estimation method for insar

8/7/2019 Estimation Method for InSAR

1/15

Estimation Method for InSAR

Interferometric Phase Based

on Generalized Correlation

Steering Vector

GUISHENG LIAO, Member IEEE

Xidian University

China

HAI LI

Civil Aviation University of China

We propose a new method, based on the generalizedcorrelation steering vector, to estimate synthetic aperture radar

interferometry (InSAR) interferometric phase. In this method the

generalized correlation steering vector with a large coregistration

error is determined according to the joint data vector. The

generalized correlation steering vector is then used to estimate

the InSAR interferometric phase. The method can simultaneously

auto-coregister the synthetic aperture radar (SAR) images and

reduce the interferometric phase noise. Theoretical analysis

and computer simulation results show that the method can

provide an accurate estimate of the terrain interferometric phase

(interferogram), even if the coregistration error reaches one pixel.

Manuscript received July 23, 2007; revised April 17 and October 1,

2008 and March 25, 2009; released for publication April 3, 2009.

IEEE Log No. T-AES/46/3/937979.

Refereeing of this contribution was handled by M. Rangaswamy.

This paper was sponsored by the National Nature ScienceFoundation of China (NFSC) under Grant 60736009 and Grant

60825104. This work was also supported by the Program for

Cheung Kong Scholars and the Innovative Research Team in

University (PCSIRT, IRT0645).

Authors addresses: G. Liao, National Lab of Radar Signal

Processing, Xidian University, Xian 710071, China; H. Li, Tianjin

Key Lab for Advanced Signal Processing, Civil Aviation University

of China, Tianjin 300300, China, E-mail: ([email protected]).

0018-9251/10/$26.00 c 2010 IEEE

I. INTRODUCTION

Synthetic aperture radar interferometry (InSAR) isan important remote sensing technique to retrieve theterrain digital elevation model (DEM) [1, 2]. Imagecoregistration [16], InSAR interferometric phaseestimation (or noise filtering) and interferometricphase unwrapping [710] are the key processingprocedures of InSAR. It is well known that theperformance of interferometric phase estimationsuffers seriously from the inaccuracy of the imagecoregistration. Almost all of the conventional InSARinterferometric phase estimation methods are based oninterferogram filtering, such as pivoting mean filtering[11], pivoting median filtering [12], adaptive phasenoise filtering [13]; and adaptive contoured windowfiltering [14]. The problem here is that, when aninterferogram is very poor in quality due to a largecoregistration error, it is very difficult to retrieve thetrue terrain interferometric phases with these methods.In fact the interferometric phases we obtain arerandom quantities, with their variances being inverselyproportional to the correlation coefficients betweenthe corresponding pixel pairs of the two coregisteredsynthetic aperture radar (SAR) images [2]. Thereforethe terrain interferometric phases should be estimatedstatistically.

In previous papers [15, 16], we propose a jointsubspace projection method to estimate the InSARinterferometric phase in the presence of largecoregistration errors. However the noise subspacedimension of the covariance matrix varies with thecoregistration error. To accurately estimate the InSARinterferometric phase, the noise subspace dimensionof the covariance matrix must be made known, andthe performance of the method degrades when thenoise subspace dimension is not estimated correctly.In this paper a new method, based on the model of thegeneralized correlation steering vector, is proposed.By benefiting from the generalized correlation steeringvector, this method can make full use of the coherenceinformation of neighboring pixel pairs in order toestimate the terrain interferometric phase accuratelyin the presence of large coregistration errors. Howeverour method avoids the trouble of calculating the noisesubspace dimension before estimating the InSARinterferometric phase. The key processing proceduresof the approach are summarized as follows. Afterthe coarse coregistration the generalized correlationsteering vector is determined according to the jointdata vector, the joint data vector is used to estimatethe corresponding covariance matrix, and thenbeamforming [17] and Capon [18, 19] techniques withthe generalized correlation steering vector are used toestimate the InSAR interferometric phase. For a pairof SAR images that are not coregistered accurately,the method can auto-coregister them and accuratelyestimate the corresponding terrain interferometricphase.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 3 JULY 2010 1389


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The paper is arranged as follows. The statisticalmodel of the generalized correlation steering vector isgiven in Section II, together with the characteristicsof the steering vector. In Section III beamforming andCapon methods that use the generalized correlationsteering vector for the purpose of estimating theInSAR interferometric phase in the presence of largecoregistration errors are presented. The processingprocedures of the InSAR interferometric phase

estimations are given in detail in Section IV. InSection V we demonstrate the robustness of themethod to coregister errors by using two sets ofsimulated data and real data. Conclusions are reportedin Section VI.

II. STATISTICAL MODEL OF THE GENERALIZEDCORRELATION STEERING VECTOR

Assume that the SAR images are accuratelycoregistered and that the interferometric phasesare flattened with a zero-height reference planesurface. The complex data vector s(i) of a pixel pairi (corresponding to the same ground area) of thecoregistered SAR images can be formulated as [20]

s(i) = [s1(i), s2(i)]T = a(i) [x1(i),x2(i)]

T + n(i)

= a(i) x(i) + n(i) (1)

where

a(i) = [1, eji ]T (2)

denotes the spatial steering vector (i.e., the arraysteering vector) of the pixel i, where s1 and s2 arethe complex SAR images, superscript T denotesthe vector transpose operation, i is the terrain

interferometric phase to be estimated, denotes theHadamard product, x(i) is the complex magnitudevector (i.e., the complex reflectivity vector of thescene received by the satellites) of pixel i, and wheren(i) is the additive noise term. The complex datavector s(i) can be modeled as a joint complex circularGaussian random vector [1, 2] with zero-mean,and the corresponding covariance matrix Cs(i) isgiven by

Cs(i) =Efs(i)sH(i)g

= a(i)aH(i) Efx(i)x

H(i)g + 2n I

= 2s (i)a(i)aH(i) Rs(i) + 2n I (3)

where

Rs(i) =

r11(i, i), r12(i, i)

r21(i, i), r22(i, i)

(4)

is called the correlation coefficient matrix, I isa 2 2 identity matrix, rmn(i, i) (0 rmn(i, i) 1,n = 1,2, m = 1,2) are the normalized correlationcoefficients between satellites m and n, Ef g denotesthe statistical expectation, superscript H denotes vector

Fig. 1. Formulation of joint data vector.

conjugate-transpose, 2s (i) is the backscatter energy ofthe pixel i, and where 2n is the noise power.

