Geometry of the Phase Retrieval Problem: Graveyard of Algorithms

320 Pages·2022·8.224 MB·English

by  Alexander H. Barnett, Charles L. Epstein, Leslie Greengard, Jeremy Magland

GeometryofthePhaseRetrievalProblem
RecoveringthephaseoftheFouriertransformisaubiquitousprobleminimaging
applicationsfromastronomytonanoscaleX-raydiffractionimaging.Despitethe
effortsofamultitudeofscientists,fromastronomerstomathematicians,thereisasyet
nosatisfactorytheoreticaloralgorithmicsolutiontothisclassofproblems.Written
formathematicians,physicists,andengineersworkinginimageanalysisand
reconstruction,thisbookintroducesaconceptual,geometricframeworkforthe
analysisoftheseproblems,leadingtoadeeperunderstandingoftheessential,
algorithmicallyindependent,difficultyoftheirsolutions.Usingthisframework,the
bookstudiesstandardalgorithmsandarangeoftheoreticalissuesinphaseretrieval
andprovidesseveralnewalgorithmsandapproachestotheseproblemswiththe
potentialtoimprovethereconstructedimages.Thebookislavishlyillustratedwiththe
resultsofnumerousnumericalexperimentsthatmotivatethetheoreticaldevelopment
andplaceitinthecontextofpracticalapplications.
alexander h. barnettisGroupLeaderforNumericalAnalysisatthe
CenterforComputationalMathematicsintheFlatironInstitute.Hehaspublished
around60papersonpartialdifferentialequations,waves,fastalgorithms,integral
equations,neuroscience,imaging,signalprocessing,inverseproblems,andphysics,
andreceivedseveralresearchgrantsfromtheNationalScienceFoundation.
charles l. epsteinistheThomasA.ScottProfessorEmeritusof
MathematicsattheUniversityofPennsylvania,wherehefoundedthegraduategroup
inAppliedMathematicsandComputationalScience.HeiscurrentlyaSenior
ResearchScientistattheCentreforComputationalMathematicsattheFlatiron
Institute.Hehasworkedonawiderangeofproblemsinpureandappliedanalysisand
istheauthorofawidelyusedtextbookAnIntroductiontotheMathematicsofMedical
Imaging(SIAM2008).HesharedtheBergmanPrizein2016withFrancoisTrevesand
isafellowoftheAmericanAssociationfortheAdvancementofScienceandthe
AmericanMathematicalSociety.
leslie greengardisSilverProfessorofMathematicsandComputerScience
attheCourantInstitute,NewYorkUniversityandDirectoroftheCenterfor
ComputationalMathematicsattheFlatironInstitute.Heiscoinventorofseveralwidely
usedfastalgorithmsandamemberoftheAmericanAcademyofArtsandSciences,
theNationalAcademyofSciences,andtheNationalAcademyofEngineering.
jeremy maglandisaSeniorDataScientistattheFlatironInstitute.He
receivedhisPhDinMathematicsfromtheUniversityofPennsylvania.Priortojoining
theFlatironInstitutein2015,heworkedforaboutadecadeasaresearchscientistin
theRadiologyDepartmentoftheHospitaloftheUniversityofPennsylvania,wherehe
developedsoftwaresystemsthatdramaticallystreamlinedexperimentalworkonMRI
scanners.
The Cambridge Monographs on Applied and Computational Mathematics series
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Geometry of the Phase Retrieval Problem
ALEXANDER H. BARNETT
FlatironInstitute
CHARLES L. EPSTEIN
FlatironInstitute
LESLIE GREENGARD
CourantInstitute
JEREMY MAGLAND
FlatironInstitute
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781316518878
DOI:10.1017/9781009003919
©AlexanderH.Barnett,CharlesL.Epstein,LeslieGreengard,andJeremyMagland2022
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2022
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Barnett,Alex,1972December7-author.|Epstein,CharlesL.,
author.|Greengard,Leslie,author.|Magland,Jeremy,author.
Title:Geometryofthephaseretrievalproblem:graveyardofalgorithms/
AlexanderH.Barnett,FlatironInstitute,CharlesL.Epstein,Flatiron
Institute,LeslieGreengard,CourantInstitute,andJeremyMagland,FlatironInstitute.
Description:NewYork,NY:CambridgeUniversityPress,2022.|
Series:Cambridgemonographsonappliedandcomputational|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2021044734(print)|LCCN2021044735(ebook)|
ISBN9781316518878(hardback)|ISBN9781009003919(epub)
Subjects:LCSH:Geometry.|Algorithms.
Classification:LCCQC20.7.G44B362022(print)|LCCQC20.7.G44(ebook)|
DDC530.15/6–dc23/eng/20211117
LCrecordavailableathttps://lccn.loc.gov/2021044734
LCebookrecordavailableathttps://lccn.loc.gov/2021044735
ISBN978-1-316-51887-8Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
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Contents
Preface pageix
Acknowledgments xii
1 Introduction 1
1.1 Discrete,PhaseRetrievalProblems 8
1.2 ConditioningandIll-PosednessoftheDiscrete,Classical,Phase
RetrievalProblem 13
1.3 AlgorithmsforFindingIntersectionsofSets 16
1.4 NumericalExperiments 21
1.5 ComparisontotheContinuumPhaseRetrievalProblem 28
1.6 OutlineoftheBook 31
1.A Appendix:FactoringPolynomialsinSeveralVariables 31
1.B Appendix:TheConditionNumberofaProblem 33
PARTI THEORETICALFOUNDATIONS 37
2 TheGeometryNearanIntersection 39
2.1 TheTangentSpacetotheMagnitudeTorus 42
2.2 TheIntersectionoftheTangentBundleandtheSupportConstraint 48
2.3 NumericalExamples 54
2.A Appendix:TheTangentandNormalBundlesforSubmanifolds
ofRN 60
2.B Appendix:FastProjectionsontotheTangentandNormalBundles 63
3 Well-Posedness 65
3.1 ConditioningandTransversality 68
3.2 ExamplesofIll-Posedness 76
v
vi Contents
4 UniquenessandtheNonnegativityConstraint 85
4.1 SupportandtheAutocorrelationImage 87
4.2 UniquenessforNonnegativeImages 90
4.3 NonnegativeImagesandthe1-Norm 93
4.4 The1-NormontheTangentSpace 94
4.5 TransversalityofAa∩∂B+andAa∩∂Br11 97
5 SomePreliminaryConclusions 103
PARTII ANALYSISOFALGORITHMSFORPHASE
RETRIEVAL 107
6 IntroductiontoPartII 109
7 AlgorithmsforPhaseRetrieval 115
7.1 ClassicalAlternatingProjection 116
7.2 HybridIterativeMaps 118
7.3 NonlinearSubmanifolds 129
7.4 ANoniterativeApproachtoPhaseRetrieval 134
7.A Appendix:AlternatingProjectionandGradientFlows 140
8 TheDiscrete,Classical,PhaseRetrievalProblem 147
8.1 HybridIterativeMapsinModelProblems 149
8.2 LinearizationofHybridIterativeMapsAlongtheCenterManifold 165
8.3 FurtherNumericalExamples 170
9 PhaseRetrievalwiththeNonnegativityConstraint 191
9.1 HybridIterativeMapsUsingNonnegativity 192
9.2 NumericalExamples 194
9.3 AlgorithmsBasedonMinimizationinthe1-Norm 199
9.A Appendix:AnEfficientMethodforProjectionontoaBallin
the1-Norm 203
10 AsymptoticsofHybridIterativeMaps 205
10.1 Stagnation 206
10.2 NumericalExamples 211
PARTIII FURTHERPROPERTIESOFHYBRIDITERATIVE
ALGORITHMSANDSUGGESTIONSFORIMPROVEMENT 221
11 IntroductiontoPartIII 223
12 StatisticsofAlgorithms 226
12.1 StatisticsofPhases 228
Contents vii
12.2 StatisticsofEnsembles 230
12.3 AveragingtoImproveReconstructions 239
12.4 SomeConclusions 247
13 SuggestionsforImprovements 249
13.1 UseofaSharpCutoffs 250
13.2 ExternalHolography 255
13.3 AGeometricNewton’sMethodforPhaseRetrieval 264
13.4 ImplementationoftheHolographicHilbertTransformMethod 269
13.A Appendix:ProofofTheorem13.6 289
14 ConcludingRemarks 292
15 NotationalConventions 297
References 300
Index 305

