A Schrodinger Wave Mechanics Formalism for the Eikonal Problem and Its Associated Gradient Density Computation

MISSING IMAGE

Material Information

Title:
A Schrodinger Wave Mechanics Formalism for the Eikonal Problem and Its Associated Gradient Density Computation
Physical Description:
1 online resource (126 p.)
Language:
english
Creator:
Gurumoorthy,Karthik S
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering, Computer and Information Science and Engineering
Committee Chair:
Banerjee, Arunava
Committee Co-Chair:
Rangarajan, Anand
Committee Members:
Ho, Jeffrey
Vemuri, Baba C
Gader, Paul D
Shabanov, Sergei

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:
Many computational techniques based on classical mechanics exist but surprisingly there isn't a concomitant borrowing from quantum mechanics. Our work shows an application of the Schrodinger formalism to solve the classical eikonal problem---a nonlinear, first order, partial differential equation of the form gradient S =f, where the forcing function f is a positive valued bounded function. Hamiltonian Jacobi based solvers like the fast marching and fast sweeping methods solve for S by the Godunov upwind discretization scheme. In sharp contrast to that, we present a Schrodinger wave mechanics formalism to solve the eikonal equation by recasting it as a limiting case of a quantum wave equation. We show that a solution to the non-linear eikonal equation is obtained in the limit as Planck's constant hbar (treated as a free parameter) tends to zero of the solution to the corresponding linear Schrodinger equation. We begin with, by considering the Euclidean distance function problem, a special case of the eikonal equation where the forcing function is everywhere identically equal to one. We show that the solution to the Schrodinger wave function can be expressed as a discrete convolution between two functions efficiently computable by the Fast Fourier Transforms (FFT). The Euclidean distance function can then be recovered from the exponent of the wave function. Since the wave function is computed for a small but non-zero hbar, the obtained solution is an approximation. We show convergence of our approximate closed form solution for the Euclidean distance function problem to the true solution as hbar approaches 0 and also bound the error for a given value of hbar. Moreover the differentiability of our solution allows us to compute its first and second derivatives in closed form, also computable by a series of convolutions. In order to determine the sign of the distance function (positive inside a close region and negative outside), we compute the winding number in 2D and topological degree in 3D, by explicitly showing that their computations can also be done via convolutions. We show an application our of method by computing the medial axes for a set of 2D silhouettes. A major advantage of our approach over the other classical methods is that, we do not require a spatial discretization of gradient operators as we obtain a closed-form solution for the wave function. For the general eikonal problem where the forcing can be an arbitrary but positive and bounded function, the Schrodinger equation turns out to be a generalized, screened Poisson equation. Despite being linear, it does not have a closed-form solution. We use a standard perturbation analysis approach to compute the solution which is guaranteed to converge for all positive and bounded forcing functions. The perturbation technique requires a sequence of discrete convolutions which can be performed using the FFT. Finally using stationary phase approximations we establish a mathematical result relating the density of the gradient(s) of distance function S and the scaled power spectrum of the wave function for small values of hbar, when the scalar field S appears as the phase of the wave function. By providing rigorous mathematical proofs, we justify our result for an arbitrary thrice differentiable function in one dimension and for distance transforms in two dimensions. We also furnish anecdotal visual evidences to corroborate our claim. Our result gives a new signature for the distance transforms and potentially serve as its gradient density estimator.
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 Karthik S Gurumoorthy.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Banerjee, Arunava.
Local:
Co-adviser: Rangarajan, Anand.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2011
System ID:
UFE0043099:00001


This item is only available as the following downloads:


Full Text

PAGE 1

ASCHRODINGERWAVEMECHANICSFORMALISMFORTHEEIKONALPROBLEMANDITSASSOCIATEDGRADIENTDENSITYCOMPUTATIONByKARTHIKS.GURUMOORTHYADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

PAGE 2

c2011KarthikS.Gurumoorthy 2

PAGE 3

Tomyever-lovingandever-caringAmmaandAppaandthepersonwithwhomIwillbesharingtherestofmylife,mybelovedanceeBhavya. 3

PAGE 4

ACKNOWLEDGMENTS OneofthehardestthingtowriteinmyPh.D.thesisisthisacknowledgementsection.NotthatIdonothaveanybodytoacknowledge,onthecontraryIhavescoresofpeopletowhomIshouldbegratefulforlendingtheirsupportiveshouldersonthisawesomebutpainful,fantasticyetsometimesfrustratingPh.D.journey.Thoughitmaysoundmundanetothankone'sadvisor(s)(everyPh.D.student'sacknowledgementsectiondenitelyhasit),theybyallmeansdeserveit.WithoutmyadvisorsDr.ArunavaBanerjeeandDr.AnandRangarajan,Iwouldn'tbewritingthisacknowledgementsectionintherstplace(andofcourseyouwon'tbereadingit).Togiveasenseofhowinformaltheyarewithme,thisprobablymightbetherst(andthelast)timeIamaddressingthemwiththeinitialsDr.infrontoftheirnames.Theyaresimplyawesomepeopletoworkwith.Thoughitmaysoundalittlebombastictosaythattheykneweverythingunderthesun,theyactuallydid!!!.Thankyouguysforbearingwithmeoverthepastsixyears(Iknowthatsoundstoolong)andmakeeachdaycountandamemorableonetocherish.Ithankmycommitteemembers,Dr.JefferyHo,Dr.PaulGader,Dr.BabaVemuriandDr.SergeiShabanovforagreeingtobeapartofmyPh.D.supervisorycommitteeandspendingtheirinvaluabletimeoverthenumeroushelpfuldiscussionsIhadwiththem.IamsincerelygratefultoDr.PaulGaderwho,amongotherthings,hasbeengenerousenoughtosupportmeandfundmyeducationrightthroughthisPh.D.voyage.IamalsoindebtedtomygraduatecoordinatorsMr.JohnBowersandMs.JoanCrismanforlendingtheirhelpinghandatalltimesofneed.IfIfailtothankmylab-matesandmyroom-mates,theymightcomeaftermesayingthatIfwehadn'tbeenhelpingyououtwithyourresearchproblems,youwouldn'tbehavingathesistowriteandifwehadspentourtimeinourresearchratherthandiscussingwithyou,wecouldhavegraduatedmuchearlier.Itakethisopportunitytothankeachoneofthemparticularly,AjitRajwade,VenkatakrishnanRamaswamy,Amit 4

PAGE 5

Dhurandhur,GnanaSundarRajendiran,BhupinderSingh,MansiPrasad,LokeshKumarBhoobalan,SubhajitSengupta,ShahedNejhum,ChiYu-Tseh,MohsenAli,ManuSethi,NathanVanderkraats,NicholasFisherandAmitVerma.Iowemygratitudetoallmyfamilymembers,myamma(meansmotherinmynativelanguageTamil),appa(meansfather),brother,sister-in-law,uncleandmyanceeBhavyaLakshmananforconstantlymotivatingandassuringmethatIwillgraduateoneday.FinallyIthankyoureaderwhoiscuriousenoughandwillingtoembarkonknowingwhatIdidinthepastsixyearsasagraduatestudentatUniversityofFlorida. 5

PAGE 6

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1EikonalEquation ................................ 13 1.1.1TheClassicalApproach ........................ 14 1.1.2TheQuantumMechanicalApproach ................. 16 1.2EuclideanDistanceFunctions ......................... 19 1.3GradientDensityEstimation .......................... 20 2CLASSICALANDQUANTUMFORMULATIONFORTHEEIKONALEQUATION 22 2.1Hamilton-JacobiFormalismfortheEikonalEquation ............ 22 2.1.1Fermat'sPrincipleofLeastTime ................... 22 2.1.2AnEquivalentVariationalPrinciple .................. 25 2.2SchrodingerWaveEquationFormalismfortheEikonalEquation ..... 27 2.2.1APathIntegralDerivationoftheSchrodingerEquation ....... 27 2.2.2ObtainingtheEikonalEquationfromtheSchrodingerEquation .. 32 3EUCLIDEANDISTANCEFUNCTIONS ....................... 34 3.1Closed-FormSolutionsforConstantForcingFunctions ........... 34 3.2ProofsofConvergencetotheTrueDistanceFunction ........... 36 3.3ModiedGreen'sFunction ........................... 38 3.4ErrorBoundBetweentheObtainedandtheTrueDistanceFunction ... 40 3.5EfcientComputationoftheApproximateDistanceFunction ........ 41 3.5.1SolutionfortheDistanceFunctioninHigherDimensions ...... 41 3.5.2NumericalIssuesandExactComputationalComplexity ...... 42 4SIGNEDDISTANCEFUNCTIONANDITSDERIVATIVES ............ 43 4.1ConvolutionBasedMethodforComputingtheWindingNumber ...... 43 4.2ConvolutionBasedMethodforComputingtheTopologicalDegree .... 45 4.3FastComputationoftheDerivativesoftheDistanceFunction ....... 46 6

PAGE 7

5GENERALEIKONALEQUATION .......................... 49 5.1PerturbationTheory .............................. 49 5.2DerivingaBoundfortheConvergenceofthePerturbationSeries ..... 52 5.3EfcientComputationoftheWaveFunction ................. 54 5.3.1NumericalIssues ............................ 55 5.3.2ExactComputationalComplexity ................... 56 6GRADIENTDENSITYESTIMATIONINONEDIMENSION ........... 57 6.1MotivationfromQuantumMechanics:TheCorrespondencePrinciple ... 57 6.2PhaseRelationshipBetweenandS .................... 58 6.3DensityFunctionfortheGradients ...................... 59 6.4ExistenceoftheDensityFunction ....................... 60 6.5EquivalenceoftheGradientDensityandthePowerSpectrum ....... 61 6.5.1BriefExpositionoftheResult ..................... 63 6.5.2FormalProof .............................. 64 6.5.3SignicanceoftheResult ....................... 70 7DENSITYESTIMATIONFORTHEDISTANCETRANSFORMS ......... 71 7.1DensityFunctionfortheDistanceTransforms ................ 72 7.2PropertiesoftheFourierTransformofCWR ................. 74 7.2.1BriefExpositionoftheTheorem .................... 76 7.2.2FormalProof .............................. 77 7.3SpatialFrequenciesasGradientHistogramBins .............. 80 7.3.1BriefExpositionoftheTheorem .................... 81 7.3.2FormalProof .............................. 82 7.4SignicanceoftheResult ........................... 96 8EXPERIMENTALRESULTS ............................. 98 8.1EuclideanDistanceFunctions ......................... 98 8.1.12DExperiments ............................. 98 8.1.2Medialaxiscomputations ....................... 101 8.1.33DExperiments ............................. 104 8.2TheGeneralEikonalEquation ........................ 107 8.2.1ComparisonwiththeTrueSolution .................. 107 8.2.2ComparisonwithFastSweeping ................... 109 8.3TopologicalDegreeExperiments ....................... 112 8.4EmpiricalResultsfortheGradientDensityestimationinOneDimension 113 8.5EmpiricalResultsfortheDensityFunctionsoftheDistanceTransforms 114 8.5.1CWRanditsFourierTransform .................... 114 8.5.2ComparisonResults .......................... 115 7

PAGE 8

9DISCUSSIONANDFUTUREWORK ........................ 119 9.1Conclusion ................................... 119 9.2FutureWork ................................... 120 REFERENCES ....................................... 122 BIOGRAPHICALSKETCH ................................ 126 8

PAGE 9

LISTOFTABLES Table page 3-1AlgorithmfortheapproximateEuclideandistancefunction ............ 41 5-1Algorithmfortheapproximatesolutiontotheeikonalequation .......... 55 8-1Maximumpercentageerrorfordifferentvaluesof~. ............... 99 8-2PercentageerroroftheEuclideandistancefunctioncomputedusingthegridpointsoftheshapesasdatapoints ......................... 101 8-3PercentageerrorandthemaximumdifferencefortheSchrodingermethodoverdifferentiterations ................................ 108 8-4PercentageerrorandthemaximumdifferencefortheSchrodingermethodincomparisontofastsweeping ............................ 109 9

PAGE 10

LISTOFFIGURES Figure page 7-1VoronoidiagramofthegivenKpoints.EachVoronoiboundaryismadeofstraightlinesegments. ................................ 73 7-2RegionthatexcludesboththesourcepointandtheVoronoiboundary ..... 74 7-3Plotoftheboundarybetweenthetwoangles. ................... 89 7-4Plotofradiallengthvsangle. ............................ 90 7-5Threedisconnectedregionsfortheangle. ..................... 90 8-1Percentageerrorversus~. ............................. 99 8-2PercentageerrorbetweenthetrueandcomputedEuclideandistancefunctions. 100 8-3Shapes ........................................ 100 8-4Shapecontourplots. ................................. 101 8-5Aquiverplotof5S=(Sx,Sy)(bestviewedincolor). .............. 102 8-6Zoomedquiverplot .................................. 103 8-7Medialaxisplots ................................... 105 8-8Bunnyisosurfaces. .................................. 106 8-9Dragonisosurfaces. ................................. 107 8-10Contourplotsshowingcomparisonwiththetruesolution. ............ 108 8-11Contourplotsshowingcomparisonwithfastsweeping(experiment1). ..... 110 8-12Contourplotsshowingcomparisonwithfastsweeping(experiment2). ..... 110 8-13Contourplotsshowingcomparisonwithfastsweepingonalargergrid. ..... 112 8-14Topologicaldegreeexperiments. .......................... 113 8-15Comparisonresultswiththetruegradientdensityin1D. ............. 114 8-16CWRanditsFouriertransform. ........................... 116 8-17ComparisonresultswiththetrueorientationdensitiesofEuclideandistancefunctions. ....................................... 118 8-18PlotofL1errorvs~fortheorientationdensityfunctions. ............. 118 10

PAGE 11

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyASCHRODINGERWAVEMECHANICSFORMALISMFORTHEEIKONALPROBLEMANDITSASSOCIATEDGRADIENTDENSITYCOMPUTATIONByKarthikS.GurumoorthyAugust2011Chair:ArunavaBanerjeeCochair:AnandRangarajanMajor:ComputerEngineeringManycomputationaltechniquesbasedonclassicalmechanicsexistbutsurprisinglythereisn'taconcomitantborrowingfromquantummechanics.OurworkshowsanapplicationoftheSchrodingerformalismtosolvetheclassicaleikonalproblemanonlinear,rstorder,partialdifferentialequationoftheformk5Sk=f,wheretheforcingfunctionf(X)isapositivevaluedboundedfunctionand5denotesthegradientoperator.HamiltonianJacobibasedsolverslikethefastmarchingandfastsweepingmethodssolveforSbytheGodunovupwinddiscretizationscheme.Insharpcontrasttothat,wepresentaSchrodingerwavemechanicsformalismtosolvetheeikonalequationbyrecastingitasalimitingcaseofaquantumwaveequation.Weshowthatasolutiontothenon-lineareikonalequationisobtainedinthelimitasPlanck'sconstant~(treatedasafreeparameter)tendstozeroofthesolutiontothecorrespondinglinearSchrodingerequation.Webeginwith,byconsideringtheEuclideandistancefunctionproblem,aspecialcaseoftheeikonalequationwheretheforcingfunctioniseverywhereidenticallyequaltoone.WeshowthatthesolutiontotheSchrodingerwavefunctioncanbeexpressedasadiscreteconvolutionbetweentwofunctionsefcientlycomputablebytheFastFourierTransforms(FFT).TheEuclideandistancefunctioncanthenberecoveredfromtheexponentofthewavefunction.Sincethewavefunctioniscomputedforasmallbut 11

PAGE 12

non-zero~,theobtainedsolutionisanapproximation.WeshowconvergenceofourapproximateclosedformsolutionfortheEuclideandistancefunctionproblemtothetruesolutionas~!0andalsoboundtheerrorforagivenvalueof~.Moreoverthedifferentiabilityofoursolutionallowsustocomputeitsrstandsecondderivativesinclosedform,alsocomputablebyaseriesofconvolutions.Inordertodeterminethesignofthedistancefunction(positiveinsideacloseregionandnegativeoutside),wecomputethewindingnumberin2Dandtopologicaldegreein3D,byexplicitlyshowingthattheircomputationscanalsobedoneviaconvolutions.Weshowanapplicationourofmethodbycomputingthemedialaxesforasetof2Dsilhouettes.Amajoradvantageofourapproachovertheotherclassicalmethodsisthat,wedonotrequireaspatialdiscretizationofgradientoperatorsasweobtainaclosed-formsolutionforthewavefunction.Forthegeneraleikonalproblemwheretheforcingcanbeanarbitrarybutpositiveandboundedfunction,theSchrodingerequationturnsouttobeageneralized,screenedPoissonequation.Despitebeinglinear,itdoesnothaveaclosed-formsolution.Weuseastandardperturbationanalysisapproachtocomputethesolutionwhichisguaranteedtoconvergeforallpositiveandboundedforcingfunctions.TheperturbationtechniquerequiresasequenceofdiscreteconvolutionswhichcanbeperformedusingtheFFT.Finallyusingstationaryphaseapproximationsweestablishamathematicalresultrelatingthedensityofthegradient(s)ofdistancefunctionSandthescaledpowerspectrumofthewavefunctionforsmallvaluesof~,whenthescalareldSappearsasthephaseofthewavefunction.Byprovidingrigorousmathematicalproofs,wejustifyourresultforanarbitrarythricedifferentiablefunctioninonedimensionandfordistancetransformsintwodimensions.Wealsofurnishanecdotalvisualevidencestocorroborateourclaim.Ourresultgivesanewsignatureforthedistancetransformsandpotentiallyserveasitsgradientdensityestimator. 12

PAGE 13

CHAPTER1INTRODUCTIONComputationaltechniquesadaptedfromclassicalmechanicsrunthegamutfromLagrangianactionprinciplestoHamilton-Jacobieldequations[ 22 ]:witnessisthepopularityofthefastmarching[ 34 41 ]andfastsweepingmethods[ 50 ]whichareessentiallyfastHamilton-Jacobisolvers.Insharpcontrast,thereareveryfewapplicationsofquantummechanics-inspiredcomputationalmethods.Despitethewellknownfactthatmostofclassicalmechanicscanbeobtainedasalimitingcaseofquantummechanics(asPlanck'sconstant~tendstozero)[ 24 ]andthatthelinearSchrodingerequations[ 19 24 ]arethequantumcounterparttonon-linearHamilton-Jacobiequations[ 9 ],thispaucityissomewhatsurprising.Ratherthanspeculateonthereasonsforthisdearthofapplications,wewishtopointoutthatinthiswork,weareprimarilyinterestedinexploitingaconcreterelationshipbetweentheclassical,non-linearHamilton-Jacobiequation[ 22 ]andthequantum,linearSchrodingerequation[ 24 ].Wefeelthatfocusingmorenarrowlyonthisrelationship(whichwillbecomemoreobviousasweproceed)ismoreproductivethandwellingonthemoremysteriousandspecicallyquantummechanicalissuesofi)interpretationofthewavefunction,ii)roleofprobabilitiesand,iii)theproblemofmeasurement[ 19 24 ].Whiletheseissuesarecertainlyimportant,theydonotplayanyroleinthiswork.Thecurrentworkdemonstratesanapplicabilityofthisrelationshipforaveryspeciccase,namelytheeikonalequationandtheestimationofitsgradientdensities. 1.1EikonalEquationTheeikonal(fromtheGreekwordoorimage)equationistraditionallyencounteredinthewaveandgeometricopticsliteraturewheretheprincipalconcernisthepropagationoflightraysinaninhomogeneousmedium[ 12 ].Itstwinrootsareinwavepropagationtheoryandingeometricoptics.Inwavepropagationtheory,itisobtainedwhenthewaveisapproximatedusingtheWentzelKramersBrillouin(WKB) 13

PAGE 14

approximation[ 35 ].Ingeometricoptics,itcanbederivedfromHuygen'sprinciple[ 3 ].Inthepresentday,theeikonalequationhasoutgrownitshumbleopticsoriginsandnowndsapplicationinfarungareassuchaselectromagnetics[ 35 ],robotmotionpathplanning[ 10 ]andimageanalysis[ 28 33 ].Theeikonalequationisanonlinear,rstorder,partialdifferentialequation[ 46 ]oftheform k5S(X)k=f(X),X2(1)subjecttotheboundaryconditionSj@=U(X),whereisaboundedsubsetofRD.Theforcingfunctionf(X)isapositivevaluedboundedfunctionand5denotesthegradientoperator.Detaileddiscussionsontheexistenceanduniquenessofthesolutioncanbefoundin[ 16 ].ThepresentworkconcernswithsolvingtheeikonalequationonadiscretizedspatialgridconsistingofNgridlocationsfromasetofKpointsourcesfYkgKk=1.Lightwavessimultaneouslyemanatefromthegivenpointsourcesandpropagatewithavelocityof1 f(X)atthegridlocationX.ThevalueofSatagridpointX0(S(X0))correspondstothetimetakenbytherstlightwave(outoftheKlightwaves)toreachthegridlocationX0. 1.1.1TheClassicalApproachWhiletheeikonalequationisvenerableandclassical,itisonlyinthelasttwentyyearsthatwehaveseentheadventofnumericalmethodsaimedatsolvingthisproblem.Tonameafewarethepioneeringfastmarching[ 34 41 ]andfastsweeping[ 50 ]methods.AlgorithmsbasedondiscretestructuressuchasthewellknownDijkstrasinglesourceshortestpathalgorithm[ 15 ]canalsobeadaptedtosolvethisproblem.WhenweseeksolutionsonadiscretizedspatialgridwithNpoints,thecomplexityofthefastmarchingmethodisO(NlogN)oating-pointoperationswhilethatofthefastsweepingmethodisO(N)andthereforebothoftheseefcientalgorithmshaveseenwidespreadusesincetheirinception.Recently,theingeniousworkofSapiroet.al.providedanO(N)implementationofthefastmarchingmethod[ 49 ].Fastsweeping 14

PAGE 15

methodshavealsobeenextendedtothemoregeneralstaticHamilton-Jacobiequation[ 27 ]andalsofortheeikonalequationonnon-regulargrids[ 26 36 ].AnHamiltonianapproachtosolvetheeikonalequationcanbefoundin[ 42 ].Theeikonalequationcanalsobederivedfromavariationalprinciple[ 21 ],namely,Fermat'sprincipleofleasttimewhichstatesthatNaturealwaysactsbytheshortestpaths[ 4 ].ThevariationalproblemisdenotedbythesymbolL,calledtheLagrangian.TheintegraloftheLagrangianiscalledtheactionofthephysicalsystem[ 22 ]givenby S(X,t)Ztt0Ldt.(1)ForthecasewhentheLagrangiandoesn'thaveanexplicitdependenceontime,theactionScanbeseparatedasS(X,t)=S(X))]TJ /F4 11.955 Tf 13.02 0 Td[(Et.ThequantitySiscalledtheHamilton'scharacteristicfunctionandEdenotestheenergyofthesystemrepresentingtheconstantsofmotion[ 22 ].Fromthisvariationalprinciple,theclassicalphysicsdevelopmentalsequenceforderivingtheeikonalequationproceedsasfollows:TherstorderHamilton'sequationsofmotionarederivedusingaLegendretransformationoftheLagrangianwhereinnewmomentumvariablesareintroduced.Subsequently,acanonicaltransformationconvertsthetimevaryingmomentaintoconstantsofthemotion.TheHamilton-Jacobiequationemergesfromthecanonicaltransformation[ 22 ].IntheHamilton-Jacobiformalismspecializedtotheeikonalproblem,thesolutiontoitisactuallytheHamilton'scharacteristicfunctionS.Hereweseekasurfacesuchthatitsincrementsareproportionaltothespeedofthelightrays1 f(X).ThisiscloselyrelatedtoHuygen'sprincipleandthusmarkstherapprochementbetweengeometricandwaveoptics[ 3 ].Itisthisnexusthatdrivesnumericalanalysismethods[ 34 50 ](focusedonsolvingtheeikonalequation)tobasetheirsolutionsaroundtheHamilton-Jacobiformalism. 15

PAGE 16

1.1.2TheQuantumMechanicalApproachTheimportanceofLagrangianLandtheactionSinquantummechanicswasenvisionedbyDiracintheearly1930s,whenheobservedthattheshort-timepropagator[ 11 ](denedinSection 2.2 )istheexponentialofiS ~,whereSistheclassicalaction.ThiseventuallyledFeynmantotheinventionofquantum-mechanicalpathintegralapproachtoderivingtheSchrodingerwaveequation[ 19 ].Sincetheclassicalandquantummechanicsaresointimatelyrelated[ 9 ],itisnaturaltoaskthesefollowingquestions,namely HowdoestheSchrodingerwaveequationfortheeikonalequationlooklike? OncewederivedthecorrespondingSchrodingerwaveequation,howdowesolveforthewavefunction? Fromthecomputationalperspective,howdoweefcientlycomputeit? Howtoretrievethesolutiontotheoriginaleikonalequation(Equation 1 )fromthewavefunction?Thepresentworkseekstoanswerthesequestions.Beforeweproceed,wewouldliketogiveaverybriefintroductiontotheSchrodingerwaveequation.Schrodingerwaveequation,formulatedbytheAustrianphysicistErwinSchrodingerintheyear1926,isanequationthatdescribeshowthequantumstateorthewavefunctionofaphysicalsystemevolvesintime.ItisascentraltoquantummechanicsasNewton'slawsaretoclassicalmechanics.Aquantumstateisamostcompletedescriptionthatcanbegiventoaphysicalsystem.SolutionstotheSchrodingerwaveequationcandescribequantitiesrangingfromthemolecular,subatomicparticlestothatofthewholeuniverse.Themostgeneralformisthetime-dependentSchrodingerequation,describingasystemthatevolveswithtime.Itiswrittenas i~@ @t=^H (1)where, 16

PAGE 17

(X,t)isthewavefunctionwith carryingaprobabilityinterpretationofndingaparticleatpositionXattimet. i~@ @tistheenergyoperatorwhere~h 2iscalledthereducedPlanck'sconstant. ^HistheHamiltonianoperatorwhichdeterminestheevolutionofthesystemwithtime.IftheHamiltonianoperator^Hisindependentoftime,Equation 1 canbefactoredtoobtaintime-independentSchrodingerwaveequation ^Hn=Enn(1)where n(X)isthestationarystatewaverepresentingthestateofthesystemthatdoesnotchangewithtime. Eniscalledtheenergyofthesystem.Easytoseethattheabovesystemisaneigenfunctionsystemwithnbeingtheeigenfunctionofthe^HwithaneigenvalueEn.Ifweknowalltheeigenfunctionsnof^H,thesolutionforthetime-dependentwavefunction (X,t)isgivenby (X,t)=Xncne)]TJ /F16 5.978 Tf 7.78 3.53 Td[(iEnt ~n(X)(1)wherecnaredenedbytheinitialcondition 0(X,t0)=Xncne)]TJ /F16 5.978 Tf 7.78 4.39 Td[(iEnt0 ~n(X)(1)AdetaileddescriptionoftheSchrodingerwaveequation,itsoriginanditspropertiescanbefoundin[ 24 ].Sincetheadventofquantumtheory,specicallytheSchrodingerwaveequation,thecloserelationshipbetweentheSchrodingerandHamilton-Jacobiequationshasbeenintenselystudied[ 9 ].Ofparticularimportancehereisthequantumtoclassicaltransitionas~!0,wherethelawsofquantummechanicsareassumedtonaturallygiverisetothelawsofclassicalmechanics.Whenthetime-independent 17

