MULTIDIMENSIONAL SPECKLE NOISE, MODELLING AND FILTERING RELATED TO SAR DATA. Carlos López Martínez

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1 MULTIDIMENSIONAL SPECKLE NOISE, MODELLING AND FILTERING RELATED TO SAR DATA by Carlos López Martínez Xavier Fàbregas Cànovas, Thesis Advisor Ph.D. Dissertation Thesis Committee: Antoni Broquetas i Ibars Ignasi Corbella i Sanahuja Jong-Sen Lee Eric Pottier Juan Manuel López Sánchez Barcelona, June 2, 2003

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3 A mis padres, José y María del Pilar y hermanos, Sergio y Eva.

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5 -Podrías decirme, por favor, qué camino he de tomar para salir de aquí? -Depende mucho del punto adonde quieras ir -contestó el Gato. -Me da casi igual dónde -dijo Alicia. -Entonces no importa qué camino sigas -dijo el Gato. -...siempre que llegue a alguna parte -añadió Alicia, a modo de explicación. -Ah!, seguro que lo consigues -dijo el Gato-, si andas lo suficiente. -Would you tell me, please, which way I ought to go from here? -That depends a good deal on where you want to get to -said the Cat. -I don t much care where -said Alice. -Then it doesn t matter which way you go -said the Cat. -...so long as I get somewhere -Alice added as an explanation. -Oh!, you re sure to do that -said the Cat-, if you only walk long enough. Alicia en el país de las maravillas, Alice s Adventures in Wonderland, CARROLL, Lewis (Charles Lutwidge Dodgson) ( )

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7 Preface This thesis represents the work carried out during the last four years in the Electromagnetics and Photonics Engineering Group of the Signal Theory and Communications Department at the Technical University of Catalonia (UPC), Barcelona (Spain) and the Institute of High Frequency and Radar Systems of the German Aerospace Center (DLR), Oberpfaffenhofen (Germany). Four years... is a long period of time in which one learns numerous things. But perhaps, the most important lesson for me does not have anything to do with science or technology, but with what one learns about life from the relation with other people. It is for this reason that I want to dedicate the first lines of this thesis to all those people, who, in a greater or smaller degree, have made it possible. Without doubt, this thesis would not has seen the light without the considerable support of my thesis advisor, Xavier Fàbregas Cànovas. His aid and advice, as well as his critical spirit and detailed vision about radar polarimetry have been fundamental in the development of this work. With him, I have had the possibility to maintain multitude of scientific and human discussions, from which I have learned to see things from a more calmed and rational point of view. Also, I am thankful to Antoni Broquetas i Ibars, my thesis tutor and old Master thesis director, for the opportunity he offered to me, five years ago, of becoming part of everything what surrounds SAR. His tireless encouragement to make things, to discover and to try to make things as best as possible, have been, and will be, a reference for me to look up to. At this point, I want to thank to Madhukar Chandra for the possibility he offered me to stay for a year and a half at the DLR. I do not have the smallest doubt that the stay in Germany has been one of the most fructiferous periods of these four years, and a determining factor of the results obtained in this thesis. From the logistics point of view, I must thank Josep Maria Haro and the group of scholarship holders in charge of everything what has to do with computers, for all the help and patience. During these four years, I have had diverse office mates, with which I have shared many and good moments, and whose friendship is one of the most important fruits of this period of my life. From my first period at the UPC, I keep very pleasing memories of Emilio and Alfonso. Special mention deserves Eduard, since although we shared office during a pair of months, since then we maintain a good friendship. During my stay in Germany, I shared office with Vito, who introduced me to the DLR Italian community. During the last part of this thesis, I have shared office with Xavier Fàbregas Cànovas, which has caused him to transform from my thesis advisor to a work colleague and a friend. Also, I would like to mention all the people who belong to the Electromagnetics and Photonics Engineering Group. To the doctorate students who have already finished: Marc, Lluis and especially to Daniel for his advice about the meaning of making a doctorate. To who have still a way to walk: Oscar, Pau, Pablo, Gerard and Jesús, I wish them good lack. To Mercè and Nuria. And also, to Jordi Mallorquí, a.k.a. Carmele, and Jordi Romeu for his special, and sarcastic, sense of humor. Also, I would like to mention the Contubernio, since all of them have participated in and suffered this thesis. vii