Preface
Coherentdiffractionimaging(CDI)isanexperimentaltechniquefordetermin-
ingthedetailedstructureofanobjectatthenanometerlengthscale.Coherent
X-ray sources are used to illuminate the sample, and the scattered photons
are captured in the far field on a detector array. In this regime, classical
electromagnetic theory, in the Born approximation, shows that the measured
intensity at each pixel in the detector is well approximated as the square of
the modulus of the 3D Fourier transform, |ρ(cid:2)(k)|2, of the X-ray scattering
density,ρ(x).Throughoutthisbookweusethe“mathematician’s”convention
forthek vector,inwhichexponentialstaketheforme2πi(cid:3)x,k(cid:4),ratherthanthe
“physicist’s” convention, in which they are ei(cid:3)x,k(cid:4). With this convention the
forwardFouriertransformisdefinedby
(cid:3)
ρˆ(k)=F(ρ)(k)=d ρ(x)e−2πi(cid:3)x,k(cid:4)dx, (1)
Rd
withinverse
(cid:3)
ρ(x)=F−1(ρˆ)(x)=d ρˆ(k)e2πi(cid:3)x,k(cid:4)dk. (2)
Rd
We assume that the illuminating light is a plane wave, Ae2πi(cid:3)x,p0(cid:4), with
wave vector p =(0,0,p) in the above convention. If the detector array is
0
orientedorthogonaltotheilluminatinglightatadistanceDdownstreamfrom
thesource,thenthemeasuredspatialfrequency,k, isrelatedtothelocationof
themeasurementpointy=(y ,y ,D)bytheEwaldSphereconstruction:
1 2
(cid:4) (cid:5)
y
k=p −(0,0,1) . (3)
(cid:5)y(cid:5)
Here we assume that the experimental parameters are in the so-called Fraun-
hofer regime: D is much larger than R, the diameter of the support of ρ,
andfurthermoreλD(cid:6)R2,whereλ=1/p isthewavelength.Themaximum
ix