Table Of ContentKe Chen
Carola-Bibiane Schönlieb
Xue-Cheng Tai
Laurent Younes
Editors
Handbook of
Mathematical
Models and
Algorithms in
Computer Vision
and Imaging
Mathematical Imaging and Vision
Handbook of Mathematical Models
and Algorithms in Computer Vision
and Imaging
Ke Chen • Carola-Bibiane Schönlieb (cid:129)
Xue-Cheng Tai (cid:129) Laurent Younes
Editors
Handbook of
Mathematical Models
and Algorithms
in Computer Vision
and Imaging
Mathematical Imaging and Vision
With553Figuresand72Tables
Editors
KeChen Carola-BibianeSchönlieb
DepartmentofMathematicalSciences DepartmentofAppliedMathematicsand
TheUniversityofLiverpool TheoreticalPhysics
Liverpool,UK UniversityofCambridge
Cambridge,UK
Xue-ChengTai
HongKongCenterfor LaurentYounes
CerebrocardiovascularHealth DepartmentofAppliedMathematicsandStatistics
Engineering(COCHE) JohnsHopkinsUniversity
Shatin,HongKong,China Baltimore,MD,USA
ISBN978-3-030-98660-5 ISBN978-3-030-98661-2(eBook)
https://doi.org/10.1007/978-3-030-98661-2
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Preface
Therapiddevelopmentofnewimaginghardware,theadvanceinmedicalimaging,
the advent of multi-sensor data fusion and multimodal imaging, as well as the
advances in computer vision have sparked numerous research endeavours leading
to highly sophisticated and rigorous mathematical models and theories. Motivated
by the increasing use of variational models, shapes and flows, differential geome-
try, optimisation theory, numerical analysis, statistical/Bayesian graphical models,
machine learning, and deep learning, we have invited contributions from leading
researchersandpublishthishandbook toreviewandcapturethestateoftheartof
researchinComputerVisionandImaging.
This constantly improving technology that generates new demands not readily
met by existing mathematical concepts and algorithms provides a compelling
justification for such a book to meet the ever-growing challenges in applications
and to drive future development. As a consequence, new mathematical models
have to be found, analysed and realised in practice. Knowing the precise state-of-
the-art developments is key, and hence this book will serve the large community
of mathematics, imaging, computer vision, computer sciences, statistics, and, in
general,imagingandvisionresearch.Ourprimaryaudienceare
(cid:129) Graduatestudents
(cid:129) Researchers
(cid:129) Imagingandvisionpractitioners
(cid:129) Appliedmathematicians
(cid:129) Medicalimagers
(cid:129) Engineers
(cid:129) Computerscientists
Viewingdiscreteimagesasdatasampledfromfunctionalsurfacesenablestheuseof
advancedtoolsfromcalculus,functionsandcalculusofvariations,andoptimisation
and provides the basis of high-resolution imaging through variational models. No
otherframeworkcanprovidethecomparableaccuracyandprecisiontoimagingand
vision.
v
vi Preface
Although our initial emphasis is on the variational methods, which represent
the optimal solutions to class of imaging and vision problems, and on effective
algorithms,whicharenecessaryforthemethodstobetranslatedtopracticalusein
various applications, the editors recognise that the range of effective and efficient
methods for solving problems from computer vision and imaging go beyond
variationalmethodsandhaveenlargedourcoveragetoincludemathematicalmodels
and algorithms. So, the book title reflects this viewpoint and a big vision for the
referencebook.
All chapters will have introductions so that the book is readily accessible to
graduate students. We have divided the 53 chapters of this book into 3 sections,
namely
(a) ConvexandNon-convexLarge-ScaleOptimisationinImaging
(b) Model-andData-DrivenVariationalImagingApproaches
(c) ShapeSpacesandGeometricFlows
tofacilitatebrowsingthecontentlist.However,suchadivisionisartificialbecause,
these days, research becomes increasingly intra-disciplinary as well as inter-
disciplinary,andideasfromonetopicoftendirectlyorindirectlyinspireortranspire
anothertopic.Thisisveryexciting.
For newcomers to the field, the book provides a comprehensive and fast track
introductiontothecoreresearchproblems,tosavetimeandgetonwithtacklingnew
andemergingchallenges,ratherthanrunningtheriskofreproducing/comparingto
someoldworksalreadydoneorreinventingsameresults.Forresearchers,exposure
to the state of the art of research works leads to an overall view of the entire field
soastoguidenewresearchdirectionsandavoidpitfallsinmovingthefieldforward
andlookingintothenext25yearsofimagingandinformationsciences.
The dreadful Covid-19 pandemic starting from 2020 has affected lives of
everyone, of course including all researchers. We are still not out of the woods.
The editors are very much grateful to the book authors who have endured much
hardshipduringthelast3yearsandovercomemanydifficultiestohavecompleted
their chapters on time. We are also indebted to many anonymous reviewers who
provided valuable reviews and helpful criticism to improve presentations of our
chapters.
The original gathering of all editors was in 2017 when the first three editors
co-organised the prestigious Isaac Newton Institute programme titled “Variational
methods and effective algorithms for imaging and vision” (https://www.newton.
ac.uk/event/vmv/), partially supported by UK EPSRC GR/EP F005431 and Isaac
Newton Institute for Mathematical Sciences. During the programme, Mr Jan
HollandfromSpringer-Naturekindlysuggestedtheideaofabook.Wearegrateful
to his suggestion which sparked the editors’ fruitful collaboration in the last few
Preface vii
years. The large team of publishers who have offered immense help to us include
Michael Hermann (Springer), Allan Cohen (Palgrave) and Salmanul Faris Nedum
Palli(Springer).Wethankthemall.
