Table Of ContentApplied 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
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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
