheequalityofEquations 2 and 2 isprovenbyconsideringthecoordinatesseparately.First,(a(k+j))mod2=(ak+aj)mod2=((ak)mod2+(aj)mod2)mod2.(Identity 3 ) Second, 2(Identity 1 )=(a)k+j(djdk) 2(Identity 4 )=(a)k+j(dk+dj)+(2)(dj^dk) 2(Identity 6 )=(a)kdk 1 )=(a)k 8 ) 40

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2 ) Thus(k+j)r+(a)k+j Finally,bysubstitutingcforr,(k+j)c+(a)k+j Theorem2.3. Proof. and TheequalityofEquations 2 and 2 isprovenbyconsideringthecoordinatesseparately.First,((k)((aj)mod2))mod2=a^(jmod2)^(kmod2)(Identity 2 )=a^(dk^dj)=a^(kj)mod2(Identity 5 ) 41

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2 ) Second,(ka)j 1 )=(ka)jdj 2 )=(ka)jdj 8 )=(a)kjkdj+kdjdjdk 2(Identity 8 )=(a)kj(kj)mod2 2(Identity 5 )=(a)kj 1 ) Thuskjr+(ka)j Finally,bysubstitutingcforr,kjc+(ka)j 12 ]providePythonsourcecodeforsomebasiccoordinateoperations,allowingadirectcomparisonofASAtoHIPfortheseoperations.RecallfromSection 1.2 thatHIPisoneofthespiraladdressingtechniques.TheresultsinTable 2-4 arethemeanruntimesof10,000operationsperformedonrandomlyselectedaddresses.ThesourcecodeofthecomparisoncanbefoundinAppendix B .Table 2-5 comparesthecomplexityofeachoperation.The 42

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RunTimeComparisonBetweenASAandHIP(s,mean(std)) OperationHIPASARunTimeRatio Address(vector)Addition23.85(3.15)2.11(0.97)11.28Address(vector)Subtraction33.98(3.56)2.56(0.47)13.28ScalarMultiplication6652.08(4076.89)3.73(0.73)1782.20GetNearestNeighborAddresses118.94(10.49)3.31(0.75)35.89CalculateEuclideanDistance15.83(2.43)2.73(0.56)5.79ConversionfromCartesian9189.68(3784.79)4.48(1.13)2052.31ConversiontoCartesian8.30(1.85)2.12(0.93)3.92 Table2-5. ComplexityComparisonBetweenASAandHIP OperationHIPASA Address(vector)AdditionO((logN)2)O(1)Address(vector)SubtractionO((logN)2)O(1)ScalarMultiplicationO(N(logN)2)O(1)GetNearestNeighborAddressesO((logN)2)O(1)CalculateEuclideanDistanceO(logN)O(1)ConversionfromCartesianO(N(logN)2)O(1)ConversiontoCartesianO(logN)O(1) B 43

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wherethesubscriptindicatesthearraycoordinate.Figure 3-1 showsthisoperationpictorially. ASAconvolutionisnomorecomplexthanperformingaconvolutiononarectan-gularlysampledimageandtheconvolutionsontheindividualarrayscouldbedoneinparallel.Forexample,assumeanNNrectangularlysampledimageandanNNhexagonallysampledimage.Also,assumeazerovaluedbordersurroundstheimagessothatallpixelsusethefullconvolutionmask.Aconvolutionoperationontherectangularlysampledimageusingammconvolutionmaskrequiresm2N2multiplicationsand(m21)N2additions.Performingaconvolutiononthehexagonally 44

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AnexampleofhowconvolutionsareperformedusingASA. Figure3-2. Agraphcomparingthesizesofnearestneighborneighborhoods(hexagonalvs.rectangular). 45

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and(2)m2 exactlythesamenumberofoperationsastherectangularlysampledimage.However,sinceahexagonalneighborhoodis7pixels(nearestneighbors)andarectangularneighborhoodis9pixels(nearestneighbors,8-wayconnected),onewouldusea7-pointmaskinplaceofthe9-pointmask,makingtheASAconvolutionslightlyfasterforneighborhood-basedlters.Additionally,astheneighborhoodsizeincreases(including2ndnearestneighbors,then3rdnearest,etc.),thegapbetweenthesizesofthemasksincreasesaswell,asshowninFigure 3-2 .SoASAconvolutionincreasinglyimprovesinperformancecomparedtorectangularconvolutionasthesizeoftheneighborhoodincreases.Fromtheaboveanalysis,itiseasytoseethatthecomplexityofASAconvolutionisO(n),wheren=N2isthenumberofpixelsintheinputimage,assumingn>>m2. 3-3 showsanexample,wheretheblacklinesindicatetheintensityvaluesoftheneighboringpixelsandthegoalistodeterminethevaluesofthexandycoefcients,eandfrespectively,ofthebluettedplane.ThederivationparallelsthatofSnyder[ 15 ],whoderivedconvolutionmasksforlocalgradientestimationusingan 46

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Exampleofttingaplanetoahexagonalneighborhood. Figure3-4. Depictionofhexagonalneighborhoodindexing. obliqueaddressingscheme.TheconvolutionmasksresultingfromtheASAderivationareequivalenttoSnyder'sresults. Letthesubscriptkindexthepixelsofahexagonalneighborhoodwithk=0beingthecenterpixel,k=1beingitsneighborimmediatelytotheright,andkincreasingcounter-clockwisearoundthecenterasdepictedinFigure 3-4 .Thenpk=(ak,rk,ck)T2ASA,k=1...6,aretheneighboringpixelsofp0.LetB1=0B@1 201p 2p

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AndletZk=B1pk=0B@XkYk1CA=0B@ax 3-3 )anddenetheerrorfunctionas DifferentiatingEquation 3 withrespecttoAresultsin SetttingEquation 3 equaltozeroresultsin whereS=6Xk=1ZkZTk

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(3) Equation 3 canberearrangedtoyield Therefore,A=S16Xk=1IkZk=S10B@P6k=1IkXkP6k=1IkYk1CA. (3) Assumingp0=(0,0,0)T,thepkcoordinatevaluescanbesubstitutedintoXk=ak 3 resultinginS=0B@30031CA. Thefamiliarformulafortheinverseofa22matrixyieldsS1=0B@1 3001 31CA. SubstitutingS1,Xk,andYkintoEquation 3 yields 3I1+1 6I21 6I31 3I41 6I5+1 6I6p 2I2p 2I3+p 2I5+p 2I61CA.(3) 49

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3 givesexpressionsforeandfwhichareobviouslyimplementableasconvolutions.Theresultant(scaled)convolutionmasksforestimatingthelocalgradientsinthexandydirectionsareFx=0BBBB@11202111CCCCA shownpictoriallyinFigure 3-5 .Also,becauseofthenaturalsymmetryofhexagonalgrids,thesemaskscanberotatedtoprovidegradientestimatesalongsixdifferentaxes. Figure3-5. Convolutionmasksthatestimatelocalgradientsandthesixaxesalongwhichgradientscanbeestimated. 50

