A novel method to estimate interferometric coherence in synthetic aperture radar interferometry is proposed. It is demonstrated that this approach is not affected by the terrain topography, contrary to multilook techniques. In addition, since the method is based on the two-dimensional discrete wavelet packet transform, it allows recovering coherence information with a high spatial resolution. Results derived from simulated and experimental ESAR-DLR X- and L-band interferograms corroborate the performance of the proposed technique.Peer Reviewe

Fabregas Canovas, Francisco Javier

López Martínez, Carlos

Pottier, Eric

IEEE Signal Processing Letters

English

UPCommons

Wavelet Transform-Based Interferometric SAR Coherence Estimator

Lopez-Martinez, Carlos

Fabregas, Xavier

HAL-Univ-Nantes

Wavelet Transform Based Interferometric SAR Coherence Estimator.

HAL-Rennes 1

UPCommons. Portal del coneixement obert de la UPC

IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 12, DECEMBER 2005 831Wavelet Transform-Based InterferometricSAR Coherence EstimatorC. López-Martínez, Member, IEEE, X. Fàbregas, Member, IEEE, and E. Pottier, Member, IEEEAbstract—A novel method to estimate interferometric coher-ence in synthetic aperture radar interferometry is proposed. Itis demonstrated that this approach is not affected by the terraintopography, contrary to multilook techniques. In addition, sincethe method is based on the two-dimensional discrete waveletpacket transform, it allows recovering coherence informationwith a high spatial resolution. Results derived from simulatedand experimental ESAR-DLR X- and L-band interferogramscorroborate the performance of the proposed technique.Index Terms—Coherence, discrete wavelet transform (DWT), in-terferometry, synthetic aperture radar (SAR).I. INTRODUCTIONSYNTHETIC aperture radar (SAR) interferometry (InSAR)is able to retrieve the terrain topography from complex in-terferograms [1]. An interferogram consists of the complex Her-mitian product of two SAR images acquired from slightly dif-ferent spatial locations. Consequently, the complex Hermitianproduct phase contains topographic information coded in a 2modulus, which must be unwrapped to derive height informa-tion unambiguously.The correlation coefficient of a pair of SAR images andis an important source of information [2](1)where is the ensemble average and * is the complex con-jugate. The amplitude of (1), i.e., , called coherence, deter-mines the precision of the derived topographic information. Co-herence is also important since it may be employed in the phaseunwrapping process or to estimate the signal-to-noise ratio ofthe SAR system. In combination with polarimetric techniques,coherence can be also employed to derive forest height [3]. Co-herence depends on many factors, for instance, the signal-to-noise ratio of the SAR system, the co-registration process ofthe SAR images, or the so-called temporal decorrelation term,Manuscript received May 18, 2005; revised July 11, 2005. This work was sup-ported in part by the Spanish CICYT under Grant TIC 2002-04451-C02-01 andin part by the EU-RTN AMPER project under Grant HPRN-CT-2002-00205.The associate editor coordinating the review of this manuscript and approvingit for publication was Dr. Antonia Papandreou-Suppappola.C. López-Martínez and E. Pottier are with the Electronics and Telecom-munications Institute of Rennes UMR CNRS 6164, University of Rennes 1,35042 Rennes Cedex, France (e-mail: carlos.lopez@univ-rennes1.fr; eric.pot-tier@univ-rennes1.fr).X. Fàbregas is with the Department of Signal Theory and Communica-tions, Technical University of Catalonia, 08094 Barcelona, Spain (e-mail:fabregas@tsc.upc.edu).Digital Object Identifier 10.1109/LSP.2005.859513which accounts for coherence losses when SAR images are ac-quired at different times. Finally, the spatial or geometric decor-relation takes into account the fact that both SAR images do notobserve exactly the same area, since they are acquired from dif-ferent spatial locations [4]. Since SAR images cover from tensto hundreds of kilometers, with spatial resolutions from metersto tens of meters, coherence must be estimated locally since allthe previous factors are nonhomogeneous in the SAR images.Under the assumption that , and are homoge-neous and ergodic in mean, (1) may be estimated locally bymeans of the multilook coherence estimator(2)where and are the spatial coordinates. Equation (2) overes-timates low coherences in such a way that the larger the numberof averaged pixels, the lower the coherence bias [2], but also,the larger the loss of spatial resolution and spatial details [5]. Inthe case of InSAR, even if and are homogeneous, they candiffer in a deterministic and nonhomogeneous phase component, representing the topographic induced phase [2], [6]. If isconstant within the analysis window, it does not affectcoherence estimation. Nevertheless, if is not constant, it in-duces a second bias term