Table Of Content

Springer Complexity SpringerComplexityisaninterdisciplinaryprogrampublishingthebestresearchandacademic- level teaching on both fundamental and applied aspects of complex systems – cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience,socialandcomputerscience. ComplexSystemsaresystemsthatcomprisemanyinteractingpartswiththeabilitytogener- ateanewqualityofmacroscopiccollectivebehaviorthemanifestationsofwhicharethesponta- neousformationofdistinctivetemporal,spatialorfunctionalstructures.Modelsofsuchsystems canbesuccessfullymappedontoquitediverse“real-life”situationsliketheclimate,thecoherent emissionoflightfromlasers,chemicalreaction–diffusionsystems,biologicalcellularnetworks, thedynamicsofstockmarketsandoftheInternet,earthquakestatisticsandprediction,freeway traffic,thehumanbrain,ortheformationofopinionsinsocialsystems,tonamejustsomeofthe popularapplications. Althoughtheirscopeandmethodologiesoverlapsomewhat,onecandistinguishthefollow- ingmainconceptsandtools:self-organization,nonlineardynamics,synergetics,turbulence,dy- namicalsystems,catastrophes,instabilities,stochasticprocesses,chaos,graphsandnetworks, cellularautomata,adaptivesystems,geneticalgorithmsandcomputationalintelligence. ThetwomajorbookpublicationplatformsoftheSpringerComplexityprogramarethemono- graphseries“UnderstandingComplexSystems”focusingonthevariousapplicationsofcom- plexity,andthe“SpringerSeriesinSynergetics”,whichisdevotedtothequantitativetheoretical andmethodologicalfoundations.Inadditiontothebooksinthesetwocoreseries,theprogram alsoincorporatesindividualtitlesrangingfromtextbookstomajorreferenceworks. EditorialandProgrammeAdvisoryBoard Pe´terE´rdi CenterforComplexSystemsStudies,KalamazooCollege,USA andHungarianAcademyofSciences,Budapest,Hungary KarlFriston NationalHospital,InstituteforNeurology,WellcomeDept.Cogn.Neurology,London,UK HermannHaken CenterofSynergetics,UniversityofStuttgart,Stuttgart,Germany JanuszKacprzyk SystemResearch,PolishAcademyofSciences,Warsaw,Poland ScottKelso CenterforComplexSystemsandBrainSciences,FloridaAtlanticUniversity,BocaRaton,USA Ju¨rgenKurths NonlinearDynamicsGroup,UniversityofPotsdam,Potsdam,Germany LindaReichl DepartmentofPhysics,PrigogineCenterforStatisticalMechanics,UniversityofTexas,Austin,USA PeterSchuster TheoreticalChemistryandStructuralBiology,UniversityofVienna,Vienna,Austria FrankSchweitzer SystemDesign,ETHZu¨rich,Zu¨rich,Switzerland DidierSornette EntrepreneurialRisk,ETHZu¨rich,Zu¨rich,Switzerland Understanding Complex Systems FoundingEditor:J.A.ScottKelso Future scientific and technological developments in many fields will necessarily depend upon coming to grips with complex systems. Such systems are complex in both their composition – typically many different kinds of components interacting simultaneouslyandnonlinearlywitheachotherandtheirenvironmentsonmultiple levels–andintherichdiversityofbehaviorofwhichtheyarecapable. The Springer Series in Understanding Complex Systems series (UCS) promotes new strategies and paradigms for understanding and realizing applications of complex systems research in a wide variety of fields and endeavors. UCS is explicitly transdisciplinary. It has three main goals: First, to elaborate the concepts, methods and tools of complex systems at all levels of description and in all scientific fields, especiallynewlyemergingareaswithinthelife,social,behavioral,economic,neuro- andcognitivesciences(andderivativesthereof);second,toencouragenovelapplica- tionsoftheseideasinvariousfieldsofengineeringandcomputationsuchasrobotics, nano-technologyandinformatics;third,toprovideasingleforumwithinwhichcom- monalities and differences in the workings of complex systems may be discerned, henceleadingtodeeperinsightandunderstanding. UCS will publish monographs, lecture notes and selected edited contributions aimedatcommunicatingnewfindingstoalargemultidisciplinaryaudience. · · · R. Dahlhaus J. Kurths P. Maass J. Timmer (Eds.) Mathematical Methods in Signal Processing and Digital Image Analysis With96Figuresand20Tables VolumeEditors RainerDahlhaus Ju¨rgenKurths Universita¨tHeidelberg Universita¨tPotsdam Inst.AngewandteMathematik Inst.Physik,LSTheoretischePhysik ImNeuenheimerFeld294 AmNeuenPalais19 69120Heidelberg 14469Potsdam Germany Germany [email protected] [email protected] PeterMaass JensTimmer Universita¨tBremen Universita¨tFreiburg FB3Mathematik/Informatik ZentrumDatenanalyse ZentrumTechnomathematik Eckerstr.1 28334Bremen 79104Freiburg Germany Germany [email protected] [email protected] ISBN:978-3-540-75631-6 e-ISBN:978-3-540-75632-3 UnderstandingComplexSystemsISSN:1860-0832 LibraryofCongressControlNumber:2007940881 (cid:2)c 2008Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. CoverDesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper 9 8 7 6 5 4 3 2 1 springer.com Preface Interest in time series analysis and image processing has been growing very