SAR interferometric phase statistics in wavelet domain

References 1 MARTiN-NEIRA, M., and GOUTOULE, J.M.: 'MIUS - a two-dimensional aperture-synthesis radiometer for soil-moisture and ocean salinity observations', ESA Bull., 1997, (92), pp. 95-104 2 SIVESTRIN, p., BERGER, M., ERR, Y., and FONT, J.: 'ESA's Second Earth Explorer Opportunity Mission: The Soil Moisture and Ocean Salinity Mission - SMOS', IEEE Geosci. Remote Sens. Newsl., 2001, (1 18), pp. 11-14 CAMPS, A., BAe, J., CORBELLA, I., and TORRES, F.: 'The processing of hexagonally sampled signals with standard rectangular techniques: application to 2D large aperture synthesis interferometic radiometers', IEEE Trans. Geosci. Remote Sens. GRS-35, 1997, pp. 183-190 4 BA~, J., CAMPS, A., TORRES, F., and CORBELLA, I.: 'Angular resolution of two-dimensional hexagonally sampled interferometric radiometer', Radio Sci., 1998, 33, (5), pp. 1459-1473 BA~, J.: 'Reliability analysis in aperture synthesis interferometric radiometers: application to L-band MIRAS instrument', Radio Sci., 3 5 VALL-LLOSSERA, M., DUFFO, N., CAMPS, A., CORBELLA, I., TORRES, E, and 2001,36, (I), pp. 107-1 17 ~, Kurtosis KS significance level (%) SAR interferometric phase statistics in wavelet domain ~, ., Kurtosis KS Kurtosis KS significance significance level (%) level (%) C. Lopez and X. Fabregas Scale 1 1 2.98 Synthetic aperture radar (SAR) interferometry is employed to obtain topographic information. Owing to noise, interferometric information has to be filtered. The wavelet transform can be employed to filter the interferometric phase, maintaining the spatial resolution, but new signal models have to be studied in this domain for further processing. 87.93 1 2.98 1 79.83 2.94 1 98.34 Introduction: Synthetic aperture radar interferometry (InSAR) is an established technique to obtain information about the earth's surface topography. The interferometric phase is calculated as the phase difference between two complex SAR images from the same area, but taken from slightly different positions. Owing to the lack of interferometric coherence (y I between both SAR images, the interferometric phase is noisy. In addition, the interferometric phase is only known within the interval [-z, n), it being necessary to unwrap it to recover unambiguously the height information. The unwrapping process is also affected by phase noise, since it induces phase residues. Phase filtering is thus necessary to reduce noise effects. In the last decade, the wavelet transform (WT) has shown a big potential for image processing applications. In the field of SAR data processing, the use of the WT is emerging since it allows processing of SAR imagery, keeping the spatial resolution and image details. Since the physics behind SAR data is completely different from that of optical images, any data processing has to take this into account. Thus, it is necessary to review or even to define new noise models adapted to this problem. In this Letter, we provide a study of a signal model for the interferometric phase in the wavelet domain. This model is validated with real interferometric data. frequency plane. N, has a one-to-one relation with the coherence IyJ providing, thus, the same information [3,4]. For a constant interferometric phase and homogeneous noise (i.e. constant Iyl), the parameter Nc, as well as the terms COS(C$;) and sin (4:) are constant. Therefore, signal randomness is only due to v," and v,". The discrete wavelet transform (DWT) can be seen as the addition of (weighted) random variables. By the central limit theorem, the weighted sum of identically distributed random variables can be approximated by a Gaussian distribution. Therefore, v," and v," are approximately Gaussian distributed. To test it, avoiding any interference from the phase d;, a constant slope producing 20 pixel fringes, corrupted with a noise equivalent to a coherence IyI = 0.6, has been simulated. Table 1 shows a statistical test applied over the real part of the interferometric complex phase in the wavelet domain. As shown, since the useful signal is concentrated in the low frequency band (LL), the wavelet bands (HL, LH and HH) present a kurtosis close to 3 and the significance levels for the Kolmogorov-Smimov (KS) test, assuming a Gaussian distribution, are high. These results demonstrate that v," and v," can be described by a Gaussian distribution. The same agreement is observed for any other value of Iyl, and for the imaginary part of the complex interferometric phase in the wavelet domain. The LL band deserves special attention. In