Two-dimensional Signal Analysis

René Garello is a Professor at the Ecole Nationale Supérieure des Télécommunications de Bretagne, Brest, France. He has been Head of the TIME Laboratory (Information Processing for the Ocean and the Environment) of the CNRS, France since January 2003.He is also an elected AdCom senior member of the IEEE Oceanic Engineering Society and has headed several committees related to IEEE conferences.

• Introduction 13Chapter 1. Basic Elements of 2-D Signal Processing 17Claude CARIOU, Olivier ALATA and Jean-Marc LE CAILLEC1.1. Introduction 171.2. Deterministic 2-D signals 181.2.1. Definition 181.2.2. Particular 2-D signals 191.3. Random 2-D signals 221.3.1. Definition 221.3.2. Characterization up to the second order 231.3.3. Stationarity 241.3.4. Characterization of orders higher than two 261.3.5. Ergodicity 261.3.6. Specificities of random 2-D signals 271.3.7. Particular random signals 281.4. 2-D systems 311.4.1. Definition 311.4.2. Main 2-D operators 311.4.3. Main properties 321.4.4. Linear time-invariant (LTI) system 331.4.5. Example 341.4.6. Separable system 341.4.7. Stability of 2-D systems 361.4.8. Support of the impulse response – causality 371.5. Characterization of 2-D signals and systems 391.5.1. Frequency response of an LTI system 391.5.2. 2-D Fourier transform 411.5.3. Discrete 2-D Fourier transform 431.5.4. 2-D z transform 461.5.5. Frequency characterization of a random 2-D signal 551.5.6. Output of a 2-D system with random input 571.6. 2-D Wold decomposition 581.6.1. Innovation, determinism and regularity in the 2-D case 581.6.2. Total decomposition of three fields 601.6.3. Example of an outcome 611.7. Conclusion 631.8. Bibliography 63Chapter 2. 2-D Linear Stochastic Modeling 65Olivier ALATA and Claude CARIOU2.1. Introduction 652.2. 2-D ARMA models 662.2.1. Definition 662.2.2. 2-D ARMA models and prediction supports 672.3. L-Markovian fields 732.3.1. 2-D Markov fields and L-Markovian fields 732.3.2. 2-D L-Markovian fields and Gibbs fields 742.4. “Global” estimation methods 762.4.1. Maximum likelihood 762.4.2. Yule-Walker equations 792.4.3. 2-D Levinson algorithm (for the parametric 2-D AR estimation) 852.5. “Adaptive” or “recursive” estimation methods 932.5.1. Connectivity hypotheses for adaptive or recursive algorithms 932.5.2. Algorithms 932.6. Application: segmentation of textured images 1002.6.1. Textured field and segmented field 1002.6.2. Multiscale or hierarchical approach 1032.6.3. Non-supervised estimation of the parameters 1042.6.4. Examples of segmentation 1082.7. Bibliography 109Chapter 3. 2-D Spectral Analysis 115Claude CARIOU, Stéphanie ROUQUETTE and Olivier ALATA3.1. Introduction 1153.2. General concepts 1163.3. Traditional 2-D spectral estimation 1183.3.1. Periodogram technique 1183.3.2. Correlogram technique 1193.3.3. Limits of traditional spectral analysis 1203.4. Parametric 2-D spectral estimation 1213.4.1. Spectral estimation by linear stochastic models 1223.4.2. Maximum entropy method 1283.4.3. Minimum variance method 1323.5. 2-D high resolution methods 1343.5.1. 