rapidlyinrecentyears.Inputfromdifferentscientificdisciplinesandnewthe- oretical advances are matched by an increasing demand from an expanding diversity of applications. Consequently, signal and image processing has been established as an independent research direction in such different areas as electrical engineering, theoretical physics, mathematics or computer science. This has lead to some rather unstructured developments of theories, methods and algorithms. The authors of this book aim at merging some of these diverging directions and to develop a consistent framework, which combines these heterogeneous developments. The common core of the different chap- tersistheendavourtodevelopandanalyzemathematically justifiedmethods and algorithms. This book should serve as an overview of the state of the art research in this field with a focus on nonlinear and nonparametric models for time series as well as of local, adaptive methods in image processing. The presented results are in its majority the outcome of the DFG-priority programSPP1114“Mathematicalmethodsfortimeseriesanalysisanddigital imageprocessing”.Thestartingpointforthispriorityprogramwastheconsid- eration,thatthenextgenerationofalgorithmicdevelopmentsrequiresaclose cooperation of researchers from different scientific backgrounds. Accordingly, this program, which was running for 6 years from 2001 to 2007, encompassed approximately20researchteamsfromstatistics,theoreticalphysicsandmath- ematics. The intensive cooperation between teams from different specialized disciplinesismirroredbythedifferentchaptersofthisbook,whichwerejointly written by several research teams. The theoretical findings are always tested with applications of different complexity. We do hope and expect that this book serves as a background reference to the present state of the art and that it sparks exciting and creative new research in this rapidly developing field. This book, which concentrates on methodologies related to identification of dynamical systems, non- and semi-parametric models for time series, VI Preface stochastic methods, wavelet or multiscale analysis, diffusion filters and mathematical morphology, is organized as follows. The Chap. 1 describes recent developments on multivariate time series analysis. The results are obtained from combinig statistical methods with the theory of nonlinear dynamics in order to better understand time series measured from underlying complex network structures. The authors of this chapter emphasize the importance of analyzing the interrelations and causal influencesbetweendifferentprocessesandtheirapplicationtoreal-worlddata such as EEG or MEG from neurological experiments. The concept of de- termining directed influences by investigating renormalized partial directed coherence is introduced and analyzed leading to estimators of the strength of the effect of a source process on a target process. The development of surrogate methods has been one of the major driv- ing forces in statistical data analysis in recent years. The Chap. 2 discusses the mathematical foundations of surrogate data testing and examines the statistical performance in extensive simulation studies. It is shown that the performanceofthetestheavilydependsonthechosencombinationofthetest statistics, the resampling methods and the null hypothesis. The Chap. 3 concentrates on multiscale approaches to image processing. It starts with construction principles for multivariate multiwavelets and in- cludessomewaveletapplicationstoinverseproblemsinimageprocessingwith sparsityconstraints.Thechapterincludestheapplicationofthesemethodsto real life data from industrial partners. The investigation of inverse problems is also at the center of Chap. 4. Inverse problems in image processing naturally appear as parameter identification problems for certain partial differential equations. The applications treated in this chapter include the determination of heterogeneous media in subsurface structures, surface matching and morphological image matching as well as a medically motivated image blending task. This chapter includes a survey of the analytic background theory as well as illustrations of these specific applications. Recent results on nonlinear methods for analyzing bivariate coupled systems are summarized in Chap. 5. Instead of using classical linear methods based on correlation functions or spectral decompositions, the present chapter takes a look at nonlinear approaches based on investigating recurrence features. The recurrence properties of the underlying dynamical system are investigatedondifferenttimescales,whichleadstoamathematicallyjustified theoryforanalyzingnonlinearrecurrenceplots.Theinvestigation includesan analysis of synchronization effects, which have been developed into one of the most powerfull methodologies for analyzing dynamical systems. Chapter6takesanewlookatstrucutredsmoothingproceduresfordenois- ing