into the multilook estimator (2). Thus,it is necessary to compensate in order to estimate unbiasedcoherence values. The estimator bias is especially problematicfor mountainous areas, where presents a large variability dueto steep slopes. The topography compensated coherence esti-mator [6] is shown in(3)where is the estimated topographic component. The phasecomponent can be obtained by means of external digital ele-vation models (DEMs), or it can be estimated from data [5]. Theformer presents the problem that an external DEM may be notavailable, whereas the later has the handicap that topography es-timation is problematic for medium and low coherences [2].This letter presents a novel interferometric coherence estima-tion algorithm based on the 2D-DWPT [7], which adapts to thenonhomogeneous natures of both and . The necessity to1070-9908/$20.00 © 2005 IEEE832 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 12, DECEMBER 2005estimate coherence with a high spatial resolution is importantwhen SAR images reflect areas with man-made structures inorder to avoid the mixture and the loss of spatial details. Thisapproach presents three advantages with respect to classical ap-proaches, i.e., (2) and (3). It does not need both SAR imagesseparately to estimate coherence, the estimated coherence is notbiased by the topographic phase, and the algorithm allows highspatial resolution coherence estimation.II. SIGNAL MODEL AND ESTIMATION ALGORITHMThe measured phase of the Hermitian product of a pair ofSAR images is described by [5](4)where is a zero-mean noise, independent of , whose vari-ance depends on and [8]. In [9], it has been demon-strated that the real and imaginary parts of the complex phasorexp may be described by the noise models(5)(6)where and are the real and imaginary parts, respec-tively, and and are two zero-mean noises with variancesequal to(7)It can be observed that and are independent of . Con-sidering the Gaussian scattering hypothesis [8] to describemay be expressed for single look, i.e., nonaveraged, SAR data,as(8)where is the probability density function of considering[8], and represents the Gauss hyperge-ometric function. increases with , as shown in Fig. 1.Consequently, contains the same information as the orig-inal coherence. Equations (5) and (6) allow separating the com-plex phasor exp into two components: the additive termexp , which contains useful information concerning(through the parameter ) and the topographic information ,and the complex additive noise term . Hence, if the com-plex phasor exp is considered, can be obtained if isestimated by using (5) and (6). Thus, it is not necessary to con-sider both SAR images and separately.III. WAVELET DOMAIN SIGNAL MODELThe usefulness of the discrete wavelet transform (DWT) indenoising and estimation problems has been largely demon-strated. Most of the proposed algorithms and techniques arebased on hard- or soft-thresholding techniques, which are notconsidered for the problem described previously. In what fol-lows, we shall perform the 2D-DWPT of exp . If the com-plex noise component is filtered by means of thresh-Fig. 1. N as a function of jj.olding-like techniques, the topographic phase would not beretrieved correctly. As a result, parameter is not estimatedcorrectly either, presenting a large number of image artifacts.This poor performance has its origin into the fact that, sincesome complex wavelet coefficients are eliminated, the lack ofthese coefficients prevents a correct reconstruction of the com-plex signal exp . A new framework to estimate the pa-rameter , adapted to the noise models (5) and (6), is consid-ered next.If (5) and (6) are considered, the estimation of the coherenceby means of is also affected by the bias introduced by thetopographic phase component . On the contrary, if one con-siders the 2D-DWPT of the noise models (5) and (6), by as-suming ideal filters, these models translate into the followingnoise models in the wavelet domain [9]:DWT (9)DWT (10)where represents the wavelet scale, and is a filtered ver-sion, in the wavelet domain, of . The term corresponds,for each iteration of the 2D-DWPT, to a lowpass and downsam-pled version for the low-frequency band of the wavelet domain,whereas it is a filtered, downsampled, and frequency-invertedversion for the three high-frequency wavelet bands. In (9) and(10), the 2D-DWPT introduces a multiplicative factor equal toin the first additive term, i.e., the useful information term.Despite the fact that wavelet bands present a two-dimensionalbandwidth equal to one fourth of that of the original signal, the2D-DWPT also introduces a factor ( when power is consid-ered) that compensates for the bandwidth loss. Consequently,the noise terms in the wavelet domain, i.e., and , presentthe same variance values as and . The average wavelet co-efficient intensity is obtained, then, asDWT (11)Equation (11) shows that the wavelet transform is able to re-trieve coherence