BUILDING HEIGHT ESTIMATION USING MULTIBASELINE L-BAND SAR DATA AND POLARIMETRIC WEIGHTED SUBSPACE FITTING METHODS

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1 BUILDING HEIGHT ESTIMATION USING MULTIBASELINE L-BAND SAR DATA AND POLARIMETRIC WEIGHTED SUBSPACE FITTING METHODS Yue Huang, Laurent Ferro-Famil University of Rennes 1, Institute of Telecommunication and Electronics of Rennes 263 Avenue General Leclerc,3542 Rennes Cedex,France ABSTRACT Multibaseline (MB) PolinSAR techniques are applied to the analysis of urban areas to estimate the heights of scatterers, discriminate the layover contributions and extract scattering mechanisms. This paper applies the weighted subspace fitting (WSF) method to resolve the MB-PolinSAR problem in urban areas. Considering two scatterer signal models, i.e, the coherent and distributed scatterers, WSF is optimally model adaptive compared to other High-Resolution methods. A fully polarimetric extension WSF estimator is proposed to enhance the height estimation performance and determine the physical characteristics of buildings. For computation efficiency, an analytic optimization solution is developed to maintain the search dimension of the fully WSF estimation to the one of single polarization case. The effectiveness of the proposed polarimetric WSF estimator is demonstrated using multibaseline PolInSAR data acquired by DLR s E- SAR system at L-band over an urban area of Dresden. Key words: SAR image, subspace fitting. 1. INTRODUCTION Polarimetric interferometric SAR (PolInSAR) is a technique aiming to determine the height of scatterers and extract their physical properties. Several Hight-Resolution (HR) spectral analysis techniques have been applied to the estimation of building height using multibaseline interferometric SAR (MB-InSAR) data, in order to investigate problems linked to the complexity of scattering patterns over dense urban areas. A recent study by Sauer et al. [1] proposed to apply polarimetric versions of spectral estimation methods (Capon, MUSIC, Maximum Likelihood (ML)) to the case of airborne polarimetric MB- InSAR data sets acquired at L-band. Results showed that using polarimetric data could improve building height estimation, both in terms of discrimination of mixed scatterer responses (layover, vegetation...) and determination of physical characteristics of the observed media. Despite undeniable performance improvements, such an approach may have some limitations: As shown by Ferro- Famil et al. [2], scatterers in urban areas may have very different statistical properties that are not optimally handled by the methods proposed by Sauer et al. [1] and may involve estimation errors and instability; Processing MB- InSAR data acquired from irregularly distributed baselines can cause ambiguous responses and sidelobe effects that may lead to erroneous interpretations and estimations. In this paper, we propose to overcome these limitations using subspace fitting methods and unambiguous baseline analysis. In section 2, we show that over urban areas, scattered signals can be categorized in two types defined by the Conditional and Unconditional model assumptions (CM et UM) [3], which correspond to coherent scatterers (e.g. point-like or double bounce scatterers) or distributed scatterers with speckle affected (e.g. surfaces or vegetation), responses respectively. The ML estimation of building height leads to deterministic ML and stochastic ML solutions, the former being inconsistent. In order to overcome these problems, weighted subspace fitting (WSF [4]) methods are proposed to estimate building heights in section 3. The WSF estimator is formulated in a rigorous mathematical way and a fully polarimetric version (FP-WSF) is developed in section 4, which allows to refine building characterization and scattering mechanism estimation, without additional optimization computation complexity, compared to the single polarization case. Finally, experimental results of WSF and the proposed FP-WSF are shown in section 5 to estimate the height of scatterers and their physical properties using fully polarimetric MB-InSAR data over Dresden city acquired by the DLR s E-SAR system. 