PAGE 18

Hamilton-JacobiscalareldSistheexponentofthestationarystatewavefunction,specically(X)=exp()]TJ /F7 7.97 Tf 6.59 0 Td[(S(X) ~),andif(X)satisesthetime-independentSchrodingerequation,weshowthatas~!0,SsatisestheHamilton-Jacobiequation.Notethatintheabove,anonlinearHamilton-Jacobiequationisobtainedinthelimitas~!0ofalinearSchrodingerequationwhichisnovelfromanumericalanalysisperspective.Consequently,insteadofsolvingtheHamilton-Jacobiequation,onecansolveitsSchrodingercounterpart(takingadvantageofitslinearity),andcomputeanapproximateSforasuitablysmallvalueof~.ThiscomputationalprocedureisapproximatelyequivalenttosolvingtheoriginalHamilton-Jacobiequation.Surprisingly,thisrelationshiphasfoundveryfewapplicationsinthenumericalanalysisliteraturedespitebeingwellknown.Inthiswork,weleveragetheimportantdistinctionbetweentheSchrodingerandHamilton-Jacobiequations,namely,thattheformerislinearwhereasthelatterisnot.WetakeadvantageofthelinearityoftheSchrodingerequationwhileexploitingitsrelationshiptoHamilton-Jacobiandderivecomputationallyefcientsolutionstotheeikonalequation.Sincetheefcientsolutionofalinearwaveequationisthecornerstoneofourapproach,wenowbrieydescribetheactualcomputationalalgorithmused.WederivethestaticSchrodingerequationfortheeikonalproblem.Theresultisageneralized,screenedPoissonequation[ 18 ]whosesolutionisknownatKseedpoints.Thislinearequationdoesnothaveaclosed-formsolutionandthereforeweresorttoaperturbationmethod[ 17 ]ofsolutionwhichisrelatedtotheBornexpansion[ 30 ].Theperturbationmethodcomprisesasequenceofmultiplicationswithaspace-varyingforcingfunctionfollowedbyconvolutionswitha(modied)Green'sfunction(forthescreenedPoissonoperator)whichwesolveusinganefcientFastFouriertransform(FFT)-basedtechnique[ 7 14 ].Perturbationanalysisinvolvesageometricseriesapproximationforwhichweshowconvergenceforallboundedforcingfunctionsindependentofthevalueof~. 18

PAGE 19

1.2EuclideanDistanceFunctionsInthespecialcasewheref(X)equalsoneeverywhere,thesolutiontotheeikonalequationistheEuclideandistancefunction[ 33 ]andhencetheHamilton-JacobiscalareldSsatisesthedifferentialequation k5S(X)k=1.(1)WhenweseeksolutionforSfromasetofKdiscretepointsfYkgKk=1,theEuclideandistanceproblemcanbeformallystatedas:Givenapoint-setY=fYk2RD,k2f1,...,KggwhereDisthedimensionalityofthepoint-setandasetofequallyspacedCartesiangridpointsX,theEuclideandistancefunctionproblemrequiresustoassign S(X)=minkkX)]TJ /F4 11.955 Tf 11.95 0 Td[(Ykk(1)wherethenormk.kcorrespondstotheEuclideandistance.AlltheaforementionedHamilton-JacobisolverssolvesforSinEquation 1 byspatiallydiscretizingthederivativeoperatorbasedontheGodunovupwinddiscretizationscheme[ 50 ].AmajoradvantageofourSchrodingerapproachisthatitdoesnotrequirederivativediscretizationandthissometimesaccountsforimprovedaccuracyofourtechnique.MoreoverSchrodingerformalismresultsinaclosed-formsolutionfortheEuclideandistancefunctionproblemandthatitcanbeexpressedasadiscreteconvolutionandcomputedusingaFastFourierTransform(FFT)[ 7 ].However,acaveatisthatourEuclideandistancefunctionisanapproximationsinceitisobtainedforasmallbutnon-zerovalueofPlanck'sconstant~,butneverthelessconvergestothetruesolutionas~!0.TheSchrodingerequationapproachgivesusanunsigneddistancefunction.Wecomplementthisbyindependentlyndingthesignofthedistancefunction,byefcientlycomputingthewindingnumberforeachlocationinthe2Dgridanditsequivalentconcepttopologicaldegreein3D.WeshowthatjustasinthecaseoftheSchrodinger 19

PAGE 20

equation,thewindingnumberandthetopologicaldegreecomputationscanalsobewritteninclosed-form,expressedasadiscreteconvolutionandcomputedusingtheFFT.Thisappears(tousatanyrate)tobeanovelcontribution.Furthermore,weoftenseekthegradient,divergence,curvatureandmedialaxesofthesigneddistancefunctionwhicharenoteasytoobtainbytheseHamilton-Jacobiapproachesduetothelackofdifferentiabilityofthesigneddistancefunction.Butwecanleveragetheclosed-formsolutionobtainedfromtheSchrodingertocomputethesequantities.Sinceourdistancefunctionisdifferentiableeverywhere,wecanonceagainwritedownclosed-formexpressionsforthegradientsandcurvature,expressthemasdiscreteconvolutionsandagaincomputethesequantitiesusingFFTs.Wevisualizethegradientsandthemaximumcurvatureusing2Dshapesilhouettesasthesource.Themaximumcurvaturehasahauntingsimilaritytothemedialaxesasshowninourexperimentalresults.Toourknowledge,thefastcomputationofthederivativesofthedistancefunctiononaregulargridusingdiscreteconvolutionsisnew. 1.3GradientDensityEstimationSofarinourdevelopmentontherelationbetweenthewavefunctionandthescalareldS,weexpressedSintheexponentof.ButwhenSappearsasthephaseofthestationarywavefunction,insteadofappearinginitsexponent,specicallywhen (X)=expiS(X) ~,(1)asonendsintheWKBapproximationoftheeikonalequation[ 35 ],wenoticedasurprisingresultrelatingthedensityofthegradientsofS(Sx,Sy)andthescaledpowerspectrumofthewavefunctionforsmallvaluesof~.InotherwordsthesquaredmagnitudeoftheFouriertransformofthewavefunctionisapproximatelyequaltothedensityfunctionofthegradientsofSwiththeapproximationbecomingincreasinglyexactas~!0.Usingstationaryphaseapproximationsawellknowntechniqueinasymptoticanalysisandstandardintegrationtechniqueswithproperorderingoflimits, 20

PAGE 21

weestablishthisgradientfrequencyrelationforanarbitrarythricedifferentiablefunctionSinonedimensionandfordistancetransformsintwodimensions.Wealsofurnishanecdotalvisualevidencestocorroborateourclaim.Thesignicanceofourresultisthatspatialfrequenciesbecomehistogrambins.Apartfromprovidingwithanewsignatureforthedistancetransforms,ourresultcanbeusedtoserveasitsgradientdensityestimator.SincethedensityfunctionsareobtaineddirectlyfromthefunctionS,ourmethodcircumventstheneedtocomputeitsderivative(s). 21

PAGE 22

CHAPTER2CLASSICALANDQUANTUMFORMULATIONFORTHEEIKONALEQUATION 2.1Hamilton-JacobiFormalismfortheEikonalEquation 2.1.1Fermat'sPrincipleofLeastTimeItiswellknownthattheHamilton-JacobiequationformalismfortheeikonalequationcanbeobtainedbyconsideringavariationalproblembasedonFermat'sprincipleofleasttime[ 4 ].WhileweuseD=2forillustrationpurposes,theapproachisgeneralandnotrestrictedtoaparticularchoiceofdimension: I[q]=Zt1tof(q1,q2)q _q21+_q22dt.(2)Thevariationalproblemdenedabovehasitsrootsingeometricoptics[ 12 ].Consideramediumwithrefractiveindexf(q1,q2)andletaspacecurvebedescribedby(q1(t),q2(t),t),t2[t0,t1].Fermat'sleasttimeprinciplestatesthatthenecessaryconditionfortobealightrayisthatIbeanextremum.Thequantity L(q1,q2,_q1,_q2,t)=f(q1,q2)q _q21+_q22(2)iscalledtheopticalLagrangian.Thegeneralizedmomentaaredenedas[ 22 ] pi@L @_qi=f_qi p _q21+_q22.(2)SincetheLagrangianishomogeneousofdegreeonein(_q1,_q2),wecannotsolvefor_qiasafunctionofpi.Howeverthemomentumvariables(pi)satisfythefollowingrelation,namely p21+p22=f2(2)usingwhichtheeikonalequationisderived.ALegendretransformation[ 3 ]isusedtoobtaintheHamiltonian,whichisgivenby H(q1,q2,p1,p2,t)=2Xi=1pi_qi)]TJ /F4 11.955 Tf 11.95 0 Td[(L(q1,q2,_q1,_q2,t)(2) 22

PAGE 23

where_qiisassumedtobeafunctionofpi.ALegendretransformisanoperationthattransformsonereal-valuedfunctionintoanotherinawaysuchthatthederivativeoftheoriginalfunctionbecomestheargumentofthetransformedfunction.Itiseasiertoexplainwithfunctionsofonevariable.Considerafunctiong(x).ItsLegendretransformgisdenedas g(p)maxx(px)]TJ /F4 11.955 Tf 11.96 0 Td[(g(x)).(2)Foradifferentiablefunctiong,maximizingtheabovequantityoverx,givesustherelation g0(x)=p.(2)ScenariosunderwhichxcanbeexpressedintermsofpusingEquation 2 ,theLegendrefunctiongcanbeobtainedinclosedform.Letx0bethelocationwherethefunctionpx)]TJ /F4 11.955 Tf 11.95 0 Td[(g(x)attainsitsmaximum.Theng(p)satises g(x0)=g0(x0)x0)]TJ /F4 11.955 Tf 11.95 0 Td[(g(p).(2)FromEquation 2 ,g(p)canbeinterpretedasthenegativeofthey-interceptofthetangentlinetothegraphofgthathasslopep=g0(x0).Withoutdelvingintofurtherdetails,webasicallytransformedafunctiongdenedonthevariablex,toafunctiongdenedonpusingEquation 2 .Thereadermayreferto[ 3 ]togetadetailedexplanationonLegendretransformations.ToobtaintheHamiltonianformulationfromtheLagrangianformulation,wecarriedoutsomethingsimilar.TheLagrangianL,denedinEquation 2 ,isafunctionofthepositionalcoordinatesqianditstimederivative_qi.TheHamiltonianshiftsthefunctionaldependenciestothepositionsqiandmomentapibydeningpiasinEquation 2 andexpressing_qiasafunctionofp0is.UnfortunatelyinourcasetheHamiltonianHcannotbeexpressedinclosedform,as_qicannotbeexplicitlyexpressedasafunctionofp0is. 23

PAGE 24

TheHamilton-Jacobiequationisobtainedviaacanonicaltransformation[ 22 ]oftheHamiltonian.Inclassicalmechanics,acanonicaltransformationisdenedasachangeofvariableswhichleavestheformoftheHamiltonianunchanged.Foratype2canonicaltransformation,wehave 2Xi=1pi_qi)]TJ /F4 11.955 Tf 11.95 0 Td[(H(q1,q2,p1,p2,t)=2Xi=1Pi_Qi)]TJ /F4 11.955 Tf 11.95 0 Td[(K(Q1,Q2,P1,P2,t)+dF dt(2)whereF)]TJ /F12 11.955 Tf 23.91 8.96 Td[(P2i=1QiPi+F2(q,P,t)whichgives dF dt=)]TJ /F8 7.97 Tf 17.71 14.94 Td[(2Xi=1)]TJ /F6 11.955 Tf 8.87 -7.02 Td[(_QiPi+Qi_Pi+@F2 dt+2Xi=1@F2 @qi_qi+@F2 @Pi_Pi.(2)Whenwepickaparticulartype2canonicaltransformationwherein_Pi=0,i=1,2andK(Q1,Q2,P1,P2,t)=0,weget @F2 @t+H(q1,q2,@F2 @q1,@F2 @q2,t)=0(2)whereweareforcedtomaketheidentication pi=@F2 @qi,i=1,2.(2)NotethatthenewmomentaPiareconstantsofthemotion(usuallydenotedbyi,i=1,2).ChangingF2toSasincommonpractice,wehavethestandardHamilton-JacobiequationforthefunctionS(q1,q2,1,2,t),namely @S @t+H(q1,q2,@S @q1,@S @q2,t)=0.(2) 24

PAGE 25

TocompletethecirclebacktotheLagrangian,wetakethetotaltimederivativeoftheHamilton-JacobifunctionStoget dS(q1,q2,1,2,t) dt=2Xi=1@S @qi_qi+@S @t=2Xi=1pi_qi)]TJ /F4 11.955 Tf 11.95 0 Td[(H(q1,q2,@S @q1,@S @q2,t)=L(q1,q2,_q1,_q2,t). (2) ConsequentlyS(q1,q2,1,2,t)=Rtt0Ldtandtheconstantsf1,2gcannowbeinterpretedasintegrationconstants.ForamoreaccessibletreatmentoftherelationshipbetweenLagrangian'sandtheHamilton-JacobieldS(q1,q2,1,2,t),pleasesee[ 22 ].SincetheHamiltonianisnotanexplicitfunctionoftime,Equation 2 canbesimpliedtothestaticHamilton-Jacobiequation.Byseparationofvariables,weget S(q1,q2,t)=S(q1,q2))]TJ /F4 11.955 Tf 11.96 0 Td[(Et(2)whereEisthetotalenergyofthesystemandS(q1,q2)iscalledHamilton'scharacteristicfunction[ 3 ].Observingthat@S @qi=@S @qi,andusingtheEquations 2 (replacingF2byS),and 2 ,wehavek5Sk2=f2,whichisthefamiliareikonalequation(Equation 1 ). 2.1.2AnEquivalentVariationalPrincipleWetakeanidiosyncraticapproachtotheeikonalequationbyconsideringadifferentvariationalproblemwhichisstillverysimilartoFermat'sleasttimeprinciple.TheadvantageofthisvariationalformulationisthatthecorrespondingSchrodingerwaveequationcanbeeasilyobtained.Considerthefollowingvariationalproblemnamely, I[q]=Zt1to1 2(_q21+_q22)f2(q1,q2)dt(2)wheretheLagrangianLisdenedas L(q1,q2,_q1,_q2,t)1 2(_q21+_q22)f2(q1,q2).(2) 25

PAGE 26

TheabovedenitionforLisactuallythesquareoftheLagrangianconsideredinEquation 2 .SquaringtheLagrangianmakesitnottobehomogeneousofdegreeonein_qi,allowingustorewrite_qiintermsofmomentumvariablepi,givenby pi@L @_qi=f2(q1,q2)_qi.(2)ItisworthemphasizingthatminimizingI[q]inboththeEquations 2 and 2 givesthesamesolutionforq.ByapplyingtheLegendretransformation[ 3 ]asbefore,wegetaclosedformexpressionfortheHamiltonianofthesystemin2Das H(q1,q2,p1,p2,t)=1 2(p21+p22) f2(q1,q2).(2)FromacanonicaltransformationoftheHamiltonian[ 22 ],weobtainthefollowingHamilton-Jacobiequation @S @t+1 2@S @q12+@S @q22 f2(q1,q2)=0(2)SinceHisindependentoftime,bytheseparationofvariableswecanexpressS(q1,q2,t)=S(q1,q2))]TJ /F4 11.955 Tf 13.2 0 Td[(Et.Observingthat@S @qi=@S @qi,Equation 2 canberewrittenas 1 2"@S @q12+@S @q22#=Ef2.(2)ChoosingtheenergyEtobe1 2,weobtain k5Sk2=f2(2)whichistheoriginaleikonalequation(Equation 1 ).SistherequiredHamilton-Jacobiscalareldwhichisefcientlyobtainedbythefastsweeping[ 50 ]andfastmarchingmethods[ 34 ]. 26

PAGE 27

2.2SchrodingerWaveEquationFormalismfortheEikonalEquationInthissectionwederiveaSchrodingerequationforouridiosyncraticvariationalproblem(Equation 2 )fromrstprinciplesandthenrecoverthescalareldS[asinEquation 2 ]fromthewavefunction. 2.2.1APathIntegralDerivationoftheSchrodingerEquationFirstly,weconsiderthecasewheretheforcingfunctionfisconstantandequals~feverywhereandthengeneralizetospatiallyvaryingforcingfunctions.Forconstantforcingfunctions,theLagrangianLdenedinEquation 2 isgivenby L(q1,q2,_q1,_q2,t)1 2(_q21+_q22)~f2(q1,q2).(2)WefollowtheFeynmanpath-integralapproach[ 19 44 ]toderivingthedifferentialequationforthetime-dependentwavefunction andsubsequentlyarriveatthetime-independentwavefunction.WewouldliketoemphasizethatthoughtheFeynmanpathintegralapproachgivesaconstructivemechanismforderivingtheSchrodingerwaveequation,itisnotconsideredmathematicallyrigorousinthegeneralsetting.Foramoredetailedexplanationonthissubject,thereadermayreferto[ 11 19 ].Thekeyideaistoconsiderthetransitionamplitude(alsocalledtheshort-timepropagator)K(X,t2,,t1)whereKKcorrespondstotheconditionaltransitionalprobabilitydensityofaparticlegoingfrom(t1)toX(t2).ForanyspecicpathX(t)=fx1(t),x2(t)gin2D,theamplitudeisassumedtobeproportionalto expi ~Zt2t1L(X,_X,t)dt(2)wheretheLagrangianLisgivenbyEquation 2 .IftheparticlecanmovefromtoXoverasetofpaths,thetransitionamplitudeisdenedasthesumoftheamplitudesassociatedwitheachpath,so K(X,t2;,t1)Zexpi ~Zt2t1L(X,_X,t)dtDX.(2) 27

PAGE 28

NowsupposethataparticleismovingfromastartingpositionX+=(x1+1,x2+2)attimetandendsatxattimet+whiletravelingforaveryshorttimeinterval.UsingthedenitionoftheLagrangianfromEquation 2 ,thetransitionamplitudeforthiseventis K(X,t+;X+,t)=Zexpi ~Zt+t1 2(_x21+_x22)~f2(x1,x2)dt0DXZexp(i 2~"1 2+2 2#~f2(x1,x2))DX=Mexpi 2~)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(21+22~f2(x1,x2). (2) HereMRDX.Inordertoderivethewaveequationfor ,werstrecallthatthewavefunction hasaninterpretationthat =j (X,t)j2denotestheprobabilityofndingaparticleatXandattimet.SinceKbehavesmorelikeaconditionaltransitionalprobabilityfromX+toX,thewavefunctionshouldsatisfy (X,t+)=ZK(X,t+;X+,t) (X+,t)d.(2)whereKisgivenbyEquation 2 .Expandingtotherstorderintandsecondorderinweget +@ @t=ZMexpi 2~)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(21+22~f2(x1,x2) +1@ @x1+2@ @x2+21 2@2 @x21+22 2@2 @x22+12@2 @x1@x2d=I1 +I2@ @x1+I3@ @x2+I41 2@2 @x21+I51 2@2 @x22+I6@2 @x1@x2 (2) 28

PAGE 29

wheretheintegralsI1,I2,I3,I4,I5,I6aredenedas I1MZZexpi 2~)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(21+22~f2(x1,x2)d1d2,I2MZZexpi 2~)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(21+22~f2(x1,x2)1d1d2,I3MZZexpi 2~)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(21+22~f2(x1,x2)2d1d2,I4MZZexpi 2~)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(21+22~f2(x1,x2)21d1d2,I5MZZexpi 2~)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(21+22~f2(x1,x2)22d1d2,I6MZZexpi 2~)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(21+22~f2(x1,x2)12d1d2. (2) ObservingthattheintegralI2canberewrittenasaproductoftwointegrals,i.e, I2=MZexpi 2~21~f2(x1,x2)1d1Zexpi 2~22~f2(x1,x2)d2(2)andthattherstintegralisanoddfunctionof1,itfollowsthatI2=0.AsimilarargumentshowsthatI3=I6=0.Usingtherelation Z1exp(is2)ds=r i (2)andnoticingthatI1canbewrittenasaseparateproductoftwointegralsin1andin2,itfollowsthat I1=Mi2~ ~f2(x1,x2).(2)Inorderfortheequationfor tohold,I1shouldapproach1as!0.Hence, M=~f2(x1,x2) i2~.(2)LetIdenotetheintegralinEquation 2 .Then 1 i@I @=i 2p i 3 2=Zexp(is2)s2ds.(2) 29

PAGE 30

UsingtherelationsinEquations 2 and 2 with=~f2 2~andwritingI4andI5asproductoftwoseparateintegralsin1and2andsubstitutingthevalueofMfromEquation 2 ,weobtain I4=I5=i~ ~f2(x1,x2).(2)SubstitutingbackthevalueoftheseintegralsinEquation 2 ,weget +@ @t= +i 2~ ~f2@2 @x21+@2 @x22(2)fromwhichweobtaintheSchrodingerwaveequation[ 24 ] i~@ @t=^H (2)wheretheHamiltonianoperator^Hisgivenby ^H=)]TJ /F5 11.955 Tf 13.64 8.09 Td[(~2 2~f2@2 @x21+@2 @x22.(2)SincetheHamiltonianHdoesn'texplicitlydependontime,usingseparationofvariables (X,t)=(X)g(t),weget i~_g g=)]TJ /F5 11.955 Tf 13.64 8.09 Td[(~2 2~f252 =E(2)whereEistheenergystateofthesystemand52istheLaplacianoperator.Solvingforg,weget g(t)=expEt i~(2)andsatises )]TJ /F5 11.955 Tf 16.3 8.09 Td[(~2 2~f252=E.(2)ThesolutionfortheSchrodingerwave isthenoftheform (X,t)=(X)expEt i~.(2) 30

PAGE 31

WeareprimarilyinterestedinsolvingforandrelatethestationarywavefunctionandtheHamilton-JacobiscalareldStoobtainthesolutionforthelatter.WhenE>0inEquation 2 ,thesolutionsforareoscillatoryinnatureandwhenE<0thesolutionsaregeneralizedfunctions(distributions)whichareexponentialinnature.Thisnatureofthesolutionwillbecomeclearerwhenweprovidetheactualclosed-formexpressionforthewavefunctioninSection 3.1 .Foreikonalproblems,weareprimarilyinterestedonlyintheexponentialsolutionforcomputedatE=)]TJ /F8 7.97 Tf 10.49 4.7 Td[(1 2,asitallowsustoexplicitlyshowtheconvergenceofourclosed-formsolution(obtainedforconstantforcingfunctions)tothetruesolutionas~!0.ThereadermayrefertoSection 3.2 fordetailedconvergenceproofs.SettingE=)]TJ /F8 7.97 Tf 10.49 4.71 Td[(1 2inEquation 2 ,wegettheSchrodingerwaveequationwherethewavefunctionsatisesthedifferentialequation )]TJ /F5 11.955 Tf 11.96 0 Td[(~252+~f2=0.(2)Evenforanarbitrarypositive,boundedforcingfunctionf,weproposetosolveadifferentialequationverysimilartoEquation 2 byreplacingtheconstantforce~fwiththespatiallyvaryingforcingfunctionf.TheSchrodingerwaveequationforthegeneraleikonalproblemcanthenbestatedas )]TJ /F5 11.955 Tf 11.96 0 Td[(~252+f2=0.(2)Wewouldliketopointoutthattheproposedwaveequation(Equation 2 )forthegeneraleikonalequationcanbederivedfromrstprinciplesbasedontheFeynmanpathintegralapproachbyreplacing~fbyfandexactlyfollowingthestepsdelineatedabove.TheonlycaveatinthederivationbeingthattheHamiltonianoperatordenedinEquation 2 willnolongerbeself-adjointandhencemaynotbehaveasthequantummechanicaloperatorcorrespondingtototalenergyofthephysicalsystem[ 24 ].Neverthelessweshowthatthewaveequation(Equation 2 )inthelimitas~!0givesrisetotheeikonalequation. 31

PAGE 32

2.2.2ObtainingtheEikonalEquationfromtheSchrodingerEquationWhentheactionSandthewavefunctionarerelatedthroughtheexponent,specically (X)=exp)]TJ /F4 11.955 Tf 9.3 0 Td[(S(X) ~,(2)andsatisesEquation 2 ,weseethatSsatisestheeikonalequation(Equation 1 )as~!0.Thisrelationship(Equation 2 )canalsobeseenintheWKBapproximationofthewavefunctiontoobtaintheeikonalequation[ 35 ].SincesolutionstoEquation 2 arereal-valuedfunctions[ 16 ],wehadSappearingintheexponentof.FordifferentialequationswheresolutionstoarecomplexandoscillatorycorrespondingtopositiveenergyEinEquation 2 Sappearsasthephaseof,specically(X)=expi ~S(X)asonendsintheWKBapproximation.Chapter 6 isentirelydevotedtothisphaserelationshipbetweenthewavefunctionandthescalareldSanditsapplicationstoestimationofgradientdensitiesofS.When(x1,x2)=expn)]TJ /F7 7.97 Tf 6.58 0 Td[(S(x1,x2) ~o,therstpartialsofare @ @x1=)]TJ /F6 11.955 Tf 9.3 0 Td[(1 ~exp)]TJ /F4 11.955 Tf 9.3 0 Td[(S ~@S @x1,@ @x2=)]TJ /F6 11.955 Tf 9.3 0 Td[(1 ~exp)]TJ /F4 11.955 Tf 9.3 0 Td[(S ~@S @x2.(2)ThesecondpartialsrequiredfortheLaplacianare @2 @x21=1 ~2exp)]TJ /F4 11.955 Tf 9.3 0 Td[(S ~@S @x12)]TJ /F6 11.955 Tf 13.24 8.09 Td[(1 ~exp)]TJ /F4 11.955 Tf 9.3 0 Td[(S ~@2S @x21,@2 @x22=1 ~2exp)]TJ /F4 11.955 Tf 9.3 0 Td[(S ~@S @x22)]TJ /F4 11.955 Tf 14.44 8.08 Td[(i ~exp)]TJ /F4 11.955 Tf 9.3 0 Td[(S ~@2S @x22. (2) Fromthis,Equation 2 canberewrittenas @S @x12+@S @x22)]TJ /F5 11.955 Tf 11.95 0 Td[(~@2S @x21+@2S @x22=f2(2)whichinsimpliedformis k5Sk2)]TJ /F5 11.955 Tf 11.96 0 Td[(~52S=f2.(2) 32

PAGE 33

Theadditional~52Sterm(relativetoEquation 1 )isreferredtoastheviscosityterm[ 16 34 ]whichemergesnaturallyfromtheSchrodingerequationderivationanintriguingresult.Again,sincej52Sjisbounded,as~!0,Equation 2 tendsto k5Sk2=~f2(2)whichistheoriginaleikonalequation(Equation 1 ).ThisrelationshipmotivatesustosolvethelinearSchrodingerequation(Equation 2 )insteadofthenon-lineareikonalequationandthencomputethescalareldSvia S(X)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~log(X).(2) 33

PAGE 34

CHAPTER3EUCLIDEANDISTANCEFUNCTIONSTheEuclideandistancefunctionproblemmorepopularlyreferredtoasdistancetransforms,isaspecialcaseofthegeneraleikonalequationwheretheforcingfunctionf(X)isidenticallyequaltoone.HencetheHamilton-JacobiscalareldSsatisestherelation k5Sk=1.(3)WhenweseeksolutionforSfromasetofKdiscretepointsfYkgKk=1onadiscretizedspatialgrid,theunsignedEuclideandistanceproblemcanbeformallystatedas:Givenapoint-setY=fYk2RD,k2f1,...,KggwhereDisthedimensionalityofthepoint-setandasetofequallyspacedCartesiangridpointsX,theEuclideandistancefunctionproblemrequiresustoassign S(X)=minkkX)]TJ /F4 11.955 Tf 11.95 0 Td[(Ykk(3)withtheEuclideannormusedinEquation 3 .Incomputationalgeometry,thisistheVoronoiproblem[ 5 ]andthesolutionS(X)canbevisualizedasasetofcones(withthecentersbeingthepoint-setlocationsfYkg).TheanalysisfortheEuclideandistanceproblemcanbeextendedtoeikonalequationwithconstantforcingfunctionwheref=~f=const.~f=1specializesfortheEuclideandistancetransform.Inthesubsequentsectionswearriveattheclosed-formsolutionsfortheSchrodingerequationcorrespondingtoconstantforcingfunctions(Equation 2 ),showproofsofconvergencetothetruesolutioninthelimitas~!0andalsoprovideanefcientFFT-basednumericaltechniquetocomputethesolution. 3.1Closed-FormSolutionsforConstantForcingFunctionsWenowderivetheclosed-formsolutionfor(X)(in1D,2Dand3D)satisfyingEquation 2 andhenceforS(X)byEquation 2 34

