Advances in discrete tomography and its applications PDF
Preview Advances in discrete tomography and its applications
Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto UniversityofMaryland Editorial Advisory Board AkramAldroubi DouglasCochran VanderbiltUniversity ArizonaStateUniversity IngridDaubechies HansG.Feichtinger PrincetonUniversity UniversityofVienna ChristopherHeil MuratKunt GeorgiaInstituteofTechnology SwissFederalInstituteofTechnology,Lausanne JamesMcClellan WimSweldens GeorgiaInstituteofTechnology LucentTechnologies,BellLaboratories MichaelUnser MartinVetterli SwissFederalInstitute SwissFederalInstitute ofTechnology,Lausanne ofTechnology,Lausanne M.VictorWickerhauser WashingtonUniversity Advances in Discrete Tomography and Its Applications Gabor T. Herman Attila Kuba Editors Birkha¨user Boston • Basel • Berlin GaborT.Herman AttilaKuba† Ph.D.PrograminComputerScience DepartmentofImageProcessing TheGraduateCenter andComputerGraphics TheCityUniversityofNewYork UniversityofSzeged 365FifthAvenue A´rpa´dte´r2. NewYork,NY10016 H-6720Szeged U.S.A. Hungary CoverdesignbyJosephSherman. MathematicsSubjectClassification(2000):05-04,06-04,15A29,52-04,65K10,68U10,90C05,92-08 LibraryofCongressControlNumber:2006937473 ISBN-10:0-8176-3614-5 e-ISBN-10:0-8176-4543-8 ISBN-13:978-0-8176-3614-2 e-ISBN-13:978-0-8176-4543-4 Printedonacid-freepaper. (cid:2)c2007Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaLLC,233Spring Street,NewYork,NY10013,USA)andtheauthor,exceptforbriefexcerptsinconnectionwithreviews orscholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped isforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. 9 8 7 6 5 4 3 2 1 www.birkhauser.com (KeS/SB) To the memory of Alberto Del Lungo ANHA Series Preface The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientific communities with sig- nificant 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 inter- leaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonicanalysisisa wellspringofideasandapplicability thathas flour- ished, developed, and deepened over time within many disciplines and by meansofcreativecross-fertilizationwith diverseareas.The intricate andfun- damental relationship between harmonic analysis and fields such as signal processing, partial differential equations (PDEs), and image processing is re- flected in our state-of-the-art ANHA series. Ourvisionofmodernharmonicanalysisincludes mathematicalareassuch as wavelettheory, Banach algebras,classicalFourier analysis,time-frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them. For example, wavelet theory can be considered an appropriate tool to deal with some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, biomedical engineering, and tur- bulence.Theseareasimplementthelatesttechnologyfromsamplingmethods on surfaces to fast algorithms and computer vision methods. The underlying mathematicsofwavelettheorydepends notonlyonclassicalFourieranalysis, but also on ideas from abstract harmonic analysis, including von Neumann algebras and the affine group. This leads to a study of the Heisenberg group and its relationship to Gabor systems, and of the metaplectic group for a meaningful interaction of signal decomposition methods. The unifying influ- enceofwavelettheoryintheaforementionedtopicsillustratesthejustification viii ANHASeries Preface forprovidingameansforcentralizinganddisseminatinginformationfromthe broader, but still focused, area of harmonic analysis. This will be a key role of ANHA. We intend to publish with the scope and interaction that such a host of issues demands. Alongwithourcommitmenttopublishmathematicallysignificantworksat the frontiers of harmonicanalysis,wehavea comparablystrongcommitment topublishmajoradvancesinthefollowingapplicabletopicsinwhichharmonic analysis plays a substantial role: Antennatheory Predictiontheory Biomedicalsignalprocessing Radarapplications Digitalsignalprocessing Samplingtheory Fastalgorithms Spectralestimation Gabortheory andapplications Speechprocessing Imageprocessing Time-frequencyand Numerical partial differential equations time-scaleanalysis Wavelettheory The above point of view for the ANHA book series is inspired by the history of Fourier analysis itself, whose tentacles reach into so many fields. 