Table Of ContentSignaland Image Multiresolution Analysis
Signal and Image
Multiresolution Analysis
Edited by
Abdeldjalil Ouahabi
Series Editor
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Table of Contents
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
AbdeldjalilOUAHABI
Chapter1.IntroductiontoMultiresolutionAnalysis . . . . . . . . . . . . . . 1
AbdeldjalilOUAHABI
1.1.Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.Wavelettransforms:anintroductoryreview . . . . . . . . . . . . . . . . 3
1.2.1.Briefhistory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2.Continuouswavelettransforms. . . . . . . . . . . . . . . . . . . . . . 6
1.2.2.1.Wavelettransformmodulusmaxima. . . . . . . . . . . . . . . . . 9
1.2.2.2.Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3.Discretewavelettransforms. . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.Multiresolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1.Multiresolutionanalysisandwaveletbases . . . . . . . . . . . . . . 17
1.3.1.1.Approximationspaces. . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.1.2.Detailspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.2.Multiresolutionanalysis:pointstoremember . . . . . . . . . . . . . 21
1.3.3.Decompositionandreconstruction. . . . . . . . . . . . . . . . . . . . 22
1.3.3.1.Calculationofcoefficients . . . . . . . . . . . . . . . . . . . . . . 22
1.3.3.2.ImplementationofMRA:Mallatalgorithm. . . . . . . . . . . . 24
1.3.3.3.Extensiontoimages. . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.4.Waveletpackets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.5.Multiresolutionanalysissummarized . . . . . . . . . . . . . . . . . . 30
1.4.Whichwaveletstochoose? . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.4.1.Numberofvanishingmoments,regularity,
support(compactness),symmetry,etc. . . . . . . . . . . . . . . . . . . . . . 33
1.4.2.Well-knownwavelets,scalingfunctionsandassociatedfilters. . . 34
1.4.2.1.Haarwavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
vi SignalandImageMultiresolutionAnalysis
1.4.2.2.Daubechieswavelets . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.4.2.3.Symlets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.4.2.4.Coiflets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.4.2.5.Meyerwavelets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.4.2.6.Polynomialsplinewavelets . . . . . . . . . . . . . . . . . . . . . 43
1.5.Multiresolutionanalysisandbiorthogonalwaveletbases. . . . . . . . . 48
1.5.1.Whybiorthogonalbases? . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.5.2.Multiresolutioncontext . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.5.3.Exampleofbiorthogonalwavelets,scalingfunctions
andassociatedfilters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.5.4.Theconceptofwaveletlifting . . . . . . . . . . . . . . . . . . . . . . 51
1.5.4.1.Thenotionoflifting. . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.5.4.2.Significanceofstructurelifting . . . . . . . . . . . . . . . . . . . 52
1.6.Waveletchoiceataglance. . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.6.1.Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.6.2.Vanishingmoments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.6.3.Othercriteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.6.4.Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.7.Workedexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.7.1.Examplesofmultiresolutionanalysis . . . . . . . . . . . . . . . . . . 55
1.7.2.Compression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.7.3.Denoising(reductionofnoise) . . . . . . . . . . . . . . . . . . . . . . 64
1.8.Someapplications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
1.8.1.Discoveryandcontributionsofwavelets . . . . . . . . . . . . . . . . 74
1.8.2.Biomedicalengineering . . . . . . . . . . . . . . . . . . . . . . . . . . 76
1.8.2.1.ECG,EEGandBCI. . . . . . . . . . . . . . . . . . . . . . . . . . 77
1.8.2.2.Medicalimaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
1.8.3.Telecommunications. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
1.8.3.1.Adaptivecompressionforsensornetworks . . . . . . . . . . . . 110
1.8.3.2.Maskingimageencodingandtransmissionerrors . . . . . . . . 114
1.8.3.3.Suppressionofcorrelatednoise . . . . . . . . . . . . . . . . . . . 118