PAGE 35

RecallthatweareinterestedinsolvingfortheeikonalproblemonlyonadiscretizedspatialgridconsistingofNgridlocationsfromasetofKdiscretepointsourcesfYkgKk=1wherethedistancefunctionisdenedtobezero,namelyS(Yk)=0,8Yk,k2f1,...,Kg.HenceitcircumventstheneedtodeterminethesolutionforSandforthewavefunctionatthesesourcelocations.FurthermoresincetheHamiltonianoperator^H=)]TJ /F5 11.955 Tf 9.29 0 Td[(~252+1ispositivedenite,i.e,alltheeigenvalueof^Hiftheyexistarestrictlypositive,theeigensystem )]TJ /F5 11.955 Tf 11.95 0 Td[(~252=()]TJ /F6 11.955 Tf 9.39 2.66 Td[(~f2)(3)aimingatndinganon-trivialeigenfunctionwithaneigenvalueof)]TJ /F6 11.955 Tf 9.4 2.65 Td[(~f2isinsatiable.Hencewelookforsolutionswhicharegeneralizedfunctions(distributions)byconsideringtheforcedversionoftheequation,namely )]TJ /F5 11.955 Tf 11.95 0 Td[(~252+~f2=KXk=1(X)]TJ /F4 11.955 Tf 11.95 0 Td[(Yk).(3)whereweforcethedifferentialequationtobesatisedatallthegridlocationsexceptatthepointsourcelocationsfYkgKk=1whereSisa-prioriknowntobezero.SinceitismeaningfultoassumethatS(X)goestoinnityforpointsatinnity,wecanuseDirichletboundaryconditions(X)=0attheboundaryofanunboundeddomain.NowusingaGreen'sfunctionapproach[ 2 ],wecanwriteexpressionsforthesolution.TheGreen'sfunctionGsatisestherelation ()]TJ /F5 11.955 Tf 9.3 0 Td[(~252+~f2)G(X)=)]TJ /F3 11.955 Tf 9.29 0 Td[((X).(3)TheformofGvariouswithdimensionsandanditsexpression[ 2 ]in1D,2Dand3Doveranunboundeddomainwithvanishingboundaryconditionsat1isgivenby,1D: G(X,Y)=1 2~exp )]TJ /F6 11.955 Tf 9.39 2.66 Td[(~fjX)]TJ /F4 11.955 Tf 11.96 0 Td[(Yj ~!.(3) 35

PAGE 36

2D: G(X,Y)=1 2~2K0 ~fkX)]TJ /F4 11.955 Tf 11.96 0 Td[(Yk ~! (3) exp)]TJ /F8 7.97 Tf 6.62 1.77 Td[(~fkX)]TJ /F7 7.97 Tf 6.58 0 Td[(Yk ~ 2~q 2~~fkX)]TJ /F4 11.955 Tf 11.96 0 Td[(Yk,kX)]TJ /F4 11.955 Tf 11.96 0 Td[(Yk ~0.25whereK0isthemodiedBesselfunctionofthesecondkind.3D: G(X,Y)=1 4~2exp)]TJ /F8 7.97 Tf 6.62 1.77 Td[(~fkX)]TJ /F7 7.97 Tf 6.59 0 Td[(Yk ~ ~fkX)]TJ /F4 11.955 Tf 11.96 0 Td[(Yk.(3)Thesolutionsforcanthenbeobtainedbyconvolution (X)=KXk=1G(X)(X)]TJ /F4 11.955 Tf 11.96 0 Td[(Yk)=KXk=1G(X)]TJ /F4 11.955 Tf 11.96 0 Td[(Yk).(3)fromwhichScanberecoveredusingtheEquation 2 3.2ProofsofConvergencetotheTrueDistanceFunctionWenowshowthatas~!0,Sconvergestothetruesolutionr=~fminkkX)]TJ /F4 11.955 Tf 12.25 0 Td[(YkkforallgridpointsXexceptthesourcelocationsYk.1D:FromEquation 2 ,weget S(X)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~logKXk=1exp )]TJ /F6 11.955 Tf 9.4 2.66 Td[(~fkX)]TJ /F4 11.955 Tf 11.95 0 Td[(Ykj ~!+~log(2~).(3)Observethat S(X))]TJ /F5 11.955 Tf 28.56 0 Td[(~logexp)]TJ /F4 11.955 Tf 9.3 0 Td[(r ~+~log(2~)=r+~log(2~). (3) Also, S(X))]TJ /F5 11.955 Tf 28.56 0 Td[(~logKexp)]TJ /F4 11.955 Tf 9.3 0 Td[(r ~+~log(2~)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~logK+r+~log(2~). (3) 36

PAGE 37

As~!0,~logK!0and~log~!0.Furthermore,weseefromEquations 3 and 3 that lim~!0S(X)=r.(3)2D:FromEquation 2 ,weget S(X)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~logKXk=1K0 ~fkX)]TJ /F4 11.955 Tf 11.95 0 Td[(Ykk ~!+~log(2~2).(3)Then, S(X))]TJ /F5 11.955 Tf 21.92 0 Td[(~logK0r ~+~log(2~2).(3)UsingtherelationK0(r h)exp()]TJ /F16 5.978 Tf 8.05 3.26 Td[(r h) p r hwhenr h0.5,weget S(X))]TJ /F5 11.955 Tf 28.56 0 Td[(~log"r ~ rexp)]TJ /F4 11.955 Tf 9.3 0 Td[(r ~#+~log(2~2)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~logr ~ r+r+~log(2~2). (3) Moreover S(X))]TJ /F5 11.955 Tf 21.92 0 Td[(~logKK0)]TJ /F4 11.955 Tf 9.29 0 Td[(r ~+~log(2~2).(3)UsingtherelationK0(r ~)exp()]TJ /F7 7.97 Tf 6.58 0 Td[(r ~)whenr h1.5,weget S(X))]TJ /F5 11.955 Tf 28.56 0 Td[(~logKexp)]TJ /F4 11.955 Tf 9.3 0 Td[(r ~+~log(2~2)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~logK+r+~log(2~2). (3) As~!0,~logK!0,~logr!0and~log~!0.Furthermore,weseefromEquations 3 and 3 that lim~!0S(X)=r.(3)3D:FromEquation 2 S(X)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~logKXk=1exp)]TJ /F8 7.97 Tf 6.62 1.77 Td[(~fkX)]TJ /F7 7.97 Tf 6.59 0 Td[(Ykk ~ ~fkX)]TJ /F4 11.955 Tf 11.95 0 Td[(Ykk+~log)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(4~2.(3) 37

PAGE 38

Then, S(X))]TJ /F5 11.955 Tf 28.55 0 Td[(~logexp)]TJ /F10 7.97 Tf 6.67 -4.97 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(r ~ r+~log(4~2)=r+~logr+~log)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(4~2. (3) Also, S(X))]TJ /F5 11.955 Tf 28.56 0 Td[(~log"Kexp)]TJ /F10 7.97 Tf 6.68 -4.98 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(r ~ r#+~log(4~2)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~logK+r+~logr+~log(4~2). (3) As~!0,~logK!0,~logr!0and~log~!0.Furthermore,weseefromEquations 3 and 3 that lim~!0S(X)=r.(3)Hence,weseethat(in1D,2Dand3D),theclosedformsolutionforguaranteesthatSapproachesthetruefunctioninthelimit~!0. 3.3ModiedGreen'sFunctionBasedonthenatureoftheGreen'sfunctionwewouldliketohighlightonthefollowingveryimportantpoint.Inthelimitingcaseof~!0, lim~!0expn)]TJ /F8 7.97 Tf 6.62 1.77 Td[(~fkXk ~o c~dkXkp=0,forkXk6=0(3)forc,dandpbeingconstantsgreaterthanzeroandthereforeweseethatifwedene ~G(X)=Cexp )]TJ /F6 11.955 Tf 9.4 2.66 Td[(~fkXk ~!(3)forsomeconstantC, lim~!0jG(X))]TJ /F6 11.955 Tf 13.71 2.66 Td[(~G(X)j=0,forkXk6=0(3)andfurthermore,theconvergenceisuniformforkXkawayfromzero.Therefore,~G(X)providesaverygoodapproximationfortheactualGreen'sfunctionas~!0.Foraxedvalueof~andX,thedifferencebetweentheGreen'sfunctionsisOexp)]TJ /F13 5.978 Tf 5.78 1.33 Td[(~fkXk ~ ~2 38

PAGE 39

whichisrelativelyinsignicantforsmallvaluesof~andforallX6=0.Moreover,using~GalsoavoidsthesingularityattheoriginthatGhasinthe2Dand3Dcase.Theaboveobservationmotivatesustocomputethesolutionsforbyconvolvingwith~G,namely (X)=KXk=1~G(X)(X)]TJ /F4 11.955 Tf 11.95 0 Td[(Yk)=KXk=1~G(X)]TJ /F4 11.955 Tf 11.96 0 Td[(Yk)(3)insteadoftheactualGreen'sfunctionGandrecoverSusingtheEquation 2 ,givenby S(X)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~log"KXk=1exp )]TJ /F6 11.955 Tf 9.39 2.66 Td[(~fkX)]TJ /F4 11.955 Tf 11.96 0 Td[(Ykk ~!#+~log(C).(3)Since~log(C)isaadditiveconstantindependentofXandconvergesto0as~!0,itcanignoredwhilecomputingSatsmallvaluesof~itisequivalenttosettingCtobe1.HencetheSchrodingerwavefunctionforconstantforcingfunctionscanbeapproximatedby (X)=KXk=1exp )]TJ /F6 11.955 Tf 9.4 2.66 Td[(~fkX)]TJ /F4 11.955 Tf 11.95 0 Td[(Ykk ~!.(3)Itisworthemphasizingthattheabovedenedwavefunction(X)(Equation 3 ),containsallthedesirablepropertiesthatweneed.Firstly,wenoticethatas~!0,(Yk)!1atthegivenpoint-setlocationsYk.HencefromEquation 2 ,S(Yk)!0as~!0satisfyingthenecessaryinitialconditions.Secondlyas~!0,PKk=1exp)]TJ /F8 7.97 Tf 6.62 1.77 Td[(~fkX)]TJ /F7 7.97 Tf 6.58 0 Td[(Ykk ~canbeapproximatedbyexp)]TJ /F10 7.97 Tf 6.68 -4.97 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(r ~wherer=~fminkkX)]TJ /F4 11.955 Tf 12.76 0 Td[(Ykk.HenceS(X))]TJ /F5 11.955 Tf 23.91 0 Td[(~logexp)]TJ /F10 7.97 Tf 6.68 -4.98 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(r ~=r,whichisthetruevalue.Thirdly,canbeeasilycomputedusingthefastFouriertransformasdescribedunderSection 3.5 ).HenceforallcomputationalpurposesweconsiderthewavefunctiondenedinEquation 3 asthesolutiontotheSchrodingerwaveequation(Equation 2 ). 39

PAGE 40

3.4ErrorBoundBetweentheObtainedandtheTrueDistanceFunctionUsingtheEquation 2 andthemodiedGreen'sfunction(~G)wecomputetheapproximatedistancefunctionas S(X)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~log KXk=1exp )]TJ /F6 11.955 Tf 9.4 2.66 Td[(~fkX)]TJ /F4 11.955 Tf 11.95 0 Td[(Ykk ~!!.(3)Intuitively,as~!0,PKk=1exp)]TJ /F8 7.97 Tf 6.62 1.77 Td[(~fkX)]TJ /F7 7.97 Tf 6.59 0 Td[(Ykk ~canbeapproximatedbyexp)]TJ /F10 7.97 Tf 6.67 -4.98 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(r ~wherer=~fminkkX)]TJ /F4 11.955 Tf 12.62 0 Td[(Ykk.HenceS(X))]TJ /F5 11.955 Tf 23.59 0 Td[(~logexp)]TJ /F10 7.97 Tf 6.68 -4.97 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(r ~=r.TheboundderivedbelowbetweenS(X)andralsounveilstheproximitybetweenthecomputedandtheactualdistancefunction.NotefromEquation 3 that S(X))]TJ /F5 11.955 Tf 28.56 0 Td[(~logexp)]TJ /F4 11.955 Tf 9.3 0 Td[(r ~=r. (3) Also,observethat S(X))]TJ /F5 11.955 Tf 28.56 0 Td[(~logKexp)]TJ /F4 11.955 Tf 9.3 0 Td[(r ~=)]TJ /F5 11.955 Tf 9.3 0 Td[(~logK+r (3) andhence, r)]TJ /F4 11.955 Tf 11.95 0 Td[(S(X)~logK.(3)FromEquations 3 and 3 jr)]TJ /F4 11.955 Tf 11.95 0 Td[(S(X)j~logK.(3)Equation 3 showsthatas~!0,S(X)!r.Itisworthcommentingthatthebound~logKisactuallyverytightas(i)itscalesonlyasthelogarithmofthecardinalityofthepoint-set(K)and(ii)itcanbemadearbitrarilysmallbychoosingasmallbutnon-zerovalueof~. 40

PAGE 41

Table3-1. AlgorithmfortheapproximateEuclideandistancefunction 1.Computethefunction~G(X)=exp)]TJ /F8 7.97 Tf 6.62 1.77 Td[(~fkXk ~atthegridlocations.2.Denethefunctionkron(X)whichtakesthevalue1atthepoint-setlocationsand0atothergridlocations.3.ComputetheFFTof~Gandkron,namely~GFFT(U)andFFT(U)respectively.4.ComputethefunctionH(U)=~GFFT(U)FFT(U).5.ComputetheinverseFFTofHtoobtain(X)atthegridlocations.6.Takethelogarithmof(X)andmultiplyitby()]TJ /F5 11.955 Tf 9.29 0 Td[(~)togettheapproximateEuclideandistancefunctionatthegridlocations. 3.5EfcientComputationoftheApproximateDistanceFunctionInthissection,weprovidenumericaltechniquesforefcientlycomputingthewavefunction.RecallthatweareinterestedinsolvingtheeikonalequationonlyatthegivenNdiscretegridlocations.Inordertoobtainthedesiredsolutionfor(Equation 3 )computationally,wemustreplacethefunctionbytheKroneckerdeltafunctionkron(X)=8><>:1ifX=Yk;0otherwisethattakes1atthepoint-setlocations(fYkg)and0atothergridlocations.Thencanbeexactlycomputedatthegridlocationsbythediscreteconvolutionof~G(settingC=1)withtheKronecker-deltafunction.Bytheconvolutiontheorem[ 7 ],adiscreteconvolutioncanbeobtainedastheinverseFouriertransformoftheproductoftwoindividualtransformswhichfortwoO(N)sequencescanbeperformedinO(NlogN)time[ 14 ].OnejustneedstocomputethediscreteFouriertransform(DFT)of~Gandkron,computetheirpoint-wiseproductandthencomputetheinversediscreteFouriertransform.TakingthelogarithmoftheinversediscreteFouriertransformandmultiplyingitby()]TJ /F5 11.955 Tf 9.3 0 Td[(~),givestheapproximateEuclideandistancefunction.ThealgorithmisadumbratedinTable 3-1 3.5.1SolutionfortheDistanceFunctioninHigherDimensionsUsing~GinsteadoftheboundeddomainGreen'sfunctionGprovidesastraightforwardgeneralizationofourtechniquetohigherdimensions.Regardlessofthespatialdimension,theapproximatesolutionforthedistancefunctionScanbecomputed 41

PAGE 42

fromthewavefunctionusingO(NlogN)oating-pointoperationsasimplementingthediscreteconvolutionusingFFT[ 7 ]alwaysinvolvesO(NlogN)oating-pointcomputations[ 14 ]irrespectiveofthespatialdimension.Thoughthenumberofgridpoints(N)mayincreasewithdimensionthesolutionisalwaysO(NlogN)inthenumberofgridpoints.Thisspeaksforthescalabilityofourtechnique. 3.5.2NumericalIssuesandExactComputationalComplexityWerequestthereadertorefertoSection 5.3.1 togetanaccountonthenumericalissuesinvolvedincomputingthewavefunctionandtheneedforarbitraryprecisionarithmeticpackageslikeGMPandMPFR[ 20 45 ].MoreovertheO(NlogN)timecomplexityoftheFFTalgorithm[ 14 ]foranO(N)lengthsequencetakesintoaccountonlythenumberofoating-pointoperationsinvolved,barringanynumericalaccuracy.Section 5.3.2 givestheexactcomputationalcomplexity,whenonetakesintoaccountthenumberofprecisionbitsusedinoatingpointcomputations. 42

PAGE 43

CHAPTER4SIGNEDDISTANCEFUNCTIONANDITSDERIVATIVESThesolutionfortheapproximateEuclideandistancefunctionin( 3 )(with~f=1)islackinginonerespect:thereisnoinformationonthesignofthedistance.ThisistobeexpectedsincethedistancefunctionwasobtainedonlyfromasetofpointsfYkgkk=1andnotacurveorasurface.Wenowdescribeanewmethodforcomputingthesigneddistancein2Dusingwindingnumbersandin3Dusingtopologicaldegree.FurthermorejustastheapproximateEuclideandistancefunctionS(X)canbeefcientlycomputed,socanthederivatives.ThisisimportantbecausefastcomputationofthederivativesofS(X)onaregulargridcanbeveryusefulinmedialaxisandcurvaturecomputations. 4.1ConvolutionBasedMethodforComputingtheWindingNumberAssumethatwehaveaclosed,parametriccurvex(1)(t),x(2)(t),t2[0,1].WeseektodetermineifagridlocationinthesetfXi2R2,i2f1,...,Nggisinsidetheclosedcurve.ThewindingnumberisthenumberoftimesthecurvewindsaroundthepointXi(ifatall)andifthecurveisoriented,counterclockwiseturnsarecountedaspositiveandclockwiseturnsasnegative.Ifapointisinsidethecurve,thewindingnumberisanon-zerointeger.Ifthepointisoutsidethecurve,thewindingnumberiszero.Ifwecanefcientlycomputethewindingnumberforallpointsonagridw.r.t.toacurve,thenwewouldhavethesigninformation(inside/outside)forallthepoints.Wenowdescribeafastalgorithmtoachievethisgoal.IfthecurveisC1,thentheangle(t)ofthecurveiscontinuousanddifferentiableandd(t)=x(1)_x(2))]TJ /F7 7.97 Tf 6.59 0 Td[(x(2)_x(1) kxk2dt.SinceweneedtodeterminewhetherthecurvewindsaroundeachofthepointsXi,i2f1,...,Ng,dene(^x(1)i,^x(2)i)(x(1))]TJ /F4 11.955 Tf 12.74 0 Td[(X(1)i,x(2))]TJ /F4 11.955 Tf -431.33 -23.91 Td[(X(2)i),8i.ThenthewindingnumbersforallgridpointsinthesetXare i=1 2IC ^x(1)i_^x(2)i)]TJ /F6 11.955 Tf 12.14 0 Td[(^x(2)i_^x(2)i k^xik2!dt,8i2f1,...,Ng.(4) 43

PAGE 44

Asitstands,wecannotactuallycomputethewindingnumberswithoutperformingtheintegralinEquation 4 .Tothisend,wediscretizedthecurveandproduceasequenceofpointsfYk2R2,k2f1,...,KggwiththeunderstandingthatthecurveisclosedandthereforethenextpointafterYKisY1.(Thewindingnumberpropertyholdsforpiecewisecontinuouscurvesaswell.)TheintegralinEquation 4 becomesadiscretesummationandweget i=1 2KXk=1Y(1)k)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)iY(2)k1)]TJ /F4 11.955 Tf 11.95 0 Td[(Y(2)k)]TJ /F12 11.955 Tf 11.96 13.27 Td[(Y(2)k)]TJ /F4 11.955 Tf 11.96 0 Td[(X(2)iY(1)k1)]TJ /F4 11.955 Tf 11.95 0 Td[(Y(1)k kYk)]TJ /F4 11.955 Tf 11.96 0 Td[(Xik2(4)8i2f1,...,Ng,wherethenotationY()k1denotesthatY()k1=Y()k+1fork2f1,...,K)]TJ /F6 11.955 Tf 12.1 0 Td[(1gandY()K1=Y()1.WecansimplifythenotationinEquation 4 (andobtainameasureofconceptualclarityaswell)bydeningthetangentvectorfZk,k=f1,...,Kggas Z()k=Y()k1)]TJ /F4 11.955 Tf 11.95 0 Td[(Y()k,k2f1,...,Kg(4)withthe()symbolindicatingeithercoordinate.UsingthetangentvectorZ,werewriteEquation 4 as i=1 2KXk=1Y(1)k)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)iZ(2)k)]TJ /F12 11.955 Tf 11.96 13.27 Td[(Y(2)k)]TJ /F4 11.955 Tf 11.96 0 Td[(X(2)iZ(1)k kYk)]TJ /F4 11.955 Tf 11.96 0 Td[(Xik2,8i2f1,...,Ng(4)WenowmakethesomewhatsurprisingobservationthatinEquation 4 isasumoftwodiscreteconvolutions.Therstconvolutionisbetweentwofunctionsfcr(X)fc(X)fr(X)andg2(X)=PKk=1Z(2)kkronwheretheKroneckerdeltafunction(kron)isdenedinEquation 3.5 .Thesecondconvolutionisbetweentwofunctionsfsr(X)fs(X)fr(X)andg1(X)PKk=1Z(1)kkron.Thefunctionsfc(X),fs(X)andfr(X)aredenedas fc(X)X(1) kXk,fs(X)X(2) kXk,and (4) fr(X)1 kXk (4) 44

PAGE 45

withtheunderstandingthatfc(0)=fs(0)=fr(0)=0.HerewehaveabusednotationsomewhatandletX(1)(X(2))denotethex(y)-coordinateofallthepointsinthegridsetX.Armedwiththeserelationships,werewriteEquation 4 toget (X)=1 2[)]TJ /F4 11.955 Tf 9.29 0 Td[(fcr(X)g2(X)+fsr(X)g1(X)](4)whichcanbesimultaneouslycomputedforalltheNgridpointsXiusingtwoFFTs. 4.2ConvolutionBasedMethodforComputingtheTopologicalDegreeThewindingnumberconceptfor2Dadmitsastraightforwardgeneralizationto3Dandhigherdimensions.Theequivalentconceptisthetopologicaldegreewhichisbasedonnormalizeduxcomputations.Assumethatwehaveanorientedsurfacein3D[ 23 ]whichisrepresentedasasetofKtriangles.EachkthtrianglehasanoutwardpointingnormalPkandthiscaneasilybeobtainedoncethesurfaceisoriented.(Wevectorizetheedgeofeachtriangle.Sincetrianglesshareedges,ifthesurfacecanbeoriented,thenthere'saconsistentwayoflendingdirectiontoeachtriangleedge.Thetrianglenormalismerelythecrossproductofthetrianglevectoredges.)Wepickaconvenienttrianglecenter(thetriangleincenterforinstance)foreachtriangleandcallitYk.Thenormalizedux(whichisverycloselyrelatedtothetopologicaldegree)[ 1 ]determinestheratiooftheoutwarduxfromapointXitreatedastheorigin.IfXiisoutsidetheenclosedsurface,thenthetotaloutwarduxiszero.Ifthepointisinside,theoutwardnormalizeduxwillbenon-zeroandpositive.ThenormalizeduxforapointXiis i=1 4KXk=1h(Yk)]TJ /F4 11.955 Tf 11.95 0 Td[(Xi),Pki kYk)]TJ /F4 11.955 Tf 11.96 0 Td[(Xik3.(4)Thiscanbewrittenintheformofconvolutions.Toseethis,wewriteEquation 4 incomponentform, i=1 4KXk=1(Y(1)k)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)i)P(1)k+(Y(2)k)]TJ /F4 11.955 Tf 11.96 0 Td[(X(2)i)P(2)k+(Y(3)k)]TJ /F4 11.955 Tf 11.95 0 Td[(X(3)i)P(3)k kYk)]TJ /F4 11.955 Tf 11.96 0 Td[(Xik3(4) 45

PAGE 46

whichcanbesimpliedas (X)=)]TJ /F6 11.955 Tf 14.03 8.09 Td[(1 4(f1(X)g1(X)+f2(X)g2(X)+f3(X)g3(X))(4)wheref()(X)X() kXk3andg()(X)PKk=1P()kkron,wheretheKroneckerdeltafunction(kron(X))isgivenbyEquation 3.5 .ThiscanbesimultaneouslycomputedusingthreeFFTsforalltheNgridpointsXi.Forthesakeofclarity,weexplicitlyshowthegeneralizationofthewindingnumbertothetopologicaldegreebyrewritingsomeofthecalculationsinvolvedincomputingthewindingnumber.RecallthatforeverypointYkonthediscretizedcurve,wedeneditstangentvectorZkasinEquation 4 .TheoutwardpointingnormalPk=(P(1)k,P(2)k),atthepointYk(PkwillpointoutwardsprovidedY1,Y2,,Ykaretakenintheanti-clockwiseorder),isgivenbyP(1)k=Z(2)k,P(2)k=)]TJ /F4 11.955 Tf 9.3 0 Td[(Z(1)k.UsingthenormalvectorPk,Equation 4 canberewrittenas i=1 2KXk=1h(Yk)]TJ /F4 11.955 Tf 11.95 0 Td[(Xi),Pki kYk)]TJ /F4 11.955 Tf 11.96 0 Td[(Xik2.(4)NoticethesimilaritybetweentheEquations 4 and 4 .Thegeneralizationisquiteobvious.ThuswehaveshownthatthesigncomponentoftheEuclideandistancefunctioncanbeseparatelycomputed(withoutknowledgeofthedistance)inparallelinusingFFT'sonaregular2Dand3Dgrid. 4.3FastComputationoftheDerivativesoftheDistanceFunctionAsmentionedbefore,eventhederivativesofEuclideandistancefunctionS(X)canberepresentedasconvolutionsandefcientlycomputedusingFFT's.Below,wedetailhowthiscanbeachieved.Webeginwiththegradientsandforillustrationpurposes,the 46

PAGE 47

derivationsareperformedin2D: Sx(X)=PKk=1X(1))]TJ /F7 7.97 Tf 6.58 0 Td[(Y(1)k kX)]TJ /F7 7.97 Tf 6.58 0 Td[(Ykkexpn)]TJ /F10 7.97 Tf 10.49 5.7 Td[(kX)]TJ /F7 7.97 Tf 6.59 0 Td[(Ykk ~o PKk=1expn)]TJ /F10 7.97 Tf 10.49 5.7 Td[(kX)]TJ /F7 7.97 Tf 6.58 0 Td[(Ykk ~o.(4)AsimilarexpressioncanbeobtainedforSy(X).Theserstderivativescanberewrittenasdiscreteconvolutions: Sx(X)=fc(X)f(X)kron(X) f(X)kron,Sy(X)=fs(X)f(X)kron f(X)kron,(4)wherefc(X)andfs(X)areasdenedinEquation 4 ,theKroneckerdeltafunction(kron(X))isgivenbyEquation 3.5 and f(X)=exp )]TJ /F6 11.955 Tf 9.4 2.65 Td[(~fkXk ~!(4)isthemodiedGreen'sfunction~G(denedinEquation 3 )withtheconstantC=1.Thesecondderivativeformulaearesomewhatinvolved.Ratherthanhammeroutthealgebrainaturgidmanner,wemerelypresentthenalexpressionsalldiscreteconvolutionsforthethreesecondderivativesin2D: Sxx(X)=)]TJ /F8 7.97 Tf 10.5 4.71 Td[(1 ~f2c(X)+f2s(X)fr(X)f(X)g(X) f(X)g(X)+1 ~(Sx)2(X), (4) Syy(X)=)]TJ /F8 7.97 Tf 10.5 4.71 Td[(1 ~f2s(X)+f2c(X)fr(X)f(X)g(X) f(X)g(X)+1 ~(Sy)2(X),and (4) (4) Sxy(X)=)]TJ /F12 11.955 Tf 11.29 9.68 Td[(1 ~+fr(X)fc(X)fs(X)f(X)g(X) f(X)g(X)+1 ~Sx(X)Sy(X) (4) wherefr(X)isasdenedinEquation 4 .Wealsoseethat k5Sk2)]TJ /F5 11.955 Tf 11.96 0 Td[(~52S=1)]TJ /F5 11.955 Tf 11.96 0 Td[(~fr(X)f(X)g(X) f(X)g(X)(4) 47

