Two-Dimensional Signal Analysis

Two-Dimensional Signal Analysis......Page 5
Table of Contents......Page 7
Introduction......Page 15
1.1. Introduction......Page 19
1.2.1. Definition......Page 20
1.2.2. Particular 2-D signals......Page 21
1.3.1. Definition......Page 24
1.3.2. Characterization up to the second order......Page 25
1.3.3. Stationarity......Page 26
1.3.5. Ergodicity......Page 28
1.3.6. Specificities of random 2-D signals......Page 29
1.3.7.1. White noise......Page 30
1.3.7.2. Gaussian process......Page 31
1.4.2. Main 2-D operators......Page 33
1.4.3. Main properties......Page 34
1.4.4. Linear time-invariant (LTI) system......Page 35
1.4.6. Separable system......Page 36
1.4.7. Stability of 2-D systems......Page 38
1.4.8. Support of the impulse response – causality......Page 39
1.5.1. Frequency response of an LTI system......Page 41
1.5.2.1. Definition......Page 43
1.5.2.2. Properties......Page 44
1.5.3.1. Definition......Page 45
1.5.3.2. Properties......Page 46
1.5.3.3. Calculation of the 2-D DFT......Page 47
1.5.4.1. Definition......Page 48
1.5.4.2. Region of convergence......Page 49
1.5.4.3. Properties......Page 51
1.5.4.4. Transfer function of a 2-D system......Page 53
1.5.4.6. Application to the study of stability of LTI systems......Page 55
1.5.4.7. Minimum or non-minimum phase LTI system......Page 56
1.5.5. Frequency characterization of a random 2-D signal.......Page 57
1.5.6. Output of a 2-D system with random input......Page 59
1.6.1. Innovation, determinism and regularity in the 2-D case......Page 60
1.6.2. Total decomposition of three fields......Page 62
1.6.3. Example of an outcome......Page 63
1.8. Bibliography......Page 65
2.1. Introduction......Page 67
2.2.1. Definition......Page 68
2.2.2.1. Causal models......Page 69
2.2.2.2. Causal quarter plane model......Page 72
2.2.2.3. Causal model whose support is delimited by any two NSHPs......Page 73
2.2.2.5. Non-causal model......Page 74
2.3.1. 2-D Markov fields and L-Markovian fields......Page 75
2.3.2. 2-D L-Markovian fields and Gibbs fields......Page 76
2.4.1.1. Estimation criteria by supposing the fixed order......Page 78
2.4.1.2. Probability criteria “penalized” to estimate the order of the model......Page 80
2.4.2. Yule-Walker equations......Page 81
2.4.2.1. Representation of minimum variance and formulation......Page 82
2.4.2.2. Non-causal support and L-Markovian fields......Page 83
2.4.2.3. Causal support and 2-D AR model......Page 85
2.4.3. 2-D Levinson algorithm (for the parametric 2-D AR estimation)......Page 87
2.4.3.1. Recalling the 1-D case......Page 88
2.4.3.2. Approach for 2-D causal and non-causal prediction models......Page 91
2.4.3.3. Multichannel approach and 2-D QP AR model......Page 93
2.4.3.4. Other approaches......Page 94
2.5.2.1. Methods based on a stochastic gradient......Page 95
2.5.2.2. Methods based on the recursive least squares criterion......Page 97
2.5.2.3. Methods based on the geometric approach of the RLS criterion......Page 98
2.6.1. Textured field and segmented field......Page 102
2.6.2. Multiscale or hierarchical approach......Page 105
2.6.3.1. Multinominal distribution......Page 106
2.6.3.2. Information criterion......Page 107
2.6.4.1. Synthesis textures......Page 110
2.7. Bibliography......Page 111
3.1. Introduction......Page 117
3.2. General concepts......Page 118
3.3.1. Periodogram technique......Page 120
3.3.2. Correlogram technique......Page 121
3.3.3. Limits of traditional spectral analysis......Page 122
3.4. Parametric 2-D spectral estimation......Page 123
3.4.1.1. AR model......Page 124
3.4.1.2. ARMA model......Page 128
3.4.1.3. Gauss-Markov model......Page 129
3.4.2. Maximum entropy method......Page 130
3.4.2.2. Implementation......Page 131
3.4.2.3. Example: alternate projection methods......Page 132
3.4.3. Minimum variance method......Page 134
3.5. 2-D high resolution methods......Page 136
3.5.2. Calculation of a pseudo-spectrum......Page 137
3.5.3. Pseudo-spectrum estimation......Page 139
3.7. Comparative study of some techniques......Page 140
3.7.1. Analysis of 2-D harmonic components......Page 141
3.7.1.1. Synthesis of 2-D sinusoidal signals......Page 142
3.7.1.2. General behavior of PSD estimates......Page 143
3.7.1.3. Statistics of frequency estimation......Page 152
3.7.1.4. Separability of two frequency components......Page 155