2. MB INSAR SIGNAL MODEL The MB InSAR signals received by m sensors, reflected by d backscattering sources with different heights and located in the same azimuth-range resolution cell, can be formulated as follows: y(l) = A(z)x(l) + n(l) (1) where l = 1,..., L represents a realization of the signal acquisition and L is the number of independent looks. Proc. of 4th Int. Workshop on Science and Applications of SAR Polarimetry and Polarimetric Interferometry PolInSAR 29, 26 3 January 29, Frascati, Italy (ESA SP-668, April 29)

2 The noisy observed data vector is denoted y(l) C m 1 where x(l) C d 1 stands for the unknown source (reflected) signal vector and n(l) C m 1 is the additive noise, assumed to be gaussian with zero mean and power σ 2, i.e. n(l) N (, σ 2 I). The steering matrix A(z) consists of d steering vectors corresponding each to a backscattering source: A(z) = [a(z 1 ),..., a(z d )] (2) with z, a vector of unknown parameters (heights) z = [z 1,..., z d ] T. The column vectors a(z) account for the interferometric phase variation from one acquisition to another, and depend on MB InSAR array configuration as shown in Figure 1 a(z) = [1, exp(jk z2 z),..., exp(jk zm z)] T (3) = 4π B (j) λ rsinα where k zj is the vertical wavenumber of jth image pair and z, r, α are a scatterer s height, slant range position and incidence angle. Figure 1: MB-InSAR configuration The problem of estimating the unknown height and scattering parameters of each source from the observed noisy data y(l) is the main concern of this paper. Different assumptions on the backscattering source signal x(l) result in different expected statistical properties of y(l) as shown in the following Unconditional MB InSAR Signal Model Considering the d sources as distributed scatterers with speckle affected responses, the source signals x(l) are assumed to be stochastic and the signal model (1) leads to the one of [5] that has been developed for an MB InSAR configuration: y u (l) = d x i (l) a(z i ) + n(l) (4) i=1 with being Schur-Hadamard product. In this case, the source signal can be expressed as x i (l) = τ i x ui (l) C d 1 where the multiplicative speckle term x ui (l) is a gaussian random vector with zero mean and covariance matrix C i, i.e. x ui (l) N d (, C i ) and τ i represents the reflectivity of the source. Stochasic signals {x ui (l)} L l=1 are independently and identically distributed. This type of source signal model can be applied to the surface or roof reflexion and volume scattering. Under the unconditional model (UM) assumption, the received data vector y u (l) is a gaussian random process, i.e. y u (l) N m (, R). The scatterer s reflectivity τ i and height z i are the unknown parameters to be estimated from R. The steering vector a(z i ) is formulated as (3) in MB InSAR configuration Conditional MB InSAR Signal Model In case of coherent scatterers, the backscattering source signal x(l) is deterministic and is thus frozen for all the observations, i.e. E{x(l)} = x(l). Therefore the model (1) under the conditional model (CM) assumption is given by: y c (l) = d x i (l) a(z i ) + n(l) (5) i=1 Compared to the unconditional model, the backscattering source signal x i (l) = τ i x ci (l) C d 1 is frozen over the different realizations, i.e. E{x ci (l)} = x ci (l). This behavior is observed for the point-like scatterer or coherent double bounce reflexion. The received data vector in (5) is then as a gaussian random process with the mean A(z)x(l) and covariance matrix equal to the additive noise one, i.e. y c (l) N m (A(z)x(l), σ 2 I). The reflectivity and height τ i, z i can be estimated from A(z)x(l) Hybrid MB InSAR Signal Model A hybrid MB InSAR signal model, introduced by Sauer et al [1], includes both coherent and distributed (UM and CM) scattering assumptions: y(l) = y c (l) + y u (l) d1 = τi x ci (l) a(z i ) i=1 d2 + τi x ui (l) a(z i ) + n(l) i=1 Applying common HR estimation methods to this hybrid MB InSAR Signal Model leads to various estimation performance. MUSIC is adapted to the case of distributed scatterers, but its performance may degrade significantly in the presence of the coherent scatterers since the source signal covariance matrix may be singular. Deterministic ML is adapted to coherent scatterers and cannot accurately localize distributed scatterers. As to stochastic ML, it is asymptotically efficient to estimate distributed scatterers, but it cannot handle with coherent (6)