PAGE 48

[sincef2c(X)+f2s(X)=1]withtherightsidegoingtooneas~!0forpointsXawayfrompointsintheseedpoint-setfYkgKk=1.ThisisinaccordancewithEquation 2 andvindicatesourchoiceofthereplacementGreen'sfunctioninEquation 3 .SincewecanefcientlycomputetherstandsecondderivativesoftheapproximateEuclideandistancefunctioneverywhereonaregulargrid,wecanalsocomputederivedquantitiessuchascurvature(Gaussian,meanandprincipalcurvatures)forthetwo-dimensionalsurfaceS(X)computedatthegridlocationsX.Intheexperimentalsection,wevisualizethederivativesandmaximumcurvatureforshapesilhouettesandusethemasavehicletodeterminethemedialaxisforthesesilhouettes. 48

PAGE 49

CHAPTER5GENERALEIKONALEQUATIONInthischapter,weprovidenumericaltechniquesforefcientlysolvingtheSchrodingerequation(Equation 2 )derivedforanarbitrary,spatiallyvarying,positivevalued,boundedforcingfunctionf. 5.1PerturbationTheoryWhenfisconstantasinthecaseofEuclideandistancefunctionproblem(whereitisidenticallyequaltoone),weshowedtheexistenceofclosed-formsolutionsforthewavefunctioninChapter 3 .Butforarbitraryforcingfunctionsfwhichiswhatwefaceinthecaseoftheeikonal,itisgenerallynottrue.Consequently,weproposetosolvethelinearsystemEquation 2 usingtechniquesfromperturbationtheory[ 17 ].Assumingthatfisclosetoaconstantnon-zeroforcingfunction~f,Equation 2 canberewrittenas ()]TJ /F5 11.955 Tf 9.3 0 Td[(~252+~f2)h1+()]TJ /F5 11.955 Tf 9.3 0 Td[(~252+~f2))]TJ /F8 7.97 Tf 6.59 0 Td[(1(f2)]TJ /F6 11.955 Tf 12.05 2.66 Td[(~f2)i=0.(5)Now,deningtheoperatorLas L()]TJ /F5 11.955 Tf 9.3 0 Td[(~252+~f2))]TJ /F8 7.97 Tf 6.59 0 Td[(1(f2)]TJ /F6 11.955 Tf 12.05 2.66 Td[(~f2)(5)and0as 0(1+L)(5)weseethat0satises ()]TJ /F5 11.955 Tf 9.3 0 Td[(~252+~f2)0=0(5)and =(1+L))]TJ /F8 7.97 Tf 6.59 0 Td[(10.(5)Noticethatinthedifferentialequationfor0(Equation 5 ),theforcingfunctionisconstantandequals~feverywhere.Hence0behaveslikethewavefunction 49

PAGE 50

correspondingtotheconstantforcingfunction~fandcanbeapproximatedby 0(X)=KXk=1exp )]TJ /F6 11.955 Tf 9.4 2.65 Td[(~fkX)]TJ /F4 11.955 Tf 11.96 0 Td[(Ykk ~!(5)(referEquation 3 ).WenowsolveforinEquation 5 usingageometricseriesapproximationfor(1+L))]TJ /F8 7.97 Tf 6.58 0 Td[(1.Firstly,observethattheapproximatesolutionfor0inEquation 5 isasquare-integrablefunctionwhichisnecessaryforthesubsequentsteps.LetHdenotethespaceofsquareintegrablefunctionsonRD,i.e,g2Hiff Zg2d<1.(5)Thefunctionnormkgkforafunctiong2Hisgivenby kgk2=Zg2d.(5)LetBdenoteaclosedunitballintheHilbertspaceH,i.e B=fg2H:kgk1g.(5)Letc0kLkoptheoperatornormdenedby c0=supfkLgk,8functionsg2Bg.(5)Ifc0<1,wecanapproximate(1+L))]TJ /F8 7.97 Tf 6.59 0 Td[(1usingtherstfewT+1termsofthegeometricseriestoget (1+L))]TJ /F8 7.97 Tf 6.58 0 Td[(11)]TJ /F4 11.955 Tf 11.95 0 Td[(L+L2)]TJ /F4 11.955 Tf 11.96 0 Td[(L3+...+()]TJ /F6 11.955 Tf 9.3 0 Td[(1)TLT(5)wheretheoperatornormofthedifferencecanbeboundedby k(1+L))]TJ /F8 7.97 Tf 6.58 0 Td[(1)]TJ /F7 7.97 Tf 16.99 14.95 Td[(TXi=0()]TJ /F6 11.955 Tf 9.29 0 Td[(1)iLikop1Xi=T+1kLikop1Xi=T+1ci0=cT+10 1)]TJ /F4 11.955 Tf 11.95 0 Td[(c0(5) 50

PAGE 51

whichconvergesto0exponentiallyinT.WewouldliketopointoutthattheabovegeometricseriesapproximationissimilartoaBornexpansionusedinscatteringtheory[ 30 ].Wenowderiveanupperboundforc0.LetL=A1A2whereA1()]TJ /F5 11.955 Tf 9.3 0 Td[(~252+~f2))]TJ /F8 7.97 Tf 6.58 0 Td[(1andA2f2)]TJ /F6 11.955 Tf 12.25 2.66 Td[(~f2.WenowprovideanupperboundforkA1kop.Foragiveng2B,letz=A1(g),i.ezsatisestherelation ()]TJ /F5 11.955 Tf 9.3 0 Td[(~252+~f2)z=g(5)withvanishingDirichletboundaryconditionsat1.Then k()]TJ /F5 11.955 Tf 9.3 0 Td[(~252+~f2)zk2=k)]TJ /F5 11.955 Tf 20.59 0 Td[(~252zk2+k~f2zk2+2~2~f2h)-222(52z,zi=kgk21. (5) Wenowusetherelation 5.(z5z)=z52z+j5zj2(5)tocompute h)-222(52z,zi=)]TJ /F12 11.955 Tf 11.29 16.28 Td[(Zz52zd=)]TJ /F12 11.955 Tf 11.29 16.28 Td[(Z5.(z5z)d+Zj5zj2d.(5)Nowfromthedivergencetheoremwehave )]TJ /F12 11.955 Tf 11.95 16.27 Td[(Z5.(z5z)d=0(5)andhence h)-222(52z,zi=Zj5zj2d0.(5)UsingtheaboverelationinEquation 5 ,wethenobservethat kzk=kA1(g)k1 ~f2,8g2B.(5) 51

PAGE 52

SinceweshowedA1(B)isboundedandlessthanorequalto1 ~f2,weimmediatelyhave kA1kop1 ~f2.(5)Now,letM=supfjf2)]TJ /F6 11.955 Tf 12.05 2.66 Td[(~f2jg.Then,foranyg2B k(f2)]TJ /F6 11.955 Tf 12.05 2.65 Td[(~f2)gk2=Z(f2)]TJ /F6 11.955 Tf 12.05 2.65 Td[(~f2)2g2dM2kgk2M2(5)andhencefromEquation 5 kA2kopM=supfjf2)]TJ /F6 11.955 Tf 12.06 2.66 Td[(~f2jg.(5)SincekLkopkA1kopkA2kop,fromEquations 5 and 5 ,weobservethat c0=kLkopsupfjf2)]TJ /F6 11.955 Tf 12.05 2.66 Td[(~f2jg ~f2.(5)Itisworthcommentingthattheboundforc0isindependentof~.So,ifweguaranteethatsupfjf2)]TJ /F8 7.97 Tf 6.63 1.77 Td[(~f2jg ~f2<1,thegeometricseriesapproximationfor(1+L))]TJ /F8 7.97 Tf 6.59 0 Td[(1(Equation 5 )convergesforallvaluesof~. 5.2DerivingaBoundfortheConvergenceofthePerturbationSeriesInterestingly,foranypositive,upperboundedforcingfunctionfboundedawayfromzero,i.ef(X)>forsome>01,bydening~f=supff(X)g,weobservethatjf2)]TJ /F6 11.955 Tf 12.53 2.66 Td[(~f2j<~f2.FromEquation 5 ,weimmediatelyseethatc0<1.Thisprovestheexistenceof~fforwhichthegeometricseriesapproximation(Equation 5 )isalwaysguaranteedtoconvergeforanypositiveboundedforcingfunctionfboundedawayfromzero.Thechoiceof~fcanthenbemadeprudentlybydeningittobethevaluethatminimizes F(~f)=supfjf2)]TJ /F6 11.955 Tf 12.06 2.65 Td[(~f2jg ~f2.(5) 1Iff(X)=0,thenthevelocityv(X)=1 f(X)becomes1atX.Henceitisreasonabletoassumef(X)>0. 52

PAGE 53

Thisinturnminimizestheoperatornormc0,therebyprovidingabettergeometricseriesapproximationfortheinverse(Equation 5 ).Letfmin=infff(X)gandletfmax=supff(X)g.WenowshowthatF(~f)attainsitsminimumat ~f==r f2min+f2max 2.(5)Case(i):If~f<,thensupfjf2)]TJ /F6 11.955 Tf 12.05 2.66 Td[(~f2jg=f2max)]TJ /F6 11.955 Tf 12.05 2.66 Td[(~f2.Clearly, f2max)]TJ /F6 11.955 Tf 12.05 2.66 Td[(~f2 ~f2>f2max)]TJ /F3 11.955 Tf 11.96 0 Td[(2 2.(5)Case(ii):If~f>,thensupfjf2)]TJ /F6 11.955 Tf 12.06 2.66 Td[(~f2jg=~f2)]TJ /F4 11.955 Tf 11.96 0 Td[(f2min.Itfollowsthat ~f2)]TJ /F4 11.955 Tf 11.95 0 Td[(f2min ~f2=1)]TJ /F4 11.955 Tf 13.15 8.09 Td[(f2min ~f2>1)]TJ /F4 11.955 Tf 13.15 8.09 Td[(f2min 2.(5)Wethereforeseethat~f==q f2min+f2max 2istheoptimalvalue.Usingtheaboveapproximationfor(1+L))]TJ /F8 7.97 Tf 6.58 0 Td[(1fromEquation 5 andthedenitionofLfromEquation 5 ,weobtainthesolutionforas =0)]TJ /F3 11.955 Tf 11.95 0 Td[(1+2)]TJ /F3 11.955 Tf 11.96 0 Td[(3+...+()]TJ /F6 11.955 Tf 9.3 0 Td[(1)TT(5)whereisatisestherecurrencerelation ()]TJ /F5 11.955 Tf 9.3 0 Td[(~252+~f2)i=(f2)]TJ /F6 11.955 Tf 12.06 2.66 Td[(~f2)i)]TJ /F8 7.97 Tf 6.59 0 Td[(1,8i2f1,2,...,Tg.(5)ObservethatEquation 5 isaninhomogeneous,screenedPoissonequationwithaconstantforcingfunction~f.FollowingaGreen'sfunctionapproach[ 2 ],eachicanbeobtainedbyconvolution i=Gh(f2)]TJ /F6 11.955 Tf 12.05 2.66 Td[(~f2)i)]TJ /F8 7.97 Tf 6.59 0 Td[(1i(5)whereGisgivenbyEquations 3 3 or 3 dependinguponthespatialdimension.Oncethei'sarecomputed,thewavefunctioncanthenbedeterminedusingtheapproximation(Equation 5 ).Thesolutionfortheeikonalequationcanberecovered 53

PAGE 54

usingtheEquation 2 .Noticethatiff=~feverywhere,thenalli'sexcept0isidenticallyequaltozeroandweget=0asdescribedintheChapter 3 5.3EfcientComputationoftheWaveFunctionInthissection,weprovidenumericaltechniquesforefcientlycomputingthewavefunction.AsdescribedinChapter 3 ,inordertoobtainthedesiredsolutionfor0computationally,wemustreplacethefunctionbytheKroneckerdeltafunctionkron(X)=8><>:1ifX=Yk;0otherwisethattakes1atthepoint-setlocations(fYkg)and0atothergridlocations.Then0canbeexactlycomputedatthegridlocationsbythediscreteconvolutionof~G(settingC=1)withtheKronecker-deltafunction.Tocomputei,wereplaceeachoftheconvolutionsinEquation 5 withthediscreteconvolutionbetweenthefunctionscomputedattheNgridlocations.AsdiscreteconvolutioncanbedoneusingFastFourierTransforms,thevaluesofeachiattheNgridlocationscanbeefcientlycomputedinO(NlogN)makinguseofthevaluesofi)]TJ /F8 7.97 Tf 6.58 0 Td[(1determinedattheearlierstep.Thus,theoveralltimecomplexitytocomputetheapproximateusingtherstfewT+1termsisthenO(TNlogN).Takingthelogarithmofthenprovidesanapproximatesolutiontotheeikonalequation.ThealgorithmisadumbratedinTable 5-1 .Wewouldliketoemphasizethatthenumberofterms(T)usedinthegeometricseriesapproximationof(1+L))]TJ /F8 7.97 Tf 6.58 0 Td[(1inEquation 5 isindependentofN.Usingmoretermsonlyimprovestheapproximationofthistruncatedgeometricseriesasshownintheexperimentalsection.FromEquation 5 ,itisevidentthattheerrorincurredduetothisapproximationconvergestozeroexponentiallyinTandhenceevenwithasmallvalueofT,weshouldbeabletoachievegoodaccuracy. 54

PAGE 55

Table5-1. Algorithmfortheapproximatesolutiontotheeikonalequation 1.Computethefunction~G(X)=exp)]TJ /F8 7.97 Tf 6.62 1.77 Td[(~fkXk ~atthegridlocations.2.Denethefunctionkron(X)whichtakesthevalue1atthepoint-setlocationsand0atothergridlocations.3.ComputetheFFTof~Gandkron,namely~GFFT(U)andFFT(U)respectively.4.ComputethefunctionH(U)=~GFFT(U)FFT(U).5.ComputetheinverseFFTofHtoobtain0(X)atthegridlocations.6.Initialize(X)to0(X).7.ConsidertheGreen'sfunctionGcorrespondingtothespatialdimensionandcomputeitsFFT,namelyGFFT(U).8.Fori=1toTdo9.DeneP(X)=hf2(X))]TJ /F6 11.955 Tf 12.05 2.66 Td[(~f2ii)]TJ /F8 7.97 Tf 6.59 0 Td[(1(X).10.ComputetheFFTofPnamelyPFFT(U).11.ComputethefunctionH(U)=GFFT(U)PFFT(U).12.ComputetheinverseFFTofHandmultiplyitwiththegridwidtharea/volumetocomputei(X)atthegridlocations.13.Update(X)=(X)+()]TJ /F6 11.955 Tf 9.3 0 Td[(1)ii(X).14.End15.Takethelogarithmof(X)andmultiplyitby()]TJ /F5 11.955 Tf 9.3 0 Td[(~)togettheapproximatesolutionfortheeikonalequationatthegridlocations. 5.3.1NumericalIssuesInprinciple,weshouldbeabletoapplyourtechniqueatverysmallvaluesof~andobtainhighlyaccurateresults.Butwenoticedthatanavedoubleprecision-basedimplementationtendstodeterioratefor~valuesveryclosetozero.Thisisduetothefactthatatsmallvaluesof~(andalsoatlargevaluesof~f),exp)]TJ /F8 7.97 Tf 6.62 1.78 Td[(~fkXk ~dropsoffveryquicklyandhenceforgridlocationswhicharefarawayfromthepoint-set,theconvolutiondoneusingFFTmaynotbeaccurate.Tothisend,weturnedtotheGNUMPFRmultiple-precisionarithmeticlibrarywhichprovidesarbitraryprecisionarithmeticwithcorrectrounding[ 20 ].MPFRisbasedontheGNUmultiple-precisionlibrary(GMP)[ 45 ].Itenabledustorunourtechniqueatverysmallvaluesof~givinghighlyaccurateresults.Wecorroborateourclaimanddemonstratetheusefulnessofourmethodwiththesetofexperimentsdescribedinthesubsequentsection. 55

PAGE 56

5.3.2ExactComputationalComplexityMorethenumberofprecisionbitspusedintheGNUMPFRlibrary,betteristheaccuracyofourtechnique,astheerrorincurredintheoatingpointoperationscanbeboundedbyO(2)]TJ /F7 7.97 Tf 6.58 0 Td[(p).Butusingmorebitshasanadverseeffectofslowingdowntherunningtime.TheO(NlogN)timecomplexityoftheFFTalgorithm[ 14 ]foranO(N)lengthsequencetakesintoaccountonlythenumberofoating-pointoperationsinvolved,barringanynumericalaccuracy.TheaccuracyoftheFFTalgorithmandourtechniqueentirelydependsonthenumberofprecisionbitsusedforcomputingelementaryfunctionslikeexp,log,sinandcosandhenceshouldbetakenintoaccountwhilecalculatingthetimecomplexityofouralgorithm.Ifpprecisionbitsareused,thetimecomplexityforcomputingtheseelementaryfunctionscanbeshowntobeO(M(p)logp)[ 8 39 43 ],whereM(p)isthecomputationalcomplexityofmultiplyingtwop-digitnumbers.TheSchonhage-Strassenalgorithm[ 40 ]givesanasymptoticupperboundonthetimecomplexityformultiplyingtwop-digitnumbers.Therun-timebitcomplexityisM(p)=O(plogploglogp).Thentakingthesepprecisionbitsintoaccount,thetimecomplexityofouralgorithmforcomputingSatthegivenNgridlocations,usingtherstT+1termsinthegeometricseriesapproximationof(Equation 5 ),isO(TNlog(N)p(logp)2log(logp))bit-wiseoperations. 56

PAGE 57

CHAPTER6GRADIENTDENSITYESTIMATIONINONEDIMENSIONThischapterisfocussedonusingthephaserelationshipbetweentheSchrodingerwaveequationandtheHamilton-JacobiscalareldStorelatethedensityofthegradientofSwiththepowerspectrumofthewavefunctioninonedimension. 6.1MotivationfromQuantumMechanics:TheCorrespondencePrincipleTherulesimposedbyquantummechanicshasbeenverysuccessfulindescribingmicroscopicobjectslikemolecules,atomsandsubatomicparticleslikeelectrons.ButtheworldofmacroscopicobjectsarelargelygovernedbyclassicalmechanicssayforexampleNewton'slawsofmotion.Ifquantumtheoryisassumedtobemorefundamentalthanitsclassicalcounterpartandshouldbealsobecapableofexplainingmacroscopicphenomenon,thenthereshouldbealimitwherethelawsofquantummechanicsbecomeclosetothelawsofclassicalmechanics.Theconditionsunderwhichthequantumandclassicallawsagreeiscalledthecorrespondencelimitortheclassicallimit.ThisprinciplewasformulatedbyBohrinearly1920sandroughlystatesthatclassicalandquantumphysicsshouldgivethesameanswerwhenthesystembecomeslarge.Itisassumedthatas~!0,thelawsofquantummechanicsnaturallygivesrisetothelawsofclassicalmechanics,conspicuousintheFeynman'spathintegralderivationoftheSchrodingerwaveequation[ 11 ],wherethetransitionalamplitudeortheshort-timepropagatorKisassumedtobetheexponentofiS ~,whereSrepresentstheclassicalphysicalactionofthesystem(seeSection 2.2.1 ).Henceas~!0,wecanexpectthatthewavefunctionofaparticlewillbehavelike (X,t)expi ~S(X,t),=constant.(6)If~!0,theexponentinEquation 6 becomesrapidlyoscillatingandthenthetrajectoryoftheparticleisdenedbytheminimumoftheactionSasinthecaseoftheclassicalmechanics. 57

PAGE 58

6.2PhaseRelationshipBetweenandSInSection 2.2.2 ,whenwerelatedthetime-independentSchrodingerwaveandtheHamilton'scharacteristicfunctionS[ 22 ](referSection 2.1 throughtheexponent,specically(X)=exp()]TJ /F7 7.97 Tf 6.59 0 Td[(S(X) ~),weshowedthatwhensatisesEquation 2 ,Sasymptoticallysatisestheeikonalequation(Equation 1 )as~!0.RecallthatinderivingtheequationforweconsideredthenegativevaluesfortheenergyE,namelyE=)]TJ /F8 7.97 Tf 6.59 0 Td[(1 2,forwhichthedifferentialequationsatisedbythewavefunction(Equation 2 )hadgeneralizedfunctions(distributions)assolutions,whichwerereal-valuedfunctions.WealsobrieymentionedthatwhenEispositive,forwhichsolutionsofarecomplex-valued,byrelatingandSthroughthephaseasinEquation 6 ,specically (X)=expi ~S(X),(6)asonendsintheWKBapproximationfortheeikonalequation[ 35 ],Scanbeshowntosatisfytheeikonalequationinthelimitas~!0.SofarweignoredthisphaserelationshipbetweenandS.Buttoouramazementweobservedthat,thepowerspectrum[ 7 ]ofinEquation 6 hasanhauntingsimilaritytothedensityofgradientsofS.InotherwordsthesquaredmagnitudeoftheFouriertransformofthewavefunctionisapproximatelyequaltothedensityfunctionofthegradientofSwiththeapproximationbecomingincreasinglyexactas~!0.ThegradientsofScorrespondstotheclassicalmomentumofaparticle[ 22 ].Intheparlanceofquantummechanics,themagnitudesquareofthewavefunctionexpressedeitherinitspositionormomentumbasiscorrespondstoitspositionormomentumdensityrespectively.Sincetheserepresentations(eitherinthepositionormomentumbasis)aresimplythe(suitablyscaled)Fouriertransformsofeachother,themagnitudesquareoftheFouriertransformofthewavefunctionexpressedinitspositionbasis,isitsquantummomentumdensity[ 24 ].Belowwerigorouslyshowthattheclassicalmomentumdensity 58

PAGE 59

canbeexpressedasalimitingcase(as~!0)ofthequantummomentumdensity,incompleteagreementwiththecorrespondenceprinciple. 6.3DensityFunctionfortheGradientsThecurrentchapterrestrictsitselfwithestimatingthedensityofgradientsofathricedifferentiableSdenedonaboundedsetinonedimension.Thenextfewlinesplacesournotionofthegradientdensityonarmfooting.Theprobabilitydensityfunctionofthegradientscanbeobtainedviaarandomvariabletransformationofauniformlydistributedrandomvariable[ 6 ]andasillustratedin[ 37 ].Specically,letS:!Rdenoteathricedifferentiablefunctiondenedonaclosedboundedinterval=[b1,b2].LetS0denoteitsderivative.LetXdenoteauniformlydistributedrandomvariabledenedon.AfterrestrictingS0tobeacontinuousfunction,deneanewrandomvariableY=S0(X).Inotherwords,weperformarandomvariabletransformationusingS0onXtoobtainthenewrandomvariableY.ThedensityoftherandomvariableYthenrepresentsthedensityofthegradientsS0.Inthecurrentchapter,weprovideasimpleFouriertransformbasedtechniquetodeterminethedensityofY.OurmethodcomputesthegradientdensitydirectlyfromthefunctionS,circumventingtheneedtocomputeitsderivativeS0.TheapproachisbasedonexpressingthefunctionSasthephaseofawavefunction,specically(x)=expiS(x) ~forsmallvaluesof~andthenconsideringthenormalizedpowerspectrummagnitudesquaredoftheFouriertransformof[ 7 ].Usingthestationaryphaseapproximationawellknowntechniqueinasymptoticanalysisweshowthatinthelimitingcaseas~!0,thepowerspectrumofexactlymatchesthedensityofYandhencecanserveasitsdensityestimatoratsmall,non-zerovaluesof~.Thisrelationshipalsohasstrongconnotationstothecorrespondenceprinciple(Section 6.1 )wheretheclassicalmomentumdensityoftherandomvariableYisexpressedasalimitingcaseofitscorrespondingquantummomentumdensity. 59

PAGE 60

6.4ExistenceoftheDensityFunctionAsstatedabove,thedensityfunctionforthegradientsofS(denotedbyY)canbeobtainedviaarandomvariabletransformationofauniformlydistributedrandomvariableXusingthederivativeS0asthetransformationfunction,namely,Y=S0(X).WeassumethatSisthricedifferentiableonaclosed,boundedinterval=[b1,b2](withlengthL=b2)]TJ /F4 11.955 Tf 11.95 0 Td[(b1)andhasanon-vanishingsecondderivativealmosteverywhereon,i.e. (fx:S00(x)=0g)=0,(6)wheredenotestheLebesguemeasure.TheassumptioninEquation 6 ismadeinordertoensurethatthedensityfunctionofYexistsalmosteverywhere.ThisisfurtherclariedinLemma 3 below.Denethefollowingsets: Bfx:S00(x)=0g,CfS0(x):x2Bg[fS0(b1),S0(b2)g,andAufx:S0(x)=ug. (6) Here,S0(b1)=limx!b+1S0(x)andS0(b2)=limx!b)]TJ /F13 5.978 Tf -.2 -6.19 Td[(2S0(x).ThehigherderivativesofSattheendpointsb1,b2arealsodenedalongsimilarlinesusingone-sidedlimits.Themainpurposeofdeningtheseone-sidedlimitsistoexactlydeterminethesetCwherethedensityofYisnotdened.Since(B)=0,wealsohave(C)=0. Lemma1(FinitenessLemma). Auisniteforeveryu=2C. Proof. Weprovetheresultbycontradiction.FirstlyobservethatAuisasubsetofthecompactset.IfAuisnotnite,thenbytheorem(2.37)in[ 38 ],Auhasalimitpointx02.Considerasequencefxng1n=1,witheachxn2Au,convergingtox0.SinceS0(xn)=u,8n,fromthecontinuityofS0wegetS0(x0)=uandhencex02Au.Then, limn!1S0(x0))]TJ /F4 11.955 Tf 11.96 0 Td[(S0(xn) x0)]TJ /F4 11.955 Tf 11.96 0 Td[(xn=0=S00(x0).(6) 60

PAGE 61

BasedonthedenitionsgiveninEquation 6 ,wehavex02Bandhenceu2Cresultinginacontradiction. Lemma2(IntervalLemma). Foreveryu=2C,9>0andaclosedintervalJ=[u)]TJ /F3 11.955 Tf 11.95 0 Td[(,u+],suchthatJ\Cisempty. Proof. ObservethatBisclosedbecauseifx0isalimitpointofB,fromthecontinuityofS00wehaveS00(x0)=0andhencex02B.Bisalsocompactasitisaclosedsubsetof.SinceS0iscontinuous,C=S0(B)[fS0(b1),S0(b2)gisalsocompactandhenceR)-228(Cisopen.Thenforu=2C,thereexistsanopenneighborhoodNr(u)forsomer>0aroundusuchthatNr(u)\C=;.Bydening=r 2,theproofiscomplete. Lemma3(DensityLemma). TheprobabilitydensityofYonR)-222(Cexistsandisgivenby P(u0)=1 LN(u0)Xk=11 jS00(xk)j,(6)wherethesummationisoverAu0(whichisthenitesetoflocationsxk2whereS0(xk)=u0asperLemma 1 ,withjAu0j=N(u0). Proof. SincetherandomvariableXisassumedtohaveauniformdistributionon,itsdensityisgivenbyfX(x)=1 Lforeveryx2.RecallthattherandomvariableYisobtainedviaarandomvariabletransformationfromX,usingthefunctionS0.Hence,itsdensityfunctionexistsonR)-253(Cwherewehavebanishedtheimage(underS0)ofthemeasurezerosetofpointswhereS00vanishesandisgivenbyEquation 6 .Thereadermayreferto[ 6 ]foradetailedexplanation. 6.5EquivalenceoftheGradientDensityandthePowerSpectrumWenowprovethemainresultthatrelatesthenormalizedpowerspectrumofexpiS(x) ~(inthelimitas~!0)withtheprobabilitydensityoftherandomvariableY(denotedbyP). 61