8 From my stay in Germany I keep many and very pleasing memories, thanks mainly to the people who I had the opportunity to meet. First of all, I thank to José Luis for making my landing in Germany smooth and simple. To Ralf Horn and Gabrielle Herbst, Gabi, for the moments and laughter that we shared. I have to mention to Irena Hajnsek and Kostantinos P. Papathanassiou, Kostas, since they have become some of my best friends. With them I have shared good, and not so good moments, but despite everything, they have supported me until the end. To meet Kostas has been a privilege for me, since nowadays, in few occasions you have the opportunity to meet straightforward people, but who, at the same time, greatly influence your life. We have shared many discussions, more or less heated, but the most important thing I have learned from him is the secret of being happy with the work which one does. Finally, I want to thank to my beloved parents, José and María del Pilar, for being there and supporting me at any time. Carlos López Martínez Barcelona, March 2003

9 Acknowledgements The author wants to acknowledge the following institutions: DURSI (Departament d Universitats, Recerca i Societat de la Informació de la Generalitat de Catalunya) for providing the pre-doctoral fellowship (Ref. 1999FI-00541) during the development of this thesis. DLR-HR (Deutches Zentrum für Luft- und Raumfahrt e. V. Institut für Hochfrequenztechnik und Radarsysteme) for accepting the author as a guest scientist during his eighteen-month stay from 2000 to 2002, and also for their support in providing data. The European Commission for providing the necessary funding during the stay at DLR-HR in the framework of the project Radar Polarimetry: Theory and Applications (Contract number ERBFM- RXCT980211) of the Training and Mobility of Researchers Programme (TMR). CICYT (Comisión Interministerial de Ciencia y Tecnología) for providing financial support for part of the research undertaken in this thesis, under the project TIC C ix

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11 Abstract Synthetic Aperture Radar (SAR) systems have emerged, during the last decades, as a useful tool to gather and to analyze information from the Earth s surface. Owing to its coherent nature, this type of systems can collect electromagnetic scattering information with a high spatial resolution, but, on the other hand, it yields also speckle. Despite speckle is a true electromagnetic measurement, it can be only analyzed as a noise component due to the complexity associated with the scattering process. A noise model for speckle exists only for single channel SAR systems. Consequently, the work presented in this thesis concerns the definition and the comprehensive validation of a novel series of multidimensional speckle noise models, together with their application to optimal speckle noise reduction and information extraction. First, a speckle noise model for the Hermitian product complex phase component is derived in the spatial domain and translated, subsequently, to the wavelet domain. This analysis is especially relevant to interferometric SAR data. This model demonstrates, on the one hand, that the wavelet transform itself is an interferometric phase noise filter that maintains spatial resolution. On the other hand, it makes possible a high spatial resolution coherence estimation. In a second part, a speckle noise model for the complete Hermitian product is proposed. It is proved that speckle is due to two noise components, with multiplicative and additive natures, respectively. The multidimensional speckle model, relevant for polarimetric SAR data, is finally derived by extending the Hermitian product noise model. From a multidimensional speckle noise reduction point of view, this noise model allows to prove that the covariance matrix entries can be processed differently without corrupting the signal properties. On the other hand, it allows to redefine, and to extend, the principles under which an optimum multidimensional speckle noise model has to be set out. On the basis of these principles, a novel polarimetric speckle noise reduction algorithm is proposed. KEYWORDS Synthetic Aperture Radar (SAR), Multidimensional SAR imagery, SAR Interferometry, SAR Polarimetry, Polarimetric SAR Interferometry, Speckle Noise, Speckle Noise Modelling, Speckle Noise Filtering, Coherence Estimation, Wavelet Transform xi