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48 ]edgedetectoronanequivalentrectangularlysampledimage.Gradient-basededgedetectionproceedsbyestimatingthelocalgradientsinthexandydirections,calculatingthemagnitudesoftheresultantgradientvectors,andthresholdingthemagnitudetodeterminetheedges.LetIbetheASArepresentationofthehexagonallysampledinputimage.Usingthegradientestimationlters,FxandFy,fromtheprevioussection,letGx=FxI 51

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49 ]toselectedgelocations.Figures 3-6 3-7 ,and 3-8 showtheresultsofrunninggradient-basededgedetectiononthreedifferentimages.Theimageswereobtainedfromhigh-resolutionsquare-sampledimagesusingthetechniquedescribedinSection 1.5.1 .Eachgureshowstheinputimageatthetop,theresultantedgemapinthemiddle,andthegradientmagnitudeimageatthebottom.ThegradientmagnitudeimageincludesacolorbarwiththethresholdselectedbyOtsu'salgorithmgivenandindicatedbyanarrow.Thehexagonallysampledimagesareontheleftandtherectangularlysampledimagesareontheright.Theredoesnotappeartobeanysignicantdifferencesbetweentheresultantedgemapsofthesethreeexamples. 50 ]aspartofamodelofmammalianearlyvisionprocessing.Theirrststepistoselectasmoothingltertoreducetherangeofscalesofintensitychanges.TheyselectaGaussiansmoothingltertooptimizethetrade-offbetweensmooth,band-limitedfrequencyresponseandgoodspatiallocalization.Next,theydeterminethatthebestwaytodetectintensitychanges(edges)istondzero-crossingsinthesecond(directional)derivativeoftheintensityimage,intheappropriatedirection.Ofcoursetheappropriatedirectionsareunknownapriori,soratherthancomputemultipledirectionalderivatives,theyreducethecomputationalburdenbyselectingtheLaplacian(anorientation-independentsecondderivativeoperator)instead.Theproblem(forrectangularly-sampledimages)isthusreducedtondingthezero-crossingsofL(x,y)=4(G(x,y)I(x,y))=(4G(x,y))I(x,y) 52

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Hexagonallysampled(left)andrectangularlysampled(right)imagesusedingradient-basededgedetectorexample. 53

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Hexagonallysampled(left)andrectangularlysampled(right)imagesusedingradient-basededgedetectorexample. 54

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Hexagonallysampled(left)andrectangularlysampled(right)imagesusedingradient-basededgedetectorexample. 55

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@x(G(x,y)I(x,y))=@ @xZZI(u,v)G(xu,yv)dudv=ZZI(u,v)@ @xG(xu,yv)dudv=@ @xG(x,y)I(x,y). Hence,theLoGltercanbepre-computedasLoG(x,y)=4G(x,y)=@2 andtheLoGimageisobtainedthrougharegularconvolutionoperation,L(x,y)=LoG(x,y)I(x,y). Similarly,forhexagonallysampledimages,theLoGltercanbepre-computedinASAspacebysubstituting(a 3 ,LoG(a,r,c)=4G(a,r,c)=1 TheASArepresentationoftheLoGimageisobtainedthroughanASAconvolutionoperationasdescribedinSection 3.1 ,L(a,r,c)=LoG(a,r,c)I(a,r,c), andtheedgesarefoundbylocatingthezero-crossings.Thezero-crossingsoccurwhereanegativevaluedpixelhasoneormorepositivevaluedneighbors(orviceversa).Figures 3-9 3-10 ,and 3-11 showtheresultsofrunningLoGedgedetectiononthesamethreeimagesusedinthegradient-basededgedetection.Eachgureshowstheinput 56

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51 ]isbasedonthegoalofoptimizingperformancewithregardtothefollowingcriteria: 1. Lowerrorrateedgedetection 2. Goodedgelocalization 3. Singularresponsetoasingleedge Cannydevelopstheequationsandderivestheoptimalstepedgeoperatorinonedimension.ItisthennotedthattheoptimaloperatorisverysimilartotherstderivativeofaGaussian.AquantitativecomparisonshowsthattherstderivativeofaGaussianoperatorisabout20%worsethantheoptimaloperatorforcriteria 1 and 2 ,andabout10%worseforcriterion 3 .SincethisperformancedegradationisconsideredminorforrealimagesandbecausethederivativeofaGaussian(DoG)operatorcanbecomputedintwodimensionsmuchmoreeasilythantheoptimaloperator,theapproximationisselectedovertheoptimal.TheCannyedgedetectionmethodconsistsofseveralsteps: 1. FiltertheimagewithaGaussiansmoothingltertoreducenoise 2. ApplytheDoGoperatorstogetlocaldirectionalgradientestimates 3. Calculatethemagnitudeanddirectionofthelocalgradientestimates 4. Performnon-maximasuppressionusinggradientdirections 5. Applyhysteresisthresholdsusinggradientmagnitudes TherststepisperformedviaconvolutionwithaGaussiansmoothinglterIs(a,r,c)=G(a,r,c)I(a,r,c) 57

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Hexagonallysampled(left)andrectangularlysampled(right)imagesusedinLoGedgedetectorexample. 58

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Hexagonallysampled(left)andrectangularlysampled(right)imagesusedinLoGedgedetectorexample. 59

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Hexagonallysampled(left)andrectangularlysampled(right)imagesusedinLoGedgedetectorexample. 60

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whereGx(a,r,c)=1 ThenthemagnitudeanddirectionofeachlocalgradientarecalculatedbyM(a,r,c)=p respectively.ThegradientdirectionsarethenquantizedintosixdirectionsasshowninFigure 3-12 .Notethatwithrectangularlysampledimages,thegradientdirectionsarequantizedintofourdirectionsonly.Thisimprovedangularresolutionisanotherbenetofhexagonalsampling.Non-maximasuppressionisusedtothinthecandidateedgesbykeepingonlythepixelsthataregreaterthantheirneighborsonbothsidesalongthe(quantized)gradientdirection.Andlastly,thresholdsareappliedtothegradientmagnitudesusinghysteresis.Anycandidateedgepixelthatexceedsthehighthresholdisautomaticallyaddedtotheedgemapandlabeledastrongedge.Anypixelthatexceedsthelowthresholdandisconnectedtoastrongedgepixelisconsideredaweakedgepixelandisalsoaddedtotheedgemap.Inthiscontext,connectedmeans 61