information through the parameter , in sucha way that the higher the number of wavelet scales, i.e., , themore negligible the influence of the noise terms. Therefore, forthat is large enough, the wavelet coefficients’ intensity is propor-tional to , i.e., proportional to the coherence of the space-fre-quency area determined by the wavelet coefficient.As observed in (9) and (10) for ideal wavelet filters, the realand imaginary parts of all coefficients containing useful infor-mation are multiplied by 2 for every wavelet scale. This is notthe case for real filters, since the band pass is not flat and equal to2. Thus, the 2D-DWPT must be computed considering waveletLÓPEZ-MARTÍNEZ et al.: WAVELET TRANSFORM-BASED INTERFEROMETRIC SAR COHERENCE ESTIMATOR 833filters with flat frequency response, achieved by consideringfilters with a relatively high number of coefficients. The useof short filters decreases the performance of the algorithm, aswavelet coefficients containing useful information are multi-plied by values lower than 2, hence preventing their detection.IV. COHERENCE ESTIMATION IN THE WAVELET DOMAINIn order to estimate , the improving factor multiplyingit, as it is shown in (9) and (10), has to be maintained whenthe inverse 2D-DWPT is calculated. The proposed algorithmdetects, for every wavelet scale that is inversely transformed,those wavelet coefficients containing useful information, i.e.,, multiplying its real and imaginary parts by . Thedetection is performed by means of(12)where is the intensity of the wavelet coefficients, and [see(7)] is the noise power. This power may be estimated from the in-tensity of the wavelet coefficients of the high-frequency waveletscales, since they contain mainly noise [7]. Next, a threshold isapplied to (12). The wavelet coefficients presenting largerthan the threshold are processed, i.e., their real and imaginaryparts are multiplied by , whereas those below it are maintained,avoiding introducing false information. The final effect of thisprocess is that the weight of those wavelet coefficients classi-fied as coefficients containing useful information is increasedwith respect to those coefficients that are not processed. Thus,the influence of the noise terms and is reduced with re-spect to . Once the inverse 2D-DWPT is applied to the pro-cessed signal, the derived complex phasor in the original do-main presents an amplitude proportional to and a denoisedtopographic phase [9]. In order to derive the coherence infor-mation, denoted as in the following, the amplitude maybe normalized by , where represents the maximumnumber of scales of the 2D-DWPT. The selection of the numberof wavelets scales is, in principle, arbitrary. Nevertheless, ifthe number of wavelet scales is increased too much, the waveletcoefficients in the low-frequency scales will mask those coeffi-cients in the high-frequency scales. This effect will result in aloss of spatial resolution in the coherence image. Finally, (8) isemployed to invert in order to derive .V. RESULTSFirst, two phase ramps have been simulated over the completecoherence range. The first interferogram presents a phaseexcursion, i.e., a phase fringe, over 40 pixels. The second hasa 12-pixel fringe. In both cases, and have beencalculated with a 5 5 analysis window (see Fig. 2). For the40-pixel fringe phase, both approaches estimate the correctvalues, as is basically constant within the averaging window.For the 12-pixel fringe interferogram, the estimatorpresents a clear bias since cannot be considered constantover the averaging window. Nevertheless, estimates thecorrect since is compensated for. Finally, if isapplied to both cases (in a four-iteration 2D-DWPT configura-tion with 40 coefficient Daubechies filters [7]), nonbiasedvalues are retrieved without compensation of the topographicFig. 2. Estimated coherences for simulated data. (a) Forty-pixel fringeinterferogram. (b) Twelve-pixel fringe interferogram.Fig. 3. Coherence estimators’ standard deviation for simulated data.term , since the 2D-DWPT allows to decouple the estimationof the from the random phase term . As deduced in [2]and [9], a speckle-induced bias is observed for low coherences.In the case of , this bias term decreases if the numberof averaged pixels is increased but at the cost of losing spatialdetails. In the case of , this bias decreases by increasingthe maximum number of wavelets scales , which cannotbe increased arbitrarily, as indicated previously. Nevertheless,the properties of the 2D-DWPT allow minimizing the spatialresolution losses with respect to multilook techniques. Fig. 3presents the standard deviation measurements for and.The estimators (within a 5 5 pixel analysis window)and (in a three-iteration 2D-DWPT configurationwith 40 coefficient Daubechies [7]) are also applied to anESAR-DLR X-band interferogram of Mt. Etna (Italy), repre-senting steep slopes (see Fig. 4). As presented by Fig. 4(c),the image difference shows a bias, whichis