Frequency Analysis of Gradient Estimators in Volume Rendering

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Frequency Analysis of Gradient Estimators in Volume Rendering
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  Frequencyanalysisofgradientestimatorsinvolume  rendering    nalversion MarkJ.Bentum UniversityofTwente Dept.ofElectricalEngineering Lab.forNetworkTheory P.O.box217 7500AEEnschede TheNetherlands phone3153892673 fax3153334701 emailmarknt.el.utwente.nl BartholdB.A.LichtenbeltandTomMalzbender HewlettPackardLaboratories VisualComputingDepartment 1501PageMillRoad PaloAltoCA94304 UnitedStates phone14158576760 fax14158523791 emailbartholdhpl.hp.com emailtom malzbenderhpl.hp.com  November81995  KeywordsVolumerenderingvolumevisualizationgradientlters 1   Abstract  Involumerenderinggradientinformationisusedtoclassifyandcolorsamplesalongaray. Inthispaperwepresentananalysisofthetheoreticallyidealgradientestimatorandcom paresomecommonlyusedgradientestimatorswiththeidealestimator.Anewmethodis presentedtocalculatethegradientonanarbitrarypositionusingthederivativeofthein terpolationlterasthebasisforthenewgradientlter.Asanexamplewewilldiscussthe useofthederivativeofthecubicspline.Acomparisonwillbemadeshowingthedierences withothermethods.Computationaleciencycanberealizedsincepartsoftheinterpolation computationcanbeleveragedinthegradientestimation. 2   1Introduction  Visualizingagiventhreedimensionaldatasetcanbedonebysurfacerenderingalgorithms e.g.MarchingCubes8orbydirectvolumerenderingalgorithmse.g.raycasting7or splatting19.Fordirectvolumerenderingmethodsthevoxelintensityandthegradient magnitudeisoftenusedtoshadeandclassifythedataset.Forsurfacerenderingtechniques thegradientisusedtoestimatethesurfacenormal.Thisnormalisneededinordertoshade thepolygonsthatformthesurface. Theshadingofobjectsinnaturalenvironmentsprovidescuestothethreedimensionalstruc tureandpositionoftheobjects.Involumerenderingthelocalgradientisoftenusedasan approximationtothesurfacenormalforuseinashadingmodel.Phongshading11iswidely usedinsurfaceanddirectvolumerenderingalgorithms. Involumerenderingthedatasettoberenderedhastobeclassied.Classicationcalculates anopticaldensityvalueforeachvoxelinthedatasetcalledopacity.Opacityiscalculated toaccentuatesurfacesortheboundariesbetweendierentmaterials.Opacitiesaretypically calculatedusingeithervoxelintensitiesoracombinationofvoxelintensitiesandgradient information. Oncethedatasetisclassiedandtheopacityandcoloriscalculatedthedatahastobe rendered.Thecolorandopacityvaluesarecompositedtoachievethenaltwodimensional projection.Severalrenderingtechniquesareknown.Themostimportantdistinctionbetween thetechniquesistheorderinwhichthevoxelsareprocessedtocreateanimageimage orderorobjectorderalgorithms.Examplesofimageorderalgorithmsareraycasting7and Sabellasmethod14.Examplesofobjectorderalgorithmsarethesplattingalgorithm18 Vbueralgorithm16andtheSliceShearingalgorithm4. Itispossibletocalculatecolorandopacityasapreprocessingstepyieldingtwonewvoxel datasetsacolorandanopacitydataset.Duringtherenderingstepthecolorandopacity onsamplepointsgenerallynotonvoxelpositionsarecalculatedbyinterpolation.This mayreducethequalityoftheimageandanalternativeistocalculatethecolorandopacity duringrenderingonthesamplepositions.Thishassomeconsequencesforthecomplexityof thealgorithm. Thispaperconsistsof9sections.Insectiontwoidealinterpolationwillbebrieydiscussed. Understandinginterpolationisessentialtounderstandtheideasintherestofthepaper. Sectionthreediscussesthenotionofperfectgradientestimation.Theninsectionfoursome wellknowngradientestimatorsarediscussedandanalyzedwithrespecttotheperfectgra dientestimator.Sectionvediscussesthefrequencyanalysisofgradientestimators.Section sixproposesadierentviewtodesigningagradientestimatorbyusingthederivativeofthe interpolationfunctionasthegradientestimator.Anexamplewhichusesthecubicspline isgiven.Sectionsevendiscussestheimplementationofthisexampleinvolumerendering. 