Fourier Optics and Computational Imaging

314 Pages·2015·14.517 MB·English

by  Khare, Kedar

Fourier Optics
and
Computational Imaging

Fourier Optics
and
Computational Imaging

Kedar Khare
Department of Physics
Indian Institute of Technology Delhi
New Delhi, India

John Wiley & Sons Ltd.

Ane Books Pvt. Ltd.
Fourier Optics and Computational Imaging
© 2016 Kedar Khare

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ISBN: 978-11-1890-034-5

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Contents
Preface 11
1 Introduction 13
1.1 Organizationofthebook . . . . . . . . . . . . . . . 16
Part 1: Mathematical preliminaries
2 Fourierseriesandtransform 21
2.1 FourierSeries . . . . . . . . . . . . . . . . . . . . . 21
2.2 Gibbsphenomenon . . . . . . . . . . . . . . . . . . 23
2.3 FouriertransformasalimitingcaseofFourierseries 27
2.3.1 Fouriertransformoftherectangledistribution 28
2.4 Samplingbyaveraging,distributions
anddeltafunction . . . . . . . . . . . . . . . . . . . 30
2.5 Propertiesofdeltafunction . . . . . . . . . . . . . . 32
2.6 Fouriertransformofunitstepandsignfunctions . . . 33
2.7 Fouriertransformofatrainofdeltafunctions . . . . 36
2.8 FouriertransformofaGaussian . . . . . . . . . . . 36
2.9 Fouriertransformofchirpphase . . . . . . . . . . . 37
2.10 PropertiesofFouriertransform . . . . . . . . . . . 40
2.11 Fouriertransformofthe2Dcirc
function . . . . . . . . . . . . . . . . . . . . . . . . 42
2.12 Fourierslicetheorem . . . . . . . . . . . . . . . . . 43
2.13 Wignerdistribution . . . . . . . . . . . . . . . . . . 45
6FourierOpticsandComputationalImaging
3 SamplingTheorem 49
3.1 Poissonsummationformula . . . . . . . . . . . . . 50
3.2 Samplingtheoremasaspecialcase . . . . . . . . . . 51
3.3 Additionalnotesonthesampling
formula . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Samplingofcarrier-frequencysignals . . . . . . . . 53
3.5 Degrees of freedom in a signal: space bandwidth
product . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Slepian(prolatespheroidal)functions . . . . . . . . 56
3.6.1 PropertiesofmatrixA(0) . . . . . . . . . . . 59
3.7 Extrapolationofbandlimitedfunctions . . . . . . . . 63
4 OperationalintroductiontoFastFourierTransform 67
4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Usageof2DFastFourierTransformforproblemsin
Optics . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Linearsystemsformalismandintroductiontoinverseprob-
lemsinimaging 75
5.1 Space-invariantimpulseresponse . . . . . . . . . . . 77
5.2 Ill-posednessofinverseproblems . . . . . . . . . . . 78
5.3 Inversefilter . . . . . . . . . . . . . . . . . . . . . . 80
5.4 Wienerfilter . . . . . . . . . . . . . . . . . . . . . . 82
6 Constrainedoptimizationmethodsforimagerecovery 87
6.1 Imagedenoising . . . . . . . . . . . . . . . . . . . . 87
6.2 Imagede-convolutionbyoptimization . . . . . . . . 91
6.3 Blindimagedeconvolution . . . . . . . . . . . . . . 95
6.4 CompressiveImaging . . . . . . . . . . . . . . . . 97
6.4.1 Guidelines for sub-sampled data measure-
mentandimagerecovery . . . . . . . . . . . 99
6.4.2 Systemlevelimplicationsofcompressive
imagingphilosophy . . . . . . . . . . . . . . 103
6.5 Topicsforfurtherstudy . . . . . . . . . . . . . . . . 104
7 Randomprocesses 107
7.1 Probabilityandrandomvariables . . . . . . . . . . . 107
7.1.1 JointProbabilities . . . . . . . . . . . . . . 108
Contents 7
