Table Of Content

Applied and Numerical Harmonic Analysis Gitta Kutyniok Holger Rauhut Robert J. Kunsch Editors Compressed Sensing in Information Processing Applied and Numerical Harmonic Analysis SeriesEditors JohnJ.Benedetto KassoOkoudjou UniversityofMaryland DeptofMathematics,TuftsUniversity CollegePark,MD,USA Medford,MA,USA WojciechCzaja Mathematics,UniversityofMaryland CollegePark,MD,USA EditorialBoardMembers AkramAldroubi MauroMaggioni VanderbiltUniversity JohnsHopkinsUniversity Nashville,TN,USA Baltimore,MD,USA DouglasCochran ZuoweiShen ArizonaStateUniversity NationalUniversityofSingapore Phoenix,AZ,USA Singapore,Singapore HansG.Feichtinger ThomasStrohmer UniversityofVienna UniversityofCalifornia Vienna,Austria Davis,CA,USA ChristopherHeil YangWang GeorgiaInstituteofTechnology HongKongUniversityofScience& Atlanta,GA,USA Technology Kowloon,HongKong StéphaneJaffard UniversityofParisXII Paris,France GittaKutyniok LudwigMaximilianUniversityof Munich München,Bayern,Germany Gitta Kutyniok (cid:129) Holger Rauhut (cid:129) Robert J. Kunsch Editors Compressed Sensing in Information Processing Editors GittaKutyniok HolgerRauhut MathematischesInstitut LehrstuhlfürMathematik LudwigMaximilianUniversityofMunich RWTHAachenUniversity München,Bayern,Germany Aachen,Nordrhein-Westfalen,Germany RobertJ.Kunsch LehrstuhlfürMathematik RWTHAachenUniversity Aachen,Nordrhein-Westfalen,Germany ISSN2296-5009 ISSN2296-5017 (electronic) AppliedandNumericalHarmonicAnalysis ISBN978-3-031-09744-7 ISBN978-3-031-09745-4 (eBook) https://doi.org/10.1007/978-3-031-09745-4 MathematicsSubjectClassification:94Axx,65F22,68U10,90C25,15B52,86A10 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland ANHA Series Preface The Applied and Numerical Harmonic Analysis (ANHA) book series aims to providetheengineering,mathematical,andscientificcommunitieswithsignificant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series reflects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applicationsandtheircreativesymbioticevolutionareaxiomatic. Harmonicanalysisisawellspringofideasandapplicabilitythathasflourished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental rela- tionship between harmonic analysis and fields such as signal processing, partial differentialequations(PDEs),andimageprocessingisreflectedinourstate-of-the- artANHAseries. Ourvisionofmodernharmonicanalysisincludesabroadarrayofmathematical areas, e.g., wavelet theory, Banach algebras, classical Fourier analysis, time- frequencyanalysis,deeplearning,andfractalgeometry,aswellasthediversetopics thatimpingeonthem. Forexample,wavelettheorycanbeconsideredanappropriatetooltodealwith some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technology from sampling methods on surfaces to fast algorithms and computer vision methods. The underlying mathematics of wavelet theorydependsnotonlyonclassicalFourieranalysisbutalsoonideasfromabstract harmonicanalysis,includingvonNeumannalgebrasandtheaffinegroup.Thisleads toastudyoftheHeisenberggroupanditsrelationshiptoGaborsystems,andofthe metaplecticgroupforameaningfulinteractionofsignaldecompositionmethods. Theunifyinginfluenceofwavelettheoryintheaforementionedtopicsillustrates thejustificationforprovidingameansforcentralizinganddisseminatinginforma- tionfromthebroader,butstillfocused,areaofharmonicanalysis.Thiswillbeakey v vi ANHASeriesPreface roleofANHA.Weintendtopublishwiththescopeandinteractionthatsuchahost ofissuesdemands. Alongwithourcommitmenttopublishmathematicallysignificantworksatthe frontiersofharmonicanalysis,wehaveacomparablystrongcommitmenttopublish majoradvancesinthefollowingapplicabletopicsinwhichharmonicanalysisplays asubstantialrole: *AnalyticNumbertheory*AntennaTheory*ArtificialIntelligence*Biomedical SignalProcessing*ClassicalFourierAnalysis*CodingTheory* CommunicationsTheory*CompressedSensing*Crystallographyand Quasi-Crystals*DataMining*DataScience*DeepLearning*DigitalSignal Processing*DimensionReductionandClassification*FastAlgorithms*Frame TheoryandApplications*GaborTheoryandApplications*Geophysics*Image Processing*MachineLearning*ManifoldLearning*NumericalPartial DifferentialEquations*NeuralNetworks*PhaselessReconstruction*Prediction Theory*QuantumInformationTheory*RadarApplications*SamplingTheory (UniformandNon-uniform)andApplications*SpectralEstimation*Speech Processing*StatisticalSignalProcessing*Super-resolution*TimeSeries* Time-FrequencyandTime-ScaleAnalysis*Tomography*Turbulence* UncertaintyPrinciples*Waveformdesign*WaveletTheoryandApplications TheabovepointofviewfortheANHAbookseriesisinspiredbythehistoryof Fourieranalysisitself,whosetentaclesreachintosomanyfields. In the last two centuries, Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientificphenomena,andonthesolutionofsomeofthemostimportantproblems in mathematics and the sciences. Historically, Fourier series were developed in the analysis of some of the classical PDEs of mathematical physics; these series were used to solve such equations. In order to understand Fourier series and the kindsofsolutionstheycouldrepresent,someofthemostbasicnotionsofanalysis were defined, e.g., the concept of “function.” Since the coefficients of Fourier seriesareintegrals,itisnosurprisethatRiemannintegralswereconceivedtodeal with uniqueness properties of trigonometric series. Cantor’s set theory was also