1.8.4.“Compressivesensing”,ICA,PCAandMRA. . . . . . . . . . . . . 119
1.8.4.1.Principalcomponentanalysis . . . . . . . . . . . . . . . . . . . . 120
1.8.4.2.Independentcomponentanalysis . . . . . . . . . . . . . . . . . . 121
1.8.4.3.Compressivesensing . . . . . . . . . . . . . . . . . . . . . . . . . 122
1.8.5.Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
1.9.Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Chapter2.DiscreteWaveletTransform-BasedMultifractalAnalysis . . . 135
AbdeldjalilOUAHABI
2.1.Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
2.1.1.Fractalsandwavelets:ahappymarriage? . . . . . . . . . . . . . . . 135
TableofContents vii
2.1.2.Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2.1.3.Mono/multifractalprocesses . . . . . . . . . . . . . . . . . . . . . . . 137
2.1.4.Chapteroutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.2.Fractality,variabilityandcomplexity . . . . . . . . . . . . . . . . . . . . 139
2.2.1.Systemcomplexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.2.2.Complexphenomenaproperties . . . . . . . . . . . . . . . . . . . . . 141
2.2.2.1.Tendencyofautonomousagentstoself-organize . . . . . . . . 141
2.2.2.2.Variabilityandadaptability . . . . . . . . . . . . . . . . . . . . . 142
2.2.2.3.Bifurcationconceptandchaoticmodel. . . . . . . . . . . . . . . 143
2.2.2.4.Hierarchyandscaleinvariance . . . . . . . . . . . . . . . . . . . 146
2.2.2.5.Self-organizedcriticalphenomena . . . . . . . . . . . . . . . . . 146
2.2.2.6.Highlyoptimizedtolerance. . . . . . . . . . . . . . . . . . . . . . 147
2.2.3.Fractality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
2.3.Multifractalanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
2.3.1.Point-wiseregularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
2.3.2.Hölderexponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
2.3.3.Signalclassificationaccordingtotheregularityproperties . . . . . 152
2.3.3.1.Monofractalsignal. . . . . . . . . . . . . . . . . . . . . . . . . . . 152
2.3.3.2.Multifractalsignal. . . . . . . . . . . . . . . . . . . . . . . . . . . 152
2.3.4.Hausdorffdimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
2.3.4.1.Theoreticalapproach . . . . . . . . . . . . . . . . . . . . . . . . . 155
2.3.4.2.Qualitativeapproachandmultifractalspectrum . . . . . . . . . 155
2.4.Multifractalformalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
2.4.1.Reminderonwaveletdecomposition . . . . . . . . . . . . . . . . . . 156
2.4.2.Point-wiseregularitycharacterization. . . . . . . . . . . . . . . . . . 157
2.4.3.Structurefunctionandpowerlawbehavior. . . . . . . . . . . . . . . 158
2.4.4.Linkbetweenscalingexponentsandsingularityspectrum. . . . . . 159
2.4.5.Useofwaveletleaders. . . . . . . . . . . . . . . . . . . . . . . . . . . 160
2.4.5.1.Indexingadyadicsquareandwaveletleaders . . . . . . . . . . 161
2.4.5.2.Polynomialexpansionandlog-cumulants. . . . . . . . . . . . . 162
2.4.6.Waveletleadersvariant:“maximum”coefficients . . . . . . . . . . 165
2.5.Algorithmandperformances . . . . . . . . . . . . . . . . . . . . . . . . . 165
2.5.1.Singularityspectrumestimationalgorithm. . . . . . . . . . . . . . . 165
2.5.2.Analysisofafewwidelyusedprocesses . . . . . . . . . . . . . . . . 167
2.5.2.1.fBm:amonofractalprocess . . . . . . . . . . . . . . . . . . . . . 167
2.5.2.2.CMC:amultifractalprocess. . . . . . . . . . . . . . . . . . . . . 170
2.5.2.3.BMC:anotherclassofmultifractalprocesses. . . . . . . . . . . 173
2.5.3.Estimationperformances . . . . . . . . . . . . . . . . . . . . . . . . . 176
2.5.3.1.fBm,CMC-LNandCMC-LPsimulation . . . . . . . . . . . . . 176
2.5.3.2.Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
2.5.3.3.Interpretationandrecommendations . . . . . . . . . . . . . . . . 182
2.6.Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
2.6.1.Turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
viii SignalandImageMultiresolutionAnalysis
2.6.1.1.FromLeonardodaVincitoKolmogorov . . . . . . . . . . . . . 186
2.6.1.2.Multifractalprocessinturbulence. . . . . . . . . . . . . . . . . . 191
2.6.2.Multifractalprocessinfinance . . . . . . . . . . . . . . . . . . . . . . 194
2.6.2.1.Stockmarketcomplexitymodeling. . . . . . . . . . . . . . . . . 194
2.6.2.2.Marketturbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
