Study and Applications of POLSAR Data Time-Frequency Correlation Properties

Study and Applications of POLSAR Data Time-Frequency Correlation Properties L. Ferro-Famil 1, A. Reigber 2 and E. Pottier 1 1 University of Rennes 1, Institute of Electronics and Telecommunications of Rennes. Campus de Beaulieu, Bat 11D, 263 Avenue General Leclerc, F-3542 Rennes, France 2 Berlin University of Technology, Computer Vision and Remote Sensing. Franklinstr. 28/29 FR3-1, D-1587 Berlin, Germany. Email: anderl@cs.tu-berlin.de ABSTRACT This article presents a new approach for the study of PolSAR images using Time-Frequency (TF) correlation properties. PolSAR information is analyzed using a linear time-frequency (TF) decomposition which permits to describe a scene polarimetric behavior for different azimuth angles of observation and frequencies of illumination. A TF signal model is proposed and studied using two statistical descriptors related to the signal stationary aspect and coherence in the time-frequency domain. These indicators are shown to provide complementary information for an enhanced description of the scene. A TF based classification is proposed and applied to fully polarimetric SAR data acquired by the ESAR sensor at L- band. Finally, some description of PolSAR data TF correlation properties are proposed and commented. 1 Introduction Conventional scattering analysis and geophysical parameter retrieval techniques from strip-map synthetic aperture radar (SAR) data generally assume that scenes are observed in the direction perpendicular to the flight track and at a fixed frequency, equal to the emitted signal carrier frequency. These assumptions may lead to erroneous interpretations over complex targets, characterized by anisotropic geometrical structures, showing varying electromagnetic behavior as they are illuminated from different positions and at different frequency components during SAR integration. This paper presents a new approach for the study of PolSAR data, using a TF analysis based on second order statistics evaluated in the azimuth direction. Section 2 introduces the principle of TF analysis. In part 3, a TF signal model is proposed and tested by means of two statistical descriptors. Applications are presented in section 4 and a refined description of PolSAR TF correlation properties and their stability are introduced in 5. 2 TF analysis of PolSAR images 2.1 Principle of SAR images TF analysis The time-frequency approach developed in this study is based on the use of a two-dimensional windowed Fourier transform, or 2D Gabor transform. This kind of transformation permits to decompose of a two-dimensional signal, s(l), with l = [x, y], into different spectral components, using a convolution with an analyzing function g(l), as follows [1]: s(l ; ω ) = s(l)g(l l )exp (jω (l l )) dl (1) where ω = [ω x, ω y ] indicates a position in frequency, and s(l ; ω ) represents the decomposition result around the spatial and frequency locations l and ω. The application of a Fourier transform to (1) shows that the spectrum of s(l ; ω ) is given by the following relation: S(ω; ω ) = S(ω)G(ω ω ) (2) Equations (1) and (2) indicate that this TF approach may be used to characterize, in the spatial domain, behaviors corresponding to particular spectral components of the signal under analysis, selected by the analyzing function g(). The time and frequency resolutions of the TF analysis are fixed by the analysis functions and verify the following Heisenberg inequality [1]: l ω r (3) where r is a constant term. The expression given in (3) implies a compromise between space and frequency resolutions in order to perform a selective frequency scanning of the signal while maintaining the spatial resolution to values that permit details discrimination. SAR images are built from measurements of a scene electromagnetic response over a large frequency band and a wide range of observation angles. In the case of already synthesized SLC SAR images,