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Localgradientdirectionsarequantizedintosixdirections. thatthereisapathfromtheweakedgepixeltoastrongedgepixelconsistingofweakedgepixels.Recallthatthereisnoambiguityaboutconnectednesswithhexagonallysampledimages,thussimplifyingthisstepcomparedtorectangularlysampledimages.Thishysteresisapproachreducesthestreakingphenomenonthatiscommonwithedgedetectionmethodsthatusethresholding.Cannyusesaxedpercentageofthehistogramtoselectthehighthresholdandclaimsitshouldbe2or3timesthelowthreshold.Otsu'salgorithm[ 49 ]canalsobeusedtoselectthehighthresholdwiththelowthresholdsetto40%ofthehigh.Figures 3-13 3-14 ,and 3-15 showtheresultsofrunningCannyedgedetectiononthesamethreeimagesusedinthepreviousedgedetectionmethods.Eachgureshowstheinputimageatthetop,theresultantedgemapinthemiddle,andthemagnitudeimagefollowingnon-maximasuppressionatthebottomwithacolorbar.Thehexagonallysampledimagesareontheleftandtherectangularlysampledimagesareontheright.FortheCannymethod,therearedifferencesbetweenthehexagonallyandrectangularlysamplededgemaps.Namely,thehexagonallysampledversiondetectsnoticeablymoreedges.ComparethelocationscircledinredontheedgemapsinFigures 3-13 3-14 ,and 3-15 toseesomeexamples. 62

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Hexagonallysampled(left)andrectangularlysampled(right)imagesusedinCannyedgedetectorexample. 63

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Hexagonallysampled(left)andrectangularlysampled(right)imagesusedinCannyedgedetectorexample. 64

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Hexagonallysampled(left)andrectangularlysampled(right)imagesusedinCannyedgedetectorexample. 65

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12 ],downsamplingproducesa7reductioninresolutionandintroducesa40.9rotationoftheimageperiteration.DownsamplinginASAproducesa4reductioninresolutionandintroducesnorotation.Itisanefcientoperationconsistingofaconvolutionwithananti-aliasinglterfollowedbysubsamplingtoformthedownsampledimageasshowninFigure 3-16 .Notethatforwaveletdecompositions,theanti-aliasinglterisgenerallyreplacedbyanFIRlterbank.LetIbetheNNintensityimagetobedownsampledandletF=0BBBB@1 81 81 81 41 81 81 81CCCCA. betheanti-aliasinglter.Then,H=FIisthelteredimage.LetDbethedownsampledimageformedfromtheselectedpixelsofH.Thefollowingcodesnipetshowsthemethodforsubsamplingthesmoothedimage.

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Figure3-16. DownsamplingwithASA.Afterconvolvingtheoriginalimagewiththeanti-aliasinglter,thelightgraypixelsformthe0-arrayandthedarkgraypixelsformthe1-array. 67

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4.1.1TheHexagonalDiscreteFourierTransform(HDFT) 7 ],MersereaudenestheHDFTanditsinverseforahexagonallysampledtwo-dimensional(2-D)signalx(n1,n2)as 2N1+N2+n2k2 and 2N1+N2+n2k2 whereR(N1,N2)istheregionofsupportforx(n1,n2),(n1,n2),(k1,k2)2R(N1,N2),K=N2(2N1+N2),and(n1,n2)and(k1,k2)arecoordinatesofobliqueaddressingschemes.N1andN2arethehexagonalperiodsofthehexagonallyperiodicextendedversionofx(n1,n2).UsingthemethodofSection 2.6.2 toconvertfromobliquetoASAcoordinates,itiseasytoshowthatn1=a+r+cn2=a+2r 68

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Therefore,equations 4 and 4 canberewrittenas 2m+(a+2r)(b+2s) and 2mnXbXsXdX(b,s,d)expj(a+2c)(b+2d) 2m+(a+2r)(b+2s) whereR(A,R,C)istheregionofsupportforx(a,r,c)and(a,r,c),(b,s,d)2R(A,R,C).Thus,equations 4 and 4 aretheHDFTanditsinverseforhexagonallysampledimagesaddressedwithASA. 7 ]whenconsideringthesameapproachsincetheFourierkernelisnolongerseparableasinequations 4 and 4 ,duetotheproducttermcontainingbothcoordinatesintheexponent.However,noticeinequations 4 and 4 thattherowcoordinates,rands,areseparatefromthecolumncoordinates,candd,meaningthatthecoordinatesdonotappeartogetherinaproducttermintheexponent.Therefore,theFourierkernelisseparableinASAcoordinates!Someresearchershavedevisedothermethodstomake 69

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52 ]enforcedconstraintsonthesamplingpointgeneratingandperiodicextensionvectorsystemstoderivecoordinatesystemswhichleadtoaseparablekernelintheHDFT.However,theconstraintsleadalsotocoordinatesystemsthathaveobliquebasisvectorswhichexacerbatetheproblemsofobliqueaddressingschemesasdescribedinSection 1.2 .AlimandMoeller[ 53 ]achieveseparabilitybyassumingrectangularperiodicityinthespatialdomainwhichcorrespondstorectangularsamplinginthefrequencydomain.Hence,thisapproachdisallowstakingadvantageofthebenetsprovidedbythehexagonalgridinthefrequencydomain.ASA,ontheotherhand,providesanelegantcoordinatesystemthatusesthehexagonalgridinbothspatialandfrequencydomainsandleadstoanefcientFFTbasedontheseparabilityoftheFourierkernel. LetgKa(b,r,d)=Xcx(a,r,c)expj2(c)(b+2d) Then,X(b,s,d)=XaXrXcx(a,r,c)expj(a+2c)(b+2d) 2m+(a+2r)(b+2s) 2m+(2r)(b+2s) 2m+(1+2r)(b+2s) 2m#expj(1+2r)(b+2s)

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2mXrg1(b,r,d)expj(1+2r)(b+2s) nXrg1(b,r,d)expj(2r)(b+2s) LetfKa(b,s,d)=Xrga(b,r,d)expj2(r)(b+2s) Then, nf1(b,s,d).(4) EachofthefunctionsgaandfacanbecalculatedinO(logN)time,makingequation 4 anHFFT. 2m=m1Xc=0x(a,r,c)expj2(c)(d) (4) Equation 4 showsthatga(0,r,d)isthestandard1-DFFToftherowsofx.Therefore,anystandardFFTalgorithmmaybeusedtocalculatethe0arraysofg0andg1fromtherowsofx.LetWxK=expj2 andassumen,m2f2i:iisapositiveintegerg.Applyingthestandarddecimation-in-space(DIS)radix-2approach(usuallycalledadecimation-in-timeapproach,butsincetheseareimages,DISisusedhereinstead)resultsinthestandardDISradix-2FFT 71