2.6.3.Internettraffic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
2.6.3.1.TheInternetrevolution . . . . . . . . . . . . . . . . . . . . . . . . 203
2.6.3.2.MultifractalnatureofInternettraffic? . . . . . . . . . . . . . . . 204
2.6.4.Biomedicalfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
2.6.4.1.Analyzingimagetexturethroughamultifractalapproach . . . 205
2.6.4.2.Multifractalityinmedicalimaging . . . . . . . . . . . . . . . . . 211
2.7.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
2.8.Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Chapter3.MultimodalCompressionUsingJPEG2000:
SupervisedInsertionApproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
RégisFOURNIERandAmineNAÏT-ALI
3.1.Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
3.2.TheJPEG2000standard. . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
3.3.Multimodalcompressionbyunsupervisedinsertion. . . . . . . . . . . . 227
3.3.1.Principleofinsertioninthewavelettransformdomain. . . . . . . . 228
3.3.2.Principleofinsertioninthespatialdomain. . . . . . . . . . . . . . . 229
3.4.Multimodalcompressionbysupervisedinsertion . . . . . . . . . . . . . 231
3.4.1.Choiceofinsertionzone. . . . . . . . . . . . . . . . . . . . . . . . . . 232
3.4.2.Insertionandseparationfunction. . . . . . . . . . . . . . . . . . . . . 233
3.4.2.1.Insertionfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
3.4.2.2.Separationfunction . . . . . . . . . . . . . . . . . . . . . . . . . . 235
3.5.Criteriaforqualityevaluation. . . . . . . . . . . . . . . . . . . . . . . . . 236
3.5.1.Peaksignal-to-noiseratio . . . . . . . . . . . . . . . . . . . . . . . . . 236
3.5.2.Percentresidualdifference . . . . . . . . . . . . . . . . . . . . . . . . 237
3.6.Somepreliminaryresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
3.7.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
3.8.Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Chapter4.CerebralMicroembolismSynchronousDetection
withWaveletPackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Jean-MarcGIRAULT
4.1.Issueandstakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
4.2.Priorinformationresearch . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
4.2.1.Dopplerultrasoundbloodflowsignal. . . . . . . . . . . . . . . . . . 247
4.2.2.EmbolicDopplerultrasoundsignal . . . . . . . . . . . . . . . . . . . 250
4.3.Dopplerultrasoundbloodembolisignalmodeling. . . . . . . . . . . . . 251
TableofContents ix
4.3.1.“Physical”model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
4.3.2.“Signal”model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
4.3.3.Statisticaltests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
4.3.3.1.Stationaritytest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
4.3.3.2.Cyclostationaritytest . . . . . . . . . . . . . . . . . . . . . . . . . 261
4.3.3.3.“Gaussian”testandcardiaccycleregularity . . . . . . . . . . . 262
4.4.Energydetection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
4.4.1.State-of-the-artandstandarddetection . . . . . . . . . . . . . . . . . 263
4.4.2.Synchronousdetection. . . . . . . . . . . . . . . . . . . . . . . . . . . 266
4.5.Waveletpacketenergydetection . . . . . . . . . . . . . . . . . . . . . . . 269
4.5.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
4.5.2.Multiresolutionanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . 271
4.5.3.Waveletpacketsubbanddetection. . . . . . . . . . . . . . . . . . . . 274
4.5.4.Waveletpacketsynchronousdetection . . . . . . . . . . . . . . . . . 277
4.6.Resultsanddiscussions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
4.6.1.Insimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
4.6.2.Invivo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
4.7.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
4.8.Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
ListofAuthors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Introduction
Wavelet-based multiresolution analysis can be applied for analysis, processing
andsynthesisofmono-ormultidimensionalsignalsatdifferentlevelsofresolution,
by decomposing them into (orthonormal) “scaling” and (orthonormal) “wavelet”
functions. Therefore, multiresolution analysis provides a family of orthonormal
wavelets,andreducesanyredundancytonil.