the spectral position, ω = [ωrg, ωaz ] can be linked to the frequency of illumination, f, and to the angle of observation in azimuth, φ, as : ωrg = 4π (f fc ), c ωaz = 4π fc sin(φ), c (4) where fc is the carrier frequency and c the light velocity. Variations of the backscattered signal with ω may be interpreted as anisotropic or frequency selective behaviors of particular media or scatterers and linked to some of their physical properties [2 6]. This paper concentrates on the TF analysis of SAR data characteristics in the azimuth direction which provides the most useful information for building characterization. 2.2 Application to PolSAR data 1 H π/4 α( ) Figure 1 shows a color-coded polarimetric SAR image of the city of Dresden acquired by DLR E-SAR sensor data at L-band. The scene is mainly composed of built-up areas including vegetation spots. A forest and a park can be seen on the left part of the image and a river with smooth banks is located on the right part. azimuth range Figure 2: H (top) and α parameters over Dresden city double bounce reflexions. Buildings which do not face the radar track are characterized by a strong cross-polarization component and high entropy and can hardly be discriminated from vegetated areas. Figure 1: Color coded image of the Dresden test site (Pauli basis) The entropy image shown in Fig.2 reveals that the polarimetric behavior of most of the scene is highly random. Over urban areas, the polarimetric response is composed of a large number of different polarimetric contributions originating from complex building structures as wall as from surrounding vegetation. The resulting high entropy involves that an interpretation of polarimetric indicators may not be relevant. Over buildings aligned with the flight track direction, the entropy has intermediate values and the α parameter reveals the presence of dominant single and 2.3 Polarimetric continuous TF analysis Figure 3 presents a continuous TF analysis in the azimuth direction of three different media : a building facing the radar, an oriented building, and a forested area. The SPAN, corresponding to the sum of the intensities in all polarimetric channels, and the polarimetric α angle are computed for the different media at each frequency location and mean values are then estimated over pixels belonging to the object. A non stationary behavior is clearly visible in Fig.3(a) with a sudden a large variation of both SPAN and α levels with the observation angle in azimuth φ. This anisotropic behavior is due to the highly directional patterns of coherent scattering mechanisms

SPAN (db) α( ) 3 TF analysis using PolSAR second order statistics 3.1 PolSAR data TF model Azimuth look angle ( ) (a) Building facing the radar The proposed TF signal model is given by the following expression, where the spatial coordinates, l, have be omitted : s(ω) = t(ω) + c(ω) (5) SPAN (db) α( ) The signal s(ω) contains the full coherent polarimetric polarization information and can be associated to a well-known scattering vector [8] : k(ω) = 1 2 [S hh (ω) + S vv (ω), S hh (ω) + S vv (ω), 2S hv (ω)] T SPAN (db) Azimuth look angle ( ) (b) Oriented building α( ) (6) where S pq (ω) represents an element of the (2 2) scattering matrix S sampled at the frequency coordinates ω. The signal described in (5) is composed of two contributions : Azimuth look angle ( ) (c) Forested area Figure 3: Continuous TF analysis in the azimuth direction (SPAN in red, α in blue) which may occur as the radar faces a large artificial structure, such as a building [7]. This particular effect can only be observed if the building orientation with respect to the radar flight track falls within the processed antenna azimuth aperture. On the contrary, oriented buildings and vegetated areas, like forest patches, show stationary behaviors. The identification of buildings from their TF response thus requires an additional criterion to complement the stationarity information. It is known that man-made objects are likely to have a coherent response, whereas natural media may be considered as random. The discrimination of such responses can be achieved by studying the coherence of the backscattered polarimetric signal in the Time- Frequency domain and requires the use of an adequate TF polarimetric SAR (PolSAR) signal model. The term t(ω) is highly coherent and can be associated to a deterministic or almost deterministic target response. Depending on the structure of the observed object, the response can remain constant during the SAR acquisition, or can be non-stationary if the backscattering behavior is sensitive to the azimuth angle of observation or illumination frequency. The second term, c(ω), represents the response of distributed environments. It is uncorrelated, but may follow a non-stationary behavior in particular cases, e.g. vegetated terrains with a strong topography, very dense environments whose response results from the sum of a large number of uncorrelated contributions. This composite model may be tested using s(ω) second order statistics : The coherence of s(ω) can be used to determine the dominant component within the pixel under consideration. A high value indicates that t(ω) is the most important term in (5), and a low one corresponds to scattering from an incoherent, distributed, medium. The stability of the dominant component can then be tested by studying the stationarity of the variance of s(ω) 3.2 TF second order statistics Due to the signal high dimensionality, usual scalar tools are not well adapted to the study of secondorder TF polarimetric statistics. A polarimetric TF