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wheregma,oddiscomprisedoftheoddcolumnsofg2maand,similarly,gma,eveniscomprisedoftheevencolumnsofg2ma.Noticethatequation 4 isarecursivedenitionofga(0,r,d)thatcanbecomputedinO(logN)time. Next,considerthe1arrayofga,ga(1,r,d)=m1Xc=0x(a,r,c)expj2(c)(2d+1) 2m=m1Xc=0x(a,r,c)expj2(c)d+1 2 (4) Equation 4 isnotthestandardFouriertransformduetotheconstantoffsetappliedtothedvariable.Equation 4 willbereferredtoasthenon-standardtransform1(NST1).TheDISradix-2approachcanstillbeappliedasfollows,ga(1,r,d)=m1Xc=0x(a,r,c)W(c)(d+1 2)m=Xcoddx(a,r,c)W(c)(d+1 2)m+Xcevenx(a,r,c)W(c)(d+1 2)m=m 2)m+m 2)m=W(d+1 2)mm 2)m 2)m 2)mgm (4) NotethatW2knK=expj2 4 isagainarecursivedenitionofga(1,r,d)thatcanbecomputedinO(logN)time.Similarly, 72

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Equations 4 and 4 arenotstandardFouriertransformsandwillbereferredtoasnon-standardtransforms2and3(NST2andNST3),respectively.UsingthesameapproachasthederivationofEquation 4 ,fa(0,s,d)andfa(1,s,d)canbedenedasfa(0,s,d)=W2snfn Again,thesearerecursivedenitionsoffa(b,r,d)thatcanbecomputedinO(logN)time. 54 ]tovisualizetherequiredcomputations,whichcanalsosimplifythecountingofthecomplexadditionsandcomplexmultiplicationsrequiredtocalculatetheFFT.Figure 4-1 showsastandardDISbutterydiagramforan8-point1-Dsignalwithallcomputations.Noticetheinputisconsideredtobepresentedinbit-reversedorder.ItisobviousfromthediagramthatthefullcomputationversionrequiresNlogNcomplexmultiplicationsandNlogNcomplexadditionsforanN-point1-Dsignal.However,itiscommontoexploittheperiodicity(WkN=W(k+N)N)andsymmetry(WkN=W(k+N=2)N)ofthecomplexexponentialinordertoreducethesecomputations.Althoughadditionaloptimizationsarepossible,thisanalysiswillbelimitedtotheonesmentionedabove.Figure 4-2 showstheresultofapplyingtheseoptimizationstoan8-point1-Dsignal.Thenumberofcomplexmultiplicationsarereducedby1 2andhalfofthecomplexadditionshavebecomesubtractions.Considera2nmrectangularlysampledimagewheren,m2f2i:i2Ng.Thestandardradix-2DISFFTusingtheabovementioned 73

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ComputationsRequiredforHexagonalFastFourierTransformArrays ComplexComplexArrayInputTransformAdditionsMultiplications ThecomputationsrequiredforthevariousarraysusedduringthecalculationoftheHFFTareslightlydifferentthanthestandardFFTformostofthearrays.Table 4-1 givesthenumberofoperationsrequiredtocalculateeacharrayneededinthecomputationoftheHFFT.ThesumofthesevaluesresultsinthenumberofoperationsrequiredtocomputetheHFFTasdescribedabove.TheHFFTrequires2nmlog2nmcomplexmultiplications,and2nmlog2nm2complexadditions. ThisislessthantwicethenumberoftotaloperationsrequiredforthestandardFFT.TheHFFTcansimilarlybedescribedusingsignalowdiagramsjustlikeorsimilartothebutterydiagrams.Figures 4-3 4-5 ,and 4-7 showthefullcomputationdiagramsforNS1,NS2,andNS3,respectively.Figures 4-4 4-6 ,and 4-8 showthereducedcomputationdiagramsforNS1,NS2,andNS3,respectively. 74

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AstandardFFTbutterydiagramwithallcomputations.Thearrowheadsindicatecomplexmultiplicationsandthedotsindicatecomplexadditions. Figure4-2. AstandardFFTbutterydiagramwithoptimizations.Thearrowheadsindicatecomplexmultiplicationsandthedotsindicatecomplexadditions. 75

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NST1butterydiagramwithallcomputations.Thearrowheadsindicatecomplexmultiplicationsandthedotsindicatecomplexadditions. Figure4-4. NST1butterydiagramwithoptimizations.Thearrowheadsindicatecomplexmultiplicationsandthedotsindicatecomplexadditions. 76

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NST2butterydiagramwithallcomputations.Thearrowheadsindicatecomplexmultiplicationsandthedotsindicatecomplexadditions. Figure4-6. NST2butterydiagramwithoptimizations.Thearrowheadsindicatecomplexmultiplicationsandthedotsindicatecomplexadditions. 77

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NST3butterydiagramwithallcomputations.Thearrowheadsindicatecomplexmultiplicationsandthedotsindicatecomplexadditions. Figure4-8. NST3butterydiagramwithoptimizations.Thearrowheadsindicatecomplexmultiplicationsandthedotsindicatecomplexadditions. 78

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35 ],imagesaddressedwiththeGBTapproacharehighlyamenabletoafastFouriertransformsincethenumberofpixelsisalwaysapowerof7.FastalgorithmsforcomputingtheHDFTusingspiraladdressinghavebeendeveloped[ 12 ],[ 35 ],[ 41 ]thatrequireO(NlogN)operations.However,thespiraladdressingtechniquesemployradix-7decimationratherthanradix-2decimation,resultinginaslightspeedup.Asshownabove,theASAHFFTrequiresNlog2Ncomplexmultiplications;Middleton[ 12 ]showsthattheHIPHFFTrequiresNlog7Ncomplexmultiplications.ThereforetheASAHFFTrequiresaboutlog272.8timesmoreoperationsthantheHIPHFFT. where(i,j)isthetwo-dimensionalDiracdeltafunction(2DDiraccomb).Equation 4 canberewrittenasf(a,r,c)=f(0,r,c)+f(1,r,c)=Z1Z1fr(x,y)(xc,yp 2,yp 2dxdy. TakingtheDSFTyieldsF(!1,!2)=F0(!1,!2)+F1(!1,!2)=1Xr=1Xc=f(0,r,c)exphj!1c+!2p

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2+!2p 2=1Xr=1Xc=f(0,r,c)exphj!1c+!2p 2!#1Xr=1Xc=f(1,r,c)exphj!1c+!2p (4) thehexagonaldiscrete-spaceFouriertransform(HDSFT). where(i,j)isa2DDiraccomb.TakingtheLaplacetransformoffq(x,y)resultsinFq(s1,s2)=Z1Z1fq(x,y)exp[(s1x+s2y)]dxdy=Z1Z11Xa=01Xr=1Xc=fr(x,y)hxc+a

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wheref(a,r,c)isthesetofdiscretesamplesoffr(x,y).Letz1=exp(s1)andz2=exps2p Equation 4 isthehexagonalz-transform.Sincethez-transformisageneralizationoftheDSFT,thecompatibilityofthisresultwiththatofSection 4.1.3 canbeveriedbyevaluatingEquation 4 ontheunitsphere,i.e.withz1=exp(j!1)andz2=expj!2p 4 ,F(!1,!2)=Fzexp(j!1),expj!2p 2+!2p 2=1Xr=1Xc=f(0,r,c)exphj!1c+!2p 2!#1Xr=1Xc=f(1,r,c)exphj!1c+!2p 81

