Discrete wavelet transform: a signal processing approach

325 Pages·2015·6.629 MB·English

by  Sundararajan, D

DISCRETE WAVELET
TRANSFORM
DISCRETE WAVELET
TRANSFORM
A SIGNAL PROCESSING APPROACH
D.Sundararajan
AdhiyamaanCollegeofEngineering,India
Thiseditionfirstpublished2015
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Contents
Preface xi
ListofAbbreviations xiii
1 Introduction 1
1.1 TheOrganizationofThisBook 2
2 Signals 5
2.1 SignalClassifications 5
2.1.1 PeriodicandAperiodicSignals 5
2.1.2 EvenandOddSignals 6
2.1.3 EnergySignals 7
2.1.4 CausalandNoncausalSignals 9
2.2 BasicSignals 9
2.2.1 Unit-ImpulseSignal 9
2.2.2 Unit-StepSignal 10
2.2.3 TheSinusoid 10
2.3 TheSamplingTheoremandtheAliasingEffect 12
2.4 SignalOperations 13
2.4.1 TimeShifting 13
2.4.2 TimeReversal 14
2.4.3 TimeScaling 14
2.5 Summary 17
Exercises 17
3 ConvolutionandCorrelation 21
3.1 Convolution 21
3.1.1 TheLinearConvolution 21
3.1.2 PropertiesofConvolution 24
3.1.3 ThePeriodicConvolution 25
3.1.4 TheBorderProblem 25
3.1.5 ConvolutionintheDWT 26
vi Contents
3.2 Correlation 28
3.2.1 TheLinearCorrelation 28
3.2.2 CorrelationandFourierAnalysis 29
3.2.3 CorrelationintheDWT 30
3.3 Summary 31
Exercises 31
4 FourierAnalysisofDiscreteSignals 37
4.1 TransformAnalysis 37
4.2 TheDiscreteFourierTransform 38
4.2.1 Parseval’sTheorem 43
4.3 TheDiscrete-TimeFourierTransform 44
4.3.1 Convolution 48
4.3.2 ConvolutionintheDWT 48
4.3.3 Correlation 50
4.3.4 CorrelationintheDWT 50
4.3.5 TimeExpansion 52
4.3.6 SamplingTheorem 52
4.3.7 Parseval’sTheorem 54
4.4 ApproximationoftheDTFT 55
4.5 TheFourierTransform 56
4.6 Summary 56
Exercises 57
5 Thez-Transform 59
5.1 Thez-Transform 59
5.2 Propertiesofthez-Transform 60
5.2.1 Linearity 60
5.2.2 TimeShiftofaSequence 61
5.2.3 Convolution 61
5.3 Summary 62
Exercises 62
6 FiniteImpulseResponseFilters 63
6.1 Characterization 63
6.1.1 IdealLowpassFilters 64
6.1.2 IdealHighpassFilters 65
6.1.3 IdealBandpassFilters 66
6.2 LinearPhaseResponse 66
6.2.1 Even-SymmetricFIRFilterswithOddNumberofCoefficients 67
6.2.2 Even-SymmetricFIRFilterswithEvenNumberofCoefficients 68
6.3 Summary 69
Exercises 69
7 MultirateDigitalSignalProcessing 71
7.1 Decimation 72
Contents vii
7.1.1 DownsamplingintheFrequency-Domain 72
7.1.2 DownsamplingFollowedbyFiltering 75
7.2 Interpolation 77
7.2.1 UpsamplingintheFrequency-Domain 77
7.2.2 FilteringFollowedbyUpsampling 78
7.3 Two-ChannelFilterBank 79
7.3.1 PerfectReconstructionConditions 81
7.4 PolyphaseFormoftheTwo-ChannelFilterBank 84
7.4.1 Decimation 84
7.4.2 Interpolation 87
7.4.3 PolyphaseFormoftheFilterBank 91
7.5 Summary 94
Exercises 94
8 TheHaarDiscreteWaveletTransform 97
8.1 Introduction 97
8.1.1 SignalRepresentation 97
8.1.2 TheWaveletTransformConcept 98
8.1.3 FourierandWaveletTransformAnalyses 98
8.1.4 Time-FrequencyDomain 99
8.2 TheHaarDiscreteWaveletTransform 100
8.2.1 TheHaarDWTandthe2-PointDFT 102
8.2.2 TheHaarTransformMatrix 103
8.3 TheTime-FrequencyPlane 107
8.4 WaveletsfromtheFilterCoefficients 111
8.4.1 TwoScaleRelations 116
8.5 The2-DHaarDiscreteWaveletTransform 118
8.6 DiscontinuityDetection 126
8.7 Summary 127
Exercises 128
9 OrthogonalFilterBanks 131
9.1 HaarFilter 132
9.2 DaubechiesFilter 135
9.3 OrthogonalityConditions 146
9.3.1 CharacteristicsofDaubechiesLowpassFilters 149
9.4 CoifletFilter 150
9.5 Summary 154
Exercises 155
10 BiorthogonalFilterBanks 159
10.1 BiorthogonalFilters 159
10.2 5/3SplineFilter 163
10.2.1 DaubechiesFormulation 170
10.3 4/4SplineFilter 170
10.3.1 DaubechiesFormulation 177
viii Contents
10.4 CDF9/7Filter 178
10.5 Summary 183
Exercises 184
11 ImplementationoftheDiscreteWaveletTransform 189
11.1 ImplementationoftheDWTwithHaarFilters 190
11.1.1 1-LevelHaarDWT 190
11.1.2 2-LevelHaarDWT 191
11.1.3 1-LevelHaar2-DDWT 193
11.1.4 TheSignal-FlowGraphoftheFastHaarDWTAlgorithms 194
11.1.5 HaarDWTinPlace 196
11.2 SymmetricalExtensionoftheData 198
11.3 ImplementationoftheDWTwiththeD4Filter 200
