Table Of ContentVishal Monga Editor
Handbook of Convex
Optimization
Methods in Imaging
Science
Handbook of Convex Optimization
Methods in Imaging Science
Vishal Monga
Editor
Handbook of Convex
Optimization Methods
in Imaging Science
123
Editor
VishalMonga
ThePennsylvaniaStateUniversity
UniversityPark,PA,USA
ISBN978-3-319-61608-7 ISBN978-3-319-61609-4 (eBook)
DOI10.1007/978-3-319-61609-4
LibraryofCongressControlNumber:2017952608
©SpringerInternationalPublishingAG2017
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ThisbookisdedicatedtoMr.VijayChaudhary,
whoremainsaconstantsourceofsupport
andencouragement.
Preface
The value of mathematical optimization to engineering cannot possibly
be overstated. Engineering and applied math problems invariably involve
minimizationofanerrororcostfunction,oralternativelythemaximizationof
thelikelihoodofsuccess.Theoptimalchoiceofreal-worldphysicalvariables
or design parameters that enable the best minimization/maximization is
hence crucial. Often these parameters are constrained by physical laws,
the available memory and/or computation, the requirements of fast or real-
timeoperation,etc.Thisbookstudiesoptimizationtechniques,inparticular
convexoptimizationmethodsandalgorithms,forthehighimpactengineering
domainofimagingscience.Imagingscienceconcernsitselfwithallaspects
ofimageacquisition,processing,rendering,andtransmission.Theeconomic
footprint of imaging and vision systems is quite significant and continues
togrow.Technologyfueledbyoptimizationalgorithmsinimageprocessing
andvisioncanbefoundinsmartphones,cameras,displays,medicalimaging
devicesandsoftware,remote-sensingmachineryandsatellites,robotics,and
defenseandaerospaceproductstonameafew.
This book is an outgrowth of my individual research and teaching in
the intersection of convex optimization and imaging science. Truthfully, I
havebeencajoledintowritingitbyseveraldistinguishedcolleaguesengaged
in computational aspects of image processing and vision, whose work I
admiregreatly.Theareasofimageprocessingandcomputervisionbenefitted
tremendouslyfromtheappliedmathsubsetsoflinearalgebraandprobability
in the 1980s to early 2000s. I am firm on the view that an in-depth under-
standing of optimization and the opportunities it presents in solving real-
worldimagingproblemsisimperativeforthemodernresearcherinthisarea.
ThebookishenceaimedatgraduateandundergraduatestudentsinElectrical
Engineering,ComputerScience,AppliedPhysics,andMathematics.Because
thebookisaneditedvolumewithstate-of-the-artcoverageofmanyimaging
scienceproblemsfromacomputationalviewpoint,itshouldalsobeofvalue
toindustrialresearchersandpractitioners.
UniversityPark,PA,USA VishalMonga
vii
Acknowledgments
Iamgratefultoallmycolleagueswhohavekindlycontributedhigh-quality
andthoroughchapterstothisbook.Intermsofwritingandorganization,this
wasunchartedterritory,whichmakestheireffortsevenmorecommendable.
Other than the contributors, many notable senior researchers in the signal
andimageprocessingareahavebeenimmenselysupportiveandencouraging
and have made this book possible; this includes (but is not limited to)
Charlie Bouman at Purdue, Trac Tran at Johns Hopkins, P.P. Vaidyanathan
atCalTech,andMuralidharRangaswamyattheUSAirForceResearchLab.
The book would not have come about if not for the excellent research and
educational efforts of my doctoral students and alumni of the Information
Processing and Algorithms Laboratory (iPAL) (http://signal.ee.psu.edu) at
Penn State. Umamahesh Srinivas and Hojjat Mousavi have helped cowrite
a chapter with me on sparsity-constrained optimization and estimation and
havealsohelpedmeorganize,plan,andultimatelyexecutethevisionforthis
book—myspecialthankstothem.
I would greatly appreciate hearing of any errors that may be found
in the book for reasons beyond my control and that of the publisher. I
take responsibility for any limitations that the book may have and invite
[email protected].