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,andonthesolutionofsomeofthemostimportantprob- lems in mathematics and the sciences. Historically, Fourier series were devel- oped in the analysis of some of the classical PDEs of mathematical physics; theseserieswereusedto solvesuchequations.Inorderto understandFourier series and the kinds of solutions they could represent, some of the most basic notions of analysis were defined, e.g., the concept of “function.” Since the coefficients of Fourierseries are integrals,it is no surprise that Riemann inte- gralswereconceivedtodealwithuniquenesspropertiesoftrigonometricseries. Cantor’s set theory was also developed because of such uniqueness questions. A basic problem in Fourier analysis is to show how complicated phenom- ena,suchassoundwaves,canbedescribedintermsofelementaryharmonics. There are two aspects of this problem: first, to find, or even define properly, the harmonics or spectrum of a given phenomenon, e.g., the spectroscopy probleminoptics;second,todeterminewhichphenomenacanbeconstructed fromgivenclassesofharmonics,asdone,forexample,by the mechanicalsyn- thesizers in tidal analysis. Fourier analysis is also the natural setting for many other problems in engineering,mathematics,andthesciences.Forexample,Wiener’sTauberian theorem in Fourier analysis not only characterizes the behavior of the prime numbers,butalsoprovidesthepropernotionofspectrumforphenomenasuch aswhitelight;thislatterprocessleadstothe Fourieranalysisassociatedwith correlationfunctionsinfilteringandpredictionproblems,andtheseproblems, in turn, deal naturally with Hardy spaces in the theory of complex variables. ANHASeries Preface ix 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 aboundin signalprocessing,whether with the Fast Fouriertransform(FFT), or filter design, or the adaptive modeling inherent in time-frequency-scale methods suchaswavelettheory.Thecoherentstates ofmathematicalphysics aretranslated and modulated Fourier transforms,and these are used, in con- junctionwiththe uncertaintyprinciple,fordealingwithsignalreconstruction in communications theory. We are back to the raison d’ˆetre of the ANHA series! John J. Benedetto Series Editor University of Maryland College Park Preface Sevenyearshavepassedsincewefinishededitingthefirstbookondiscreteto- mography: Discrete Tomography: Foundations, Algorithms, and Applications (Birkha¨user,Boston,1999).There has been a floweringof the field since that time. New researchgroupshave started,new theoreticaland practicalresults have been presented, and new applications have developed. There have been about 200 papers published on discrete tomography since 1999. The current book reports on present advances in discrete tomography.Its structureisthesameasthatofthepreviousone:afteranintroduction(Chap- ter 1) there are chapters on new theoretical foundations (Chapters 2–7), re- constructionalgorithms(Chapters8–11),andselectedapplications(Chapters 12–16).Thelevelofpresentationaimsatapotentialreadershipofmathemati- cians, programmers, engineers, researchers working in the application areas, andstudentsinappliedmathematics,computerimaging,biomedicalimaging, computer engineering, and/or image processing. Acknowledgments We thank the National Science Foundation and the Graduate Center of City UniversityofNewYorkforsponsoringthe2005WorkshoponDiscreteTomog- raphyandItsApplications,June13–15,NewYorkCity.Wewishtothankthe Electronic Notes on Discrete Mathematics, its publisher (Elsevier, Inc.), and itseditors(P.L.HammerandV.Lozin)forpublishingaspecialissuebasedon the presentations given at the workshop and for giving permission to extend some of that material into this book. We are grateful to a number of people for technical help; among them we wishtomentionLajosRodek,La´szl´oCsernetics,L´aszl´oG.Nyu´l,AntalNagy, Ka´lma´n Pala´gyi,Stuart Rowland, and Alain Daurat. Gabor T. Herman Attila Kuba New York, New York Szeged, Hungary August 2006
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