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13 Contents Preface Acknowledgements Abstract Contents List of Figures List of Tables vii ix xi xvi xxiv xxvi 1 Introduction Radar Remote Sensing Scope and Organization of the Thesis SAR Remote Sensing Synthetic Aperture Radar Basic Concepts on SAR SAR Impulse Response SAR Imaging System Model SAR Image Statistics Speckle Second Order Statistics SAR Speckle Multiplicative Noise Model SAR Interferometry SAR Interferometry Geometric Approach Interferometric SAR System Model Interferometric SAR Data Statistics Interferometric SAR Coherence Phase Difference Noise Model SAR Polarimetry Wave Polarization xiii

14 xiv CONTENTS Wave Scattering The Scattering Matrix The Covariance and Coherency Matrices Covariance Matrix Statistics Polarimetric SAR Speckle Change of Polarization Basis Polarimetric SAR Interferometry Vector Interferometry Information Extraction Wavelet Analysis Introduction Fourier Analysis The Spaces L 1 (R) and L 2 (R) Fourier Series Fourier Transform Time-Frequency Analysis Wavelet Analysis Continuous Wavelet Transform Discrete Wavelet Transform Multiresolution Analysis Fast Discrete Wavelet Transform Two-dimensional Wavelet Transform Discrete Wavelet Packet Transform Construction of Wavelets Wavelet Function s Properties Examples of Wavelets Interferometric Phasor Noise Model Introduction Interferometric Phasor Noise Model Analysis of v Analysis of v Interferometric Phasor Noise Model Validation Multilook Interferometric Phasor Noise Model Wavelet Interferometric Phasor Noise Model True Interferometric Phasor Model One-dimensional Wavelet Interferometric Phasor Noise Model Two-dimensional Wavelet Interferometric Phasor Noise Model

15 CONTENTS xv 4.4 Generalized Wavelet Interferometric Phasor Noise Model Wavelet Interferometric Phasor Statistics A Study Case: Mt. Etna (Italy) Summary Multidimensional Speckle Noise Model Introduction Preliminaries Phase Difference Phasor Noise Model Multidimensional SAR Speckle Noise Model Analysis of zn c exp(jφ x ) Analysis of zv 1 exp(jφ x) Analysis of jzv 2 exp(jφ x) Joint Moments Hermitian Product Speckle Noise Model Generalized Speckle Noise Model Multidimensional Speckle Noise Model Validation and Interpretation Hermitian Product Speckle Noise Model Validation: Simulated PolSAR Data Hermitian Product Speckle Noise Model Validation: Real PolSAR Data Hermitian Product Speckle Noise Model Validation: Extended Validation Polarimetric SAR Interferometric Speckle Noise Model Multifrequency SAR Speckle Noise Model Summary Interferometric Phasor Noise Reduction Introduction State of the Art Interferometric Phase Noise Reduction Interferometric Coherence Estimation Modulated Coherence Estimation: Theory Wavelet Domain Estimation Modulated Coherence Estimation: Algorithm Working Principles Signal Detection Algorithm Description Modulated Coherence Estimation: Results on Int. Phase Interferometric Phase Estimation: Simulated InSAR Data Interferometric Phase Estimation: Real InSAR Data

16 xvi CONTENTS 6.6 Modulated Coherence Estimation: Results on Int. Coherence Interferometric Coherence Estimation: Simulated InSAR Data Interferometric Coherence Estimation: Real InSAR Data Summary Multidimensional Speckle Noise Reduction Introduction State of the Art Multidimensional Coherence Estimation Normalized Covariance Matrix Estimation Normalized Covariance Matrix Estimation: Real PolSAR Data Modulated Coherence Estimation: Concluding Remarks Multidimensional Speckle Noise Filtering: Theory LMMSE Speckle Noise Filter Speckle Noise Terms Separation Multidimensional Speckle Noise Reduction: Algorithm Polarimetric Speckle Noise Reduction: Results Polarimetric Speckle Noise Reduction: Simulated PolSAR Data Polarimetric Speckle Noise Reduction: Real PolSAR Data Multidimensional SAR Data Speckle Filtering: Principles Summary Conclusions and Future Work 227 A Calculation of p v1 (v 1 ) and p v2 (v 2 ) 233 B Calculation of N c 235 C Combination of Variance Expressions 239 D Linear Least Squares Regression Analysis 241 E Calculation of E{z 2 cos(φ)}, E{z c z 2 } and E{z 1 z 2 } 243

