Twodimensional Signal AnalysisISBN: 9781848210189
352 pages
June 2008, WileyISTE

Description
Concentrating its coverage on those 2D signals coming from physical sensors (such as radars and sonars), the discussion explores a 2D spectral approach but develops the modeling of 2D signals and proposes several dataoriented analysis techniques for dealing with them. Coverage is also given to potential future developments in this area.
Table of Contents
Introduction 13
Chapter 1. Basic Elements of 2D Signal Processing 17
Claude CARIOU, Olivier ALATA and JeanMarc LE CAILLEC
1.1. Introduction 17
1.2. Deterministic 2D signals 18
1.2.1. Definition 18
1.2.2. Particular 2D signals 19
1.3. Random 2D signals 22
1.3.1. Definition 22
1.3.2. Characterization up to the second order 23
1.3.3. Stationarity 24
1.3.4. Characterization of orders higher than two 26
1.3.5. Ergodicity 26
1.3.6. Specificities of random 2D signals 27
1.3.7. Particular random signals 28
1.4. 2D systems 31
1.4.1. Definition 31
1.4.2. Main 2D operators 31
1.4.3. Main properties 32
1.4.4. Linear timeinvariant (LTI) system 33
1.4.5. Example 34
1.4.6. Separable system 34
1.4.7. Stability of 2D systems 36
1.4.8. Support of the impulse response – causality 37
1.5. Characterization of 2D signals and systems 39
1.5.1. Frequency response of an LTI system 39
1.5.2. 2D Fourier transform 41
1.5.3. Discrete 2D Fourier transform 43
1.5.4. 2D z transform 46
1.5.5. Frequency characterization of a random 2D signal 55
1.5.6. Output of a 2D system with random input 57
1.6. 2D Wold decomposition 58
1.6.1. Innovation, determinism and regularity in the 2D case 58
1.6.2. Total decomposition of three fields 60
1.6.3. Example of an outcome 61
1.7. Conclusion 63
1.8. Bibliography 63
Chapter 2. 2D Linear Stochastic Modeling 65
Olivier ALATA and Claude CARIOU
2.1. Introduction 65
2.2. 2D ARMA models 66
2.2.1. Definition 66
2.2.2. 2D ARMA models and prediction supports 67
2.3. LMarkovian fields 73
2.3.1. 2D Markov fields and LMarkovian fields 73
2.3.2. 2D LMarkovian fields and Gibbs fields 74
2.4. “Global” estimation methods 76
2.4.1. Maximum likelihood 76
2.4.2. YuleWalker equations 79
2.4.3. 2D Levinson algorithm (for the parametric 2D AR estimation) 85
2.5. “Adaptive” or “recursive” estimation methods 93
2.5.1. Connectivity hypotheses for adaptive or recursive algorithms 93
2.5.2. Algorithms 93
2.6. Application: segmentation of textured images 100
2.6.1. Textured field and segmented field 100
2.6.2. Multiscale or hierarchical approach 103
2.6.3. Nonsupervised estimation of the parameters 104
2.6.4. Examples of segmentation 108
2.7. Bibliography 109
Chapter 3. 2D Spectral Analysis 115
Claude CARIOU, Stéphanie ROUQUETTE and Olivier ALATA
3.1. Introduction 115
3.2. General concepts 116
3.3. Traditional 2D spectral estimation 118
3.3.1. Periodogram technique 118
3.3.2. Correlogram technique 119
3.3.3. Limits of traditional spectral analysis 120
3.4. Parametric 2D spectral estimation 121
3.4.1. Spectral estimation by linear stochastic models 122
3.4.2. Maximum entropy method 128
3.4.3. Minimum variance method 132
3.5. 2D high resolution methods 134
3.5.1. 2D MUSIC 135
3.5.2. Calculation of a pseudospectrum 135
3.5.3. Pseudospectrum estimation 137
3.6. Other techniques 138
3.7. Comparative study of some techniques 138
3.7.1. Analysis of 2D harmonic components 139
3.7.2. Analysis of random fields 159
3.7.3. Conclusion 163
3.8. Application: spectral analysis of remote sensing images 165
3.8.1. Position of the problem 165
3.8.2. Stochastic modeling of a radar image 166
3.8.3. Example of application 167
3.9. Conclusion 169
3.10. Bibliography 170
Chapter 4. Bispectral Analysis of 2D Signals 175
JeanMarc LE CAILLEC and René GARELLO
4.1. Introduction 175
4.1.1. Higher order moments and cumulants 175