Inpractice,ontheonehand,thefieldofactivityawaitedtheworkofMallatand
Daubechies, which focused on implementation adapted to the pyramidal algorithms
of Burt and Adelson, for a concrete use of the wavelets to be generated, using the
fast wavelet transform. On the other hand, multiresolution analysis was also
contingent on the benefits of sub-band coding, introduced in 1977 by Esteban and
Galand.
The concept of multiresolution analysis provides a framework for the
decomposition (and reconstruction) of a signal in the form of a series of
approximationsofdecreasingscale,completedbyaseriesofdetails.
To illustrate this idea, let us take the case of an image constructed of a
succession of approximations; details improve this image. Therefore, coarse vision
becomesfinerandmoreprecise.
In fact, in multiresolution analysis, fast wavelet transform is achieved
without wavelets or scaling functions! All that is required are the coefficients of a
low-pass filter h (and of a high-pass filter derived from h). That is the wonder of
Mallat’s fast wavelet transform. In practice, such filters are directly linked to the
chosenwavelet.
IntroductionwrittenbyAbdeldjalilOUAHABI.
xii SignalandImageMultiresolutionAnalysis
Researchers, engineers and practitioners of a diverse range of disciplines
(multimedia, telecommunications, medicine and biology, signal and image
processing, fracture and fluid mechanics, thermodynamics, astrophysics, finance,
etc.) are confronted daily with increasingly challenging technological problems
at multiple levels of analysis, such as in classification, segmentation, detection
(of contours or parameters of interest), reduction or even elimination of noise,
compressioninpreparationfortransmissionorstorage,synthesisandreconstruction.
The concept of multiresolution analysis constitutes an efficient tool, often
without a priori knowledge, that is universally applicable to the domains noted
above. This tool, sometimes described as miraculous, produces an immediate and
readily interpretable and exploitable result. However, for specific applications that
require the extraction of targeted information, it is amply clear that it will be
necessary to develop and “merge” advanced methods that use existing techniques
or optimize analyses (for example, in compression) by taking into account edges
orborders,usingthird-generationwaveformssuchasridgelets,curvelets,andstrips.
In effect, these anisotropic wavelets are automatically oriented and expanded by
unifyingthegeometryofapotentialedge.
Since the concept of multiresolution analysis has been well-known for two
decades,itisnowappropriatetoquestiontheutilityoffurtherwork.
This book aims to provide a simple formalization and new clarity for
multiresolution analysis, rendering accessible obscure techniques, and merging,
unifyingorcompletingthetechniquewithencoding,featureextraction,compressive
sensing,multifractalanalysisandtextureanalysis.
This book is aimed at industrial engineers, medical researchers, university lab
attendants, lecturer-researchers and researchers from various specialisms. It is
also intended to contribute to the studies of graduate students in engineering,
particularly in medical imaging, intelligent instrumentation, telecommunications,
and signal and image processing. Given the diversity of the problems posed and
addressed, this book paves the way for the development of new research themes,
such as brain–computer interface (BCI), compressive sensing, functional magnetic
resonance imaging (fMRI), tissue characterization (bones, skin, etc.) and the
analysisofcomplexphenomenaingeneral.
Throughout the chapters, informative illustrations assist the uninitiated reader
in better conceptualizing certain concepts, taking the form of numerous figures
and recent applications in biomedical engineering, communication, multimedia,
finance,etc.