developedbecauseofsuchuniquenessquestions. A basic problem in Fourier analysis is to show how complicated phenomena, suchassoundwaves,canbedescribedintermsofelementaryharmonics.Thereare twoaspectsofthisproblem:first,tofind,orevendefineproperly,theharmonicsor spectrumofagivenphenomenon,e.g.,thespectroscopyprobleminoptics;second, todeterminewhichphenomenacanbeconstructedfromgivenclassesofharmonics, asdone,forexample,bythemechanicalsynthesizersintidalanalysis. Fourieranalysisisalsothenaturalsettingformanyotherproblemsinengineer- ing, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in Fourier analysis not only characterizes the behavior of the prime numbers but is a fundamental tool for analyzing the ideal structures of Banach algebras. It also provides the proper notion of spectrum for phenomena such as white light. This ANHASeriesPreface vii latter process leads tothe Fourier analysis associated withcorrelation functions in filteringandpredictionproblems.Theseproblems,inturn,dealnaturallywithHardy spacesincomplexanalysis,aswellasinspiringWienertoconsidercommunications engineeringintermsoffeedbackandstability,creatinghiscybernetics.Thislatter theory develops concepts to understand complex systems such as learning and cognitionandneuralnetworks,anditisarguablyaprecursorofdeeplearningand itsspectacularinteractionswithdatascienceandAI. Nowadays, some of the theory of PDEs has given way to the study of Fourier integral operators. Problems in antenna theory are studied in terms of unimodular trigonometric polynomials. Applications of Fourier analysis abound in signal processing, whether with the fast Fourier transform (FFT), or filter design, or the adaptivemodelinginherentintime-frequency-scalemethodssuchaswavelettheory. ThecoherentstatesofmathematicalphysicsaretranslatedandmodulatedFourier transforms, and these are used, in conjunction with the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raisond’etreoftheANHAseries! CollegePark,MD JohnBenedetto WojciechCzaja Boston,MA KassoOkoudjou Preface In April 2014, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)establishedthePriorityProgram1798“CompressedSensinginInfor- mation Processing” (CoSIP). The objective of this volume is to offer a compre- hensiveoverviewofthescientifichighlightsobtainedinthecourseofthisPriority Program,mainlyduringthesecondphasethatstartedinJuly2018. Compressed sensing is an area of research with broad applications in electrical engineering, computer science, and physics. It refers to situations where few measurementsalreadysufficetoreconstructasignalorimage,despitethefactthat the acquired information leads to an underdetermined system of linear equations. The key insight here is that most real-world signals are inherently sparse, that is, for many natural classes of signals, there exist building blocks such that decompositions of such signals with respect to these building blocks exhibit only asmallnumberofnon-zerocoefficients.Itisremarkablethatrandomnesshasbeen proven most successful in the acquisition step, enabling for a minimal number of measurements. Furthermore, there exist efficient reconstruction algorithms which makethisapproachfeasibleinpractice. The area of compressed sensing has attracted great interest of researchers in mathematicsandappliedsciencessincearound2004.Alotofrecentresearch–both intheoryandapplication–aremotivatedbywirelesscommunicationandmultiple- inputmultiple-outputchannels(MIMO),whichgainincreasingimportancewiththe adventofdigitaltechnologiesliketheInternetofThings.InparticularChaps.10,11, and 13 present an application-driven view point on wireless networks, while Chap.12 brings MIMO in context with radar imaging. These applications also push forward the development of theory on different models of sparsity such as hierarchical sparsity (see Chap.1) or low-rank matrix recovery (Chap.2), as well as theory on covariance estimation (see Chaps.3 and 4) and recovery algorithms (see Chaps.5–8). The rise of machine learning and deep neural networks likewise leaves its imprint on compressed sensing-related topics in theory-driven research (seeChaps.7–9)aswellasinresearchmotivatedbyapplications(seeChaps.10,13, and 14). Last but not least, the problem of effectively acquiring compressive ix x Preface measurementsisstillachallengeinparticularapplications,seeChap.15onmoving microphonesandChap.16onsphericalnear-fieldantennameasurements. Overall, the network of SPP 1798 comprised more than 60 scientists, and altogether 13 projects were funded in the second period and contributed to this volume (Chaps.8 and 9 are from the same project, the same holds for Chaps.10 and11).WithChap.16,wealsowelcomeacontributionfromaprojectthathasbeen associated to CoSIP. The aim of this volume is of course not to give a complete presentation of all results that have been obtained by participants of the Priority Program but rather to collect the scientific highlights in order to demonstrate the impact of CoSIP on further researches. The editors and authors hope that this volumewillarouseinterestinthereaderonthevariousnewdevelopmentsrelatedto compressed sensing that have been promoted by the Priority Program. For further informationconcerningSPP1798,pleasevisithttps://www.mathc.rwth-aachen.de/ spp1798ii/. München,Germany GittaKutyniok Aachen,Germany HolgerRauhut Aachen,Germany RobertJ.Kunsch October2021