PAGE 62

LetL=b2)]TJ /F4 11.955 Tf 11.96 0 Td[(b1.DeneafunctionF:RR+!Cas F(u,~)1 p 2~LZb2b1expiS(x) ~exp)]TJ /F4 11.955 Tf 9.3 0 Td[(iux ~dx.(6)Foraxedvalueof~,deneafunctionF~:R!Cas F~(u)F(u,~).(6)ObservethatF~iscloselyrelatedtotheFouriertransformofexpiS(x) ~.Thescalefactor1 p 2~ListhenormalizingtermsuchthattheL2normofF~isoneasseeninthefollowinglemma. Lemma4. WithF~denedasabove,F~2L2(R)andkF~k=1. Proof. DeneafunctionH(x)byH(x)8><>:1:ifx2[b1,b2];0:otherwiseLetf(x)=H(x)expiS(x) ~.Then, F~(u)=1 p 2~LZ1f(x)exp)]TJ /F4 11.955 Tf 9.3 0 Td[(iux ~dx.(6)Letu ~=vandG(v)=F~(v~).Then, p ~LG(v)=1 p 2Z1f(x)exp()]TJ /F4 11.955 Tf 9.3 0 Td[(ivx)dx.(6)SincefisL1integrable,byParseval'stheorem[ 7 ]wehave, Z1jf(x)j2dx=Z1jp ~LG(v)j2dv=~LZ1jF~(v~)j2dv.(6)Bylettingu=v~andobservingthat Z1jf(x)j2dx=Zb2b1expiS(x) ~2dx=L,(6) 62

PAGE 63

weget LZ1jF~(u)j2du=L.(6)Hence Z1jF~(u)j2du=1,(6)whichcompletestheproof. DeneafunctionP~:R)-222(C!R+as P~(u)jF~(u)j2=F~(u) F~(u).(6)Bydenition,P~0.Since(C)=0,fromLemma 4 ,R1P~(u)du=1.Hence,treatingP~(u)asadensityfunction,wehavethefollowingtheoremstatement. Theorem6.1. IfPandP~aredenedasabove,then lim!01 lim~!0Zu0+u0P~(u)du=P(u0),8u0=2C.(6)Beforeembarkingontheproof,wewouldliketoemphasizethattheorderingofthelimitsandtheintegralasgiveninthetheoremstatementiscrucialandcannotbearbitrarilyinterchanged.Topressthispointhome,weshowbelowthataftersolvingforP~,thelimitlim~!0P~doesnotexist.Hence,theorderoftheintegralfollowedbythelimit~!0cannotbeinterchanged.Furthermore,whenweswapthelimitsbetweenand~,weget lim~!0lim!01 Zu0+u0P~(u)du=lim~!0P~(u0)(6)whichdoesnotexist.Hence,thetheoremstatementcanbevalidonlyforthespeciedsequenceoflimitsandtheintegral. 6.5.1BriefExpositionoftheResultTounderstandtheresultinsimplerterms,letusreconsiderthedenitionofthescaledFouriertransformgiveninEquation 6 .Therstexponentialexp)]TJ /F7 7.97 Tf 6.67 -4.98 Td[(iS ~isavaryingcomplexsinusoid,whereasthesecondexponentialexp)]TJ /F10 7.97 Tf 6.68 -4.98 Td[()]TJ /F7 7.97 Tf 6.58 0 Td[(iux ~isaxedcomplexsinusoid 63

PAGE 64

atfrequencyu ~.Whenwemultiplythesetwocomplexexponentials,atlowvaluesof~,thetwosinusoidsareusuallynotinsyncandtendtocanceleachotherout.However,aroundthelocationswhereS0(x)=u,thetwosinusoidsareinperfectsync(asthecombinedexponentisstationary)withtheapproximatedurationofthisresonancedependingonS00(x).ThevalueoftheintegralinEquation 6 canbeapproximatedviathestationaryphaseapproximation[ 31 ]]as F~(u)1 p Lexpi 4N(u)Xk=1expi ~(S(xk))]TJ /F4 11.955 Tf 11.96 0 Td[(uxk)1 p S00(xk)(6)whereN(u)=jAuj.Theapproximationisincreasinglytightas~!0.ThesquaredFouriertransform(P~)givesustherequiredresult1 LPN(u)k=11 jS00(xk)jexceptforthecrossphasefactorsS(xk))]TJ /F4 11.955 Tf 13.12 0 Td[(S(xl))]TJ /F4 11.955 Tf 13.12 0 Td[(u(xk)]TJ /F4 11.955 Tf 13.12 0 Td[(xl)obtainedasabyproductoftwoormoreremotelocationsxkandxlindexingintothesamefrequencybinu,i.e,xk6=xl,butS0(xk)=S0(xl)=u.IntegratingthesquaredFouriertransformoverasmallfrequencyrange[u,u+]removesthesecrossphasefactorsandweobtainthedesiredresult. 6.5.2FormalProofWeshallnowprovidetheproofbyconsideringdifferentcases. Proof. case(i):Letusconsiderthecaseinwhichnostationarypointsexistforthegivenu0,i.e,thereisnox2forwhichS0(x)=u0.Lett(x)=S(x))]TJ /F4 11.955 Tf 12.34 0 Td[(u0x.Then,t0(x)isofconstantsignin[b1,b2]andhencet(x)isstrictlymonotonic.Deningv=t(x),wehavefromEquation 6 F~(u0)=1 p 2~LZt(b2)t(b1)expiv ~g(v)dv.(6)Here,g(v)=1 t0(x)wherex=t)]TJ /F8 7.97 Tf 6.59 0 Td[(1(v).Integratingbyparts,weget F~(u0)p 2~L=~ iexpit(b2) ~g(t(b2)))]TJ /F6 11.955 Tf 11.95 0 Td[(expit(b1) ~g(t(b1)))]TJ /F5 11.955 Tf 10.49 8.09 Td[(~ iZt(b2)t(b1)expiv ~g0(v)dv. (6) 64

PAGE 65

Then jF~(u0)jp 2~L~ 1 jS0(b2))]TJ /F4 11.955 Tf 11.96 0 Td[(u0j+1 jS0(b1))]TJ /F4 11.955 Tf 11.96 0 Td[(u0j+Zt(b2)t(b1)jg0(v)jdv!.(6)Hence,jF~(u0)j(u0)p ~,where(u0)>0issomecontinuousfunctionofu0.ThenP~(u0)2(u0)~.SinceS0(x)iscontinuousandu0=2C,wecannda>0suchthatforeveryu2[u0)]TJ /F3 11.955 Tf 12.13 0 Td[(,u0+],nostationarypointsexist.Thevaluecanalsobechosenappropriatelysuchthat[u0)]TJ /F3 11.955 Tf 11.96 0 Td[(,u0+]\C=;.Ifjj<,then lim~!0Zu0+u0P~(u)du=0.(6)Furthermore,fromEquation 6 wehaveP(u0)=0.Theresultimmediatelyfollows.case(ii):Wenowconsiderthecasewherestationarypointsexist.Sinceweareinterestedonlyinthesituationas~!0,thestationaryphasemethodin[ 31 32 ]canbeusedtoobtainagoodapproximationforF~(u0)denedinEquations 6 and 7 .Thephaseterminthisfunction,S(x))]TJ /F7 7.97 Tf 6.59 0 Td[(u0x ~,isstationaryonlywhenS0(x)=u0.ConsiderthesetAu0denedinEquation 6 .SinceitisnitebyLemma 1 ,letAu0=fx1,x2,...,xN(u0)gwithxk
PAGE 66

NotethattheintegralsG1andG2donothaveanystationarypoints.Fromcase(i)above,weget G1+G2=1(u0,~)=O(~)(6)as~!0.Furthermore,1(u0,~)2(u0)~,where2(u0)>0isacontinuousfunctionofu0.InordertoevaluateKkand~Kk,observethatwhenweexpandthephasetermuptosecondorder,t(x))]TJ /F4 11.955 Tf 12.78 0 Td[(t(xk)!Q(xk)(x)]TJ /F4 11.955 Tf 12.78 0 Td[(xk)2asx!xk,whereQ(xk)=S00(xk) 2.Furthermore,intheopenintervals(ck,xk)and(xk,ck+1),t0(x)=S0(x))]TJ /F4 11.955 Tf 11.92 0 Td[(u0iscontinuousandisofconstantsign.Fromtheorem(13.1)in[ 31 ],weget ~Kk=Kk=1 2expi 4)]TJ /F12 11.955 Tf 8.77 16.86 Td[(1 2expit(xk) ~s 2~ jS00(xk)j+2(u0,~).(6)Fromlemma(12.3)in[ 31 ],itcanbeveriedthat2(u0,~)=o(p ~)as~!0andcanalsobeuniformlyboundedbyafunctionofu0(independentof~)forsmallvaluesof~.InEquation 6 ,)]TJ /F1 11.955 Tf 10.1 0 Td[(istheGammafunctionandthesigninthephasetermdependsonwhetherS00(xk)>0orS00(xk)<0.PluggingthevaluesoftheseintegralsinEquation 6 andnotingthat)]TJ /F12 11.955 Tf 8.77 9.69 Td[()]TJ /F8 7.97 Tf 6.67 -4.98 Td[(1 2=p ,weget F~(u0)p 2~L=N(u0)Xk=1expi ~[S(xk))]TJ /F4 11.955 Tf 11.95 0 Td[(u0xk]s 2~ jS00(xk)jexpi 4+1(u0,~)+2(u0,~). (6) Hence, F~(u0)=1 p LN(u0)Xk=1exp)]TJ /F7 7.97 Tf 7.49 -4.98 Td[(i ~[S(xk))]TJ /F4 11.955 Tf 11.96 0 Td[(u0xk] p jS00(xk)jexpi 4 (6) +3(u0,~), (6) where3(u0,~)=1(u0,~)+2(u0,~) p 2~L=o(1)as~!0. 66

PAGE 67

FromthedenitionofP~(u)inEquation 7 ,wehave P~(u0)=1 LN(u0)Xk=11 jS00(xk)j+1 LN(u0)Xk=1N(u0)Xl=1;l6=kcos)]TJ /F8 7.97 Tf 6.68 -4.97 Td[(1 ~[S(xk))]TJ /F4 11.955 Tf 11.96 0 Td[(S(xl))]TJ /F4 11.955 Tf 11.95 0 Td[(u0(xk)]TJ /F4 11.955 Tf 11.96 0 Td[(xl)]+(xk,xl) jS00(xk)j1 2jS00(xl)j1 2+4(u0,~), (6) where4(u0,~)includesboththemagnitudesquareof3(u0,~)andthecrossproductbetweenthemain(rst)terminEquation 6 and3(u0,~).NoticethatthemainterminEquation 6 canbeboundedbyafunctionofu0independentof~as expi ~[S(xk))]TJ /F4 11.955 Tf 11.96 0 Td[(u0xk]=1,8~(6)andS00(xk)6=0,8k.Since3(u0,~)=o(1),weget4(u0,~)=o(1)as~!0.Additionally,(xk,xl)=0, 2(or))]TJ /F11 7.97 Tf 10.49 4.71 Td[( 2and(xl,xk)=)]TJ /F3 11.955 Tf 9.3 0 Td[((xk,xl).NoticethattherstterminEquation 6 exactlymatchestheexpressionforP(u0)asseenfromLemma 3 .But,sincelim~!0cos)]TJ /F8 7.97 Tf 6.67 -4.98 Td[(1 ~isnotdened,lim~!0P~(u0)doesnotexistandhencethecrosscosinetermsdonotvanishwhenwetakethelimit.WenowshowthatintegratingP~(u)overasmallnon-zerointerval[u0,u0+]andthentakingthelimitwithrespectto~(followedbythelimitwithrespectto)doesyieldthedensityofY.FromLemmas 1 and 2 ,weseethatforagivena2Au0,whenuisvariedovertheintervalJ=[u0)]TJ /F3 11.955 Tf 12.73 0 Td[(,u0+],theinversefunction(S0))]TJ /F8 7.97 Tf 6.59 0 Td[(1(u)iswelldenedwith(S0))]TJ /F8 7.97 Tf 6.59 0 Td[(1(u)2Na,whereNaissomesmallneighborhoodarounda.Foreacha2Au0,denetheinversefunction(S0a))]TJ /F8 7.97 Tf 6.59 0 Td[(1(u):J!Naas (S0a))]TJ /F8 7.97 Tf 6.58 0 Td[(1(u)=xiffu=S0(x)andx2Na.(6)WhenwemovefromP~(u0)toP~(u),thelocationsxkandxlinEquation 6 becomefunctionsofu.UsingtheinversefunctionsdenedinEquation 6 anddeningakxk(u0)andalxl(u0),xk(u)=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(S0ak)]TJ /F8 7.97 Tf 6.59 0 Td[(1(u)andxl(u)=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(S0al)]TJ /F8 7.97 Tf 6.58 0 Td[(1(u)foru2J.Dene 67

PAGE 68

thefunctionspkl(u)andqkl(u)overJas pkl(u)S(xk(u)))]TJ /F4 11.955 Tf 11.96 0 Td[(S(xl(u)))]TJ /F4 11.955 Tf 11.96 0 Td[(u(xk(u))]TJ /F4 11.955 Tf 11.96 0 Td[(xl(u)),and (6) qkl(u)1 jS00(xk(u))j1 2jS00(xl(u))j1 2. (6) Observethat p0kl(u)=S0(xk)x0k(u))]TJ /F4 11.955 Tf 11.96 0 Td[(S0(xl)x0l(u))]TJ /F6 11.955 Tf 11.95 0 Td[((xk(u))]TJ /F4 11.955 Tf 11.95 0 Td[(xl(u)))]TJ /F4 11.955 Tf 11.96 0 Td[(u(x0k(u))]TJ /F4 11.955 Tf 11.96 0 Td[(x0l(u))=xl(u))]TJ /F4 11.955 Tf 11.95 0 Td[(xk(u) (6) asu=S0(xk(u))=S0(xl(u)).Inparticular,ifxl(u0)>xk(u0),thenxl(u)>xk(u)andviceversa.Hence,p0kl(u)nevervanishesandisalsoofconstantsignoverJ.Then,itfollowsthatpkl(u)isstrictlymonotonicandspecicallybijectiveonJ.Wewillusethisresultinthesubsequentsteps.Now,letjj<.Sincetheadditionalerrorterm4(u0,~)inEquation 6 convergestozeroas~!0andcanalsobeuniformlyboundedbyafunctionofu0forsmallvaluesof~,wehave lim~!0Zu0+u04(u0,~)du=0.(6)Then,weget lim~!0Zu0+u0P~(u)du=I1+I2(6)where I11 LN(u0)Xk=1Zu0+u01 jS00(xk(u))jdu,and (6) I21 LN(u0)Xk=1N(u0)Xl=1;l6=klim~!0I3(k,l). (6) Here,I3(k,l)isgivenby I3(k,l)Zu0+u0qkl(u)cospkl(u) ~+(xk(u),xl(u))du.(6) 68

PAGE 69

Whenjj<,thesignofS00(xk(u))aroundxk(u0)andthesignofS00(xl(u))aroundxl(u0)donotchangeovertheinterval[u0,u0+].Since(xk,xl)dependsonthesignofS00,isconstanton[u0,u0+]andequalskl=(xk(u0),xl(u0)).Now,expandingthecosineterminEquation 6 ,weget I3(k,l)=cos(kl)Zu0+u0qkl(u)cospkl(u) ~du)]TJ /F6 11.955 Tf 11.29 0 Td[(sin(kl)Zu0+u0qkl(u)sinpkl(u) ~du. (6) Sincepkl(u)isbijective,bydeningvkl=pkl(u),weget I3(k,l)=cos(kl)I4(k,l))]TJ /F6 11.955 Tf 11.96 0 Td[(sin(kl)I5(k,l)(6)where I4(k,l)=Z(2)kl(1)klcosv ~gkl(v)dv,and (6) I5(k,l)=Z(2)kl(1)klsinv ~gkl(v)dv. (6) Here,(1)kl=pkl(u0),(2)kl=pkl(u0+)andgkl(v)=qkl(u) p0kl(u)whereu=p)]TJ /F8 7.97 Tf 6.59 0 Td[(1kl(v).IntegratingI4(k,l)byparts,weget I4(k,l)=~sin (2)kl ~!gkl(2)kl)]TJ /F5 11.955 Tf 11.96 0 Td[(~sin (1)kl ~!gkl(1)kl)]TJ /F5 11.955 Tf 9.3 0 Td[(~Z(2)kl(1)klsinv ~g0kl(v)dv. (6) Then, jI4(k,l)j~ gkl(2)kl+gkl(1)kl+Z(2)kl(1)kljg0kl(v)jdv!.(6)ItisworthmentioningthatqklandhencegklaredifferentiableovertheirrespectiveintervalsasthesignofS00(xk(u))S00(xl(u))doesnotchangeovertheinterval[u0,u0+].WethenhavejI4(k,l)j~MwhereMissomeconstantindependentof~.Hence,lim~!0I4(k,l)=0,8k,l.Byasimilarargument,lim~!0I5(k,l)=0,8k,l.From 69

PAGE 70

Equations 6 and 6 wegetI2=0.Since lim!01 I1=1 LN(u0)Xk=11 jS00(xk)j=P(u0),(6)theresultfollows. 6.5.3SignicanceoftheResultObservethattheintegrals I~(u0)=Zu0+u0P~(u)du,I(u0)=Zu0+u0P(u)du(6)givetheintervalmeasuresofthedensityfunctionsP~andPrespectively.Theorem 6.1 statesthatatsmallvaluesof~,boththeintervalmeasuresareapproximatelyequal,withthedifferencebetweenthembeingo(1).Recallthatbydenition,P~isthenormalizedpowerspectrum[ 7 ]ofthewavefunction(x)=expiS(x) ~.Hence,weconcludethatthepowerspectrumof(x)canpotentiallyserveasadensityestimatorforthegradientsofSatsmallvaluesof~.TheexperimentalresultsshownunderSection 8.4 serveasademonstration,anecdotallyattestingtotheverityoftheresult. 70

PAGE 71

CHAPTER7DENSITYESTIMATIONFORTHEDISTANCETRANSFORMSWedescribedinChapter 3 thattheEuclideandistancetransformssatisesthestatic,non-linearHamilton-Jacobiequation k5Sk=1.(7)InChapters 2 and 6 webrieyexplainedthatwhenweconsiderpositivevaluesfortheenergyEinEquation 2 (sayE=1 2),theaforementionednon-linearHamilton-Jacobiequation(Equation 7 )canbeembeddedinthelinearSchrodingerwaveequation )]TJ /F5 11.955 Tf 11.95 0 Td[(~252=,(7)wherebyrelatingSandthroughthephaseasinEquation 6 ,Scanbeshowntosatisfytheeikonalequationinthelimitas~!0.WecallthiswavefunctionsatisfyingthephaserelationwithS(Equation 6 )asthecomplexwaverepresentation(CWR)ofthedistancetransforms.AsmentionedinChapter 6 ,weobservedapeculiarfactbetweenthepowerspectrum(squaredFouriertransform)ofthiswavefunctionandthedensityfunctioncorrespondingtothegradients5S=(Sx,Sy)ofthedistancefunctionintwospatialdimensions.Asthenormofthegradient5Sisdenedtobe1everywhere(referEquation 7 ),wenoticedthattheFouriertransformvaluesliesmainlyontheunitcircleandthisbehaviortightensas~!0.Inthesubsequentsectionsusingstationaryphaseapproximationswedescribeanempiricaldiscoverycorroboratedbytheoreticalanalysis.WeshowthatthesquaredmagnitudeoftheFouriertransformwhenpolledontheunitcircleisapproximatelyequaltothedensityfunctionofthedistancetransformgradientswiththeapproximationbecomingincreasinglyexactas~!0. 71

PAGE 72

7.1DensityFunctionfortheDistanceTransformsThegeometryofthedistancetransformcorrespondstoasetofintersectingconeswiththeoriginsattheVoronoicenters[ 5 ].Asmentionedabove,thegradientsofthedistancetransform(whichexistgloballyexceptattheconeintersectionsandorigins)areunitvectorsandsatisfyEquation 7 .Thereforethegradientdensityfunctionisone-dimensionalanddenedoverthespaceoforientations.Theorientationsareconstantanduniquealongeachrayofeachcone.Itsprobabilitydistributionfunctionisgivenby F(+)1 LZZarctanSy Sx+dxdy(7)wherewehaveexpressedtheorientationrandomvariableas=arctanSy Sx.Theprobabilitydistributionfunctionalsoinducesaclosed-formexpressionforitsdensityfunctionasshownbelow.Letdenoteapolygonalgridsuchthatitsboundary@iscomposedofanitesequenceofstraightlinesegments.ThereasonforconsideringapolygonaldomainandnototherdomainslikeacircularregionwillbecomeclearerwhenwediscussTheorem 7.1 .LetthesetY=fYk2R2,k2f1,...,Kggbethegivenpoint-setlocations.ThentheEuclideandistancetransformatapointX=(x,y)2isgivenby S(X)minkkX)]TJ /F4 11.955 Tf 11.96 0 Td[(Ykk=mink(p (x)]TJ /F4 11.955 Tf 11.96 0 Td[(xk)2+(y)]TJ /F4 11.955 Tf 11.95 0 Td[(yk)2).(7)LetDk,centeredatYk,denotethekthVoronoiregioncorrespondingtotheinputpointYk.Dkcanberepresentedbythecartesianproduct[0,2)[0,Rk()]whereRk()isthelengthoftherayofthekthconeatorientation.IfagridpointX=(x,y)2Yk+Dk,thenS(X)=kX)]TJ /F4 11.955 Tf 12.79 0 Td[(Ykk.EachDkisaconvexpolygonwhoseboundary@DkisalsocomposedofanitesequenceofstraightlinesegmentsasshowninFigure 7-1 .NotethatevenforpointsthatlieontheVoronoiboundarywheretheradiallengthequalsRk()thedistancetransformiswelldened.TheareaLofthepolygonalgrid 72

PAGE 73

Figure7-1. VoronoidiagramofthegivenKpoints.EachVoronoiboundaryismadeofstraightlinesegments. isgivenby, L=KXk=1Z20ZRk()0rdrd=KXk=1Z20R2k() 2d.(7)Letl=p L.Withtheaboveset-upinplace,byrecognizingtheconegeometryateachVoronoicenterYk,Equation 7 canbesimpliedas F(+)1 LKXk=1Z+ZRk()0rdrd=1 LKXk=1Z+R2k() 2d.(7)Followingthisdrasticsimplication,wecanwritetheclosed-formexpressionforthedensityfunctionoftheunitvectordistancetransformgradientsas P()lim!0F(+) =1 LKXk=1R2k() 2.(7)BasedontheexpressionforLinEquation 7 itiseasytoseethat Z20P()d=1.(7)SincetheVoronoicellsareconvexpolygons[ 5 ],eachcellcontributesexactlyoneconicalraytothedensityfunctiononorientation.WhenSistreatedastheHamilton-Jacobiscalareld,P()carriesaninterpretationasitsclassicalmomentumdensity[ 22 ]. 73

PAGE 74

7.2PropertiesoftheFourierTransformofCWRSincethedistancetransformisnotdifferentiableatthepoint-setlocationsfYkgKk=1andalsoalongtheVoronoiboundaries@Dk,8k(ameasurezerosetin2D),wewouldliketoconsidertheregionwhichexcludesbothofthem.Tothisend,let0<<1 2begiven.LettheregionDkcenteredatYkberepresentedbythecartesianproduct[0,2)[R(1)k(),R(2)k()]where, R(1)k()=Rk()R(2)k()=(1)]TJ /F3 11.955 Tf 11.96 0 Td[()Rk(). (7) Asbefore,thelengthoftherayatanorientationineachDk,givenbyR(2)k())]TJ /F4 11.955 Tf 12.12 0 Td[(R(1)k()dependson.NotethatinthedenitionofDkweexplicitlyremovedthesourcepointYkwheretheraylengthr()=0andtheboundaryofthekthVoronoicellwherer()=Rk(). Figure7-2. RegionthatexcludesboththesourcepointandtheVoronoiboundary. Denethegrid K[k=1Yk+Dk.(7)ItisworthemphasizingthatalsoexcludesallthesourcepointsfYkgKk=1andtheVoronoiboundaries.ItsareaLequals LKXk=1Z20ZR(2)k()R(1)k()rdrd=(1)]TJ /F6 11.955 Tf 11.96 0 Td[(2)KXk=1Z20R2k() 2d.(7)FromEquation 7 wehaveL=(1)]TJ /F6 11.955 Tf 11.95 0 Td[(2)Landhencelim!0L=L. 74

PAGE 75

Letl=p L.DeneafunctionF:RRR+!Cby F(u,v,~)1 2~lZZexpiS(x,y) ~exp)]TJ /F4 11.955 Tf 9.3 0 Td[(i(ux+vy) ~dxdy.(7)Foraxedvalueof~,deneafunctionF~:RR!Cby F~(u,v)F(u,v,~).(7)NotethatF~iscloselyrelatedtotheFouriertransformoftheCWR,=exp)]TJ /F7 7.97 Tf 6.67 -4.98 Td[(iS ~[ 7 ].Thescalefactor1 2~listhenormalizingtermsuchthattheL2normofF~is1asseeninthefollowinglemma. Lemma5. WithF~denedasabove,F~2L2(R2)andkF~k=1. Proof. DeneafunctionH(x,y)byH(x,y)8><>:1:if(x,y)2;0:otherwiseLetf(x,y)=H(x,y)expiS(x,y) ~.Then, F~(u,v)=1 2~lZZf(x,y)exp)]TJ /F4 11.955 Tf 9.29 0 Td[(i(ux+vy) ~dxdy.(7)Letu ~=s,v ~=tandG(s,t)=F~(s~,t~).Then, ~lG(s,t)=1 2ZZf(x,y)exp()]TJ /F4 11.955 Tf 9.3 0 Td[(i(sx+ty))dxdy.(7)SincefisL1integrable,byParseval'stheorem[ 7 ]wehave, ZZjf(x,y)j2dxdy=ZZj~lG(s,t)j2dsdt=(~l)2ZZjF~(s~,t~)j2dsdt.(7)Bylettingu=s~,v=t~andobservingthat ZZjf(x,y)j2dxdy=ZZexpiS(x,y) ~2dxdy=L,(7) 75

PAGE 76

weget (l)2ZZjF~(u,v)j2dudv=L.(7)Hence ZZjF~(u,v)j2dudv=1,(7)whichcompletestheproof. Considerthepolarrepresentationofthespatialfrequencies(u,v)namelyu=~rcos(!)andv=~rsin(!)where~r>0.For(x,y)2Yk+Dk,letx)]TJ /F4 11.955 Tf 12.36 0 Td[(xk=rcos()andy)]TJ /F4 11.955 Tf 11.96 0 Td[(yk=rsin()withr2[R(1)k(),R(2)k()].ThenEquation 7 canberewrittenas F~(~r,!)=KXk=1CkIk(~r,!)(7)where, Ck=exp)]TJ /F4 11.955 Tf 9.29 0 Td[(i ~[~rcos(!)xk+~rsin(!)yk](7)and Ik(~r,!)=1 2~lZ20ZR(2)k()R(1)k()expi ~r(1)]TJ /F6 11.955 Tf 11.57 0 Td[(~rcos()]TJ /F3 11.955 Tf 11.96 0 Td[(!))rdrd.(7)Withtheaboveset-upinplace,wehavethefollowingtheoremnamely, Theorem7.1(CircleTheorem). If~r6=1,then, lim~!0F~(~r,!)=0,(7)forany0<<1 2. 7.2.1BriefExpositionoftheTheoremBeforewefurnisharigorousprooffortheaforementionedtheorem,letustrytogivesimpler,intuitivepictureofwhythestatementistrue.ObservethattherstexponentialexpiS(x,y) ~inEquation 7 isavaryingcomplexsinusoidandthesecondexponentialexp)]TJ /F7 7.97 Tf 6.59 0 Td[(i(ux+vy) ~inEquation 7 isaxedcomplexsinusoidatfrequenciesu ~andv ~respectively.Whenwemultiplythesetwocomplexexponentials,atlowvaluesof~,thetwosinusoidsareusuallynotinsyncandcancellationsoccurin 76