17 List of Figures 2.1 Synthetic Aperture Radar concept SAR stripmap geometry SAR raw data focusing steps for a point scatterer. (a) Raw data (phase). (b) Data after range compression. (c) Range cell migration detail. (d) Data after azimuth compression (SAR two-dimensional impulse response) Distributed scatterer scheme Distributed scatterer imaging geometry Two-dimensional random walk modelling the returned echo from a distributed scatterer Complex SAR image distributions for several values of σ. (a) Amplitude. (b) Intensity. (c) Phase Interferometric SAR system geometry InSAR geometry InSAR geometry Interferometric phase statistics. (a) Effect of coherence ρ. (b) Effect of φ x for ρ = Polarization ellipse Reference system. (a) FSA convention. (b) BSA convention Representation of the time-frequency plane and the time-frequency atoms associated with three different functions Tiling of the time-frequency plane carried out by the Fourier transform (a) and the short time Fourier transform (b) Time-frequency plane tiling done by the continuous wavelet transform at discrete positions of the translation parameter b and the dilation parameter a Iterated two branch filter bank to calculate the DWT. (a) Fast DWT. (b) Fast IDWT. This scheme calculates the wavelet transform with two scales: a j represent the coarse approximation coefficients whereas d j are the detail or wavelet coefficients Separable two-dimensional filter bank which calculates the two-dimensional DWT for separable dimensions. (a) Fast two-dimensional DWT. (b) Fast two-dimensional IDWT. The coefficients a j represent a coarse approximation of the original signal whereas d k j are the wavelet coefficients Two-branch filter bank which calculates the biorthogonal wavelet transform. The fast biorthogonal wavelet transform is calculated by the pair of filters (h, g). The fast inverse biorthogonal wavelet transform is calculated by the filters ( h, g) xvii

18 xviii LIST OF FIGURES 4.1 (a) Evolution of p v1 (v 1 ) as a function of the coherence value ρ. (b) Mean value of v 1, N c (a) Actual and approximated values for σ 2 v 1. (b) Approximation absolute error. In both cases α = Evolution of p v2 (v 2 ) as a function of coherence value ρ (a) Exact and approximated values for σ 2 v 2. (b) Approximation absolute error. In both cases α = Curves of σv 2 and σ 2 1 v and the approximated values of σ 2 2 v c and σv 2 s Representation of the interferometric phasor noise model Monte-Carlo analysis to test the validity of the interferometric phasor noise model. Solid lines represent actual values, whereas dashed lines represent the approximated values for the variances σv 2 and σ 2 1 v Error bars represent the variances of the calculated statistics. (a) 2. Real part mean value (N c cos(φ x )). (b) Imaginary part mean value (N c sin(φ x )). (c) Real part variance. (d) Imaginary part variance. It is important to notice that the mean values (a) and (b) present a maximum value close to 0.7 as a consequence of the homogeneous interferometric phase Representation of the term σ vc as a function of the mean value N c cos(φ x ) for a 1024 by 1024 interferometric phase with an slope producing 400-pixel fringes, with a coherence equal to Mt. Etna X-band interferometric phase Interferometric phasor noise model validation over Mt. Etna data. Dashed lines represent the theoretical relation between the mean and the variance values of the interferometric phasor components. The clouds of points represent the real values calculated by the 7 by 7 pixel sample estimators. (a) Real part interferometric phasor components. (b) Imaginary part interferometric phasor components Evolution of the parameter N c as a function of the coherence ρ and the number of looks N. The dashed line represents the coherence, whereas solid lines represent N c for a given number of looks, indicated by the number at each curve Variances of terms v 1 and v 2 as a function on the number of looks (indicated by the numbers) and the coherence. (a) Real part noise variance σv 2 (b) Imaginary part noise 1. variance σ 2 v Example of an ideal one-dimensional interferometric phase with a 25-pixel period. (a) Wrapped interferometric phase φ x. (b) Fourier transform amplitude of φ x. (c) Fourier transform amplitude of cos(φ x ) Equivalent iterated filters for the one-dimensional DWT at the wavelet scale 2 j. (a) Iteration process to derive the residue coefficients {a j } j Z and the equivalent filter, ĥt,j(ω). (b) Iteration process to derive the wavelet coefficients {d j } j Z and the equivalent filter, ĝ T,j (ω). In each case, the index T,j refers to the equivalent filter response at the scale 2 j Equivalent iterated filters for the two-dimensional DWT at the wavelet scale 2 j. The index m refers to the row dimension, whereas n refers to the column dimension Effect of the number of wavelet scales for the DWT on the intensity of the coefficients at the wavelet domain. In this case, the intensity is normalized by 2 2j. The numbers indicate the number of wavelets scales j. The infinite indicates an infinite number of scales, hence, the intensity is directly equal to N 2 c Hierarchical relation among the coefficients of the wavelet domain at different wavelet scales, but containing information about the same area of the original image