3 ones. In order to overcome these limitations, we propose to use a model adaptive method named weighted subspace method (WSF). 3. MB-WSF SPECTRAL ESTIMATION In this section, we present the basic subspace fitting problem and we introduce an asymptotically efficient parametric method called Weighted Subspace Fitting for estimating the heights of different scatterers in the same azimuth-range resolution cell. Considering d scatterers with heights z = [z 1,..., z d ], the covariance matrix of data vector y(l), R = E{y(l)y H (l)}, can be decomposed as R = AR x A H + σ 2 I = E s Λ s E H s + σ 2 E n E H n (7) where E s is the matrix formed by eigenvectors associated to the d largest eigenvalues of R, Λ s is the diagonal matrix of the d largest eigenvalues, and E n contains the remaining (m d) eigenvectors. The signal subspace is spanned by E s, whereas E n represents the noise subspace with equal eigenvalues. R x represents the source signal covariance matrix. In practice, the signal subspace Ê s and the noise subspace Ên are estimated from the data sample covariance matrix ˆR = 1 L L l=1 y(l)yh (l) Weighted subspace fitting principle The cost function of weighted subspace fitting (1) is formed from the orthogonality of noise subspace and steering matrix space, E H n A(z) =. Different choices for the weighting matrix correspond to different estimation solutions [4]: W = I: leads to MUSIC-like solution that adapted to distributed scatterers. W = ˆΛ s ˆσ 2 I = Λ: Deterministic ML adapted to coherent scatterers. W opt = (A H Ê s (Λ s σ 2 I) 2 Λ s Ê H s A) 1 : WSF using an optimal weighting matrix minimizes the estimation error covariance C(W opt ) C(W) and is adapted to distributed and coherent scatterers. WSF is equivalent to Stochastic ML for distributed scatterers Thus, WSF is optimally model adaptive technique. Under the CM assumption, WSF is optimal in the sense that it asymptotically acheives a lower estimation error variance than deterministic maximum likelihood, whereas when the UM assumption is verified, WSF is equivalent to the stochastic maximum likelihood method and attains the CRLB. 4. POLARIMETRIC WSF ESTIMATOR 3.1. Subspace fitting problem The basic subspace fitting problem is formed in the least square sense as ẑ, ˆT = arg min z,t E sw 1/2 A(z)T 2 F (8) where the full rank matrix T is used to fit the steering matrix space A(z) to the signal subspace E s that contains unknown parameters, such as the heights. The subspace fitting problem (8) is seperable in A(z) and T. By substituting the Total Least Square Solution of T, ˆT = (A(z) H A(z)) 1 A(z) H E s W 1/2 (9) we obtain the following minimizer of unknown parameters ẑ = arg min z tr{a(z) H Ê n Ê H n A(z)W} (1) The weighting matrix W can be considered as an additional degree of freedom and can be tuned to obtain specific estimation properties. For a polarimetric MB InSAR configuration, the array response can be decomposed into several parts, each related to a polarization state a(z, φ) = a 1 (z)φ 1 + a 2 (z)φ 2 + a 3 (z)φ 3 (11) where φ = [φ 1, φ 2, φ 3 ] T C 3 1 represents a arbitrary target vector. The fully polarimetric steering matrix A(z, Φ) C 3m d is given by A(z, Φ) = A 1 (z)φ 1 + A 2 (z)φ 2 + A 3 (z)φ 3 (12) with A i (z) steering matrix corresponding to a polarization channel A 1 = [ A(z) ] A 2 = [ A(z) ] A 3 = [ A(z) For one particular polarization channel, the weighting coefficients for d source signals form a diagonal matrix Φ p = diag{φ p1,..., φ pd } C d d with p = 1, 2, 3. The polarimetric WSF estimator may be extended from (1) as (ẑ, ˆΦ) = arg min tr{a H (z, Φ)ÊnÊH n A(z, Φ)W} (13) ]