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26 ]. TheDWTsusedmostcommonlyforrectangularlysampled2Dsignalsuseseparablelterbanks[ 26 ],whichsimplifyimplementation.Separablelterbanksconsistofalow-passlter,twohigh-passltersthatareequivalentaftera90degreerotation,andathirdhigh-passlterthatcoverstheremainingdisjointregionsofthefrequencyspace[ 55 ],asshownontheleftsideofFigure 4-9 .Thesub-bandresultingfromthisthirdhigh-passltercontainsmixedorientations[ 56 ]andthereforedoesnothaveastraightforwarddirectionalinterpretation[ 26 ].Thesub-bandisconsideredtobetroublesome[ 55 ]andadeciencyofseparableimplementationsoftheDWT[ 26 ].Incontrast,lterbanksdesignedforcomputingtheDWTofhexagonallysampledimagesaregenerallynotseparableandcantakeadvantageofthehighersymmetryofthehexagonalgrid.HexagonalDWTlterbanksconsistofalow-passlterandthreehigh-passltersthatareequivalentafterrotationsof120degreesand240degrees,asshownontherightsideofFigure 4-9 .Sinceallofthehigh-passsub-bandshavestraightforwarddirectionalinterpretations,itseemshexagonalDWTlterbankscanbedesignedtohavebetterpropertiesthananypossiblelterbankdesignedfortherectangulargrid[ 55 ]. AspointedoutbyMiddleton[ 12 ]andJiang[ 27 ],researchintheareaofhexagonalwaveletshasbeenslow.However,therehavebeensomeresearchersthathaveaddressedthedesignofDWTlterbanksforhexagonallysampledimages.Earlyefforts[ 57 ][ 59 ]focusedonthenotionoftheimagepyramidwithfairlysimpleltering,whichledtoperfectreconstructiontransformsthatwereovercompleteandlacked 82

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IdealizedWaveletTransformRegionsofSupport(Hexagonalvs.Rectangular). orientationinformation[ 60 ].Quadraturemirrorlters(QMFs)provideorientedpartitionsandperfectreconstructionwhileproducingcodesthatarethesamesizeastheoriginalimage[ 60 ],andseveralresearchers[ 56 ],[ 60 ][ 62 ]haveusedQMFstodesignlterbanksforimagepyramidsonthehexagonallattice.AnotherapproachbyAllen[ 55 ],[ 63 ]wasbasedonanovelblockstructurethatledtolterbankswithcertainsymmetry[ 27 ].Morerecently,Jianghaspublishedseveralpapers[ 27 ][ 29 ]onthedesignofhexagonallterbankswithvariousorthogonality,symmetry,andrenementproperties.Duetotheavailabilityofdesignsanddesignmethodsforhexagonallterbanks,lterbankdesignusingASAwasnotpursued. AtypicalDWT(regardlessofsamplinggeometry)proceedsbyapplyingeachlterofalterbanktothesignalthendownsampling(rening)thelteredimagesasshowninFigure 4-10 .Thisiscalledtheanalysisstep,whichisrecursivelyappliedtothelow-passsub-bandandleadstothecoefcientsofthewavelettransform.Theinverseoftheanalysisstepiscalledthesynthesisstep,whichproceedsbyupsamplingeachofthelowestlevelsub-bands,lteringtheupsampledsub-bandswiththesynthesislterbanklters,andsummingtheresultstoproducethenexthigherlevellow-pass 83

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4-11 .Notethatthenumberofltersinthelterbankandtherenementhavethefollowingrelationship:iftherearenltersinthelterbank,ap 29 ].Ofthethree,adyadicrenementisthemostappropriateforhexagonalgridsthatarerepresentedusingASAsincethedownsampledimagecanalsobeaddresseddirectlywithASA.Theothertworenementsrequirerotationsofthecoordinatesystemateachrenementlevel.DyadicrenementalsoparallelswhatistypicallydonewithrectangularDWTs. Figure4-10. WaveletTransformAnalysisFilterBank. AhighlydesirablecharacteristicofDWTsisknownasperfectreconstruction(PR),wherethesynthesisstepperfectlyreconstructstheinputtotheanalysisstepfromtheDWTcoefcients.TherearetwotypesoflterbanksthatleadtoPRwavelets,theseareknownasorthogonalandbi-orthogonal.Theorthogonallterbanksusethesamelters(exceptforapossiblerotation)forboththeanalysisandsynthesissteps,whereasthebi-orthogonallterbanksusedifferentltersforthetwosteps.Amuchmorecomplete,detailed,andmathematicalexplanationcanbefoundinStrang'sbook[ 64 ]. 84

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WaveletTransformSynthesisFilterBank. TheimplementationofhexagonalDWTsusingASAisstraightforwardandefcient.TheltersareallappliedusingconvolutionwhichwasdescribedandshowntobeefcientinSection 3.1 .DownsamplinginASAisdescribedinSection 3.4 ,howevertheanti-aliasinglterhasbeenreplacedbythelterbank.Upsamplingforthesynthesisstepisjusttheexactoppositeofdownsampling.Asanexample,theorthogonalPRlterbankpublishedbyAllen[ 55 ]wasrepresentedinASAandusedtoperformaDWTontheASAaddressed,hexagonallysampledimageshowninFigure 4-12 .Thedecomposition(waveletcoefcientsimage)afterthreeiterationsisshowninFigure 4-13 .Thecoefcientswherethenfedintothesynthesisstepusingthesamelterbank(rotatedby180)andtheoriginalimagewasrecovered.Perfectreconstructionwasveriedbytakingtherootmeansquareerrorofthereconstructedimage,whichwasErms=2.8x1016forthisexample.ThetotalruntimeforboththeanalysisandsynthesisstepsofaMatlabimplementationofthisexampleaveraged0.5017secondswithastandarddeviationof0.0077secondsover100runs.Incomparison,thetotalruntimeforbothstepsofaMatlabimplementationoftheCohen-Daubechies-Feauveau(CDF) 85

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Figure4-12. LiliesImage(HexagonallySampled). InordertocompareanASADWTimplementationtoaHIPDWTimplementation,thetwocomponentoperations,convolutionanddownsampling/upsampling,arecompared.AssumeanN-pointinputimageandlterbanksconsistingofm-pointlters.AnASAconvolutioncanbecomputedasJ(x)=m1Xi=0I(xni)M(Cni) 86

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HexagonalDiscreteWaveletTransformofLiliesImage. 4(4) operationsforacompletedecomposition.Similarly,aHIPconvolutioncanbecomputedasJ(x)=m1Xi=0I(xni)M(0ni) 87