target vector is built by gathering the PolSAR information sampled at R spectral coordinates ω i, i = 1,...,R. k TF = [ k T (ω 1 ),...,k T (ω R ) ] T (7) The sampling coordinates, ω i, and the frequency domain resolution of the analyzing function g are chosen so that the the R sub-spectra do not overlap and span the whole full resolution spectrum [4]. A polarimetric TF sample covariance matrix, T TF Pol, is then computed as follows T 11 T 1R T TF Pol = k TF k TF =..... T R1 T RR where T ij = k(ω i )k(ω j ) (8) 3.2.1 Indicator of stationary behavior Stationarity is assessed by testing the fluctuations of the variance of the signal at the the different spectral locations. In the polarimetric case, the signal sample variance is given by a (3 3) polarimetric coherency matrix, i.e. by the diagonal terms of the T TF Pol matrix : {T ii } i=1..r. The polarimetric TF response is considered as stationary if the sample T ii matrices, assumed to follow independent complex Wishart distributions T ii W C (n i,σ ii ) with n i looks, have the same expectation Σ. The corresponding hypothesis is given by : H : Σ 11 = = Σ RR = Σ (9) The corresponding Maximum Likelihood (ML) ratio is : max Σ L(Σ,...,Σ) Λ = (1) max Σii L(Σ 11,...,Σ RR ) The likelihood terms in (1) are maximized by replacing the expectation matrices by the ML estimates and the hypothesis is tested using the resulting ML test [2], which takes the following form : R i=1 Λ = T i ni (11) T t nt R i=1 where T t = n it ii R R i=1 n and n t = n i i i=1 Fig.4 presents a log-image of the Λ parameter on the Dresden test site, obtained with 4 spectral coordinates in the azimuth direction over the Dresden test site. The Λ parameter reache high values over natural areas indicating a stationary spectral behavior. Over buildings, Λ decreases, pointing out the invalidity of the stationary hypothesis over such objects. Highly log Λ min log Λ log Λ max Figure 4: Maximum likelihood log-ratio image anisotropic pixels, such as those corresponding to the wall-ground dihedral reflection or specular reflection from oriented roofs are clearly identified in Fig.4 due to their very low stationary aspect. 3.2.2 Indicator of coherent behavior In [5], the eigenvalues of a single-polarization covariance matrix have been used to derive a coherency indicator. These eigenvalues carry information on the correlation structure, but are also sensitive to potential PolSAR fluctuations due to non-stationarity. Under the hypothesis of uncorrelated spectral responses, the off-diagonal terms of the TF covariance matrix verify: H : Σ ij = i j (12) The corresponding ML ratio is given by : Θ = max Σ ii L(Σ 11,...,Σ RR ) max ΣT F L(Σ TF ) = T TF ni R i=1 T ii ni (13) This ML ratio expression can be rewritten as: Θ = T TF n i (14) with I Γ 12 Γ 1R T TF Pol = Γ. 12 I...... Γ 1R I where Γ ij = T ii 1/2 T ij T jj 1/2 (15)

The normalized covariance matrix, T TF Pol results from the whitening of the TF polarimetric covariance matrix by the separate polarimetric information at each frequency location. This representation is then insensitive to spectral polarimetric intensity variations and is characterized by its off-diagonal matrices Γ ij which can be viewed as an extension of the scalar normalized correlation coefficient to the polarimetric case. The ML ratio in (13) is a function of the eigenvalues of T TF Pol, which reflect the correlation structure : flat for decorrelated responses ( T TF Pol I d ), heterogeneous for correlated ones. Taking into account T TF Pol peculiar form, a correlation indicator, named TF-Pol coherence, can be defined as : 1 ρ TF Pol = 1 T TF Pol 3R (16) Figure 5 presents an image of ρ TF Pol over the Dresden test site, computed from 4 spectral locations in the azimuth direction. log Λ The stationarity and coherence indicators derived above can be merged to classify the scene. Both ρ TF Pol and Λ parameters are thresholded and comρtf Pol Figure 6: TF-Pol classification. (Top) Classification obtained on the Dresden test site. (Bottom) Associated 2D histogram bined into four classes. The application of the fusion strategy over the Dresden site is shown in Fig.6. The resulting map permits a good estimation of building locations. A physical interpretation can be given for each of the four classes : Coherent and stationary pixels (white class): The t term in (5) is dominant and constant during the SAR acquisition. This kind of behavior corresponds to strong scatterers with an isotropic response, like oriented buildings, lampposts,.... ρ 1 TF Pol Figure 5: ρ TF Pol parameter As expected, the TF-Pol coherence is high over buildings due to the presence of strong coherent reflectors. It can be also noticed that buildings are identified independently of their orientation. 