this case, as there is signal content, the signal model will be represented by the real and imaginary parts of exp (j&') plus a Gaussian noise. Therefore, in this case, the amplitude lpl +jp21 has a Rice distribution. This result is equally valid for the rest of the wavelet bands. The amplitude in the wavelet domain will be Rayleigh distributed for IyI = 0, in a particular space-frequency region, and Rice distributed for IyJ > 0 Table 1: Kurtosis and KS significance levels (Gaussian assumption) for real part of simulated complex interferometric phase ramp in wavelet domain I Horizontal band (HW I Vertical band (LH) I Diaaonal band (LW I I I I I I I Scale 3 I 3.08 I 82.87 I 2.95 I 85.32 I 3.00 I 82.87 Complex interferometricphase signal model: The topographic model assumed previously (i.e. constant slope) does not take into account spatial details, which are important, for instance, in urban areas. Since no information is available about the distribution of the 'true' topographic phase 4x, an a priori model for the spatial details is not available in the spatial domain. This drawback can be overcome in the wavelet domain. As mentioned in the preceding Section, the DWT can be interpreted as a weighted sum of random variables. Therefore, DWT2D{N, cos(&)} and DWT2D{Nc sin(&)} can be supposed to be determined as a first approximation, by a Gaussian distribution. Tests with real data (see Tables 2 and 3) show that wavelet statistics have kurtosis higher than 3. To take into account this deviation from Gaussian behaviour, a double stochastic model is proposed for the wavelet coefficients x: Complex interferometricphase noise model: Earth topography can be represented locally by a constant slope [I], thus the interferometric phase 4, can be assumed to be a constant phase ramp. In the spatial domain the measured interferometric phase complies with the model dZ = q5x + v [2], where v is a phase noise term. The real and imaginary (5) parts of the measured phase 4z coded in the unit circle, defined as the complex interferometric phase, can be modelled by [3, 41: where P,(X~~~) represents the Gaussian distribution of the wavelet coefficients and ~,d(r~) is a generalised gamma distribution (GGD) (1) (2) modelling the vahabili6 of the-variance through the phase image. px(x) cannot be obtained in a general form. Numerical integration of (9, see Fig. 1, indicates that px(x) can be assumed to be a GGD model. To test the wavelet transforms of which are [3]: the validity of this model, two real interferometric phase images taken with the German sensor E-SAR from DLR have been employed. The first image is an X-band interferogram of Mount Etna (Italy) and the second one is an L-band interferogram of the Oberpfaffenhoffen test site (Germany), were man-made structures are present. These two interferometric phases were filtered with the algorithm presented in [3], which is based on maintaining the spatial resolution. Tables 2 and 3 present the kurtosis and the KS test significance levels, assuming a GGD model, applied to the real part of the filtered complex interfero- /I, = DWT,,{COS(~,)] = 2'Nc COS(^:) + V: p2 = DWT2D{sin(&)] = 2'Nc sin(+,") + vy (3) (4) where i represents the wavelet scale. v," and v," are noise terms independent from the wavelet scale. The phase term 4: represents the interferometric phase in the wavelet domain, which contains the same information as q5= The WT is able to localise dx in the spaceELECTRONICS LETTERS 26th September 2002 Vol. 38 No. 20 1207 metric phase in the wavelet domain. These results show that the GGD model is well adapted to real data. Tables 2 and 3 also show that the Oberpfaffenhofen image presents higher kurtosis values. This result is explained by the fact that this image contains more spatial details (as buildings or roads) than the Mount Etna image, which contains only topographic information. Horizontal band (HL) Kurtosis KS significance -1 I I I I -0.4 -0.2 0 0.2 0.4 a r20.0 Vertical band (LH) Diagonal band (HH) Kurtosis KS Kurtosis KS simificance significance I I I I , I I I I -0.100-0.075 -0.050-0.025 0 0.025 0.050 0.075 0.100 b Fig. 1 Distribution comparison between GGD model and numerical integration of double stochastic model a Low kurtosis b High kurtosis __~ GGD model ~ double stochastic model Horizontal band (HL) Kurtosis KS significance level (“A) Vertical band (LH) Diagonal band (LL)? Kurtosis KS Kurtosis KS significance significance level (“A) level (“A) I l;vel(%) I I Gvel(%) I I I;vel(%) Scale 1 I 10.51 I 70.61 I 11.30 I 77.16 I 12.88 I 56.61 Scale 2 I 6.39 I 86.51 