19 LIST OF FIGURES xix 4.18 Effect of the number of wavelet scales j over the wavelet coefficients distributions. In all the cases the quotient N c /σ corresponds to a coherence equal to 0.6, which is approximately equal to As observed, the larger the wavelet scale the higher the mean amplitude and the lower the phase noise content. (a) Amplitude distribution. (b) Phase distribution Mt. Etna Interferogram. (a) Phase. (b) Coherence DWT of the interferometric phasor corresponding to the Mt. Etna data. The DWT consists of three transformed scales calculated with the Daubechies wavelet filter of the coefficients. (a) DWT of the interferometric phasor s real part, R{DWT 2D {e jφ }}. (b) Zoom corresponding to the third wavelet scale. (c) Wavelet coefficients amplitude, DWT 2D {e jφxw }. (d) Zoom corresponding to the third wavelet scale. (e) Wavelet interferometric phase arg { DWT 2D {e jφ } }. (f) Zoom corresponding to the third wavelet scale Contour plots of the joint pdf p zr,z I (z R,z I ) for the following values of φ x, (a) 0 rad (b) π/4 rad (c) π/2 rad and ρ = 0.5. It can be observed that φ x only introduces a rotation in the complex plane. In all the cases, the maximum is located in the axes origin Evolution, as a function of ρ, of the distributions: (a) p z (z) and (b) p zc (z c ). In the second case, plots have been truncated for visualization convenience (a) E { R{ze jφ } 1 } as a function of coherence ρ. (b) Standard deviation as a function of the mean for the Hermitian product amplitude and the term R{ze jφ } 1. In both cases ψ = 1 and cos(φ x ) = 1. The direction of increasing ρ is indicated Absolute error introduced by the approximation of the standard deviation of the first additive term of the Hermitian product real part, std { R{ze jφ } 1 } = E { R{ze jφ } 1 } std { R{ze jφ } 1 }. The parameters ψ and cos(φx ) are assumed to be equal to Distribution p z1 (z 1 ) for the whole coherence range (a) E { R{ze jφ } 2 } a as function of coherence ρ. (b) Actual and approximated values of var { R{ze jφ } 2 } for the whole coherence range. A detail around ρ = 0.5 is also presented. In both cases one assumes ψ = 1 and cos(φ x ) = Absolute error of the approximated value of var { R{ze jφ } 2 } obtained as var { R{ze jφ } 2 } = 1 2 ψ(1 ρ 2 ) 1.64 cos 2 (φ x ) var { R{ze jφ } 2 }. Also for this case the parameters ψ and cos(φx ) are assumed to be equal to Pdf p z2 (z 2 ) as a function of coherence. The power parameter ψ, and the phase parameter cos(φ x ) are assumed to be equal to Covariances considering correlation properties between the first and the second additive terms of the Hermitian product. In this case: ψ = 1 and cos(φ x ) = Graphics corresponding to the variances σ 2 n a1, σ 2 n a2, σ 2 n ar and σ 2 n ai. These variances are described by curves of the type σ 2 = (1/2)(1 ρ 2 ) α, where α is 1.64, 1, 1.32 and 1.32 for each variance, respectively Graphical representation of the complex Hermitian product speckle noise model Monte-Carlo analysis to test the validity of the speckle noise model for the real part of the Hermitian product of a pair of SAR images. (a) Mean value for the multiplicative term. (b) Standard deviation for the multiplicative term. (c) Mean value for the additive term. (d) Standard deviation for the additive term. Solid lines represent theoretical values as a function of ρ, whereas dashed lines represent the approximated values. Crosses represent the estimated values. In all the cases φ x = 0 rad and ψ =