3.7.1.6. Summary......Page 159
3.7.2. Analysis of random fields......Page 161
3.7.2.2. Spectral estimation of a GM field......Page 162
3.7.3. Conclusion......Page 165
3.8.1. Position of the problem......Page 167
3.8.2. Stochastic modeling of a radar image......Page 168
3.8.3. Example of application......Page 169
3.9. Conclusion......Page 171
3.10. Bibliography......Page 172
4.1.1. Higher order moments and cumulants......Page 177
4.1.2. Properties of moments and cumulants......Page 181
4.1.3. Polyspectra of stationary signals......Page 183
4.2. Moments and spectra of order p for linear signals......Page 187
4.2.1. Moments and cumulants of order p for linear signals......Page 188
4.2.3. General properties of the bispectra of linear signals......Page 189
4.2.4. Polyspectrum of a linear signal......Page 190
4.3. Signals in quadratic phase coupling, non-linearity and the Volterra system......Page 191
4.3.1. Bispectrum of a signal in quadratic phase coupling......Page 192
4.3.2.1. General principles......Page 194
4.3.2.2. Bispectrum of a Volterra model with Gaussian input data......Page 196
4.4. Bispectral estimators for 2-D signals......Page 197
4.4.1. Indirect method......Page 198
4.4.2. Direct method......Page 201
4.4.3. Autoregressive model......Page 202
4.4.4. ARMA modeling......Page 204
4.5.1. Hypothesis tests......Page 206
4.5.2. Bicoherence tables......Page 209
4.6.2. Artifact removal......Page 212
4.7. Bibliography......Page 213
5.1.1. Bilinear time-frequency representation......Page 217
5.1.2. Four spaces of representation......Page 218
5.1.3. Restriction to bilinear representation......Page 219
5.1.4.1. Bilinear representations in time-delay space......Page 220
5.2.1. TFR expression of discrete images......Page 221
5.2.1.1. Autocorrelation function of a discrete image......Page 222
5.2.1.2. Time-frequency representation of a discrete image......Page 224
5.3. Minimum properties and constraints on the kernel......Page 225
5.3.1. Compatibility with reversible linear transformations......Page 226
5.3.4. Conservation of energy......Page 227
5.3.5. Spectral estimation......Page 228
5.3.5.2. Case of signals with bilinear frequency modulation......Page 229
5.3.6.1. Pseudo-smoothed version......Page 230
5.3.6.3. Simply masked version......Page 231
5.4.1. Formulation of the problem for the images......Page 232
5.4.2. Traditional solution......Page 233
5.4.4. Solution with a non-symmetric half-plane......Page 235
5.4.5. Choice of spectral division......Page 239
5.5. Spectral analysis application of SAR images......Page 243
5.5.1.1. Characteristics of the image and preprocessing......Page 245
5.5.1.2. Analysis method......Page 246
5.5.1.3. Presentation of the results and interpretation......Page 248
5.5.3. Analysis of a small area internal wave field......Page 251
5.5.4. Prospects......Page 252
5.6.2. Problem resolution......Page 254
5.6.3.1. Overview of the benefit of this adequacy......Page 257
5.6.3.2. Verification of the adequacy hypothesis......Page 258
5.7. Bibliography......Page 259
6.1. Introduction......Page 261
6.2.1. Multiresolution analysis......Page 262
6.2.2. Wavelets and filter banks......Page 264
6.2.3. Wavelet packets......Page 266
6.2.4. 2-D extension by the simple product......Page 268
6.2.5. Non-separable 2-D wavelets......Page 274
6.2.6. Non-decimated decomposition......Page 280
6.3.1.1. Window function......Page 284
6.3.1.2. Local trigonometric bases......Page 285
6.3.2. Folding operator......Page 286
6.3.3. Windowed orthonormal base......Page 289
6.3.4. Extension of Malvar wavelets to 2-D......Page 290
6.4. Transform by frequency slicing......Page 294
6.4.1. Continuous theory of 1-D Meyer wavelets......Page 295
6.4.3. Numerical outcome of decomposition in 1-D Meyer wavelet packets......Page 297
6.4.3.1. Restriction on positive frequencies......Page 298
6.4.3.3. Definition of window wIs+1/2i for i>0......Page 299
6.4.3.5. Calculation of the coefficients of Meyer wavelet packets for i > 0......Page 301
6.4.3.6. Calculation of wavelet packet coefficients related to Ψs+1 0......Page 304
6.4.3.7. Algorithm and 1-D reconstruction......Page 306
6.4.4. Extension of Meyer wavelet packets to 2-D......Page 308
6.5. Conclusion......Page 310
6.6. Bibliography......Page 311
List of Authors......Page 315
Index......Page 317