11.4 ImplementationoftheDWTwithSymmetricalFilters 203
11.4.1 5/3SplineFilter 203
11.4.2 CDF9/7Filter 205
11.4.3 4/4SplineFilter 208
11.5 ImplementationoftheDWTusingFactorizedPolyphaseMatrix 210
11.5.1 HaarFilter 211
11.5.2 D4Filter 213
11.5.3 5/3SplineFilter 216
11.6 Summary 219
Exercises 219
12 TheDiscreteWaveletPacketTransform 223
12.1 TheDiscreteWaveletPacketTransform 223
12.1.1 NumberofRepresentations 226
12.2 BestRepresentation 227
12.2.1 CostFunctions 230
12.3 Summary 233
Exercises 233
13 TheDiscreteStationaryWaveletTransform 235
13.1 TheDiscreteStationaryWaveletTransform 235
13.1.1 TheSWT 235
13.1.2 TheISWT 236
13.1.3 AlgorithmsforComputingtheSWTandtheISWT 238
13.1.4 2-DSWT 243
13.2 Summary 244
Exercises 244
14 TheDual-TreeDiscreteWaveletTransform 247
14.1 TheDual-TreeDiscreteWaveletTransform 248
14.1.1 Parseval’sTheorem 248
14.2 TheScalingandWaveletFunctions 252
14.3 ComputationoftheDTDWT 253
Contents ix
14.4 Summary 262
Exercises 263
15 ImageCompression 265
15.1 LossyImageCompression 266
15.1.1 Transformation 266
15.1.2 Quantization 268
15.1.3 Coding 270
15.1.4 CompressionAlgorithm 273
15.1.5 ImageReconstruction 277
15.2 LosslessImageCompression 284
15.3 RecentTrendsinImageCompression 289
15.3.1 TheJPEG2000ImageCompressionStandard 290
15.4 Summary 290
Exercises 291
16 Denoising 295
16.1 Denoising 295
16.1.1 SoftThresholding 296
16.1.2 StatisticalMeasures 297
16.2 VisuShrinkDenoisingAlgorithm 298
16.3 Summary 303
Exercises 303
Bibliography 305
AnswerstoSelectedExercises 307
Index 319
Preface
The discrete wavelet transform, a generalization of the Fourier analysis, is widely used in
many applications of science and engineering. The primary objective of writing this book
is to present the essentials of the discrete wavelet transform – theory, implementation, and
applications–fromapracticalviewpoint.Thediscretewavelettransformispresentedfroma
digitalsignalprocessingpointofview.Physicalexplanations,numerousexamples,plentyof
figures,tables,andprogramsenablethereadertounderstandthetheoryandalgorithmsofthis
relativelydifficulttransformwithminimaleffort.
This book is intended to be a textbook for senior-undergraduate-level and graduate-level
discretewavelettransformcoursesorasupplementarytextbookfordigitalsignal/imagepro-
cessingcoursesinengineeringdisciplines.Forsignalandimageprocessingprofessionals,this
bookwillbeusefulforself-study.Inaddition,thisbookwillbeareferenceforanyone,student
orprofessional,specializinginsignalandimageprocessing.Theprerequisiteforreadingthis
bookisagoodknowledgeofcalculus,linearalgebra,signalsandsystems,anddigitalsignal
processingattheundergraduatelevel.Thelasttwoofthesetopicsareadequatelycoveredin
thefirstfewchaptersofthisbook.
MATLAB® programs are available at the website of the book, www.wiley.com/go/
sundararajan/wavelet. Programming is an important component in learning this subject.
Answers to selected exercises marked with * are given at the end of the book. A Solutions
Manualandslidesareavailableforinstructorsatthewebsiteofthebook.
Iassumetheresponsibilityforalltheerrorsinthisbookandwouldverymuchappreciate
receivingreaders’[email protected]
his team at Wiley for their help and encouragement in completing this project. I thank my
familyfortheirsupportduringthisendeavor.
D.Sundararajan
List of Abbreviations
bpp bitsperpixel
DFT discreteFouriertransform
DTDWT dual-treediscretewavelettransform
DTFT discrete-timeFouriertransform
DWPT discretewaveletpackettransform
DWT discretewavelettransform
FIR finiteimpulseresponse
FS Fourierseries
FT Fouriertransform
IDFT inversediscreteFouriertransform
IDTDWT inversedual-treediscretewavelettransform
IDWPT inversediscretewaveletpackettransform
IDWT inversediscretewavelettransform
ISWT inversediscretestationarywavelettransform
PR perfectreconstruction
SWT discretestationarywavelettransform
1-D one-dimensional
2-D two-dimensional