20 xx LIST OF FIGURES 5.13 Monte-Carlo analysis to test the validity of the speckle noise model for the imaginary part of the Hermitian product of a pair of SAR images. (a) Mean value for the multiplicative term. (b) Standard deviation for the multiplicative term. (c) Mean value for the additive term. (d) Standard deviation for the additive term. Solid lines represent theoretical values as a function of ρ, whereas dashed lines represent the approximated values. Crosses represent the estimated values. In all the cases φ x = 0 rad and ψ = Monte-Carlo analysis to test the parameter C 12 for the Hermitian product. (a) Real part. (b) Imaginary part. Solid lines represent theoretical values. Crosses represent the estimated values. In all the cases φ x = 0 rad and ψ = Monte-Carlo analysis to test the speckle noise model for the Hermitian product. (a) Standard deviation value for the multiplicative component of the real part. (b) Standard deviation value for the additive component of the real part. (c) Standard deviation value for the multiplicative component of the imaginary part. (d) Standard deviation value for the additive component of the imaginary part. Solid lines represent theoretical values as a function of ρ, whereas, dashed lines represent the approximated values. Crosses represent estimated values. In all the cases ρ = rad and ψ = Oberpfaffenhofen test site. Complex correlation coefficient ρ corresponding to the covariance matrix element S hh S vv. (a) Amplitude ρ. (b) Phase φ x (rad) Global scatter diagram for R{S hh Svv } in the case of the Oberpfaffenhofen test site Scatter diagrams for R{S hh Svv } in the case of the Oberfapfenhoffen test site Comparison between real and theoretical values for the different components of the speckle noise model for a high coherence area, ρ = exp(j0.331). (a) Hermitian product real part R{S hh S vv}. (b) Hermitian product imaginary part I{S hh S vv}. Solid and dashed lines represent actual and approximated values respectively. Points indicate data statistics where color refers to density, ranging from low densities (blue) to high densities (red) Comparison between real and theoretical values for the different components of the speckle noise model for a low coherence area, ρ = 0.389exp( j0.528). (a) Hermitian product real part R{S hh S vv}. (b) Hermitian product imaginary part I{S hh S vv}. Solid and dashed lines represent actual and approximated values respectively. Points indicate data statistics where color refers to density, ranging from low densities (blue) to high densities (red) Oberpfaffenhofen test site. Complex correlation coefficient ρ corresponding to the extended covariance matrix element S hh1 S vv 2. (a) Amplitude ρ. (b) Phase φ x rad Global scatter diagram for R{S hh1 S vv 2 } in the case of the Oberpfaffenhofen test site Scatter diagrams for R{S hh1 S vv 2 } in the case of the Oberfapfenhoffen test site Standard deviations for the Hermitian product multiplicative and additive terms in the case of S hh1 S vv 1, a polarimetric covariance matrix term, and S hh1 S vv 2, a polarimetric interferometric covariance matrix term Relative power increment between signal and noise Qualitative representation of the measured interferometric phasor DWT. Vector lengths are drawn similar to facilitate visualization. a j denotes the coarse approximation wavelet coefficients and d j represent the detail wavelet coefficients Values of Γ Wiener and Γ sig for the first three wavelet scales, j {1,2,3}. In the case of Γ sig, the maximum wavelet scale is 2 j = Scheme of the algorithm defined to estimate the modulated coherence term in the wavelet domain

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