4 (a) Small urban area (b) Building (a) Pauli color coded image of Dresden Figure 3: Zones under investigation where M(z) C 3d 3d is given by 2 A H 1 ΠA 1 W A H 1 ΠA 2 W A H 1 ΠA 3 W 3 4 A H 2 ΠA 1 W A H 2 ΠA 2 W A H 2 ΠA 3 W 5 A H 3 ΠA 1 W A H 3 ΠA 2 W A H 3 ΠA 3 W (17) where Π = ÊnÊH n and M(z) contains 3 3 matrices defined as {M ij } C d d with i, j = 1, 2, 3. Normalizing Ψ, the polarization parameter vector is simplified to Ψ = [e T, Ψ T 2, Ψ T 3 ] T with e = [1,..., 1] T. The estimator is given by (ẑ, ˆΨ 2, ˆΨ 3) = arg min[e T, Ψ T 2, Ψ3 T ]M(z) e Ψ 2 Ψ 3 (18) (b) Time-frequency classification of type of coherent behavior (CM or UM) Figure 2: Coherent and distributed scatterers over the Dresden site Note that any symmetric, positive definite W will yield consistent parameter estimates [6]. The optimal weight matrix that makes the estimation error covariance attain CRLB is given by W opt = (A H (z, Φ)Ês(Λ s σ 2 I) 2 Λ s Ê H s A(z, Φ)) 1 (14) The minimization of (13) requires an optimization search 5 d unknown parameters as in a 5 d-dimensional space formed by θ = {z, R(Φ 2), I(Φ 2), R(Φ 3), I(Φ 3)} R 5d (15) where Φ j = Φj Φ 1. To minimize (13), we extend the algorithm of [6] to the fully polarimetric case. Ψ C 3d 1 is a vector formed from diagonal elements of Φ 1, Φ 2, Φ 3, i.e. Ψ = [Ψ T 1, Ψ T 2, Ψ T 3 ] T C 3d 1. The WSF cost function is defined by: V (z, Ψ) = Ψ H M(z)Ψ (16) Solving V/ Ψ 2 = and V/ Ψ 3 = yields: ˆΨ 2(z) = M 1 22 (M 21 + M 23 G)e ˆΨ 3(z) = Ge (19) with G = (M H 23M 1 22 M 23 M 33 ) 1 (M 31 M 32 M 1 22 M 21). Once the optimal polarization configuration is found, the WSF configuration depends only on z and may be formulated as V (z) = [e T, ˆΨ T 2, ˆΨ T 3 ]M(z) ẑ = arg min V (z) ˆΨ 2 = ˆΨ 2(ẑ) ˆΨ 3 = ˆΨ 3(ẑ) e ˆΨ 2 ˆΨ 3 (2) Using this analytic solution, the optimization search dimension is reduced to 1 d. The fully polarimetric WSF algorithm maintains the computation efficiency equal to the one of single polarization case.

5 5. EXPERIMENTAL RESULTS The proposed methods are applied to dualbaseline PolIn- SAR data acquired by the DLR s E-SAR system at L- band over the urban area of Dresden. A Pauli color-coded image of the scene under investigation is shown in fig. 2a. The map given in fig. 2b represents the result of a classification of the Time-Frequency behavior of the pixels of the image, as described in [2]. From the classification results, we can distinguish scatterers showing a coherent response, whose response follows the CM assumption. Such scatterers are located at the wall-ground interaction or on the edge of buildings, whereas in coherent ones, that can be associated to the UM assumption, correspond to natural or distributed environments, like soils, roofs, the river and forested areas. (a) MUSIC tomogram (slant range) The different proposed estimation methods are applied over a particular area of the image or over a specific set of range bins with the same azimuth position, as shown in fig. 3 Figure 4 compares the pseudo-tomograms in slant range using the single polarization (SP) MUSIC and WSF methods, with an order (number of sources) set to 1. The SP-MUSIC tomogram shows some significant sidelobes, whose distance in elevation from the mainlobe response is about 15m and corresponds exactly to the height ambiguity for the acquisition configuration. The SP-WSF tomogram has no sidelobes. (b) MUSIC discrete tomogram (slant range) 5.1. Results of SP-MUSIC and SP-WSF Results obtained with an order set to 2 and depicted in fig. 5, reveal that the MUSIC tomogram is still affected by sidelobe effects and cannot discriminate 2 coherent scatterers corresponding to the wall-ground interaction. Due to its better resolution for coherent scatterers, the SP- WSF method can accurately locate the two contributions. One may note that this method is still not affected by spurious sidelobe contributions. From the WSF tomograms in ground range, the building shape can be perceived easily by drawing simple lines between the estimated locations after a projection in the ground range geometry. (c) WSF tomogram (slant range) (d) WSF tomogram (ground range) Figure 4: Single polarization tomograms at order Results of FP-MUSIC and FP-WSF