highly noticeable in the steepest slopes and located at thecenter of the image. Nevertheless, if is estimated from thedata, the image difference does not show bias[see Fig. 4(d)]. A wavelet technique is employed to estimate[9]. Table I presents the average value of the differenceimages in the areas selected in Fig. 4(c) and (d). In area A,it is observed that is able to reduce a small bias. Theproposed approach estimates coherence without a bias also834 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 12, DECEMBER 2005Fig. 4. (a) ESAR-DLR X-band Mt. Etna interferometric phase. (b)ESAR-DLR Mt. Etna interferometric coherence j j. (c) Differenceimage j j   j j. (d) Difference image j j   j j.TABLE IDIFFERENCE IMAGES AVERAGE VALUESin steep slopes areas, as observed in area B. Hence,estimates coherence without a bias respect to . As shown,the algorithm estimates a correct coherence in the case of steepslopes. Nevertheless, this estimation may be problematic forhighly steep slopes. In this case, useful information appearsin the first wavelet scales. Considering (9) and (10), one canobserve that the real and imaginary parts of those waveletcoefficients containing useful information are only multipliedby 2. The limited increase of amplitude due to the 2D-DWPTcan provoke that low-coherence values may not be estimatedcorrectly since the amplitude of the wavelet coefficients ismainly due to noise. Nevertheless, high coherences may beestimated correctly since they are basically proportional to[see (11)].The 2D-DWPT was selected to exploit its capability toperform high spatial resolution coherence estimation [7]. Topresent this property, an ESAR-DLR L-band interferogramof the Oberfapfenhoffen test site (Germany) is considered.The area is characterized by a flat topography that does notbias coherence estimation. Nevertheless, it contains man-madestructures which spatial resolution must be maintained.Fig. 5(a) corresponds to , whereas Fig. 5(b) correspondsto . The horizontal lines correspond to a parking lot. Inthis area, high coherence values are expected for the cars dueto their metallic nature, whereas low values are expected forthe lanes separating them. The lanes are dominated by specularreflection; hence, they contain mainly uncorrelated additivenoise. Fig. 5(c) presents the profiles of and .The profile corresponding to the image is also included butFig. 5. ESAR-DLR L-band SAR interferometric dataset. (a) j j. (b)j j. (c) Images profiles.conveniently scaled. In the profile of , the separation lanesbetween the parked cars are visible. Due to the spatial aver-aging, these lanes disappear in , whereas they becomevisible for .VI. CONCLUSIONA novel InSAR high spatial resolution coherence estimatorbased on the 2D-DWPT, significantly departing from classicalthresholding-like techniques, is presented. As it is shown, theproperties of the 2D-DWPT allow separating coherence estima-tion from the phase component, which results in a coherenceestimation process not affected by topographic biases. In addi-tion, the high spatial resolution capabilities of the 2D-DWPTallow increasing the amount of information contained in the co-herence parameter.REFERENCES[1] R. Bamler and P. Hartl, “Synthetic aperture radar interferometry,” In-verse Problems, vol. 14, pp. R1–R54, 1998.[2] R. Touzi, A. Lopes, J. Bruniquel, and P. W. Vachon, “Coherence estima-tion for SAR imagery,” IEEE Trans. Geosci. Remote Sens., vol. 37, no.1, pp. 135–149, Jan. 1999.[3] S. R. Cloude and K. P. Papathanassiou, “Polarimetric SAR interferom-etry,” IEEE Trans. Geosci. Remote Sens., vol. 36, no. 5, pp. 1551–1565,Sep. 1998.[4] F. Gatelli, A. M. Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca,“The wavenumber shift in SAR interferometry,” IEEE Trans. Geosci.Remote Sens., vol. 32, no. 4, pp. 855–865, Jul. 1994.[5] J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A.Reigber, “A new technique for noise filtering of SAR interferometricphase images,” IEEE Trans. Geosci. Remote Sens., vol. 36, no. 5, pp.1456–1465, Sep. 1998.[6] A. M. Guarnieri and C. Prati, “SAR interferometry: A quick and dirtycoherence estimator for data browsing,” IEEE Trans. Geosci. RemoteSens., vol. 35, no. 3, pp. 660–669, May 1997.[7] S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. San Diego,CA: Academic, 1998.[8] J. S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, “Intensity andphase statistics of multilook polarimetric interferometric SAR imagery,”IEEE Trans. Geosci. Remote Sens., vol. 32, no. 5, pp. 1017–1028, Sep.1994.[9] C. López-Martínez and X. Fábregas, “Modeling and reduction of SARinterferometric phase noise in the wavelet domain,” IEEE Trans. Geosci.Remote Sens., vol. 40, no. 12, pp. 2553–2566, Dec. 2002.

Wavelet transform-based interferometric SAR coherence estimator

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Wavelet Transform-Based Interferometric SAR Coherence Estimator

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