4.1.2. Properties of moments and cumulants 179
4.1.3. Polyspectra of stationary signals 181
4.1.4. Polyspectra 185
4.1.5. Definition of the coherence of order p 185
4.2. Moments and spectra of order p for linear signals 185
4.2.1. Moments and cumulants of order p for linear signals 186
4.2.2. Spectrum of order p for a linear signal 187
4.2.3. General properties of the bispectra of linear signals 187
4.2.4. Polyspectrum of a linear signal 188
4.2.5. Coherence of order p for linear signals 189
4.3. Signals in quadratic phase coupling, nonlinearity and the Volterra system 189
4.3.1. Bispectrum of a signal in quadratic phase coupling 190
4.3.2. Volterra models and decomposition of nonlinear systems 192
4.4. Bispectral estimators for 2D signals 195
4.4.1. Indirect method 196
4.4.2. Direct method 199
4.4.3. Autoregressive model 200
4.4.4. ARMA modeling 202
4.4.5. Measure of bias and variance of estimators 204
4.5. Hypothesis test for nonlinearity and bicoherence tables 204
4.5.1. Hypothesis tests 204
4.5.2. Bicoherence tables 207
4.6. Applications 210
4.6.1. Image restoration 210
4.6.2. Artifact removal 210
4.7. Bibliography 211
Chapter 5. Timefrequency Representation of 2D Signals 215
Stéphane GRASSIN and René GARELLO
5.1. Introduction 215
5.1.1. Bilinear timefrequency representation 215
5.1.2. Four spaces of representation 216
5.1.3. Restriction to bilinear representation 217
5.1.4. Spectral description using bilinear representations 218
5.2. TFR application to sampled images 219
5.2.1. TFR expression of discrete images 219
5.2.2. Support of the sums 223
5.3. Minimum properties and constraints on the kernel 223
5.3.1. Compatibility with reversible linear transformations 224
5.3.2. Positivity 225
5.3.3. TFR with real values 225
5.3.4. Conservation of energy 225
5.3.5. Spectral estimation 226
5.3.6. Evolution of properties of a modified kernel 228
5.4. Notion of analytic images 230
5.4.1. Formulation of the problem for the images 230
5.4.2. Traditional solution 231
5.4.3. Symmetric solution with reference to a hyperplane 233
5.4.4. Solution with a nonsymmetric halfplane 233
5.4.5. Choice of spectral division 237
5.5. Spectral analysis application of SAR images 241
5.5.1. Analysis of an internal waveform 243
5.5.2. Analysis of an internal wave field with superimposition 249
5.5.3. Analysis of a small area internal wave field 249
5.5.4. Prospects 250
5.6. Approximation of an internal wave train 252
5.6.1. Benefit of approximation of the frequency law 252
5.6.2. Problem resolution 252
5.6.3. Adequacy of bilinear modulation with instantaneous frequency estimation 255
5.7. Bibliography 257
Chapter 6. 2D Wavelet Representation 259
Philippe CARRÉ, Noël RICHARD and Christine FERNANDEZ
6.1. Introduction 259
6.2. Dyadic wavelet transform: from 1D to 2D 260
6.2.1. Multiresolution analysis 260
6.2.2. Wavelets and filter banks 262
6.2.3. Wavelet packets 264
6.2.4. 2D extension by the simple product 266
6.2.5. Nonseparable 2D wavelets 272
6.2.6. Nondecimated decomposition 278
6.3. Trigonometric transform to adaptive windows 282
6.3.1. Malvar wavelets 282
6.3.2. Folding operator 284
6.3.3. Windowed orthonormal base 287
6.3.4. Extension of Malvar wavelets to 2D 288
6.4. Transform by frequency slicing 292
6.4.1. Continuous theory of 1D Meyer wavelets 293
6.4.2. Definition of Meyer wavelet packets 295
6.4.3. Numerical outcome of decomposition in 1D Meyer wavelet packets 295
6.4.4. Extension of Meyer wavelet packets to 2D 306
6.5. Conclusion 308
6.6. Bibliography 309
List of Authors 313
Index 315
Author Information
He is also an elected AdCom senior member of the IEEE Oceanic Engineering Society and has headed several committees related to IEEE conferences.
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