4 Applications to urban area remote sensing 4.1 PolSAR TF classification Coherent and non stationary pixels (yellow class): The t contribution dominates but varies during the measure, causing fluctuation of the signal with the azimuth angle of observation. This anisotropic effect is characteristic of buildings facing the radar track, whose response is affected by a strong and highly directive pattern, mainly due double bounce reflections or specular single bounce reflection over roofs tilted toward the radar. Incoherent and stationarity pixels (green class): The uncorrelated component c dominates and has stable second order statistics. This class corresponds to natural environments (forests, fields, grass areas,... ) of distributed artificial media such as roads, roof tops, terraces,... Incoherent and non-stationary pixels (red class): This class indicates the presence of complex

ary signal s(t) is defined as : ϕss ( t) = hs(t)s (t + t)it (17) where the averaging is performed over the whole domain of definition of s(t). For non-stationary signals, i.e. signals whose statistical properties vary with t, this definition is generally unapropriate and is replaced by the local autocorrelation : ϕss (t, t) = hs(t + t)s (t + t + t)it Dt π/2 Figure 7: αt F parameter over pixels detected as coherent or non-stationary scattering contributions, which change during the SAR integration, and sum-up in an incoherent way, like in layover areas. 4.2 Polarimetric analysis enhancement As it was shown in Fig.2, the full resolution PolSAR information can hardly be used to analyze the scene geophysical properties due to a very high entropy inherent to the study of dense environments. The proposed PolSAR TF analysis technique can also be used to improve in a significant way the interpretation of polarimetric indicators. The most coherent TF scattering mechanisms is described by the first eigenvector of T TF Pol, which can be transformed back to the H-V polarimetric basis using a matrix P, satisfying T TF Pol = PTTF Pol P. From this eigenvector, one can extract an αt F parameter which shows a much more contrasted and relevant information than the original full resolution parameter α, Fig.7. 5 5.1 PolSAR TF correlation properties PolSAR TF autocorrelation function PolSAR TF correlation characteristics have been partially studied in section 3 by testing the presence of correlation structure between different frequency locations. The analysis of PolSAR TF correlation characteristics over pixels detected as coherent may provide more information on the nature of scattering mechanisms and require the use of TF autocorrelation function. The sample autocorrelation function of a station- (18) where the correlation is computed around the time location t, within a restricted domain given by Dt. The autocorrelation of a TF signal s(l; ω) can be defined in several different ways, depending on the domain of study. In this paper, we concentrate on the analysis of the signal spectral correlation properties, and use the following autocorrelation expression: ϕss (l; ω, ω) = hs(l; ω )s (l; ω + ω)il Dl (19) where the sample mean is performed in the space domain, whereas the signal shift is realize in the frequency domain. This expression may be normalized by the signal energy around the frequency locations ω and ω + ω : ϕss (l; ω, ω) νss (l; ω, ω) = p ϕss (l; ω, ) ϕss (l; ω + ω, ) (2) where νss 1 The expression given in (2) can be generalized to the polarimetric case using : ϕss (wω, wω + ω ; l; ω, ω) = wω T(l; ω, ω)wω + ω (21) where wω and wω + ω represent 3-element projection vectors [9] at the analysis frequency locations and T (l; ω, ω) is a TF (3 3) correlation matrix defined as : T(l; ω, ω) = s(l; ω )s (l; ω + ω) (22) Figure 8 represents the mean and maximal normalized correlation values in the hh polarization channel, evaluated over a range of spactral locations ω for a given frequency lag ω. The images in Fig.8 clearly indicate that even on pixels detected as coherent, the correlation information is not constant over the SAR acquisition, due to changes in both coherence level and type of scattering mechanism. The coherence optimization procedure described in [9] in the PolinSAR case may be applied to the autocorrelation function defined in