I 6.42 I 79.89 I 6.12 I 98.33 Scale 3 I 4.42 I 97.34 I 5.62 I 60.95 I 3.92 I 80.70 Table 3: Kurtosis and KS significance levels (GGD assumption) for real part of Oberpfaffenhofen complex interferometric phase in wavelet domain. Scale 1 I 22.50 I 10.53 1 21.44 I 31.14 I 26.95 I 14.76 Scale 2 I 13.82 I 16.24 I 11.51 I 53.28 I 12.68 I 91.26 I I I Scale 3 I 9.25 I 85.52 I 13.09 I 37.32 I 8.56 I 11.75 Conclusions: We have developed a stochastic model for the complex interferometric phase in the wavelet domain. In the first part of this Letter, assuming the topography locally as a constant slope, a reference signal model is proposed and validated. To increase its flexibility, a double stochastic signal model, which leads to a GGD model, is tested for the wavelet coefficients. In the wavelet domain, complex phase noise is Gaussian distributed, whereas the useful signal complies with a GGD model. The GGD assumption for wavelet coefficients agrees with observations reported by other researchers. Finally, this representation can be employed as a priori information for further processing inside the wavelet domain. Acknowledgments: This work was funded by the Spanish government in the CICYT project TIC1999-1050-C03-01 and the Catalonian government in a DURSI fellowship. 0 IEE 2002 Electronics Letters Online No: 20020820 DOI: 10.1049/e1:20020820 C. Lopez and X. Fabregas (Signal Theoty and Communications Department, Electromagnetics & Photonics Engineering Group, Universitat Politkcnica de Catalunya (UPC), Building 03 Room 11 8, Jordi Girona 1-3, E-08034 Barcelona, Spain) E-mail: [email protected] 27 July 2002 References GATELLI, E, GUARNERI, A.M., PARIZZI, E, PASQUALI, P, PRATI, C., and ROCCA, F.: ‘The wavenumber shift in SAR interferometry’, IEEE Trans. Geosci. Remote Sens., 1994,32, (4), pp. 855-865 LEE, J.s., PAPATHANASSIOU, K.P., AMSWORTH, T.L., GRUNES, M.R., and REIGBER, A,: ‘A new technique for noise filtering of SAR interferometric phase images’, IEEE Trans. Geosci. Remote Sens., LOPEZ-MART~EZ, C., and F~REGAS-CANOVAS, EX.: ‘Modeling and reduction of SAR interferometric phase noise in the wavelet domain’, Accepted for publication IEEE Trans. Geosci. Remote Sens., 2001 LOPEZ MART~EZ, C., and FABREGAS CANOVAS, X.: ‘Results on SAR interferometric phase noise reduction using wavelet transform’. EUSAR’02, Cologne, Germany, 2002, Vol. 1 1998,36, (5), pp. 1456-1465 Adaptive array antenna based on radial basis function network as multiuser detection for WCDMA Chang-Jun Ahn and Iwao Sasase An adaptive array antenna is proposed based on the radial basis function (RBF) network as a multiuser detector for a WCDMA system. The proposed system calculates the optimal combining weight coefficients using sample matrix inversion with a common correlation matrix algorithm and obtains the channel response vector using the RBF output signal. Introduction: Wideband code division multiple access (WCDMA) communication systems have recently attracted considerable attention as mobile cellular and IMT-2000 communication systems due to their ability to suppress a wide variety of interfering signals including narrowband interference, multiple access interference (MAI), and multipath interference (MPI). One of the approaches for improving WCDMA system performance is the use of spatial filtering at a base station with an adaptive antenna array. The adaptive array antenna is widely accepted, since it provides many promising features such as high capacity, high spectrum efficiency, and more degrees of freedom to adjust cell coverage characteristics, leading to more efficient use of radio resources. To obtain the optimal combining weight coefficients used in this technique, several adaptive algorithms such as the sample matrix inversion (SMI) and least mean square (LMS) algorithm have been proposed. In array antennas using these conventional algorithms, weight processing is individually carried out for each user. Therefore, the base station must have many independent weight processors, which increase computational complexity. To solve this problem, a sample matrix inversion with a common correlation matrix (CCM-SMI) has been proposed. This algorithm has low computational complexity, fast weight convergence, and good BER performance at the base station in a multiuser environment. However, the CCM-SMI-based adaptive array antenna system has a poor channel response vector (as does SMI) due to the active multiuser. Since the MF output signal includes MA1 with increasing users, the system 1208 ELECTRONICS LETTERS 26th September 2002 Vol. 38 No. 20