Finally,wewishallreadersahappyreading.
Theeditorialteam:
Liverpool,UK KeChen(Lead)
Cambridge,UK Carola-BibianeSchönlieb
Shatin,HongKong Xue-ChengTai
Baltimore,USA LaurentYounes
February2023
Contents
Volume1
PartI Convex and Non-convex Large-Scale Optimization
inImaging............................................. 1
1 ConvexNon-convexVariationalModels........................ 3
Alessandro Lanza, Serena Morigi, Ivan W. Selesnick,
andFiorellaSgallari
2 Subsampled First-Order Optimization Methods with
ApplicationsinImaging ..................................... 61
Stefania Bellavia, Tommaso Bianconcini, Nataša Krejic´,
andBenedettaMorini
3 Bregman Methods for Large-Scale Optimization with
ApplicationsinImaging ..................................... 97
MartinBenningandErlendSkaldehaugRiis
4 FastIterativeAlgorithmsforBlindPhaseRetrieval:ASurvey .... 139
HuibinChang,LiYang,andStefanoMarchesini
5 Modular ADMM-Based Strategies for Optimized
Compression,Restoration,andDistributedRepresentations
ofVisualData.............................................. 175
YehudaDarandAlfredM.Bruckstein
6 ConnectingHamilton-JacobiPartialDifferentialEquations
withMaximumaPosterioriandPosteriorMeanEstimators
forSomeNon-convexPriors.................................. 209
JérômeDarbon,GabrielP.Langlois,andTingweiMeng
7 Multi-modality Imaging with Structure-Promoting
Regularizers ............................................... 235
MatthiasJ.Ehrhardt
ix
x Contents
8 DiffractionTomography,FourierReconstruction,andFull
WaveformInversion ........................................ 273
Florian Faucher, Clemens Kirisits, Michael Quellmalz,
OtmarScherzer,andEricSetterqvist
9 ModelsforMultiplicativeNoiseRemoval ...................... 313
XiangchuFengandXiaolongZhu
10 RecentApproachestoMetalArtifactReductioninX-RayCT
Imaging ................................................... 347
SoominJeonandChang-OckLee
11 DomainDecompositionforNon-smooth(inParticularTV)
Minimization .............................................. 379
AndreasLanger
12 FastNumericalMethodsforImageSegmentationModels ........ 427
NoorBadshah
13 OnVariableSplittingandAugmentedLagrangianMethod
forTotalVariation-RelatedImageRestorationModels........... 503
Zhifang Liu, Yuping Duan, Chunlin Wu, and
Xue-ChengTai
14 Sparse Regularized CT Reconstruction: An Optimization
Perspective ................................................ 551
ElenaMorottiandElenaLoliPiccolomini
15 RecentApproachesforImageColorization..................... 585
FabienPierreandJean-FrançoisAujol
16 NumericalSolutionforSparsePDEConstrainedOptimization ... 623
XiaoliangSongandBoYu
17 GameTheoryandItsApplicationsinImagingandVision........ 677
Anis Theljani, Abderrahmane Habbal, Moez Kallel,
andKeChen
18 First-OrderPrimal–DualMethodsforNonsmoothNon-convex
Optimization............................................... 707
TuomoValkonen
Volume2
PartII Model- and Data-Driven Variational
ImagingApproaches................................... 749
19 LearnedIterativeReconstruction ............................. 751
JonasAdler
Contents xi
20 An Analysis of Generative Methods for Multiple Image
Inpainting ................................................. 773
Coloma Ballester, Aurélie Bugeau, Samuel Hurault,
SimoneParisotto,andPatriciaVitoria
21 Analysis of Different Losses for Deep Learning Image
Colorization ............................................... 821
Coloma Ballester, Hernan Carrillo, Michaël Clément,
andPatricia Vitoria
22 Influence of Color Spaces for Deep Learning Image
Colorization ............................................... 847
AurélieBugeau,RémiGiraud,andLaraRaad
23 VariationalModel-BasedDeepNeuralNetworksforImage
Reconstruction ............................................. 879
YunmeiChen,XiaojingYe,andQingchaoZhang
24 BilevelOptimizationMethodsinImaging...................... 909
JuanCarlosDelosReyesandDavidVillacís
25 Multi-parameterApproachesinImageProcessing .............. 943
MarkusGrasmairandValeriyaNaumova
26 GenerativeAdversarialNetworksforRobustCryo-EMImage
Denoising.................................................. 969
HanlinGu,YinXian,IlonaChristyUnarta,andYuanYao
27 Variational Models and Their Combinations with Deep
LearninginMedicalImageSegmentation:ASurvey ............ 1001
LuyingGui,JunMaandXiaopingYang
28 BidirectionalTextureFunctionModeling ...................... 1023
MichalHaindl
29 RegularizationofInverseProblemsbyNeuralNetworks ......... 1065
MarkusHaltmeierandLinhNguyen
30 Shearlets:FromTheorytoDeepLearning ..................... 1095
GittaKutyniok
31 LearnedRegularizersforInverseProblems .................... 1133
SebastianLunz
32 Filter Design for Image Decomposition and Applications
toForensics................................................ 1155
Robin Richter, Duy H. Thai, Carsten Gottschlich,
andStephanF.Huckemann