signals and images. Different techniques from stochastic kernel smoother to anisotropic variational approaches and wavelet based techniques are analyzed and compared. The common feature of these methods is their local and Preface VII adaptivenature.Astrongemphasizeisgiventothecomparisonwithstandard methods. Chapter 7 presents a novel framework for the detection and accurate quantification of motion, orientation, and symmetry in images and image sequences. It focuses on those aspects of motion and orientation that can- not be handled successfully and reliably by existing methods, for example, motion superposition (due to transparency, reflection or occlusion), illumina- tion changes, temporal and/or spatial motion discontinuities, and dispersive nonrigid motion. The performance of the presented algorithms is character- ized and their applicability is demonstrated by several key application areas including environmental physics, botany, physiology, medical imaging, and technical applications. TheauthorsofthisbookaswellasallparticipantsoftheSPP1114“Math- ematicalmethodsfortimeseriesanalysisanddigitalimageprocessing”would like to express their sincere thanks to the German Science Foundation for the generous support over the last 6 years. This support has generated and sparked exciting research and ongoing scientific discussions, it has lead to a large diversity of scientific publications and – most importantly- has allowed us to educate a generation of highly talented and ambitious young scientists, whicharenowspreadallovertheworld.Furthermore,itisourgreatpleasure to acknowledge the impact of the referees, which accompangnied and shaped the developments of this priority program during its different phases. Finally, we want to express our gratitude to Mrs. Sabine Pfarr, who prepared this manuscript in an seemingly endless procedure of proof reading, adjusting images, tables, indices and bibliographies while still keeping a friendly level of communicationwithallauthorsconcerningthosenastydetailsscientisteasily forget. Bremen, Rainer Dahlhaus, Ju¨rgen Kurths, November 2007 Peter Maass, Jens Timmer Contents 1 Multivariate Time Series Analysis Bjo¨rn Schelter, Rainer Dahlhaus, Lutz Leistritz, Wolfram Hesse, Ba¨rbel Schack, Ju¨rgen Kurths, Jens Timmer, Herbert Witte ........... 1 2 Surrogate Data – A Qualitative and Quantitative Analysis Thomas Maiwald, Enno Mammen, Swagata Nandi, Jens Timmer ...... 41 3 Multiscale Approximation Stephan Dahlke, Peter Maass, Gerd Teschke, Karsten Koch, Dirk Lorenz, Stephan Mu¨ller, Stefan Schiffler, Andreas Sta¨mpfli, Herbert Thiele, Manuel Werner ................................... 75 4 Inverse Problems and Parameter Identification in Image Processing JensF.Acker,BenjaminBerkels,KristianBredies,MamadouS.Diallo, Marc Droske, Christoph S. Garbe, Matthias Holschneider, Jaroslav Hron, Claudia Kondermann, Michail Kulesh, Peter Maass, Nadine Olischla¨ger, Heinz-Otto Peitgen, Tobias Preusser, Martin Rumpf, Karl Schaller, Frank Scherbaum, Stefan Turek ........111 5 Analysis of Bivariate Coupling by Means of Recurrence Christoph Bandt, Andreas Groth, Norbert Marwan, M. Carmen Romano, Marco Thiel, Michael Rosenblum, Ju¨rgen Kurths ............153 6 Structural Adaptive Smoothing Procedures Ju¨rgen Franke, Rainer Dahlhaus, Jo¨rg Polzehl, Vladimir Spokoiny, Gabriele Steidl, Joachim Weickert, Anatoly Berdychevski, Stephan Didas, Siana Halim, Pavel Mra´zek, Suhasini Subba Rao, Joseph Tadjuidje.................................................183 X Contents 7 Nonlinear Analysis of Multi-Dimensional Signals Christoph S. Garbe, Kai Krajsek, Pavel Pavlov, Bjo¨rn Andres, Matthias Mu¨hlich, Ingo Stuke, Cicero Mota, Martin Bo¨hme, Martin Haker, Tobias Schuchert, Hanno Scharr, Til Aach, Erhardt Barth, Rudolf Mester, Bernd Ja¨hne ......................................231 Index..........................................................289 List of Contributors Til Aach Benjamin Berkels RWTH Aachen University, Aachen, University of Bonn, Bonn, Germany Germany [email protected] [email protected] Martin Bo¨hme University of Lu¨beck, Lu¨beck, Jens F. Acker Germany University of Dortmund, Dortmund, [email protected] Germany [email protected] Kristian Bredies University of Bremen, Bremen, Björn Andres Germany University of Heidelberg, Heidelberg, [email protected] Germany bjoern.andres Rainer Dahlhaus @iwr.uni-heidelberg.de University of Heidelberg, Heidelberg, Germany Christoph Bandt [email protected] University of Greifswald, Greifswald, Stephan Dahlke Germany University of Marburg, Marburg, [email protected] Germany [email protected] Erhardt Barth University of Lu¨beck, Lu¨beck, Mamadou S. Diallo Germany ExxonMobil, Houston, TX, USA [email protected] [email protected] Anatoly Berdychevski Stephan Didas Weierstraß-Institut Berlin, Berlin, Saarland University, Saarland, Germany Germany [email protected] [email protected]