PAGE 77

theintegral.Exceptionstothecancellationhappenatlocationswhere5S=(Sx,Sy)=(u,v),asaroundtheselocations,thetwosinusoidsareinperfectsync.Sincek5Sk=1fordistancetransforms,strongresonanceoccursonlywhenu2+v2=1(~r=1).When~r6=1,thetwosinusoidstendtocanceleachotheroutas~!0,resultinginF~becomingzeroatthoselocations. 7.2.2FormalProofHavinggiventheintuitivepictureofwhytheTheorem 7.1 holdsgood,weshallnowprovidetheformalproof. Proof. AseachCkisbounded,itsufcestoshowthatif~r6=1,thenlim~!Ik(~r,!)=0forallIk.Consider I=1 2~lZ20ZR(2)()R(1)()expi ~r(1)]TJ /F6 11.955 Tf 11.57 0 Td[(~rcos()]TJ /F3 11.955 Tf 11.96 0 Td[(!))rdrd,(7)whereR(1)()=R()andR(2)()=(1)]TJ /F3 11.955 Tf 10.59 0 Td[()R().Lettheregion[0,2)[R(1)(),R(2)()]bedenotedbyD.R()isdenedinsuchawaythattheboundaryofDconsistsofnitesequenceofstraightlinesegmentsasinthecaseofeachDk.NoticethatDdoesn'tcontaintheorigin(0,0).Letp(r,)=r(1)]TJ /F6 11.955 Tf 11.87 0 Td[(~rcos()]TJ /F3 11.955 Tf 12.24 0 Td[(!))denotethephasetermofIinEquation 7 foragiven~rand!.Thepartialgradientsofp(r,)aregivenby @p @r=1)]TJ /F6 11.955 Tf 11.57 0 Td[(~rcos()]TJ /F3 11.955 Tf 11.96 0 Td[(!)@p @=r~rsin()]TJ /F3 11.955 Tf 11.95 0 Td[(!).(7)SinceDisboundedawayfromtheorigin(0,0)5piswell-denedandboundedanditequalszeroonlywhen~r=1and=!.Since~r6=1byassumption,nostationarypointsexist(5p6=0)andhencewecanexpectI!0as~!0[ 13 25 48 ].Weshowtheresultmoreexplicitlybelow.Deneavectoreldu(r,)=5p k5pk2r.Thenitiseasytoseethat 5.u(r,)expip(r,) ~=(5.u(r,))expip(r,) ~+i ~expip(r,) ~r(7) 77

PAGE 78

wherethegradientoperator5=@ @r,@ @.UsingtherelationfromEquation 7 inEquation 7 ,weget I=I(1))]TJ /F4 11.955 Tf 11.96 0 Td[(I(2)(7)where I(1)=1 2ilZZD5.u(r,)expip(r,) ~drdI(2)=1 2ilZZD(5.u(r,))expip(r,) ~drd. (7) ConsidertheintegralI(1).Fromthedivergencetheoremwehave I(1)=1 2ilZ)]TJ /F6 11.955 Tf 5.32 10.79 Td[((uTn)expip(r,) ~ds(7)where)]TJ /F1 11.955 Tf 10.09 0 Td[(isthepositivelyorientedboundaryofD,sisthearclengthof)]TJ /F1 11.955 Tf 10.1 0 Td[(andnistheunitoutwardnormalof)]TJ /F1 11.955 Tf 6.78 0 Td[(.Theboundary)]TJ /F1 11.955 Tf 10.1 0 Td[(consistsoftwodisjointregionsonealongr()=R(1)()andanotheralongr()=R(2)().Ifthelevelcurvesofp(r,)istangentialto)]TJ /F1 11.955 Tf 10.1 0 Td[(onlyatadiscretesetoflocationsgivingrisetostationarypointsofthesecondkind[ 29 47 48 ],inotherwordsifvaluesofpalongtheboundary)]TJ /F1 11.955 Tf 10.1 0 Td[(isnotconstantoveranycontiguousintervalof,thenusingonedimensionalstationaryphaseapproximations[ 31 32 ]I(1)canbeshowntobeO(p ~)andhenceconvergestozeroas~!0.SincetheboundaryofDismadeofstraightlinesegments(specicallynotarc-like),wecanshowthatthelevelcurvesofp(r,)cannotoverlapwith)]TJ /F1 11.955 Tf 10.1 0 Td[(foranon-zeroniteinterval.Thesubsequentparagraphtakescareofthistechnicalissueandthereadermaydecidetoomitituponrstreading.Thelevelcurvesofp(r,)aregivenbyR()(1)]TJ /F6 11.955 Tf 12.1 0 Td[(~rcos()]TJ /F3 11.955 Tf 12.48 0 Td[(!))=c,wherec6=0issomeconstant.Recallthateachofthetwodisjointregionsof)]TJ /F1 11.955 Tf 10.1 0 Td[(ismadeofanitesequenceoflinesegments.Forthelevelcurvesofp(r,)tocoincidewith)]TJ /F1 11.955 Tf 10.1 0 Td[(overanon-zeroniteinterval,y()=csin() 1)]TJ /F8 7.97 Tf 6.29 0 Td[(~rcos()]TJ /F11 7.97 Tf 6.59 0 Td[(!)andx()=ccos() 1)]TJ /F8 7.97 Tf 6.3 0 Td[(~rcos()]TJ /F11 7.97 Tf 6.58 0 Td[(!)shouldsatisfythelineequationy=mx+bforsomeslopemandslope-interceptb,whenvariesoversome 78

PAGE 79

contiguousinterval2[1,2].Plugginginthevalueofyandxtothelineequationandexpandingcos()]TJ /F3 11.955 Tf 11.96 0 Td[(!)weget csin=mccos()+b)]TJ /F4 11.955 Tf 11.96 0 Td[(b~r[cos()cos(!)+sin()sin(!)].(7)Combiningtermsweget sin()[c+b~rsin(!)])]TJ /F6 11.955 Tf 11.96 0 Td[(cos()[mc)]TJ /F4 11.955 Tf 11.95 0 Td[(b~rcos(!)]=b.(7)Bydening1=c+b~rsin(!)and2=)]TJ /F6 11.955 Tf 9.29 0 Td[((mc)]TJ /F4 11.955 Tf 12.03 0 Td[(b~rcos(!))weseethatsin()andcos()needstosatisfythelinearrelation 1sin()+2cos()=b(7)for2[1,2]inorderforthelevelcurvesofp(r,)tooverlapwiththepiece-wiselinearboundary)]TJ /F1 11.955 Tf 6.78 0 Td[(.AsEquation 7 cannotbetrueforaniteintervalof,I(1)=O(p ~)as~!0andhenceconvergestozerointhelimit.NowI(2)hasthesimilarformastheoriginalIinEquation 7 withrreplacedbyg1=(5.u).Bylettingu1(r,)=5p k5pk2g1,fromEquation 7 andthedivergencetheoremweget I(2)=)]TJ /F5 11.955 Tf 9.3 0 Td[(~ 2lZ)]TJ /F6 11.955 Tf 5.32 10.79 Td[((u1Tn)expip(r,) ~ds+~ 2lZZD(5.u1(r,))expip(r,) ~drd. (7) AsI(2)=O(~),itconvergestozeroas~!0.ApplyingtheobtainedresultstoEquation 7 weseethatI(andhenceIk)!0as~!0whichcompletestheproof. SincetheTheorem 7.1 istrueforany0<<1 2,italsoholdsgoodas!0.Henceasacorollarywehavethefollowingresultnamely, 79

PAGE 80

Corollary1. If~r6=1,then lim!0lim~!0F~(~r,!)=0.(7) 7.3SpatialFrequenciesasGradientHistogramBinsWenowshowthatthesquaremagnitudeoftheFouriertransformoftheCWR()whenpolledclosetotheunitcircle(~r=1)isapproximatelyequaltothedensityfunctionofthedistancetransformgradientsPwiththeapproximationtighteningupas~!0.ThesquaredmagnitudeoftheFouriertransformalsocalleditspowerspectrum[ 7 ]isgivenby P~(~r,!)jF~(~r,!)j2=F~(~r,!) F~(~r,!).(7)BydenitionP~(~r,!)0.FromLemma 5 ,wehave Z20Z10P~(~r,!)~rd~rd!=1(7)independentof~.HenceP~(~r,!)~rcanbetreatedasadensityfunctionforallvaluesof~.Intheparlanceofquantummechanics,thisquantityP~carriesaninterpretationofbeingthequantummomentumdensityofthewavefunction(referSection 6.2 ).ThecorrespondenceprincipledescribedunderSection 6.1 dictatesthatthequantumdensityP~mustapproachitsclassicalcounterpartPinthelimitas~!0.Weearlierobservedthatthegradientdensityfunctionoftheunitvectordistancetransformgradientsisone-dimensionalanddenedonlyoverthespaceoforientations.ForP~(~r,!)~rtobehaveasanorientationdensityfunction,itneedstobeintegratedalongtheradialdirection~r.SinceTheorem 7.1 statesthattheFouriertransformvaluesareconcentratedonlyontheunitcircle~r=1andconvergestozeroelsewhereas~!0,itshouldbesufcientiftheintegrationfor~risdoneoveraregionveryclosetoone.Thefollowingtheoremconrmsourcurrentobservations. 80

PAGE 81

Theorem7.2(Quantummeetsclassical). Foranygiven0<<1 2,0<<1,!02[0,2)and0<<2, lim~!0Z!0+!0Z1+1)]TJ /F11 7.97 Tf 6.59 0 Td[(P~(~r,!)~rd~rd!=Z!0+!0P(!)d!.(7) 7.3.1BriefExpositionoftheTheoremBeforeweproceedwiththeformalanalysisoftheproof,weonceagaintrytogiveaintuitivereasoningofwhythetheoremstatementistrue.TheFouriertransformoftheCWRdenedinEquation 7 involvestwospatialintegrals(overxandy)whichareconvertedintopolarcoordinatedomainintegrals.ThesquaredmagnitudeoftheFouriertransform(powerspectrum),P~(~r,!),involvesmultiplyingtheFouriertransformwithitscomplexconjugate.Thecomplexconjugateisyetanother2Dintegralwhichwewillperforminthepolarcoordinatedomain.Asthegradientdensityfunctionisone-dimensionalanddenedoverthespaceoforientations,weintegratethepowerspectrumalongtheradialdirectionclosetotheunitcircle~r=1(as!0).Thisisafthintegral.WhenwepollthepowerspectrumP~(~r,!)closeto~r=1,thetwosinusoidsnamelyexpiS(x,y) ~andexp)]TJ /F7 7.97 Tf 6.59 0 Td[(i(ux+vy) ~inEquation 7 areinresonanceonlywhenthereisaperfectmatchbetweentheorientationofeachrayofthedistancetransformS(x,y)andtheangleofthe2Dspatialfrequency(!=arctan)]TJ /F7 7.97 Tf 6.68 -4.98 Td[(v u).Allthegridlocations(x,y)havingthesamegradientorientation arctanSy Sx=arctanv u(7)castsavoteonlyatitscorrespondingspatialfrequencyhistogrambin!.Sincethehistogrambinisgenerallyindexedbymultiplegridlocations,itleadstocrossphasefactors.Integratingthepowerspectrumoverasmallrangeontheorientationhelpsincancellingoutthesephasefactorsgivingusthedesiredresultwhenwetakethelimitas~!0.Thisintegralandlimitcannotbeexchangedbecausethephasefactorswillnototherwisecancel.Theproofmainlydealswithmanagingthesesixintegrals. 81

PAGE 82

7.3.2FormalProofWenowprovidetheformalproofofourTheorem 7.2 Proof. Firstlyobservethat F~(~r,!)=KXk=1 Ck 2~lZ20ZR(2)k(0)R(1)k(0)exp)]TJ /F4 11.955 Tf 9.3 0 Td[(ir0 ~[1)]TJ /F6 11.955 Tf 11.57 0 Td[(~rcos(0)]TJ /F3 11.955 Tf 11.95 0 Td[(!)]r0dr0d0.(7)Dene I(!)Z1+1)]TJ /F11 7.97 Tf 6.58 0 Td[(P~(~r,!)~rd~r.(7)As~!0,I(!)willapproachthedensityfunctionofthegradientsofS(x,y).NotethattheintegralinEquation 7 isovertheinterval[1)]TJ /F3 11.955 Tf 12.14 0 Td[(,1+]where>0canbemadearbitrarilysmall(as~!0)duetoTheorem 7.1 .SinceP~(~r,!)equalsF~(~r,!) F~(~r,!),wehave I(!)=KXj=1KXk=11 (2~l)2Gjk(!),(7)where Gjk(!)=Z1+1)]TJ /F11 7.97 Tf 6.58 0 Td[(Z20ZR(2)k(0)R(1)k(0)Z20ZR(2)k()R(1)k()expi ~bjkf1drddr0d0d~r.(7)Here, bjk(r,,r0,0,~r)=r[1)]TJ /F6 11.955 Tf 11.57 0 Td[(~rcos()]TJ /F3 11.955 Tf 11.95 0 Td[(!)])]TJ /F4 11.955 Tf 11.95 0 Td[(r0[1)]TJ /F6 11.955 Tf 11.58 0 Td[(~rcos(0)]TJ /F3 11.955 Tf 11.96 0 Td[(!)])]TJ /F6 11.955 Tf 18.88 0 Td[(~r[cos(!)(xj)]TJ /F4 11.955 Tf 11.96 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 11.95 0 Td[(yk)] (7) and f1(r,r0,~r)=rr0~r.(7)NoticethatthephasetermofthequantityCj Ck,namely )]TJ /F6 11.955 Tf 11.57 0 Td[(~r[cos(!)(xj)]TJ /F4 11.955 Tf 11.96 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 11.95 0 Td[(yk)](7)isabsorbedinbjk.Sinceweareinterestedonlyinthelimitas~!0,essentialcontributiontoGjk(!)comesonlyfromthestationary(critical)point(s)ofbjk[ 47 ]. 82

PAGE 83

Thepartialderivativesofbjk(r,,r0,0,~r)aregivenby @bjk @r=1)]TJ /F6 11.955 Tf 11.57 0 Td[(~rcos()]TJ /F3 11.955 Tf 11.96 0 Td[(!),@bjk @=r~rsin()]TJ /F3 11.955 Tf 11.95 0 Td[(!),@bjk @r0=)]TJ /F6 11.955 Tf 9.3 0 Td[(1+~rcos(0)]TJ /F3 11.955 Tf 11.95 0 Td[(!),@bjk @0=)]TJ /F4 11.955 Tf 9.3 0 Td[(r0~rsin(0)]TJ /F3 11.955 Tf 11.95 0 Td[(!),@bjk @~r=)]TJ /F4 11.955 Tf 9.3 0 Td[(rcos()]TJ /F3 11.955 Tf 11.96 0 Td[(!)+r0cos(0)]TJ /F3 11.955 Tf 11.95 0 Td[(!))]TJ /F6 11.955 Tf 11.95 0 Td[([cos(!)(xj)]TJ /F4 11.955 Tf 11.96 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 11.95 0 Td[(yk)]. (7) Asr,r0and~r>0,itiseasytoseethatfor5bjk=0(stationary),wemusthave ~r=1,=0=!,r=r0)]TJ /F6 11.955 Tf 11.96 0 Td[([cos(!)(xj)]TJ /F4 11.955 Tf 11.95 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 11.96 0 Td[(yk)].(7)Lett0denotethestationarypoint.TheHessianmatrixWofbjkatt0isgivenbyW(r,,r0,0,~r)jt0=2666666666640000)]TJ /F6 11.955 Tf 9.3 0 Td[(10rt000000001000)]TJ /F4 11.955 Tf 9.29 0 Td[(r00)]TJ /F6 11.955 Tf 9.3 0 Td[(10100377777777775wherert0=r0)]TJ /F6 11.955 Tf 12.76 0 Td[([cos(!)(xj)]TJ /F4 11.955 Tf 12.77 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 12.77 0 Td[(yk)].Unfortunately,thedeterminantofWatthestationarypointt0equals0astherstandthirdrowscorrespondingtorandr0respectivelyarescalarmultiplesofeachother.Alsonotethatthevalueofr0isnotdeterminedatthestationarypoint.Thisimpedesusfromdirectlyapplyingthe5Dstationaryphaseapproximation[ 47 ].HenceweproposetosolveforI(!)inEquation 7 bymanualsymmetrybreaking,wherewextheconjugatevariablesr0and0andperformtheintegrationonlywithrespecttotheotherthreevariablesnamelyr,and~r.Tothisend,let I(!)=KXj=1KXk=11 (2~l)2Z20ZR(2)k(0)R(1)k(0)exp)]TJ /F4 11.955 Tf 9.29 0 Td[(ir0 ~gjk(r0,0)r0dr0d0,(7) 83

PAGE 84

where gjk(r0,0)=Z1+1)]TJ /F11 7.97 Tf 6.58 0 Td[(Z20ZR(2)j()R(1)j()expi ~jk(r,,~r)f2(r,,~r)drdd~r.(7)Here, jk(r,,~r)=r[1)]TJ /F6 11.955 Tf 11.57 0 Td[(~rcos()]TJ /F3 11.955 Tf 11.96 0 Td[(!)]+r0~rcos(0)]TJ /F3 11.955 Tf 11.95 0 Td[(!))]TJ /F6 11.955 Tf 11.57 0 Td[(~r[cos(!)(xj)]TJ /F4 11.955 Tf 11.96 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 11.95 0 Td[(yk)](7)and f2(r,,~r)=r~r.(7)Inthedenitionforjk(r,,~r)inEquation 7 ,!,r0and0areheldxedandforgjk(r0,0)inEquation 7 ,!isheldconstant.ThephasetermofthequantityCj Ck(Equation 7 )isabsorbedinjkandpursuanttoFubini'stheorem,theintegrationwithrespectto~risconsideredbeforetheintegrationforr0and0.Asstatedbefore,essentialcontributiontogjk(r0,0)comesonlyfromthestationarypointsofjkas~!0[ 47 ].Thepartialderivativesofjk(r,,~r)aregivenby @jk @r=1)]TJ /F6 11.955 Tf 11.57 0 Td[(~rcos()]TJ /F3 11.955 Tf 11.96 0 Td[(!),@jk @=r~rsin()]TJ /F3 11.955 Tf 11.95 0 Td[(!)@jk @~r=)]TJ /F4 11.955 Tf 9.3 0 Td[(rcos()]TJ /F3 11.955 Tf 11.96 0 Td[(!)+r0cos(0)]TJ /F3 11.955 Tf 11.96 0 Td[(!))]TJ /F6 11.955 Tf 11.95 0 Td[([cos(!)(xj)]TJ /F4 11.955 Tf 11.96 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 11.95 0 Td[(yk)]. (7) Asbothrand~r>0,for5jk=0(stationary),wemusthave ~r=1,=!,andr=r0cos(0)]TJ /F3 11.955 Tf 11.96 0 Td[(!))]TJ /F6 11.955 Tf 11.95 0 Td[([cos(!)(xj)]TJ /F4 11.955 Tf 11.96 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 11.96 0 Td[(yk)].(7)Letp0denotethestationarypoint.Then jk(p0)=r0cos(0)]TJ /F3 11.955 Tf 11.96 0 Td[(!))]TJ /F6 11.955 Tf 11.96 0 Td[([cos(!)(xj)]TJ /F4 11.955 Tf 11.95 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 11.96 0 Td[(yk)]=rp0,f2(p0)=rp0=jk(p0) (7) 84

PAGE 85

andtheHessianmatrixHofjkatp0isgivenbyH(r,,~r)jp0=26666400)]TJ /F6 11.955 Tf 9.3 0 Td[(10jk(p0)0)]TJ /F6 11.955 Tf 9.3 0 Td[(100377775ThedeterminantofHequals)]TJ /F3 11.955 Tf 9.3 0 Td[(jk(p0).Sincer(specicallyrp0)>0,weseethatthedeterminantofHisstrictlynegativeanditssignaturedifferencebetweenthenumberofpositiveandnegativeeigenvaluesisone.Thenfromhigherdimensionalstationaryphaseapproximation[ 47 ],wehave gjk(r0,0)=(2~)3 2p jk(p0)expijk(p0) ~+i 4+1(r0,0,!,~)(7)as~!0,where1(r0,0,!,~)includesthecontributionsfromtheboundaryinEquation 7 .Herewehaveassumedthatthestationarypointp0doesn'toccurontheboundarybutliestoitsinterior,i.e,R(1)j(!)
PAGE 86

arecomposedofanitesequenceofstraightlinesegments.Considerthesurface)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(1.Thevalueofjkonthesurface)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(1atagivenand~requals )]TJ /F13 5.978 Tf 4.82 -1.18 Td[(1jk(,~r)=R(1)j()[1)]TJ /F6 11.955 Tf 11.57 0 Td[(~rcos()]TJ /F3 11.955 Tf 11.96 0 Td[(!)]+~r,(7)where =r0cos(0)]TJ /F3 11.955 Tf 11.96 0 Td[(!))]TJ /F6 11.955 Tf 11.95 0 Td[([cos(!)(xj)]TJ /F4 11.955 Tf 11.95 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 11.96 0 Td[(yk)].(7)FollowingthelinesofTheorem 7.1 ,weobservethatforagiven~r,)]TJ /F13 5.978 Tf 4.82 -1.18 Td[(1jk(,~r)cannotbeconstantforacontiguousintervalofastheEquation 7 cannotbesatisedforanyniteinterval.BysimilarargumenttherecanexistatmostmostonlyanitediscretesetofforwhichR(1)j()cos()]TJ /F3 11.955 Tf 12.01 0 Td[(!)=.LetZdenotethisniteset.Thenforagiven=2Z,)]TJ /F13 5.978 Tf 4.82 -1.17 Td[(1jkvarieslinearlyin~randspecically,itsderivativewithrespectto~rdoesn'tvanish.Fromtheaboveobservationswecanconcludethattheredoesn'texista2Dpatchon)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(1onwhich)]TJ /F13 5.978 Tf 4.82 -1.18 Td[(1jkisconstant.Similarconclusioncanbeobtainedevenforthesurface)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(2.Hencejkcannotbeconstantontheboundary)]TJ /F1 11.955 Tf 10.09 0 Td[(overa2Dregionhavinganitenon-zeromeasure.Now,pluggingthevalueofg(r0,0)inEquation 7 weget I(!)=KXj=1KXk=1jk(!) LI(1)jk(!)+I(2)jk(!)(7)where, I(1)jk(!)=1 p 2~Z20ZR(2)k(0)R(1)k(0)expip(r0,0,!) ~q(r0,0,!)dr0d0,I(2)jk(!)=Z20ZR(2)k(0)R(1)k(0)exp)]TJ /F4 11.955 Tf 9.3 0 Td[(ir0 ~r02(r0,0,!,~)dr0d0,jk(!)=exp)]TJ /F4 11.955 Tf 9.3 0 Td[(ijk(!) ~+i 4,jk(!)=cos(!)(xj)]TJ /F4 11.955 Tf 11.95 0 Td[(xk)+sin(!)(yj)]TJ /F4 11.955 Tf 11.95 0 Td[(yk),2(r0,0,!,~)=1(r0,0,!,~) (2~l)2=1 (2l)2~)]TJ /F8 7.97 Tf 6.58 0 Td[(2(r0,0,!). (7) 86

PAGE 87

Thefunctionsp(r0,0,!)andq(r0,0,!)aregivenby p(r0,0,!)=)]TJ /F4 11.955 Tf 9.3 0 Td[(r0[1)]TJ /F6 11.955 Tf 11.95 0 Td[(cos(0)]TJ /F3 11.955 Tf 11.95 0 Td[(!)]andq(r0,0,!)=r0p r0cos(0)]TJ /F3 11.955 Tf 11.96 0 Td[(!))]TJ /F3 11.955 Tf 11.96 0 Td[(jk(!). (7) Since2,wecanconcludethatforsmallvaluesof~2(r0,0,!,~)canbeboundedby(r0,0,!)andpursuanttotheRiemann-Lesbeguelemma,lim~!0I(2)jk=0.MoreoverfromtheLesbeguedominatedconvergencetheoremitfollowsthat lim~!0Z!0+!0KXj=1KXk=1I(2)jk(!)=KXj=1KXk=1Z20lim~!0I(2)jk(!)=0.(7)UsingtheaboveresultinEquation 7 weget, lim~!0Z!0+!0I(!)d!=KXj=1KXk=1lim~!0Z!0+!0jk LI(1)jk(!)d!,(7)whichleavesustoshowthat KXj=1KXk=1lim~!0Z!0+!0jk(!) LI(1)jk(!)d!=Z!0+!0P(!)d!.(7)ConsidertheintegralI(1)jk(!).Fixa>0.Dividingtheintegralrange[0,2)for0intothreedisjointregionsnamely[0,!)]TJ /F3 11.955 Tf 11.96 0 Td[(),[!)]TJ /F3 11.955 Tf 11.96 0 Td[(,!+]and(!+,2),weget I(1)jk(!)=J(1)jk(,!)+J(2)jk(,!)+J(3)jk(,!)(7)where, J(1)jk(,!)=1 p 2~Z!+!)]TJ /F11 7.97 Tf 6.59 0 Td[(ZR(2)k(0)R(1)k(0)expip(r0,0,!) ~q(r0,0,!)dr0d0,J(2)jk(,!)=1 p 2~Z!)]TJ /F11 7.97 Tf 6.59 0 Td[(0ZR(2)k(0)R(1)k(0)expip(r0,0,!) ~q(r0,0,!)dr0d0,J(3)jk(,!)=1 p 2~Z2!+ZR(2)k(0)R(1)k(0)expip(r0,0,!) ~q(r0,0,!)dr0d0. (7) 87

PAGE 88

Sinceitistrueforany>0,wecanconsiderthecaseas!0.Fixacloseenoughtozeroandconsidertheaboveintegralsas~!0.Asessentialcontributionstotheaboveintegralscomesonlyfromthestationarypointsofp(r0,0,!)[ 13 25 48 ](with!heldxed),werstdetermineitscritical(stationary)point(s).Thegradientsofp(r0,0,!)ataxed!aregivenby @p @r0=)]TJ /F6 11.955 Tf 9.3 0 Td[(1+cos(0)]TJ /F3 11.955 Tf 11.96 0 Td[(!)@p @0=)]TJ /F4 11.955 Tf 9.3 0 Td[(r0sin(0)]TJ /F3 11.955 Tf 11.95 0 Td[(!).(7)For5p=0,wemusthave0=!.ByconstructiontheintegralsJ(2)jk(,!)andJ(3)jk(,!)donotincludethestationarypoint0=!andhence5p6=0intheseintegrals.FollowingthelinesofTheorem 7.1 ,bydeningthevectoreldu=5p k5pk2qandthenapplyingthedivergencetheorem,bothJ(2)jk(,!)andJ(3)jk(,!)canbeshowntobe~2(2)(,!)and~3(3)(,!)respectivelywhereboth2and30.5and(2)and(3)aresomecontinuousboundedfunctionofand!.Hencewecanconcludethat lim~!0Z20jk LJ(2)jk(,!)d!lim~!0~2 LZ20j(2)(,!)jd!=0(7)asjjk=1jandsimilarlyforJ(3)jk(,!)foranyxed>0.Itfollowsthattheresultalsoholdsas!0providedthelimitforisconsiderafterthelimitfor~,i.e, lim!0lim~!0Z!0+!0jk LJ(2)jk(,!)d!=0lim!0lim~!0Z!0+!0jk LJ(3)jk(,!)d!=0. (7) HenceI(1)jk(!)inEquation 7 canbeapproximatedbyJ(1)jk(,!)as!0andas~!0.UsingthisresultinEquation 7 leavesustoprovethat KXj=1KXk=1lim!0lim~!0Z!0+!0jk(!) LJ(1)jk(,!)d!=Z!0+!0P(!)d!.(7)WenowevaluateJ(1)jk(,!)byinterchangingtheorderofintegrationbetweenr0and0whichrequiresustorewrite0asafunctionofr0.Recallthatforeachdata 88