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2.8 ,HIPadditionisaO((logN)2)operation.Therefore,HIPconvolutionrequires2mNN+2mN(log7N)2operationsperlter.BecauseofthehierarchicalnatureofHIPaddressing,ap operationsforacompletedecomposition.ComparingtheexpressionsinEquations 4 and 4 ,onecanseethatanASA-basedDWTisanO(N)operationandaHIP-basedDWTisanO(N(logN)2)operation,makingtheASA-basedDWTsignicantlyfaster.NotethatthesynthesisstepoftheDWTshouldhavethesamecomplexityastheanalysisstepforbothcoordinatesystems. 88

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ThetrapezoidalimagerdescribedinSection 1.5.2 andshowninFigure 5-1 wasusedtoconductanexperimentthatwasdesignedtoconcretelydemonstratetheadvantagesofhexagonalsamplingusingarealsensor.Recallthatthetrapezoidalimagerhastheuniqueabilitytosimultaneouslysampletheexactsamescenebothrectangularlyandhexagonally.Theexperimentwasconductedbypresentingaseriesofradialsinewavestotheimager,withthefrequencyofthesinewaveincreasingastheexperimentprogressed.ThereasonforchoosingaradialsinewavewasthatthemagnitudeofitsFouriertransformisacircleinFourierspacewiththeradiusofthecirclebeingproportionaltothefrequencyofthesinewave.Figure 5-2 showsanexampleimageofaradialsinewaveandthemagnitudeofitsFouriertransform.Thischaracteristicallowsonetovisuallydeterminethefrequencyatwhichaliasingbeginstooccur(whenthecirclesbegintooverlapinFourierspace).ThedatawascollectedbydisplayingtheinputimagesonanLCDmonitorwiththesensormountedsolidlyinfrontofthemonitor.Onelaptopwasusedtodisplaytheimagesandanotherwasusedtocontrolthesensorandcollectthesensor'soutput. Figure5-1. TrapezoidalImager.ImageCourtesyofCenteye,Inc. 89

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Exampleofaradialsinewave(left)anditsFouriertransform(right). Thecollecteddatasetwasprocessedtoproducebothrectangularlyandhexagonal-lysampledimagesfromeachtrapezoidalimagebyappropriatelysumminggroupsoffourtrapezoidalunitstoformtheindividualpixels.Eachimagewasthencroppeddowntothecenter128128imagetomaintainsquareimageswithsidelengthsequaltoapoweroftwo.TheFouriertransformoftheresultantimageswascalculatedandplottedside-by-side.Recallthatbothimageshavethesamenumberofpixelsandareimagingthesamescene,theonlydifferenceisthesamplinggeometry.Theresultsclearlyshowthatthehexagonallysampledimageshaveahigherbandwidththantherectangularlysampledimages.Figure 5-3 showsafewoftheframesfromtheimagesequence.ThefrequencyatwhichthecirclesinFourierspacebegintotouchistheonsetofaliasingandhencethebandwidthofthesamplingapproach.Noticethattheratioofbandwidthsisveryclosetothetheoreticallimit,0.268=0.3090.867p 90

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Experimentalresultsfromthetrapezoidalimager.Frequency(F)isgiveninradiansperpixel.

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1. Coordinatedenition 2. EfcientcoordinateconversionbetweenASAandCartesianspace 3. DenitionofdistancemeasuresusingASA 4. DenitionofbasicASAvectoroperations(addition,negation,subtraction,scalarmultiplication) 5. ProofthatASAsatisesthe8propertiesofaZ-module 6. DenitionofefcientandcompactdigitalmemorystorageofASAimages. Additionally,severalbasicimageprocessingtechniqueshavebeendevelopedforASAandshowntobeefcient.AmethodforperformingconvolutiononanASAimagewasdevelopedandshowntorequiretheexactsamenumberofoperationsasaconvolutiononarectangularlysampledimageforthesamesizeconvolutionmask.However,itwasalsoshownthathexagonalneighborhoodsareactuallysmallerthanrectangularones,soaconvolutionmaskforaneighborhoodoperationissmallerinASArepresentationthanitsrectangularcounterpart,makingconvolutioninASAfasterthanrectangularconvolution.ThespeedupisfurtherimprovedsinceanimagecanberepresentedinASAwith13.4%fewerpixelsthaninrectangularcoordinates.Theconvolutionmasksforlocalgradientestimationwerederiveddirectlyusingaleast-squarestofaplanetotheimageintensityvaluesinalocalhexagonalneighborhood.Sincelocalgradientestimationisaneighborhoodoperationanditcanbeperformedviaconvolution,gradientestimationinASAisfasterthanitwouldbeusingrectangularcoordinates.Itwasalsoshownthattheperformanceofcommonedgedetectiontechniquesisthesame,orslightlybetterinsomecases,usingASAthanwhenusingrectangularrepresentation. 92

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Thisdocumenthasintroducedthearraysetaddressingapproachtoaddressinghexagonallysampledimages.Theadvantagesofhexagonalsamplingfordigitalimageshavebeenpresentedandtheshortcomingsoftheexistingaddressingmethodshavebeenpointedout.IthasbeenshownthatASAcanbeusedtoperformcommonimageprocessingtasksonhexagonallysampledimageswithnomorecomplexitythanwouldberequiredforarectangularlysampledimage.TheseparabilityoftheFourierkernelinASAcoordinatescorroboratestheeleganceofthissolutionfortheprocessingofhexagonallysampleddigitalimages.Withthisefcientandconvenientaddressing 93

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94

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65 ].NotethattheBCClatticecanberepresentedbyasetoftworectangularcomponents(3-DCartesiangrids)andtheFCClatticecanberepresentedbyfourrectangularcomponents.Therefore,thesameideathatisexploitedbytheASAmethodcanbeappliedto3-Dimagingaswell.ThelatticepointsofboththeBCCandFCClatticescanbeaddressedbyusingtwobinaryandthreeintegercoordinates,i.e.(b0,b1,i,j,k)2f0,1gf0,1gZZZ(notethattheBCClatticeiscompletelycontainedinoneofthebinarydimensions).Itissuspectedthatthis3-DversionofASAaddressingwillprovetobeaconvenientandefcientcoordinatesystemforprocessingoptimallysampled3-Ddata.Extrapolatingthisideatoevenhigherdimensionsleadstothefollowingconjecture: Theoptimalsamplinglatticeinn-DspaceanditsdualinFourierspace(forn-Dspacesthathaveaknownoptimalsamplinglattice)canberepresentedbythesuperpositionof,atmost,2dlog2nerectangularcomponents(n-DCartesiangrids).Therefore,acoordinatesystemwhichusesdlog2nebinarycoordinatesandnintegercoordinatescanbeusedtoefcientlyaddressthelatticepointsandprocessthedata. 95

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TheoremA.1. Proof. 96

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TheoremA.2. Proof. TheoremA.3. Proof. TheoremA.4.