7.1.2 Baye’srule . . . . . . . . . . . . . . . . . . 108
7.1.3 RandomVariables . . . . . . . . . . . . . . 109
7.1.4 ExpectationsandMoments . . . . . . . . . . 110
7.1.5 Characteristicfunction . . . . . . . . . . . . 112
7.1.6 Additionoftworandomvariables . . . . . . 113
7.1.7 Transformationofrandomvariables . . . . . 113
7.1.8 GaussianorNormaldistribution . . . . . . . 114
7.1.9 CentralLimitTheorem . . . . . . . . . . . . 115
7.1.10 Gaussianmomenttheorem . . . . . . . . . . 116
7.2 RandomProcesses . . . . . . . . . . . . . . . . . . 117
7.2.1 ErgodicProcess . . . . . . . . . . . . . . . 118
7.2.2 Propertiesofauto-correlationfunction . . . . 119
7.2.3 SpectralDensity: Wiener-Khintchinetheorem119
7.2.4 Orthogonalseriesrepresentationofrandom
processes . . . . . . . . . . . . . . . . . . . 120
7.2.5 ComplexRepresentationofrandomprocesses121
7.2.6 Mandel’stheoremoncomplexrepresentation 123
Part 2: Concepts in optics
8 GeometricalOpticsEssentials 127
8.1 Raytransfermatrix . . . . . . . . . . . . . . . . . . 127
8.2 Stopsandpupils . . . . . . . . . . . . . . . . . . . . 130
9 Waveequationandintroductiontodiffractionoflight 133
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 133
9.2 ReviewofMaxwellequations . . . . . . . . . . . . 135
9.3 IntegraltheoremofHelmholtz
andKirchhoff . . . . . . . . . . . . . . . . . . . . . 136
9.4 Diffractionfromaplanarscreen . . . . . . . . . . . 140
9.4.1 KirchhoffSolution . . . . . . . . . . . . . . 141
9.4.2 Rayleigh-SommerfeldSolution . . . . . . . 141
10 Theangularspectrummethod 145
10.1 Angularspectrummethod . . . . . . . . . . . . . . . 145
8FourierOpticsandComputationalImaging
11 FresnelandFraunhofferdiffraction 153
11.1 Fresneldiffraction . . . . . . . . . . . . . . . . . . 153
11.1.1 ComputationofFresneldiffractionpatterns . 155
11.1.2 TransportofIntensityEquation . . . . . . . 156
11.1.3 Selfimaging: Montgomeryconditionsand
Talbotteffect . . . . . . . . . . . . . . . . . 160
11.1.4 FractionalFouriertransform . . . . . . . . . 162
11.2 FraunhofferDiffraction . . . . . . . . . . . . . . . . 163
12 Coherenceoflightfields 167
12.1 Spatialandtemporalcoherence . . . . . . . . . . . 167
12.1.1 Interferencelaw . . . . . . . . . . . . . . . 169
12.2 vanCittertandZerniketheorem . . . . . . . . . . . 169
12.3 Space-frequencyrepresentationofthecoherencefunc-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . 171
12.4 Intensityinterferometry: Hanbury
BrownandTwisseffect . . . . . . . . . . . . . . . . 173
12.5 Photoncountingformula . . . . . . . . . . . . . . . 175
12.6 Specklephenomenon . . . . . . . . . . . . . . . . . 177
13 Polarizationoflight 183
13.1 TheJonesmatrixformalism . . . . . . . . . . . . . 183
13.2 TheQHQgeometricphaseshifter . . . . . . . . . . 185
13.3 Degreeofpolarization . . . . . . . . . . . . . . . . 186
14 Analysisofopticalsystems 189
14.1 Transmissionfunctionforathinlens . . . . . . . . . 189
14.2 Fouriertransformingpropertyofthinlens . . . . . . 191
14.3 Canonicalopticalprocessor . . . . . . . . . . . . . 193
14.4 Fourierplanefilterexamples . . . . . . . . . . . . . 194
14.4.1 DCblockorcoronagraph . . . . . . . . . . 194
14.4.2 Zernike’sphasecontrastmicroscopy . . . . 195
14.4.3 Edgeenhancement: vortexfilter . . . . . . . 197
14.4.4 Apodizationfilters . . . . . . . . . . . . . . 198
14.5 Frequencyresponseofopticalimagingsystems: co-
herentandincoherentillumination . . . . . . . . . . 199