UniversityPark,PA,USA VishalMonga
ix
Contents
1 Introduction ............................................. 1
VishalMonga
1.1 ConvexOptimizationinImagingScience ................. 1
1.2 ConvexOptimizationReview ........................... 3
1.2.1 ConvexSets ................................... 3
1.2.2 ConvexFunctions .............................. 4
1.2.3 ConvexOptimization............................ 5
1.2.4 ConvexOptimizationProblems:Examples.......... 5
1.2.5 OptimizationMethods........................... 6
1.2.5.1 LagrangeMultipliersandDuality ......... 6
1.2.5.2 DescentMethods ....................... 7
1.2.5.3 IterativeThresholdingMethods ........... 8
1.2.5.4 AlternativeDirectionMethod
ofMultipliers .......................... 8
1.2.6 HandlingNon-convexProblems................... 9
1.3 BookOutline......................................... 12
References................................................ 13
2 OptimizingImageQuality ................................. 15
Dominique Brunet, Sumohana S. Channappayya,Zhou Wang,
EdwardR.Vrscay,andAlanC.Bovik
2.1 Introduction.......................................... 15
2.1.1 ImageQualityAssessmentMeasures .............. 15
2.1.2 PerceptualOptimizationFramework ............... 17
2.1.3 ChapterOverview .............................. 18
2.2 MathematicalPropertiesoftheSSIMIndex ............... 18
2.2.1 StructuralandNon-StructuralDistortions........... 18
2.2.2 ConvexityandQuasi-Convexity................... 19
2.2.3 CombinationRule .............................. 20
2.2.4 SpatialAggregation............................. 21
2.3 PerceptuallyOptimalAlgorithmDesign .................. 21
2.3.1 SSIM-OptimalEqualizerDesign .................. 21
2.3.2 EqualizationProblem ........................... 21
2.3.3 Solution....................................... 22
2.3.3.1 EqualizationProblemRedefined .......... 22
2.3.3.2 StatSSIM-OptimalLinearEqualization..... 23
xi
xii Contents
2.3.3.3 ProblemReformulation ................. 24
2.3.3.4 Quasi-ConvexOptimization ............. 25
2.3.3.5 Searchfor˛........................... 26
2.3.3.6 Application to Image Denoising and
Restoration ........................... 26
2.3.4 SSIM-OptimalSoft-Thresholding................. 26
2.3.4.1 SSIMIndexintheWaveletDomain ....... 27
2.3.4.2 ProblemFormulation ................... 30
2.3.4.3 Solution .............................. 32
2.4 LocalSSIM-OptimalApproximation .................... 32
2.4.1 L2-BasedApproximation........................ 33
2.4.1.1 OrthogonalBasis ...................... 33
2.4.1.2 LinearRedundantBasis................. 34
2.4.1.3 Non-LinearApproximation.............. 34
2.4.2 SSIM-BasedApproximation ..................... 36
2.4.2.1 LinearApproximation .................. 36
2.4.3 Non-LinearApproximation...................... 38
2.4.4 VariationalSSIM .............................. 39
2.5 Image-WideVariationalSSIMOptimization .............. 39
2.6 Conclusions ......................................... 40
References ............................................... 40
3 ComputationalColorImaging ............................. 43
RajaBala,GrahamFinlayson,andChulLee
3.1 Introduction ......................................... 43
3.2 ColorCapture........................................ 45
3.2.1 Introduction................................... 45
3.2.2 ColorCorrection............................... 45
3.2.2.1 LinearModels......................... 46
3.2.2.2 LinearColorCorrection................. 47
3.2.2.3 PolynomialColorCorrection............. 48
3.2.2.4 Root-PolynomialColorCorrection........ 48
3.2.2.5 ExperimentalResults ................... 49
3.2.2.6 OtherMethods ........................ 49
3.2.3 IlluminantEstimation........................... 50
3.2.3.1 TheRGBModelofImageFormation...... 50
3.2.3.2 Moment-BasedIlluminantEstimation ..... 51
3.2.3.3 Experiments........................... 51
3.2.3.4 Extending Moment-Based Illuminant
Estimation ............................ 52
3.2.3.5 OtherMethods ........................ 53
3.3 ColorDisplay........................................ 53
3.3.1 Introduction................................... 53
3.3.2 DisplayCharacterization ........................ 53
3.3.3 RGB-to-RGBWConversion ..................... 55
3.3.3.1 ColorDistortionModel ................. 55
3.3.3.2 PowerConsumptionModelofEmissive
Displays.............................. 56
Contents xiii
3.3.3.3 ConstrainedOptimizationProblem ........ 57
3.3.3.4 Simulation............................. 58
3.4 ColorPrinting ........................................ 59
3.4.1 Introduction ................................... 59
3.4.2 Lattice-BasedPrinterCharacterization ............. 62
3.4.2.1 ProblemSetup ......................... 64
3.4.2.2 OptimizationofNodeValues ............. 64
3.4.2.3 OptimizationofNodeLocations .......... 65
3.4.2.4 JointOptimizationofNodeValuesand
Locations ............................. 66
3.4.2.5 ExperimentalResults.................... 66
3.5 Conclusions.......................................... 68
References................................................ 68
4 OptimizationMethodsforSyntheticApertureRadar
Imaging ................................................. 71
EricMason,IlkerBayram,andBirsenYazici
4.1 Introduction.......................................... 71
4.2 SARForwardModels ................................. 72
4.2.1 GeneralizedRadonTransforms ................... 73
4.2.2 ClassicalRadonTransform....................... 75
4.3 AnalyticalOptimizationMethodsforSARImaging ........ 76
4.3.1 LinearDeterministicReconstructionMethods ....... 76
4.3.1.1 Minimum Norm Solution: Filtered
Back-ProjectionFormula ................ 76
4.3.1.2 RegularizedMinimumNormSolution ..... 77
4.3.1.3 MinimumErrorSolution:Backprojection
FilteringFormula....................... 77
4.3.2 LinearStatisticalReconstructionMethods .......... 77
4.3.2.1 BestLinearUnbiasedEstimation.......... 78
4.3.2.2 MinimumMeanSquareErrorEstimation... 79
4.3.3 NonlinearReconstructionMethods ................ 82
4.3.3.1 Iterative Re-weighted Least-Squares
Algorithm ............................. 86
4.3.3.2 IterativeShrinkageThresholding
Approach ............................. 87
4.4 NumericalOptimizationMethodsforSARImaging ........ 89
4.4.1 CompressedSensinginSARImaging.............. 91
4.4.1.1 Total Variation Regularization in SAR
Imaging............................... 93
4.4.2 Low-RankMatrixRecoveryMethods ............. 94
4.4.3 BayesianCompressedSensing.................... 95
4.5 OptimizationinSARRelatedProblems................... 96
4.6 Conclusion........................................... 98
References................................................ 98