PAGE 89

pointYk,theboundariesoftheregionDkalongr(0)=R(1)k(0)andr(0)=R(2)k(0)respectivelyiscomprisedofanitesequenceofstraightlinesegments.InordertoevaluateJ(1)jk(,!)weneedtoconsidertheseboundariesonlywithintheprecinctsoftheangles[!)]TJ /F3 11.955 Tf 12.54 0 Td[(,!+]oneachDk.Butforsufcientlysmall,weobservethatforevery!2[0,2),whenweconsidertheseboundaries(alongR(1)k(0)andR(2)k(0)respectively)withintheangles[!)]TJ /F3 11.955 Tf 12.21 0 Td[(,!+],theywillbecomprisedofatmosttwolinesegments(seeFigure 7-3 ). Figure7-3. Boundaryconsideredwithintheangles[!)]TJ /F3 11.955 Tf 11.7 0 Td[(,!+]iscomprisedofatmosttwolinesegmentsL1andL2. Overeachlinesegment,r0(0)iseitherstrictlymonotonic(strictlyincreasesorstrictlydecreases)orhasexactlyonecriticalpoint(strictlydecreases,attainsaminimumandthenstrictlyincreases)asshowninFigure 7-4 .Henceitfollowsthatforsufcientlysmall,0rewrittenasafunctionofr0maybecomposedofatmostthreedisconnectedregions(referFigure 7-5 ).LetB(r0)[!)]TJ /F3 11.955 Tf 12.01 0 Td[(,!+]denotetheintegralregionfor0(r0).Treating0asafunctionofr0andapplyingFubini'stheorem,theintegralJ(1)jk(,!)canberewrittenas J(1)jk(,!)=Zr(2)k(,!)r(1)k(,!)G(r0,!)dr0,(7) 89

PAGE 90

Figure7-4. Plotofradiallength(r)vsangle(). Figure7-5. Threedisconnectedregionsfortheangle(). where r(1)k(,!)=inffR(1)k(0)g,r(2)k(,!)=supfR(2)k(0)g (7) when02[!)]TJ /F3 11.955 Tf 11.95 0 Td[(,!+]and G(r0,!)=1 p 2~ZB(r0)expip(r0,0,!) ~q(r0,0,!)d0.(7) 90

PAGE 91

NotethatwhileevaluatingtheintegralG(r0,!),r0and!areheldxed.AscontributionstoGcomesonlyfromthestationarypointsofp(r0,0,!)(withr0and!heldxed)as~!0,weevaluate@p @0=)]TJ /F4 11.955 Tf 9.3 0 Td[(r0sin(0)]TJ /F3 11.955 Tf 11.95 0 Td[(!)andforittovanish0=!.Moreover @2p @02j!=)]TJ /F4 11.955 Tf 9.3 0 Td[(r0,p(r0,!,!)=0andq(r0,!,!)=r0p r0)]TJ /F3 11.955 Tf 11.96 0 Td[(jk(!) (7) Forthegivenr0,if!=2B(r0),thennostationarypointsexists.UsingintegrationbypartsG(r0,!)canbeshowntobe3(r0,!,~)=O(p ~)whichcanbeuniformlyboundedbyafunctionofr0forsmallvaluesof~.If!2B(r0),thenusingonedimensionalstationaryphaseapproximations[ 31 32 ]itcanbeshownthat G(r0,!)=exp)]TJ /F4 11.955 Tf 9.3 0 Td[(i 4p r0p r0)]TJ /F3 11.955 Tf 11.96 0 Td[(jk(!)+4(r0,!,~)(7)where4(r0,!,~)canbeuniformlyboundedbyafunctionofr0forsmallvaluesof~andconvergestozeroas~!0.Herewehaveassumedthatthestationarypoint0=!liestotheinteriorofB(r0)andnotontheboundaryastherecanbeatmostnite(actually2)valuesofr0(withLesbeguemeasurezero)forwhich0=!canlieintheboundaryofB(r0).PluggingthevalueofG(r0,!)inEquation 7 weget Z!0+!0jk(!) LJ(1)jk(,!)d!=1 LZ20exp)]TJ /F4 11.955 Tf 9.29 0 Td[(ijk(!) ~jk(,!)d!+Z!0+!0jk(!) L(Zr(2)k(,!)r(1)k(,!)(r0,!,~)dr0)d!, (7) where jk(,!)=Zr(+)k(,!)r()]TJ /F13 5.978 Tf 5.75 0 Td[()k(,!)p r0p r0)]TJ /F3 11.955 Tf 11.95 0 Td[(jk(!)dr0,(7) 91

PAGE 92

r()]TJ /F8 7.97 Tf 6.58 0 Td[()k(,!)r(1)k(,!)andr(+)k(,!)r(2)k(,!)arethevaluesofr0suchthatwhenr()]TJ /F8 7.97 Tf 6.58 0 Td[()k(,!)<>:4(r0,!,~);r()]TJ /F8 7.97 Tf 6.58 0 Td[()k(,!)
PAGE 93

For00jk(~!)=0,wemusthave tan(~!)=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(xj)]TJ /F4 11.955 Tf 11.96 0 Td[(xk yj)]TJ /F4 11.955 Tf 11.96 0 Td[(yk=yj)]TJ /F4 11.955 Tf 11.96 0 Td[(yk xj)]TJ /F4 11.955 Tf 11.96 0 Td[(xk,(7)wherethelastequalityisobtainedusingEquation 7 .Rewritingweget yj)]TJ /F4 11.955 Tf 11.95 0 Td[(yk xj)]TJ /F4 11.955 Tf 11.95 0 Td[(xk2=)]TJ /F6 11.955 Tf 9.29 0 Td[(1(7)whichcannotbetrue.Sincethesecondderivativecannotvanishatthestationarypoint~!,fromone-dimensionalstationaryphaseapproximation[ 31 ]wehave lim~!01 LZ!0+!0exp)]TJ /F4 11.955 Tf 9.3 0 Td[(ijk(!) ~jk(,!)d!=lim~!0O(~)=0(7)where=0.5or1dependinguponwhethertheinterval[!0,!0+)containsthestationarypoint(~!)ornot.Hencewehavejk()=0forj6=k.case(ii):Ifj=k,thenkk(!)=0and kk(,!)=Zr(+)k(,!)r()]TJ /F13 5.978 Tf 5.75 0 Td[()k(,!)r0dr0,kk()=1 LZ!0+!0kk(,!)d!. (7) Fromthedenitionsofr(1)k(,!)andr(2)k(,!)inEquation 7 ,observethat lim!0r(1)k(,!)"R(1)k(!),lim!0r(2)k(,!)#R(2)k(!). (7) Sincer()]TJ /F8 7.97 Tf 6.59 0 Td[()k(,!)!r(1)k(,!)andr(+)k(,!)!r(2)k(,!)as!0,wehave lim!0r()]TJ /F8 7.97 Tf 6.59 0 Td[()k(,!)=R(1)k(!)andlim!0r(+)k(,!)=R(2)k(!). (7) 93

PAGE 94

Sincer()]TJ /F8 7.97 Tf 6.59 0 Td[()k(,!)r(1)k(,!)andr(+)k(,!)r(2)k(,!)ataxedandr0>0,weseethatkk(,!)canbeboundedabovebypositivedecreasingfunctionof,namely kk(,!)Zr(2)k(,!)r(1)k(,!)r0dr0(7)andisalsoindependentof~.Asbothr(1)k(,!)andr(2)k(,!)arealsoboundedfunctions,bytheLesbeguedominatedconvergencetheorem, lim!0lim~!0kk()=1 LZ!0+!0lim!0kk(,!)d!=1 LZ!0+!0(ZR(2)k(!)R(1)k(!)r0dr0)d!=(1)]TJ /F6 11.955 Tf 11.96 0 Td[(2) LZ!0+!0R2k(!) 2d!. (7) RecallthatL=(1)]TJ /F6 11.955 Tf 11.95 0 Td[(2)L.Hence, KXj=1KXk=1lim!0lim~!0jk()=1 LKXk=1Z!0+!0R2k(!) 2d!=1 LKXk=1Z!0+!0R2k(!) 2d!=Z!0+!0P(!)d! (7) whichcompletestheproof. Asanimplicationoftheabovetheorem,wehavethefollowingcorollary. Corollary2. Foranygiven0<<1,!02[0,2) lim!0lim!01 lim~!0Z!0+!0Z1+1)]TJ /F11 7.97 Tf 6.58 0 Td[(P~(~r,!)~rd~rd!=P(!0).(7) Proof. FromEquation 7 wehave lim!01 Z!0+!0P(!)d!=lim!0=F(!0!!0+) =P(!0).(7) 94

PAGE 95

SinceTheorem 7.2 istrueforany0<<1 2,italsoholdsgoodas!0.Theresultthenfollowsimmediately. Theorem 7.2 alsoentailsthefollowinglemma. Lemma6. Foranygiven0<<1 2,0<<1, lim~!0Z20Z1+1)]TJ /F11 7.97 Tf 6.59 0 Td[(P~(~r,!)~rd~rd!=1.(7) Proof. SincetheresultshowninTheorem 7.2 holdsgoodforany!0and,wemaychoose!0=0and=2.UsingEquation 7 theresultfollowsimmediatelyas lim~!0Z20Z1+1)]TJ /F11 7.97 Tf 6.59 0 Td[(P~(~r,!)~rd~rd!=Z20P(!)d!=1.(7) Lemmas 6 and 5 leadstothefollowingcorollaries. Corollary3. Foranygiven0<<1 2,0<<1, lim~!0Z20Z1)]TJ /F11 7.97 Tf 6.58 0 Td[(0P~(~r,!)~rd~r+Z11+P~(~r,!)~rd~rd!=0.(7) Proof. FromLemma 5 wehaveforany~>0and0<<1 2, Z20Z10P~(~r,!)~rd~rd!=1.(7)Forthegiven0<<1,dividingtheintegralrange(0,1)for~rintothreedisjointregionsnamely(0,1)]TJ /F3 11.955 Tf 11.95 0 Td[(),[1)]TJ /F3 11.955 Tf 11.95 0 Td[(,1+]and(1+,1)andletting~!0wehave, lim~!0Z20Z1)]TJ /F11 7.97 Tf 6.58 0 Td[(0P~(~r,!)~rd~r+Z1+1)]TJ /F11 7.97 Tf 6.58 0 Td[(P~(~r,!)~rd~r+Z11+P~(~r,!)~rd~rd!=1.(7)PursuanttoLemma 6 ,thelimit lim~!0Z20Z1+1)]TJ /F11 7.97 Tf 6.59 0 Td[(P~(~r,!)~rd~rd!(7)existsandequals1.Theresultthenfollows. 95

PAGE 96

Corollary4. Foranygiven0<<1 2,0<<1,!02[0,2)and0<<2, lim~!0Z!0+!0Z1)]TJ /F11 7.97 Tf 6.59 0 Td[(0P~(~r,!)~rd~r+Z11+P~(~r,!)~rd~rd!=0.(7) Proof. LetM=b2 c.Dene!i+1!i+mod2for0iM)]TJ /F6 11.955 Tf 12.64 0 Td[(1.ThenfromCorollary 3 wehave lim~!0"M)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xi=0Z!i+1!iQ(!)d!+Z!0+2!i+1Q(!)d!#=0(7)where, Q(!)=Z1)]TJ /F11 7.97 Tf 6.58 0 Td[(0P~(~r,!)~rd~r+Z11+P~(~r,!)~rd~r.(7)SinceP~(~r,!)~r0,itfollowsthatQ(!)andeachoftheintegralinEquation 7 isnon-negativeandhenceconvergestozeroindependentlyoftheotherintegrals,givingusthedesiredresult. PursuanttoTheorem 7.2 andCorollaries 2 and 4 ,thesubsequentresultsfollowsalmostimmediately. Proposition7.1. Foranygiven0<<1 2,!02[0,2)and0<<2, lim~!0Z!0+!0Z10P~(~r,!)~rd~rd!=Z!0+!0P(!)d!.(7) Corollary5. Foranygiven!02[0,2), lim!0lim!01 lim~!0Z!0+!0Z10P~(~r,!)~rd~rd!=P(!0).(7) 7.4SignicanceoftheResultTheintegrals Z!0+!0Z1+1)]TJ /F11 7.97 Tf 6.58 0 Td[(P~(~r,!)~rd~rd!,Z!0+!0P(!)d!(7) 96

PAGE 97

givestheintervalmeasureofthedensityfunctionsP~(whenpolledclosetotheunitcircle~r=1)andPrespectively.Theorem 7.2 statesthatatsmallvaluesof~,boththeintervalmeasuresareapproximatelyequal,withthedifferencebetweenthembeingo(1).Furthermoretheresultisalsotrueas!0.RecallthatbydenitionP~isthenormalizedpowerspectrumofthewavefunction(x,y)=expiS(x,y) ~.Henceweconcludethatthepowerspectrumof(x,y)whenpolledclosetotheunitcircle~r=1(as!0inTheorem 7.2 )orwhenintegratedover~r(referProposition 7.1 ),canpotentiallyserveasadensityestimatorfortheorientationdensityofSatsmallvaluesof~and.TheempiricalresultsshownunderSection 8.5 alsoprovidevisualevidencestocorroboratetoourclaim. 97

PAGE 98

CHAPTER8EXPERIMENTALRESULTSInSection 5.3.1 wegaveanaccountonthenumericalissuesinvolvedincomputingthewavefunctionandtheneedforarbitraryprecisionarithmeticpackageslikeGMPandMPFR[ 20 45 ]inoating-pointcomputations.ForthefollowingexperimentsonEuclideandistancefunctionsandeikonalequations,weweusedp=512precisionbits. 8.1EuclideanDistanceFunctionsInthissection,weshowtheefcacyofourSchrodingermethodbycomputingtheapproximateEuclideandistancefunctionSandcomparingittotheactualEuclideandistancefunctionandthefastsweepingmethod,rstonrandomlygenerated2Dpoint-setsandthenonasetofbounded2Dand3Dgridpoints. 8.1.12DExperimentsExample1:Webeginbydemonstratingtheeffectof~onourSchrodingermethodandshowthatas~!0,theaccuracyourmethoddoesimprovesignicantly.Tothisend,weconsidereda2Dgridconsistingofpointsbetween()]TJ /F6 11.955 Tf 9.3 0 Td[(0.121,)]TJ /F6 11.955 Tf 9.29 0 Td[(0.121)and(0.121,0.121)withagridwidthof1 29.ThetotalnumberofgridpointsisthenN=125125=15,625.Weran1000experimentseachtimerandomlychoosing5000gridlocationsasdatapoints(point-set),for9differentvaluesof~rangingfrom510)]TJ /F8 7.97 Tf 6.59 0 Td[(5to4.510)]TJ /F8 7.97 Tf 6.59 0 Td[(4instepsof510)]TJ /F8 7.97 Tf 6.59 0 Td[(5.Foreachrunandeachvalueof~,wecalculatedthepercentageerroras error=100 NNXi=1i Di,(8)whereDiandiarerespectivelytheactualdistanceandtheabsolutedifferenceofthecomputeddistancetotheactualdistanceattheithgridpointTheplotinFigure 8-1 showsthemeanpercentageerrorateachvalueof~.Themaximumvalueoftheerrorateachvalueof~issummarizedinTable 8-1 .Theerrorislessthan0.6%at~=0.00005demonstratingthealgorithm'sabilitytocomputeaccurateEuclideandistances. 98

PAGE 99

Figure8-1. Percentageerrorversus~in10002Dexperiments. Table8-1. Maximumpercentageerrorfordifferentvaluesof~in10002Dexperiments. ~Maximumpercentageerror 0.000050.5728%0.000101.1482%0.000151.7461%0.000202.4046%0.000253.1550%0.000304.0146%0.000354.9959%0.000406.1033%0.000457.3380% Example2:WepittedtheSchrodingeralgorithmagainstthefastsweepingmethod[ 50 ]ona2Dgridconsistingofpointsbetween()]TJ /F6 11.955 Tf 9.29 0 Td[(0.123,)]TJ /F6 11.955 Tf 9.3 0 Td[(0.123)and(0.123,0.123)withthegridwithof1 210.ThenumberofgridpointsequalsN=253253=64,009.Weran100experiments,eachtimerandomlychoosing10,000gridpointsasdatapoints.Weset~=0.0001fortheSchrodingerandranthefastsweepingfor10iterationssufcientforittoconverge.TheplotinFigure 8-2 showstheaveragepercentageerrorcalculatedaccordingtoEquation 8 forboththesetechniquesincomparisontothetrueEuclideandistancefunction.Fromtheplot,itisclearthatwhilethefastsweepingmethodhasapercentageerrorofaround7%,Schrodingermethodgaveapercentageerroroflessthan1.5%providingmuchbetteraccuracy. 99

PAGE 100

Figure8-2. PercentageerrorbetweenthetrueandcomputedEuclideandistancefunctionforSchrodinger(inblue)andfastsweeping(inred)in1002Dexperiments. Example3:Inthisexample,wecomputedtheEuclideandistancetransformusingthegridpointsofcertainsilhouettes(Figure 8-3 )[ 42 ]1,ona2Dgridconsistingofpointsbetween()]TJ /F6 11.955 Tf 9.29 0 Td[(0.125,)]TJ /F6 11.955 Tf 9.29 0 Td[(0.125)to(0.125,0.125)withagridwidthof1 210.ThenumberofgridpointsequalsN=257257=66049.Weset~fortheSchrodingermethod0.0003.Forthesakeofcomparison,weranthefastsweepingfor10iterationswhichwassufcientforconvergence.ThepercentageerrorfortheSchrodingerandthefastsweeping(calculatedasperEquation 8 whencomparedwiththetrueEuclideandistancefunctionforeachoftheseshapesisadumbratedinTable 8-2 Figure8-3. Shapes 1WethankKaleemSiddiqiforprovidingusthesetof2Dshapesilhouettes. 100

PAGE 101

Table8-2. PercentageerroroftheEuclideandistancefunctioncomputedusingthegridpointsoftheshapesasdatapoints ShapeSchrodingerFastsweeping Hand2.182%2.572%Horse2.597%2.549%Bird2.116%2.347% ThetrueEuclideandistancefunctioncontourplotandthoseobtainedfromourmethodandfastsweepingisdelineatedinFigure 8-4 A B CFigure8-4. Shapecontourplots.A)TrueEuclideandistancefunction.B)Schrodinger.C)Fastsweeping 8.1.2MedialaxiscomputationsInordertocomputethemedialaxisfortheseshapes,werstneedtodifferentiatebetweenthegridlocationsthatareeitherinsideoroutsideofeachshape.Wediditbycomputingthewindingnumberforallthegridpointssimultaneouslyusingourconvolutionbasedwindingnumberalgorithm.Gridpointswithawindingnumbervalue 101

PAGE 102

ofgreaterthanzeroafterroundoffwheremarkedasinteriorpointsandrestweremarkedasexteriorpoints.Figure 8-5 visualizesthevectorelds(Sx,Sy)foralltheinteriorpoints(markedinblue)andtheexteriorpoints(markedinred).Clearlyweseethatourconvolutionbasedtechniqueforcomputingthewindingnumbercleanly(withalmostzeroerror)separatedtheinsidegridpointsfromthosethatareexteriortothe2Dshape. Figure8-5. Aquiverplotof5S=(Sx,Sy)(bestviewedincolor). Wechosethemaximumcurvature(denedasH+p H2)]TJ /F4 11.955 Tf 11.96 0 Td[(KwhereHandKarethemeanandGaussiancurvaturesrespectivelyoftheMongepatchgivenbyfx,y,S(x,y)g)asthevehicletovisualizethemedialaxisofeachshape.Themean 102

PAGE 103

andGaussiancurvaturescanbeexpressedintermsofthecoefcientsoftherstandsecondfundamentalforms(E,F,Gande,f,g[ 23 ])respectivelywhichareinturnexpressibleinclosed-formusingtherstandsecondderivativesofS.Asthesederivativescanbewrittenasdiscreteconvolutions(elucidatedinSection 4.3 ),themax-curvaturefortheMongepatchcanbecomputedinO(NlogN)usingFFT.Fromthemax-curvaturewecaneasilyretrievethemedialaxisasexplainedbelow.ObservefromthequiverplotsinFigure 8-5 ,thatthegradientdirectionsarepreserveduntiltheymeetwiththegradientsemanatingfromothercurvelocations.AzoomedversionofthequiverplotisshowninFigure 8-6 .Inplaceswherethegradientsmeet,theirdirectionschangesignicantlyandhencethesurfaceS(x,y)exhibitshighmax-curvaturevaluesatthoselocations.Buttheselocationsexactlycorrespondtothegridpointshavingmorethanoneclosestpointontheshape'sboundaryalsoknownastheVoronoiboundarypointsorthemedialaxispoints.Henceasimplethresholdingofthemax-curvaturegivesthemedialaxis,asdeterminedbythepointswherethemax-curvatureisgreaterthan(say)1.ThemedialaxisplotsfortheseshapesareshowninFigure 8-7 andcanalsobeeasilytracedfromthequiverplots(Figure 8-5 )whenviewedincolor. Figure8-6. Zoomedquiverplot 103

PAGE 104

Wewouldliketomentionthat,computingthemedialaxisusingthemax-curvatureishandicappedbyaminordrawback.UsingFFTtocomputethedistancetransformanditsderivatives,forcesthedatatositonaregulargrid.Noticefromthemedialaxisplots(Figure 8-7 )thatthetheboundaryoftheseshapesarenotsmoothbutrugged.Thisresultedinhighmax-curvaturevaluesatvariousspuriouslocations,especiallyatthegridlocationswhichareveryclosetotheboundarypointsYkandhencewillalsobelabelledaspointsonthemedialaxis.Tocircumventthis,weincorporatedasecondlevelofthresholdingwherebyweconsideronlythosegridlocationsXwherethedistancetransformS(X)isgreaterthan(say)2.Dependingontheshape,1wassetbetween0.09and0.12and2between3and6where=1=210isthegridwidth.Aneasierxtotheaforementionedproblemistorunourmethodonamuchnergrid,increasingthenumberofgridlocationsandhavingasmoothboundary.Butthishasanadverseeffectofslowingdowntherunningtime.Abettersolutionwouldbetoadaptthegriddependinguponthedatawithvaryinggridwidthfordifferentlocations.Extendingourtechniqueforirregulargridsisbeyondthescopeofourcurrentwork.Wewouldliketoaddressthisinourfuturework. 8.1.33DExperimentsExample4:WetooktheStanfordbunnydataset2andusedthecoordinatesofthedatapointsonthemodelasourpoint-setlocations.Sincetheinputdatalocationsneednotconformtogridlocations,wescaledthespaceuniformlyinalldimensionsandroundedoffthedatasothatthedataliesatgridlocations.Theinputdatawasalsoshiftedsothatitwasapproximatelysymmetricallylocatedwithrespecttothex,yandzaxis.Weshouldpointoutthatshiftingthedatadoesn'taffecttheEuclideandistancefunctionvalueanduniformscalingofalldimensionsisalsonotanissue,asthedistancescanbe 2Thisdatasetisavailableat http://www.cc.gatech.edu/projects/large models/bunny.html 104

PAGE 105

Figure8-7. Medialaxisplots rescaledaftertheyarecomputed.Moreover,theformulaforcalculatingthepercentageerror(Equation 8 )isinvarianttoshiftingandscaling.Afterthesebasicdatamanipulations,thecardinalityofthepointsetwasK=8948withthedataconnedtothecubicregion)]TJ /F6 11.955 Tf 9.3 0 Td[(0.117x0.117,)]TJ /F6 11.955 Tf 9.3 0 Td[(0.117y0.117and)]TJ /F6 11.955 Tf 9.3 0 Td[(0.093z0.093,withagridwidthof1 28.ThenumberofpointsonthegridequalsN=182,329.WerantheSchrodingerwith~=0.0004andtheFastsweepingfor15iterationsandlatercalculatedthepercentageerrorforboththesemethodsbycomparingwiththetruedistancefunctionaccordingtoEquation 8 .Ourmethodhadapercentageerrorofonly1.23%andfavorablycomparestofastsweepingwhichhad 105

PAGE 106

apercentageerrorof4.75%.OurFFT-basedapproachdoesnotbeginbydiscretizingthespatialdifferentialoperatorasisthecasewiththefastmarchingandfastsweepingmethodsandthiscouldhelpaccountfortheincreasedaccuracy.Theisosurfaceobtainedbyconnectingthegridpointsatadistanceof0.005fromthepointset,determinedbythetrueEuclideandistancefunction,SchrodingerandthefastsweepingisshowninFigure 8-8 .Noticethesimilaritybetweentheplots.Itprovidesanecdotalvisualevidencefortheusefulnessofourapproach. A B CFigure8-8. Bunnyisosurfaces.A)ActualEuclideandistancefunction.B)Schrodinger.C)Fastsweeping. Example5:WealsocomparedtheSchrodingerEuclideandistancefunctionalgorithmwiththefastsweepingmethod[ 50 ]andtheexactEuclideandistanceontheDragonpoint-setobtainedfromtheStanford3DScanningRepository3.Aftertheinitialshifting,scalingandroundingofthedatapointssothattheyconformtogridlocations,thecommongridforthisdatasetwas)]TJ /F6 11.955 Tf 9.3 0 Td[(0.117x0.117,)]TJ /F6 11.955 Tf 9.3 0 Td[(0.086y0.086and)]TJ /F6 11.955 Tf 9.3 0 Td[(0.047z0.047withagridwidthof1 28.WerantheSchrodingerapproachat~=0.0004andranthefastsweepingmethodfor15iterationssufcientfortheGauss-Seideliterationstoconverge.Wethencalculatedthepercentageerrorasper 3Thisdatasetisavailableat http://graphics.stanford.edu/data/3Dscanrep/. 106

PAGE 107

Equation 8 .WhiletheaveragepercentageerrorintheSchrodingerapproachwhencomparedtothetruedistancefunctionwasjust1.306%,theaveragepercentageerrorinthefastsweepingmethodwas6.84%.Theisosurfaceobtainedbyconnectingthegridpointsatadistanceof0.005fromthepointset,determinedbythetrueEuclideandistancefunction,SchrodingerandfastsweepingareshowninFigure 8-9 .Thesimilaritybetweentheplotsprovidesanecdotalvisualevidencefortheusefulnessofourapproach. A B CFigure8-9. Dragonisosurfaces.A)ActualEuclideandistancefunction.B)Schrodinger.C)Fastsweeping. 8.2TheGeneralEikonalEquationInthissectionwedemonstratetheusefulnessofourapproachbycomputingtheapproximatesolutiontothegeneraleikonalequationoveraregular2Dgrid. 8.2.1ComparisonwiththeTrueSolutionExample6:Inthisexample,wesolvetheeikonalequationforthescenariowheretheexactsolutionisknownaprioriatthegridlocations.Theexactsolutionis S(x,y)=jep x2+y2)]TJ /F6 11.955 Tf 11.96 0 Td[(1j.(8)Theboundaryconditionis,S(x,y)=0atthepointsourcelocatedat(x0,y0)=(0,0).TheforcingfunctiontheabsolutegradientjrSjis f(x,y)=j5Sj=ep x2+y2(8) 107