6 (a) MUSIC tomogram (slant range) (a) MUSIC tomogram (b) MUSIC discrete tomogram (slant range) (b) MUSIC discrete tomogram (slant range) (c) WSF tomogram (slant range) (c) WSF tomogram (slant range) (d) WSF tomogram (ground range) Figure 6: Fully polarimetric tomograms (order=1) (d) WSF tomogram (ground range) Figure 5: Single polarimetric tomograms (order=2) The results obtained using the fully polarimetric MUSIC and WSF methods and shown in fig. 6 indicate that unlike FP-MUSIC, the FP-WSF technique is not affected by sidelobes. A 3D image of the urban area under study, obtained using the FP-WSF at order 1 is shown in fig. 7. The topography of this image correspond to the estimated elevation, whereas the color is related to the polarimetric α angle, indicating the nature of the retrieved polarimet-

7 ric scattering mechanism. Double bounce reflection often occurs at the wall-ground interaction and single reflexion over roofs and grounds, whereas some exceptions might be due to the slope of roofs or the orientation of buildings. The α tomogram in ground range geometry, depicted in fig. 8 shows the wall-ground interaction at the bottom of the roof rather than at the right-hand side of the roof in slant range. At order 2, WSF-α tomograms in slant range and ground range are shown in Figure 9. From the α tomogram in ground range, the shape of building can be retrieved and describe physical scattering mechanisms, like ground scattering, wall-ground reflexion, roof and ground backscattering, from left to right. (a) α tomogram (slant range) (b) α tomogram (ground range) Figure 8: α tomogram for FP-WSF technique at order 1 (a) α (slant range) Figure 7: 3D α tomogram FP-WSF at order=1 6. CONCLUSION (b) α (ground range) Figure 9: α tomogram, FP-WSF at order 2 In this paper we have introduced a new method to estimate building height and scattering patterns in complex

8 urban environments, using polarimetric MB-InSAR data sets. Common HR estimation methods may reach serious limitations like spurious sidelobes and are generally adapted to a single model of scatterer response. The proposed method based on the Weighted Subspace Fitting spectral estimation technique is model adaptive and can be optimally applied to coherent and distributed scatterers. Compared to MUSIC, WSF has no or very low sidelobes and better resolution for coherent scatterers. We have extended the WSF estimator to the fully polarimetric case using an analytical polarimetric optimization which maintain the computation cost equal to the single polarization case. Using this fully polarimetric WSF estimator, the characterization of building height and scattering could be refined. REFERENCES [1] S. Sauer, L. Ferro-Famil, E. Pottier, and A. Reigber. Physical parameter extraction over urban areas using l-band polsar data and interferometric baseline diversity. Geoscience and Remote Sensing Symposium, 27. IGARSS 27. IEEE International, pages 1 5, July 27. [2] L. Ferro-Famil and E. Pottier. Urban area remote sensing from l-band polsar data using time-frequency techniques. IEEE Urban Remote Sensing Joint Event, pages , April 27. [3] P. Stoica and A. Nehorai. Performance study of conditional and unconditional direction-of-arrival estimation. IEEE Trans. Acoust. Speech Signal Processing, ASSP-38: , Oct 199. [4] M. Viberg and B. Ottersten. Sensor array processing based on subspace fitting. IEEE Trans. Signal processing, 39: , May [5] F. Gini and F. Lombardini. Multibaseline crosstrack sar interferometry: A signal processing perspective. IEEE trans. Aerospace Electronic Systems, 38(4): , Oct 25. [6] A. Swindlehurst and M. Viberg. Subspace fitting with diversely polarized antenna arrays. IEEE trans. Antennas and propagation, 41(12): , Dec 1993.