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andkp1+kp2=0BBBB@(ka1)mod2kr1+(a1)k (A) TheequalityofEquations A and A isprovenbyconsideringthecoordinatesseparately.First,((a1a2)(k))mod2=(a1a2)^(kmod2)(Identity 2 )=((kmod2)^a1)((kmod2)^a2)(Identity 7 )=((ka1)mod2)((ka2)mod2)(Identity 2 ). Second,(k)(a1^a2)+(a1a2)k

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6 )=(a1+a2)k 1 )=(a1+a2)k 2 ) Thuskr1+kr2+(k)(a1^a2)+(a1a2)k Finally,c1,c2canbesubstitutedforr1,r2yieldingkc1+kc2+(k)(a1^a2)+(a1a2)k TheoremA.5. Proof. 2(1)c+(a)1 21CCCCA=0BBBB@arc1CCCCA=p. 99

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[1] J.Crettez,Apseudo-cosinetransformforhexagonaltessellationwithanheptarchicalorganization,inProc.IEEEInt.Conf.onPatternRecognitionandImageProcessing,1980,pp.192. [2] C.A.Curcio,J.K.R.Sloan,O.Packer,A.E.Hendrickson,andR.E.Kalina,Distributionofconesinhumanandmonkeyretina:Individualvariabilityandradialasymmetry,Science,vol.236,no.4801,pp.579,1987. [3] D.P.PetersenandD.Middleton,Samplingandreconstructionofwave-number-limitedfunctionsinn-dimensionalEuclideanspaces,Inf.Control,vol.5,no.4,pp.279,Dec.1962. [4] B.H.McCormick,TheIllinoispatternrecognitioncomputer:ILLIACIII,IEEETrans.Electron.Comput.,vol.EC-12,no.6,pp.791,Dec.1963. [5] M.Golay,Hexagonalparallelpatterntransformations,IEEETrans.Comput.,vol.C-18,no.8,pp.733,Aug.1969. [6] E.S.Deutsch,Thinningalgorithmsonrectangular,hexagonal,andtriangulararrays,Comm.ACM,vol.15,no.9,1972. [7] R.M.Mersereau,Theprocessingofhexagonallysampledtwo-dimensionalsignals,Proc.IEEE,vol.67,no.6,pp.930,Jun.1979. [8] P.J.Burt,Treeandpyramidstructuresforcodinghexagonallysampledbinaryimages,Comput.Graph.ImageProcess.,vol.14,pp.271,1980. [9] J.W.vanRoessel,ConversionofCartesiancoordinatesfromandtogeneralizedbalancedternaryaddresses,Photogramm.Eng.RemoteSens.,vol.54,no.11,pp.1565,1988. [10] D.LucasandL.Gibson,AsystemforhierarchicaladdressinginEuclideanspace,Jan.1980,interactiveSystemsCorporation. [11] L.GibsonandC.Lenzmeier,Ahierarchicalpatternextractionsystemforhexagonallysampledimages,InteractiveSystemsCorporation,Tech.Rep.,1981,nalReportforContractF49620-81-C-0039,U.S.AirForceOfceofScienticResearch. [12] L.MiddletonandJ.Sivaswamy,HexagonalImageProcessing:APracticalAp-proach.London:Springer,2005. [13] P.Sheridan,Spiralarchitectureformachinevision,Ph.D.dissertation,Univ.ofTechnologySydney,1996. [14] I.Her,Asymmetricalcoordinateframeonthehexagonalgridforcomputergraphicsandvision,J.Mech.Des.,vol.115,no.3,pp.447,1993. 108

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W.E.Snyder,H.Qi,andW.Sander,Acoordinatesystemforhexagonalpixels,inProc.SPIEMedicalImaging:ImageProcessing,vol.3661,1999,pp.716. [16] X.He,H.Wang,N.Hur,W.Jia,Q.Wu,J.Kim,andT.Hintz,Uniformlypartitioningimagesonvirtualhexagonalstructure,inProc.9thInt.Conf.Control,Automation,RoboticsandVision.LosAlamitos,CA:IEEEComputerSocietyPress,2006,pp.891. [17] X.He,W.Jia,N.Hur,Q.Wu,andJ.Kim,Imagetranslationandrotationonhexagonalstructure,inProc.6thInt.Conf.ComputerandInformationTechnology,Seoul,Korea,Dec.2006,pp.141. [18] X.He,W.Jia,Q.Wu,andT.Hintz,Descriptionofthecardiacmovementusinghexagonalimagestructures,Comput.Med.Imag.Grap.,vol.30,no.6-7,pp.377,2006. [19] X.He,J.Li,andT.Hintz,Comparisonofimageconversionsbetweensquarestructureandhexagonalstructure,Lect.NotesComput.Sci.,vol.4678,pp.262,2007. [20] L.Condat,D.VanDeVille,andM.Unser,Efcientreconstructionofhexagonallysampleddatausingthree-directionalbox-splines,inProc.IEEEInt.Conf.ImageProcessing,Atlanta,US,Oct.2006. [21] L.CondatandD.VanDeVille,Three-directionalbox-splines:Characterizationandefcientevaluation,IEEESignalProcess.Lett.,vol.13,no.7,pp.417,Jul.2006. [22] ,Newoptimizedsplinefunctionsforinterpolationonthehexagonallattice,inProc.IEEEInt.Conf.ImageProcessing,SanDiego,CA,Dec.2008,pp.1256. [23] L.Condat,B.Forster-Heinlein,andD.VanDeVille,H2O:Reversiblehexagonal-orthogonalgridconversionby1-dltering,inProc.IEEEInt.Conf.ImageProcessing,vol.2,SanAntonio,TX,Sep.2007,pp.IIII. [24] L.Condat,D.VanDeVille,andB.Forster-Heinlein,Reversible,fastandhigh-qualitygridconversions,IEEETrans.ImageProcess.,vol.17,no.5,pp.679,May2008. [25] L.CondatandD.VanDeVille,Fullyreversibleimagerotationby1-Dltering,inProc.IEEEInt.Conf.ImageProcessing,SanDiego,CA,Dec.2008,pp.913. [26] L.Condat,B.Forster-Heinlein,andD.VanDeVille,Anewfamilyofrotation-covariantwaveletsonthehexagonallattice,inProc.SPIEOpticsPho-tonics2007Conf.MathematicalMethods:WaveletXII,vol.6701,SanDiego,CA,Aug.2007,pp.67010B/67010B. 109