PAGE 108

speciedona2Dgridconsistingofpointsbetween()]TJ /F6 11.955 Tf 9.3 0 Td[(0.125,)]TJ /F6 11.955 Tf 9.29 0 Td[(0.125)and(0.125,0.125)withagridwidthof1 210.WerantheSchrodingerfor6iterationsat~=0.006andthefastsweepingfor15iterationssufcientenoughforboththemethodstoconverge.Thepercentageerror(calculatedaccordingtoEquation 8 )andthemaximumdifferencebetweenthetrueandapproximatesolutionfordifferentiterationsissummarizedintheTable 8-3 Table8-3. PercentageerrorandthemaximumdifferencefortheSchrodingermethodoverdifferentiterations IterPercentageerrorMaxdifference 12.0810420.00265121.5147450.00214031.3905520.00214241.3632560.00212851.3578940.00212861.3568980.002128 Thefastsweepinggaveapercentageerrorof1.135%.WebelievethattheerrorincurredinourSchrodingerapproachcanbefurtherreducedbydecreasing~butattheexpenseofmorecomputationalpowerrequiringhigherprecisionoatingpointarithmetic.ThecontourplotsofthetruesolutionandthoseobtainedfromSchrodingerandfastsweepingaredisplayedbelow(Figure 8-10 ).Wecanimmediatelyobservethesimilarityofoursolutionwiththetruesolution.WedoobservesmootherisocontoursinourSchrodingermethodrelativetofastsweeping. A B CFigure8-10. Contourplots.A)Truesolution.B)Schrodinger.C)Fastsweeping 108

PAGE 109

8.2.2ComparisonwithFastSweepingInordertoverifytheaccuracyofourtechnique,wecomparedoursolutionwithfastsweepingforthefollowingsetofexamples,usingthelatterasthegroundtruthasthetruesolutionisnotavailableinclosed-form.Example7:Inthisexamplewesolvedtheeikonalequationfromapointsourcelocatedat(x0,y0)=(0,0)forthefollowingforcingfunction f(x,y)=1+2(e)]TJ /F8 7.97 Tf 6.58 0 Td[(2((x+0.05)2+(y+0.05)2))]TJ /F4 11.955 Tf 11.95 0 Td[(e)]TJ /F8 7.97 Tf 6.59 0 Td[(2((x)]TJ /F8 7.97 Tf 6.59 0 Td[(0.05)2+(y)]TJ /F8 7.97 Tf 6.59 0 Td[(0.05)2))(8)ona2Dgridconsistingofpointsbetween()]TJ /F6 11.955 Tf 9.29 0 Td[(0.125,)]TJ /F6 11.955 Tf 9.3 0 Td[(0.125)and(0.125,0.125)withagridwidthof1 210.Weranourmethodfor6iterationswith~setat0.015andfastsweepingfor15iterationssufcientforbothtechniquestoconverge.WhenwecalculatedthepercentageerrorfortheSchrodingeraccordingtoEquation 8 (withfastsweepingasthegroundtruth),theerrorwasjustaround1.245%.ThepercentageerrorandmaximumdifferencebetweenthefastsweepingandSchrodingersolutionsaftereachiterationareadumbratedinTable 8-4 Table8-4. PercentageerrorandthemaximumdifferencefortheSchrodingermethodincomparisontofastsweeping IterPercentageerrorMaxdifference 11.1446320.00869421.2690280.00827431.2238360.00579941.2463920.00656051.2448850.00636561.2459990.006413 Webelievethattheuctuationsbothinthepercentageerrorandthemaximumdifferenceareduetorepeatedapproximationsoftheintegrationinvolvedintheconvolutionwithdiscreteconvolutionandsummation,butneverthelessstabilizedafter6iterations.ThecontourplotsshowninFigure 8-11 clearlydemonstratethesimilaritiesbetweenthesemethods. 109

PAGE 110

A BFigure8-11. Contourplots.A)Schrodinger.B)Fastsweeping. Example8:Herewesolvedtheeikonalequationforthesinusoidalforcingfunction f(x,y)=1+sin((x)]TJ /F6 11.955 Tf 11.95 0 Td[(0.05))sin((y+0.05))(8)onthesame2Dgridasinthepreviousexample.Werandomlychose4gridlocationsnamely, f0,0g,f0.0488,0.0977g,f)]TJ /F6 11.955 Tf 15.28 0 Td[(0.0244,)]TJ /F6 11.955 Tf 9.3 0 Td[(0.0732g,f0.0293,)]TJ /F6 11.955 Tf 9.3 0 Td[(0.0391gasdatalocationsandranourmethodfor6iterationswith~setat0.0085andranfastsweepingfor15iterations.ThepercentageerrorbetweentheSchrodingersolution(after6iterations)andfastsweepingwas4.537%withthemaximumabsolutedifferencebetweenthembeing0.0109.ThecontourplotsareshowninFigure 8-12 .NoticethattheSchrodingercontoursaremoresmootherincomparisontothefastsweepingcontours. A BFigure8-12. Contourplots.A)Schrodinger.B)Fastsweeping. 110

PAGE 111

Example9:Herewecomparedwithfastsweepingonalarger2Dgridconsistingofpointsbetween()]TJ /F6 11.955 Tf 9.29 0 Td[(5,)]TJ /F6 11.955 Tf 9.3 0 Td[(5)and(5,5)withagridwidthof0.25.Weagainconsideredthesinusoidalforcingfunction f(x,y)=1+0.3sin((x+1))sin((y)]TJ /F6 11.955 Tf 11.96 0 Td[(2))(8)andchose4gridlocationsnamelyf0,0g,f1,1g,f)]TJ /F6 11.955 Tf 15.28 0 Td[(2,)]TJ /F6 11.955 Tf 9.3 0 Td[(3g,f3,)]TJ /F6 11.955 Tf 9.3 0 Td[(4gasdatalocations.NoticethattheGreen'sfunctionGand~Ggoestozeroexponentiallyfasterforgridlocationsawayfromzeroforsmallvaluesof~.Henceforagridlocationsay()]TJ /F6 11.955 Tf 9.3 0 Td[(4,4)whichisreasonablyfarawayfrom0,thevalueoftheGreen'sfunctionsayat~=0.001maybezeroevenwhenweusealargenumberofprecisionbitsp.Thisproblemcanbeeasilycircumventedbyrstscalingdowntheentiregridbyafactor,computingthesolutionSonthesmallerdensergridandthenrescalingitbackagainbytoobtaintheactualsolution.Itisworthemphasizingthatscalingdownthegridistantamounttoscalingdowntheforcingfunctionasclearlyseenfromthefastsweepingmethod.Infastsweeping[ 50 ],thesolutionSiscomputedusingthequantityfi,jwherefi,jisthevalueofforcingfunctionatthe(i,j)thgridlocationandisthegridwidth.Hencescalingdownbyafactorofisequivalenttoxingandscalingdownfby.Sincetheeikonalequation(Equation 1 )islinearinf,computingthesolutionforascaleddownfequivalenttoascaleddowngridandthenrescalingitbackagainisguaranteedtogivetheactualsolution.canbesettoanydesiredquantity.Forthecurrentexperimentweset=100,~=0.001andranourmethodfor6iterations.Fastsweepingwasrunfor15iterations.Thepercentageerrorbetweenthesemethodswasabout3.165%.ThecontourplotsareshowninFigure 8-13 .Again,thecontoursobtainedfromtheSchrodingeraremoresmootherthanthoseobtainedfromfastsweeping. 111

PAGE 112

A BFigure8-13. Contourplots.A)Schrodinger.B)Fastsweeping. 8.3TopologicalDegreeExperimentsWedemonstratedtheefcacyofourconvolutionbasedtechniqueforcomputingthewindingnumberin2D,whenwecomputedthemedialaxisforthesilhouettes.Wenowshowitsaccuracyforcomputingthetopologicaldegreein3D.Tothisend,weconsidereda3Dgrid,connedtotheregion)]TJ /F6 11.955 Tf 9.3 0 Td[(0.125x0.125,)]TJ /F6 11.955 Tf 9.3 0 Td[(0.125y0.125and)]TJ /F6 11.955 Tf 9.3 0 Td[(0.125z0.125withagridwidthof1 28.ThenumberofgridpointswasN=274,625.Givenasetofpointssampledfromthesurfaceofa3Dobject,wetriangulatedthesurfaceusingsomeofthebuilt-inMATLABroutines.WeconsideredtheincenterofeachtriangletorepresentthedatapointsfYkgKk=1.ThenormalPkforeachtrianglecanbecomputedfromthecrossproductofthetrianglevectoredges.ThedirectionofthenormalvectorwasdeterminedbytakingthedotproductbetweenthepositionvectorYkandthenormalvectorPk.Fornegativedotproducts,Pkwasnegatedtoobtainaoutwardpointingnormalvector.WethencomputedthetopologicaldegreeforalltheNgridlocationssimultaneouslybyrunningourconvolutionbasedalgorithm.Gridlocationswherethetopologicaldegreevalueexceeded0.7weremarkedaspointslyinginsidethegiven3Dobject.Figure 8-14 showstheinteriorpointsforthethree3Dobjectscylinder,cubeandsphere(lefttoright). 112

PAGE 113

A BFigure8-14. TopologicalDegree.A)Sampledpointsfromthesurface.B)Gridpointslyinginsidethesurface(markedinblue). 8.4EmpiricalResultsfortheGradientDensityestimationinOneDimensionBelowweshowcomparisonsbetweenourFouriertransformapproachwiththestandardhistogrammingtechniqueforestimatingthegradientdensitiesonsometrigonometricandexponentialfunctionssampledonaregulargridbetween[)]TJ /F6 11.955 Tf 9.3 0 Td[(0.125,0.125]atagridspacingof1 215.Forthesakeofconvenience,wenormalizedthefunctionssuchthatitsmaximumgradientvalueis1.Usingthesampledvalues^S,wecomputedtheFastFouriertransformofexpi^S ~at~=0.00001,tookitsmagnitude 113

PAGE 114

squareandthennormalizedittocomputethegradientdensity.WealsocomputedthediscretederivativeofSatthegridlocationsandthendetermineditsgradientdensityusingthestandardhistogrammingtechniquewith220histogrambins.TheplotsshowninFigure 8-15 provideanecdotalempiricalevidence,supportingthemathematicalresultstatedinTheorem 6.1 underChapter 6 .Noticethenear-perfectmatchbetweenthegradientdensitiescomputedviastandardhistogramming,withthegradientdensitiesdeterminedusingourFouriertransformmethod. A BFigure8-15. Comparisonresults.A)Gradientdensitiesobtainedfromhistogramming.B)GradientdensitiesobtainedfromsquaredFouriertransformofthewavefunction 8.5EmpiricalResultsfortheDensityFunctionsoftheDistanceTransforms 8.5.1CWRanditsFourierTransformOntheleftsideoftheFigure 8-16 andFigure 8-16 ,wevisualizetheCWRofthedistancetransformS,computedearlierforsomeoftheseshapesilhouettes.Sincethewavefunction=exp)]TJ /F7 7.97 Tf 6.67 -4.97 Td[(iS ~hasboththerealandtheimaginarypart,weshowonlyitsimaginarycomponent,namelysin)]TJ /F7 7.97 Tf 6.67 -4.98 Td[(iS ~forvisualclarity.Usingtheseplotswecan 114

PAGE 115

envisageawaveemanatingfromtheboundariesoftheseshapes(representedbythickblacklines).TheseCWRplotswerecomputedat~=0.5.OntherightsideoftheFigure 8-16 andFigure 8-16 ,weplottheFouriertransformofat~=0.00004.Weseeabrightbluesegmentdenedonlyontheunitcircleu2+v2=~r2=1,overaplain,non-interesting,atbackground.TheshadesofbluerepresentsthevariationinthemagnitudeoftheFouriertransform.Whilethebrightblueregionsrepresentshighvaluesinthemagnitude,thenon-bright,atregionscorrespondstoverylow,almostzeromagnitudevalue.Theorem 7.1 givenunderSection 7.2 statesthatExpectontheunitcirclegivenby~r=1,theFouriertransformofthewavefunctionshouldconvergetozeroas~!0.Thesepicturesexactlyportraysthetheoremstatement.Exceptontheunitcircle~r=1whereweobservehighvalues,themagnitudeoftheFouriertransformisalmostzeroeverywhereelse. 8.5.2ComparisonResultsBelowweshowequivalencebetweenourFouriertransformapproachandthetrueorientationdensityoftheunitvectordistancetransformgradients,determinedusingtheclosed-formexpressionderivedinEquation 7 .Usingthesampledvalues^S,sampledonthegiven2Dgrid)]TJ /F6 11.955 Tf 9.3 0 Td[(0.125x0.125,)]TJ /F6 11.955 Tf 9.3 0 Td[(0.125y0.125atintervalsof1 213,wecomputedtheFastFouriertransformofexpi^S ~at~=0.000004.Wethenshiftedthefrequencies,sothatthezero-frequencycomponentisinthemiddleofthespectrumandthentookitsmagnitudesquaretocomputethediscretepowerspectrum.Using140histogrambinsfortheangle!,wesummedupthepowerspectrumvaluesalongdiscreteradialdirections(analogoustointegratingover~r)andrenormalizeditinordertocomputetheorientationdensityfunction.NoticethesimilaritiesbetweentheplotsshowninFigure 8-17 115

PAGE 116

A BFigure8-16. CWRanditsFouriertransform.A)ComplexWaveRepresentation(CWR)ofthedistancefunction.B)FouriertransformofCWR. Furthermoreateachvalueof~,wecomputedtheL1errorbetweenthetrueandthecomputeddensityfunction,bycomputingtheabsolutedifferencebetweentheirvaluesateachhistogrambinandthenaddingupthedifferences.FromthetwoplotsshowninFigure 8-18 ,wecanvisualizetheconvergenceofL1errortozeroas~!0.Theseplotsserveasatestament,strengtheningourmathematicalresultstatedunderTheorem 7.1 116

PAGE 117

A B Figure 8-16 .Continued 117

PAGE 118

A BFigure8-17. Comparisonresults.A)Truegradientdensityfunction.B)GradientdensityfunctionobtainedfromthesquaredFouriertransformoftheCWR Figure8-18. PlotofL1errorvs~fortheorientationdensityfunctions. 118

PAGE 119

CHAPTER9DISCUSSIONANDFUTUREWORK 9.1ConclusionInthiswork,weprovidedanapplicationoftheSchrodingerformalismwherewedevelopedanewapproachtosolvingthenon-lineareikonalequation.WeprovedthatthesolutiontotheeikonalequationcanbeobtainedasalimitingcaseofthesolutiontoacorrespondinglinearSchrodingerwaveequation.Insteadofdirectlysolvingtheeikonalequation,theSchrodingerformalismresultsinageneralized,screenedPoissonequationwhichissolvedatverysmallvaluesof~.OurSchrodinger-basedapproachfollowsthepioneeringHamilton-Jacobisolverssuchasthefastsweeping[ 50 ]andfastmarching[ 34 ]methodswiththecrucialdifferencebeingitslinearity.Wedevelopedafastandefcientperturbationseriesmethodforsolvingthewaveequation(generalized,screenedPoissonequation)whichisguaranteedtoconvergeprovidedtheforcingfunctionfispositiveandbounded.UsingtheperturbationmethodandtheEquation 2 ,weobtainedthesolutiontotheEquation 2 withoutspatiallydiscretizingtheoperators.FortheEuclideanfunctionproblemaspecialcaseoftheeikonalequationwheretheforcingtermisidenticallyequaltooneeverywhereweobtainedclosed-formsolutionsfortheSchrodingerwaveequationthatcanbeefcientlycomputedusingtheFFTwhichinvolvesO(NlogN)oating-pointoperations.TheEuclideandistanceisthenrecoveredfromtheexponentofthewavefunction.Sincethewavefunctioniscomputedforasmallbutnon-zero~,theobtainedEuclideandistancefunctionisanapproximation.Wederivedanalyticboundsfortheerroroftheapproximationforagivenvalueof~andprovidedproofsofconvergencetothetruedistancefunctionas~!0.WethenleveragedthedifferentiabilityoftheSchrodingersolutiontocomputethegradientsandcurvatureofthedistancefunctionS,bygivingaclosed-formexpressionwhichcanbewrittenasconvolutions.Wealsoprovidedanefcientmechanismtodeterminethesign 119

PAGE 120

ofthedistancefunctionwithourdiscreteconvolutionbasedtechniqueforcomputingthewindingnumberin2Dandthetopologicaldegreein3Dandshowedhowthegradientandcurvatureinformationcanaidinmedicalaxescomputation,whenappliedto2Dshapesilhouettes.Ourresultsondensityestimation,directlyinspiredbymomentumdensityinquantummechanics,demonstratestheusefulnessoftheoreticalphysicsideasincontextsofdensityestimation.Usingstationaryphaseapproximationsweestablishedthatthescaledpowerspectrumofthewavefunctionapproachesthedensityofthegradient(s)ofthedistancefunctionSinthelimitas~!0,whenthescalareldSappearsasthephaseofthewavefunction.Byprovidingrigorousmathematicalproofs,weestablishedthisrelationbetweenthegradientsandthefrequenciesforanarbitrarilythricedifferentiablefunctioninonedimensionandspecicallyfordistancetransformsintwodimension.Wealsofurnishedanecdotalvisualevidencestocorroborateourclaim.Ourresultgivesanewsignatureforthedistancetransformsandcanpotentiallyserveasitsgradientdensityestimator. 9.2FutureWorkWhileHamilton-JacobisolvershavegonebeyondtheeikonalequationandregulargridsbyprovidingefcientsolutionsevenforthemoregeneralstaticHamilton-Jacobiequationonirregulargrids[ 26 27 36 ]ourSchrodingerapproachinthecurrentworkrestrictsitselfonlytocomputingtheeikonalequationonregulargrids.SinceourmethodreliesonusingFastFourierTransform(FFT's)forcomputation,wewererestrictedtodenethedataonlyonregulargridlocations.However,recentlydevelopednon-FFTbasedtechniqueslikethefastmultipolemethodsmightpavethewaytoextendourSchrodingerformalismevenforirregulargrids.Inourcurrentwork,weestablishedthemathematicalrelationbetweenthepowerspectrumofthewavefunctionanditsgradientdensitiesonlyfordistancetransforms.Butpreliminaryexperimentalresultsseemstosuggestthattheresultisgeneralizableto 120

PAGE 121

amoregeneralclassoffunctionswithappropriateboundaryconditions.Wewouldliketoinvestigatethisfurtherandifitpansout,trytosupportourempiricaldiscoverywithrigorousmathematicalproofs.Thisrepresentsafruitfulavenueforfutureresearch. 121

PAGE 122

REFERENCES [1] O.Aberth,PrecisenumericalmethodsusingC++,AcademicPress,SanDiego,CA,1998. [2] M.AbramowitzandI.A.Stegun,Handbookofmathematicalfunctionswithformulas,graphsandmathematicaltables,Dover,NewYork,NY,1964. [3] V.I.Arnold,Mathematicalmethodsofclassicalmechanics,Springer,NewYork,NY,1989. [4] J.-L.Basdevant,Variationalprinciplesinphysics,Springer,NewYork,NY,2007. [5] M.DeBerg,O.Cheong,M.VanKreveld,andM.Overmars,Computationalgeome-try:Algorithmsandapplications,Springer-Verlag,NewYork,NY,2008. [6] P.Billingsley,Probabilityandmeasure,3rded.,Wiley-Interscience,NewYork,NY,1995. [7] R.N.Bracewell,TheFouriertransformanditsapplications,3rded.,McGraw-Hill,NewYork,NY,1999. [8] R.P.Brent,Fastmultiple-precisionevaluationofelementaryfunctions,J.ACM23(1976),242. [9] J.Buttereld,OnHamilton-Jacobitheoryasaclassicalrootofquantumtheory,Quo-VadisQuantumMechanics(A.Elitzur,S.Dolev,andN.Kolenda,eds.),Springer,NewYork,NY,2005,pp.239. [10] J.F.Canny,Complexityofrobotmotionplanning,TheMITPress,Cambridge,MA,1988. [11] M.ChaichianandA.Demichev,Pathintegralsinphysics:Volume1:Stochasticprocessesandquantummechanics,InstituteofPhysicsPublishing,Philadelphia,PA,2001. [12] G.Chartier,Introductiontooptics,Springer,NewYork,NY,2005. [13] J.C.Cooke,Stationaryphaseintwodimensions,IMAJ.Appl.Math.29(1982),25. [14] J.W.CooleyandJ.W.Tukey,AnalgorithmforthemachinecalculationofcomplexFourierseries,Math.Comp.19(1965),no.90,297. [15] T.H.Cormen,C.E.Leiserson,R.L.Rivest,andC.Stein,Introductiontoalgorithms,2nded.,TheMITPress,Cambridge,MA,September2001. [16] M.G.Crandall,H.Ishii,andP.L.Lions,User'sguidetoviscositysolutionsofsecondorderpartialdifferentialequations,BulletinoftheAmericanMathematicalSociety27(1992),no.1,1. 122

PAGE 123

[17] F.M.Fernandez,Introductiontoperturbationtheoryinquantummechanics,CRCPress,BocaRaton,FL,2000. [18] A.L.FetterandJ.D.Walecka,Theoreticalmechanicsofparticlesandcontinua,Dover,NewYork,NY,2003. [19] R.P.FeynmanandA.R.Hibbs,Quantummechanicsandpathintegrals,McGraw-Hill,NewYork,NY,1965. [20] L.Fousse,G.Hanrot,V.Lefevre,P.Pelissier,andP.Zimmermann,MPFR:Amultiple-precisionbinaryoating-pointlibrarywithcorrectrounding,ACMTrans.Math.Softw.33(2007),1. [21] I.M.GelfandandS.V.Fomin,Calculusofvariations,Dover,NewYork,NY,2000. [22] H.Goldstein,C.P.Poole,andJ.L.Safko,Classicalmechanics,3rded.,AddisonWesley,Boston,MA,2001. [23] A.Gray,Moderndifferentialgeometryofcurvesandsurfaceswithmathematica,2nded.,CRCPress,BocaRaton,FL,1997. [24] D.J.Grifths,Introductiontoquantummechanics,2nded.,PrenticeHall,UpperSaddleRiver,NJ,2005. [25] D.S.JonesandM.Kline,Asymptoticexpansionsofmultipleintegralsandthemethodofstationaryphase,J.Math.Phys.37(1958),1. [26] C.-Y.Kao,S.J.Osher,andJ.Qian,Legendre-transform-basedfastsweepingmethodsforstaticHamilton-Jacobiequationsontriangulatedmeshes,J.Comp.Phys.227(2008),no.24,10209. [27] C.-Y.Kao,S.J.Osher,andY.-H.Tsai,FastsweepingmethodsforstaticHamilton-Jacobiequations,SIAMJ.Num.Anal.42(2004),no.6,2612. [28] R.KimmelandJ.A.Sethian,Optimalalgorithmforshapefromshadingandpathplanning,J.Math.ImagingVis.14(2001),237. [29] J.P.McClureandR.Wong,Two-dimensionalstationaryphaseapproximation:Stationarypointatacorner,SIAMJ.Math.Anal.22(1991),no.2,500. [30] R.G.Newton,Scatteringtheoryofwavesandparticles,2nded.,Springer-Verlag,NewYork,NY,1982. [31] F.W.JOlver,Asymptoticsandspecialfunctions,AcademicPress,NewYork,NY,1974. [32] ,Errorboundsforstationaryphaseapproximations,SIAMJ.Math.Anal.5(1974),19. 123

PAGE 124

[33] S.J.OsherandR.P.Fedkiw,Levelsetmethodsanddynamicimplicitsurfaces,Springer-Verlag,NewYork,NY,October2003. [34] S.J.OsherandJ.A.Sethian,Frontspropagatingwithcurvaturedependentspeed:AlgorithmsbasedonHamilton-Jacobiformulations,J.Comp.Phys.79(1988),no.1,12. [35] D.T.ParisandF.K.Hurd,Basicelectromagnetictheory,McGraw-Hill,NewYork,NY,1969. [36] J.Qian,Y.-T.Zhang,andH.K.Zhao,Fastsweepingmethodsforeikonalequationsontriangularmeshes,SIAMJ.Num.Anal.45(2007),no.1,83. [37] A.Rajwade,A.Banerjee,andA.Rangarajan,Probabilitydensityestimationusingisocontoursandisosurfaces:Applicationtoinformationtheoreticimageregistration,IEEET.PatternAnal.31(2009),no.3,475. [38] W.Rudin,Principlesofmathematicalanalysis,3rded.,McGraw-Hill,NewYork,NY,1976. [39] T.SasakiandY.Kanada,Practicallyfastmultiple-precisionevaluationoflog(x),J.IPSJapan5(1982),247. [40] A.SchonhageandV.Strassen,Schnellemultiplikationgroerzahlen,Computing7(1971),281. [41] J.A.Sethian,Afastmarchinglevelsetmethodformonotonicallyadvancingfronts,Proc.Nat.Acad.Sci.(1996),no.4,1591. [42] K.Siddiqi,A.Tannenbaum,andS.W.Zucker,AHamiltonianapproachtotheeikonalequation,EnergyMinimizationMethodsinComputerVisionandPatternRecognition(EMMCVPR)(NewYork,NY),vol.LNCS1654,Springer-Verlag,1999,pp.1. [43] D.M.Smith,Efcientmultiple-precisionevaluationofelementaryfunctions,Math.Comp.52(1989),no.185,131. [44] C.F.Stevens,Thesixcoretheoriesofmodernphysics,TheMITPress,Cambridge,MA,1996. [45] G.Torbjornandetal.,GNUmultipleprecisionarithmeticlibrary5.0.1,June2010. [46] G.B.Whitham,Linearandnonlinearwaves,Pureandappliedmathematics,Wiley-Interscience,NewYork,NY,1999. [47] R.Wong,Asymptoticapproximationsofintegrals,AcademicPress,NewYork,NY,1989. [48] R.WongandJ.P.McClure,Onamethodofasymptoticevaluationofmultipleintegrals,Math.Comp.37(1981),no.156,509. 124

PAGE 125

[49] L.Yatziv,A.Bartesaghi,andG.Sapiro,O(N)implementationofthefastmarchingalgorithm,J.Comp.Phys.212(2006),no.2,393. [50] H.K.Zhao,Afastsweepingmethodforeikonalequations,Math.Comp.(2005),603. 125

PAGE 126

BIOGRAPHICALSKETCH KarthikS.GurumoorthyhailsfromthesouthernpartofIndia,fromthecityofMadras(nowcalledChennai).AftercompletinghisbaccalaureatedegreeincomputerengineeringfromtheUniversityofMadrasin2004,heworkedforanyearasasoftwaredeveloperatHCL-cisco,Chennai.Havingapenchantformathematicsandresearch,hedecidedtopursuehighereducationandgotadmittedforaPh.D.programincomputerengineeringatUniversityofFlorida(UF)in2005.AmidcontinuingwithhisPh.D.,heconcurrentlysignedupforadualmaster'sprograminmathematicsatUFandearnedM.Sinmathematicsintheyear2009.HealsoreceivedhisM.Sincomputersciencein2010.Hisprimaryresearchinterestisindevelopingcomputationallyefcienttechniqueswithinspirationfromquantummechanics,exploitingtheknownrelationshipbetweentheclassicalHamilton-JacobiequationandthequantumSchrodingerequationtosolvewellknownclassicalproblemsliketheeikonalequation,estimationofthethegradientdensities,etc.Hehasalsoworkedintheeldofsparserepresentations,compressibilityofsetsthroughtherepeatediterationofapolynomialfunctionandin2-workerbucket-brigadelines. 126