1 Figure 8: TF correlation in hh polarization over coherent pixels meanω [νss hh (l; ω, ω)] (top), maxω [νss hh (l; ω, ω)] (bottom) Figure 9: Maximum polarimetric coherence over coherent detected pixels (top), correlation polarization sensitivity indicator A (bottom) 5.2 (2) : wω,wω + ω with νopt1 νopt2 νopt3 (23) Similarly to the PolinSAR case, a simple indicator may be built to assess the sensitivity of the optimal correlation information to polarization [1]: νopt1 νopt3 νopt1 PolSAR TF correlation stability metrics A way of quantifying the stability of the PolSAR TF correlation consists in studying the variation of the most coherent scattering mechanisms, provided by the first eigenvector of the optimization procedure first described in [9]: νopti (l; ω, ω) = ν(wω, wω + ω ; l; ω, ω) max A= 1 (24) Over pixels whose TF correlation remain constant over the whole polarization space A is close, whereas over scatterers having a highly varying correlation behavior with polarization, A reaches 1. {wω opt1, wω + ω opt1 } = argmax ν(wω, wω + ω ; l; ω, ω) (25) The projection vectors wω opt1 and wω + ω opt1 represent the scattering mechanisms at the analysis frequency locations wich maximize the TF correlation. These quantities can be converted to scattering matrices S(ω opt1 ) and S(ω + ω opt1 ) from which a polarimetric distance can be computed [11]. In Fig.1 are displayed the cross- and co-pol nulls components of the polarimetric distance between S(ω opt1 ) and S(ω + ω opt1 ) over coherent pixels. The difference between these distance metrics

resides in the fact that cross-pol nulls variatons can be compensated via a change of polarization basis, whereas co-pol nulls variations remain invariant under such a transformation.the generally high values of the cross-polarized distance and low co-pol nulls distance reveal tha the optimal polarimetric scattering mechanisms at two different frequency locations significantly differ and correspond to changes of polarization basis which might be due to variations of the azimuth aspect angle. larimetric sar data, IEEE Trans. Geosci. Remote Sensing, vol. 41, no. 1, pp. 2264 2276, October 23. [3] J.-C. Souyris, C. Henry, and F. Adragna, On the use of complex sar image spectral analysis for target detection: assessment of polarimetry, IEEE Trans. Geosci. Remote Sensing, vol. 41, no. 12, pp. 2725 2734, December 23. [4] L. Ferro-Famil, A. Reigber, and E. Pottier, Nonstationary natural media analysis from polarimetric sar data using a two-dimensional time-frequency decomposition approach, Can. J. Remote Sensing, vol. 31, no. 1, pp. 21 29, 25. [5] R. Zandona-Schneider, K. P. Papathanassiou, I. Hajnsek, and A. Moreira, Polarimetric and interferometric characterization of coherent scatterers in urban areas, IEEE Trans. Geosci. Remote Sensing, vol. 44, no. 4, pp. 971 984, April 26. [6] P. Leducq, L. Ferro-Famil, and E. Pottier, Analysis of polsar data of urban areas using time-frequency diversity, in EUSAR, May 26. π [7] G. Franceschetti, A. Iodice, D. Riccio, and G. Ruello, Sar raw signal simulation for urban structures, IEEE Trans. Geosci. Remote Sensing, vol. 41, no. 9, pp. 1986 1994, September 23. [8] S. R. Cloude and E. Pottier, An entropy based classification scheme for land applications of polarimetric sar, IEEE Trans. Geosci. Remote Sensing, vol. 35, no. 1, pp. 68 78, January 1997. [9] S. R. Cloude and K. P. Papathanassiou, Polarimetric sar interferometry, IEEE Trans. Geosci. Remote Sensing, vol. 36, no. 5, pp. 1551 1565, September 1998. Figure 1: Polarimetric distances between maximally coherent scattering mechanisms: cross-pol nuls distance (top) co-pol nuls distance (bottom) [1] L. Ferro-Famil, E. Pottier, and J. S. Lee, Frontiers of remote sensing information processing. Singapore: World Scientific Publishing, 23, pp. 15 137. References [11] L. Ferro-Famil and E. Pottier, Decomposition of polarimetric variations using transformation operators, in EUSAR, May 26. [1] P. Flandrin, Time-frequency/time-scale analysis, ser. Wavelet Analysis and its Applications. San Diego, CA: Academic Press Inc., 1999, vol. 1. [2] L. Ferro-Famil, A. Reigber, and E. Pottier, Scene characterization using subaperture po-