Mathematical methods in signal processing and digital image analysis

302 Pages·2008·8.138 MB·English

by  Rainer Dahlhaus, Jürgen Kurths, Peter Maass, Jens Timmer

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· · ·
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
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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, meth-
ods 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 identifica-
tion of dynamical systems, non- and semi-parametric models for time series,
VI Preface
stochastic methods, wavelet or multiscale analysis, diffusion filters and math-
ematical 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 identi-
fication 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 sys-
tems are summarized in Chap. 5. Instead of using classical linear methods
based on correlation functions or spectral decompositions, the present chap-
ter 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 ana-
lyzed 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 im-
ages, 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,
Bj¨orn 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]