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Q.T.Jiang,FIRlterbanksforhexagonaldataprocessing,IEEETrans.ImageProcess.,vol.17,no.9,pp.1512,Sep.2008. [28] ,OrthogonalandbiorthogonalFIRhexagonallterbankswithsixfoldsymmetry,IEEETrans.SignalProcess.,vol.52,no.12,pp.5861,Dec.2008. [29] ,Orthogonalandbiorthogonalp [30] F.FailleandM.Petrou,Bio-inspired,invariantimagereconstructionfromirregularlyplacedsamples,inProc.BMVC2008,Leeds,UK,Sep.2008. [31] B.Gardiner,S.Coleman,andB.Scotney,Adesignprocedureforgradientoperatorsonhexagonalimages,inProc.IrishMachineVisionandImagePro-cessingConf.,Kildare,Ireland,Sep.2007,pp.47. [32] ,Multi-scalefeatureextractioninasub-pixelvirtualhexagonalenvironment,inProc.IrishMachineVisionandImageProcessingConf.,2008,pp.47. [33] M.PuschelandM.Rotteler,Algebraicsignalprocessingtheory:2-Dhexagonalspatiallattice,IEEETrans.ImageProcess.,vol.16,no.6,pp.1506,2007. [34] ,Algebraicsignalprocessingtheory:Cooley-Tukeytypealgorithmsonthe2-Dhexagonalspatiallattice,Appl.AlgebraEng.Comm.Comput.,vol.19,no.3,pp.259,2008. [35] A.VinceandX.Zheng,ComputingthediscreteFouriertransformonahexagonallattice,J.Math.Img.Vis.,vol.28,no.2,pp.125,2007. [36] K.S.Yang,Towardrenementsofspatialsmoothers:Attendingtodetailsincomputationallyintensivesettings,Ph.D.dissertation,GeorgeMasonUniv.,1999. [37] J.R.D.Barbour,Reductionofcomplexityinpathndingusinggridbasedmethods,Master'sthesis,Univ.ofRegina,2008. [38] R.C.Boyd,GeneralizedEuclideanline-of-sightonregularhexagonalgrids,Master'sthesis,Univ.ofColorodoDenver,2008. [39] M.Guenette,Triangulationofahierarchicalhexagonmesh,Master'sthesis,Queen'sUniv.,2009. [40] G.Zhou,FourieranalysisandtruncationerrorestimatesofmultigridmethodsandconservativeJacobiansonhexagonalgrids,Ph.D.dissertation,ClarksonUniv.,2009. [41] X.Zheng,EfcientFouriertransformsonhexagonalarrays,Ph.D.dissertation,Univ.ofFlorida,2007. 110

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C.Mead,AnalogVLSIandNeuralSystems.Reading,MA:Addison-Wesley,1989. [43] R.Hauschild,B.J.Hosticka,S.Muller,andM.Schwarz,ACMOSopticalsensorsystemperformingimagesamplingonahexagonalgrid,inProc.22ndEuropeanSolid-StateCircuitsConf.,1996,pp.304. [44] A.P.FitzandR.J.Green,FingerprintclassicationusingahexagonalfastFouriertransform,PatternRecogn.,vol.29,no.10,pp.1587,1996. [45] P.Halasz.(2008)Pixelgeometries.WikimediaCommons.(accessedJanuary5,2010).[Online].Available: http://commons.wikimedia.org/wiki/File:Pixel geometry 01 Pengo.jpg [46] E.LuczakandA.Rosenfeld,Distanceonahexagonalgrid,IEEETrans.Comput.,vol.25,pp.532,1976. [47] N.I.RummeltandJ.N.Wilson,Arraysetaddressing:makingtheworldsafeforhexagonalimaging,inProc.SPIE/IS&TElectronicImaging,vol.7532,2010,p.75320D. [48] J.M.S.Prewitt,Objectenhancementandextraction,inPictureProcessingandPsychopictorics,B.S.LipkinandA.Rosenfeld,Eds.NewYork:AcademicPress,1970. [49] N.Otsu,Athresholdselectionmethodfromgraylevelhistograms,IEEETrans.Syst.,Man,Cybern.,vol.SMC-9,pp.62,1979. [50] D.C.MarrandE.Hildreth,Theoryofedgedetection,Proc.Roy.Soc.Lond.,vol.B207,pp.187,1980. [51] J.F.Canny,Acomputationalapproachtoedgedetection,IEEETrans.PatternAnal.Mach.Intell.,vol.PAMI-8,pp.679,1986. [52] Y.Morikawa,H.Hamada,andN.Yamane,SeparationofkernelinthehexagonaldiscreteFouriertransform,Electron.Comm.Jpn.,vol.64-A,no.7,pp.16,Jul.1981. [53] U.R.AlimandT.Moller,AdiscreteFouriertransformforthehexagonalandbody-centeredcubiclattices,SFUComputingScience,Tech.Rep.,2008. [54] A.V.OppenheimandR.W.Schafer,DigitalSignalProcessing.Prentice-Hall,1975. [55] J.D.Allen,Perfectreconstructionlterbanksforthehexagonalgrid,inProc.5thInt.Conf.Information,Communications.andSignalProcessing,Dec.2005,pp.73. [56] E.SimoncelliandE.Adelson,Non-separableextensionsofquadraturemirrorlterstomultipledimensions,Proc.IEEE,vol.78,no.4,pp.652,Apr.1990. 111

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S.TanimotoandT.Pavlidis,Ahierarchicaldatastructureforpictureprocessing,Comput.Graph.ImageProcess.,vol.4,pp.104,1975. [58] P.J.Burt,Fastltertransformsforimageprocessing,Comput.Graph.ImageProcess.,vol.16,pp.20,1981. [59] A.B.Watson,Idealshrinkingandexpansionofdiscretesequences,NASA,Tech.Memorandum88202,Jan.1986. [60] A.B.WatsonandJ.A.J.Ahumada,Ahexagonalorthogonal-orientedpyramidasamodelofimagerepresentationinvisualcortex,IEEETrans.Biomed.Eng.,vol.36,no.1,pp.97,Jan.1989. [61] E.A.Adelson,E.Simoncelli,andR.Hingorani,Orthogonalpyramidtransformsforimagecoding,SPIEVis.Commun.ImageProcess.,vol.845,pp.50,1987. [62] X.H,W.S.Lu,andA.Antoniou,Anewdesignof2-Dnon-separablehexagonalquadrature-mirror-lterbanks,inProc.CCECE,Vancouver,BC,Canada,Sep.1993. [63] J.D.Allen,Codingtransformsforthehexagonalgrid,RicohCalif.ResearchCtr.,MenloPark,CA,Tech.Rept.CRC-TR-9851,Aug.1998. [64] G.StrangandT.Nguyen,WaveletsandFilterBanks.Wellesley-CambridgePress,1996. [65] A.Entezari,Optimalsamplinglatticesandtrivariateboxsplines,Ph.D.dissertation,SimonFraserUniversity,Vancouver,Canada,July2007. 112

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NicholasI.RummeltwasborninMichiganandattendedMichiganStateUniversity(MSU)wherehereceivedaBachelorofSciencedegreeincomputerengineering.Aftergraduation,hebeganworkingfortheAirForceinDayton,Ohio,butsoonreturnedtoMSUtogethisMasterofSciencedegreeinelectricalengineering.ReturningtotheAirForce,butseekingachangeofscenery,heeventuallyfoundapositionwiththeAirForceResearchLaboratoryinFlorida,wherehecontinuestowork. 113