1   Sectioneightgivestheresultsofusingthisexampleandsectionninediscussestheproposed methods.  2IdealInterpolation    Theprocessofinterpolationisoneofthefundamentaloperationsindigitalsignalprocessing andcomputergraphics.Thepurposeofinterpolationistocalculateintermediatevaluesofa continuoussignal  f    xyz  fromadiscretesignal.Involumerenderinginterpolationisused tocalculatethevaluesonthesamplepositionsalongrayssinceitisunlikelythesepoints willbepositionedongridpoints. Accordingtothesamplingtheorem15asignalcanbereconstructedexactlybytheideal interpolationfunctionifitisbandwidthlimitedandsampledattheNyquistrateorhigher. Theidealinterpolationfunctionisthe  sinc  function  r    x    ideal  sinc  x   sin  x x  1 Thisideallterremovesallreplicatesofthefrequencyspectrumintroducedbysamplingthe originalcontinuousfunctionbymultiplyingwithablockfunctioninthefrequencydomain. Incomputergraphicsonehasasetofdiscretepointsanimageorvolume.Thissetofpoints isobtainedbysamplingthecontinuousfunction  f    xyz  inthethreedimensionalcase.One hastobecarefulandrealizethatthesetofdiscretepointsmightnotrepresentthecontinuous function  f    xyz  onewishestoprocessbecauseofundersamplingsampledataratelower thantheNyquistrateorbecausethiscontinuousfunctionisnotbandwidthlimited.Ifthis isthecasethissetofdiscretepointsofcoursestillcanbemanipulatedandinterpolated. The  sinc  isstilltheperfectreconstructionlterbutitwillreconstructacontinuousfunction  f  0   xyz  thatisslightlydierentfromtheoneoriginallysampled. Givenasetofdiscretepointsthe  sinc  isindeedthebestinterpolationfunction.This  sinc  functionhoweverisdenedoveraninnitespatialintervalandcanthereforenotbeused asaninterpolationfunction.Othermethodsmustbeusedsuchasthenearestneighbor linearandhigherorderinterpolationfunctions16910. 2   3IdealGradientEstimation  Thegradientornormalofasurfaceisthepartialderivativeofthesurfacewithrespectto allthreedirections.Givenafunction  f    xyz  thegradientis  r  f    xyz       f x  f y  f z    2 Involumerenderinga3Ddatasetconsistsofdiscretesamplesof  f    xyz  calledvoxels. Ifthisfunction  f    xyz  isnotknownwhichingeneralisthecasethegradienthastobe calculatedusingthesevoxels. Gradientestimationcanbeanalyzedinthesamewayasinterpolation.Whenthegradient isneededatalocationotherthanagivenvoxelpointsomekindofreconstructionlter hastobeusedtoestimatethederivativeineachdirectionof  f    xyz  .Comparethisto interpolationwhichestimates  f    xyz  itselfatasamplepoint. Sincethegradientisthepartialderivativeoftheoriginalfunction  f    xyz  andidealin terpolationwiththe  sinc  willreconstructthatfunctionthegradientcanbereconstructed exactlybyusingthederivativeofthe  sinc  asareconstructionkernel. Inonedimensiontheidealgradientreconstructionlteris  d dx    sin  x x      x  cos  x      sin  x       2  x  2   cos  x x    sin  x x  2   cos  x      sinc  x    x  3 Thisresultisconsistentwiththeresultsfoundin13. Inordertoanalyzethelterofequation3wewilllookatitsfrequencyresponse.TheFourier Transformofthe  sinc  functionisablocksignalintheinterval      .Usingthederivative theoremforFourierTransforms3wendthattheFourierTransformofthederivativeof the  sinc  is   j  timestheFourierTransformofthe  sinc  itself.Thisresultsinaconstantslope inthefrequencydomainfortheidealgradientlterintheinterval      .SeeFigure1. 3 
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