2-D MUSIC 1353.5.2. Calculation of a pseudo-spectrum 1353.5.3. Pseudo-spectrum estimation 1373.6. Other techniques 1383.7. Comparative study of some techniques 1383.7.1. Analysis of 2-D harmonic components 1393.7.2. Analysis of random fields 1593.7.3. Conclusion 1633.8. Application: spectral analysis of remote sensing images 1653.8.1. Position of the problem 1653.8.2. Stochastic modeling of a radar image 1663.8.3. Example of application 1673.9. Conclusion 1693.10. Bibliography 170Chapter 4. Bispectral Analysis of 2-D Signals 175Jean-Marc LE CAILLEC and René GARELLO4.1. Introduction 1754.1.1. Higher order moments and cumulants 1754.1.2. Properties of moments and cumulants 1794.1.3. Polyspectra of stationary signals 1814.1.4. Polyspectra 1854.1.5. Definition of the coherence of order p 1854.2. Moments and spectra of order p for linear signals 1854.2.1. Moments and cumulants of order p for linear signals 1864.2.2. Spectrum of order p for a linear signal 1874.2.3. General properties of the bispectra of linear signals 1874.2.4. Polyspectrum of a linear signal 1884.2.5. Coherence of order p for linear signals 1894.3. Signals in quadratic phase coupling, non-linearity and the Volterra system 1894.3.1. Bispectrum of a signal in quadratic phase coupling 1904.3.2. Volterra models and decomposition of non-linear systems 1924.4. Bispectral estimators for 2-D signals 1954.4.1. Indirect method 1964.4.2. Direct method 1994.4.3. Autoregressive model 2004.4.4. ARMA modeling 2024.4.5. Measure of bias and variance of estimators 2044.5. Hypothesis test for non-linearity and bicoherence tables 2044.5.1. Hypothesis tests 2044.5.2. Bicoherence tables 2074.6. Applications 2104.6.1. Image restoration 2104.6.2. Artifact removal 2104.7. Bibliography 211Chapter 5. Time-frequency Representation of 2-D Signals 215Stéphane GRASSIN and René GARELLO5.1. Introduction 2155.1.1. Bilinear time-frequency representation 2155.1.2. Four spaces of representation 2165.1.3. Restriction to bilinear representation 2175.1.4. Spectral description using bilinear representations 2185.2. TFR application to sampled images 2195.2.1. TFR expression of discrete images 2195.2.2. Support of the sums 2235.3. Minimum properties and constraints on the kernel 2235.3.1. Compatibility with reversible linear transformations 2245.3.2. Positivity 2255.3.3. TFR with real values 2255.3.4. Conservation of energy 2255.3.5. Spectral estimation 2265.3.6. Evolution of properties of a modified kernel 2285.4. Notion of analytic images 2305.4.1. Formulation of the problem for the images 2305.4.2. Traditional solution 2315.4.3. Symmetric solution with reference to a hyperplane 2335.4.4. Solution with a non-symmetric half-plane 2335.4.5. Choice of spectral division 2375.5. Spectral analysis application of SAR images 2415.5.1. Analysis of an internal waveform 2435.5.2. Analysis of an internal wave field with superimposition 2495.5.3. Analysis of a small area internal wave field 2495.5.4. Prospects 2505.6. Approximation of an internal wave train 2525.6.1. Benefit of approximation of the frequency law 2525.6.2. Problem resolution 2525.6.3. Adequacy of bilinear modulation with instantaneous frequency estimation 2555.7. Bibliography 257Chapter 6. 2-D Wavelet Representation 259Philippe CARRÉ, Noël RICHARD and Christine FERNANDEZ6.1. Introduction 2596.2. Dyadic wavelet transform: from 1-D to 2-D 2606.2.1. Multiresolution analysis 2606.2.2. Wavelets and filter banks 2626.2.3. Wavelet packets 2646.2.4. 2-D extension by the simple product 2666.2.5. Non-separable 2-D wavelets 2726.2.6. Non-decimated decomposition 2786.3. Trigonometric transform to adaptive windows 2826.3.1. Malvar wavelets 2826.3.2. Folding operator 2846.3.3. Windowed orthonormal base 2876.3.4. Extension of Malvar wavelets to 2-D 2886.4. Transform by frequency slicing 2926.4.1. Continuous theory of 1-D Meyer wavelets 2936.4.2. Definition of Meyer wavelet packets 2956.4.3. Numerical outcome of decomposition in 1-D Meyer wavelet packets 2956.4.4. Extension of Meyer wavelet packets to 2-D 3066.5. Conclusion 3086.6. Bibliography 309List of Authors 313Index 315