If the SAR images are accurately coregistered andif the normalized cross-correlation coefficients (i.e.,the nondiagonal elements) of Rs(i) are high enough,the beamforming technique can be used to estimatethe InSAR interferometric phase [19]. In this case thecost function can be represented as

J1 = aH

(i)Cs(i)a(i) (5)where a(i) = [1, e

ji ]T. The estimated InSAR

interferometric phase i maximizes the cost functionJ1.

However in the presence of a coregistration error,the steering vector cannot be formulated as in (2),so the cost function given by (5) cannot be used toestimate the InSAR interferometric phase.

In this paper we propose a new method, based onthe model of generalized correlation steering vector,to estimate the terrain interferometric phase in thepresence of large coregistration errors. By considering

the presence of the image coregistration error andby making full use of the coherence information ofneighboring pixel pairs, we use, not only the currentpixel i, but also its neighboring pixels, to jointlyconstruct the data vector si(i), as shown in Fig. 1,where circles represent SAR image pixels and idenotes the desired pixel pair whose interferometricphase is to be estimated.1 It should be noted that the(2 2, 4 4) window in Fig. 1 is not the only windowwe can use to construct the joint data vector, and infact other windows can be used to construct the jointdata vector, such as a (3 3,5 5) window or a (2 2, 2 4) window (if we have a priori knowledge of

the direction of image misregistration). A larger pixelwindow has more degrees of freedom, and, thus, itcan perform a finer coregistration of SAR images, butit suffers from computational complexity and a lackof enough samples to estimate the covariance matrix[21]. Therefore the selection of the pixel window sizeis a tradeoff between these considerations. A detailed

1As shown in Fig. 1, the horizontal is along range and the vertical

is along azimuth.

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discussion of the size of the pixel window is beyondthe scope of this paper.

The joint data vector si(i), shown in Fig. 1, can be

written as

si(i) = [s1(i 1), s2(i 6), s2(i 5), s2(i 2), s2(i 1),

s1(i), s2(i 4), s2(i 3), s2(i), s2(i +1),

s1(i +3),s2(i +2),s2(i +3),s2(i +6),s2(i +7),

s1(i +4),s2(i +4),s2(i +5),s2(i +8),s2(i +9)]T:

(6)

The corresponding covariance matrix Csi(i) is given

by

Csi(i) =Efsi(i)siH(i)g

= 2s (i)ai(i)aiH(i) + Rjj(i) +

2n I (7)

ai(i) = [T(i),

T(i),T(i),

T(i)]T [rT1 , r

T2 , r

T3 ,r

T4 ]

T

(8)

(i) = [1, eji ,eji , eji , eji ]T (9)

r1 = [1, r21(i 6, i 1),r21(i 5, i 1), r21(i 2, i 1),

r21(i 1, i 1)]T

r2 = [1, r21(i 4, i), r21(i 3, i), r21(i, i), r21(i + 1, i)]T

r3 = [1, r21(i + 2, i +3),r21(i + 3, i + 3), (10)

r21(i + 6, i +3),r21(i + 7, i +3)]T

r4 = [1, r21(i + 4, i +4),r21(i + 5, i +4),r21(i + 8, i +4),

r21(i + 9, i +4)]T

r21(m, n) =Efs2(m)s

1(n)gp

Efjs2(m)j2gEfjs1(n)j

2g,

m = i 6, : : : , i + 9, n = i 1, i, i + 3, i + 4 (11)

where ai(i) is called the generalized correlation

steering vector in the presence of large coregistration

errors of the pixel i and where Rjj(i) can be

considered the interference noise matrix.

In the following the characteristics of the

generalized correlation steering vector ai(i) fordifferent coregistration errors are discussed.

A. Accurate Coregistration

For accurate coregistration the joint data vector

si(i) is shown in Fig. 1, where circles represent

SAR image pixels and where i denotes the desired

pixel pair whose interferometric phase is to be

estimated.

Fig. 2. Joint data vector for coregistration error of one pixel.

In this case the generalized correlation steeringvector ai(i) can be given by (8), (9), and (10), where

r21(i 6, i 1) = r21(i 5, i 1) = r21(i 2, i 1)

= 0

r21(i 4, i) = r21(i 3, i) = r21(i + 1, i) = 0

r21(i + 2, i + 3) = r21(i + 6, i + 3) = r21(i + 7, i + 3)

= 0

r21(i + 5, i + 4) = r21(i + 8, i + 4) = r21(i + 9, i + 4)

= 0

r21(i 1, i 1) = r21(i, i) = r21(i + 3, i + 3)

= r21(i + 4, i + 4) = 1:

(12)

So the generalized correlation steering vector can begiven as

ai(i)

= [1,0,0,0, eji ,1,0,0, eji ,0,1,0, eji ,0,0,1, eji ,0,0,0]T:

(13)

B. Coregistration Error of One Pixel

When the azimuth coregistration error is one pixeland when its direction is upwards (i.e., the pixel of theimage from the second satellite is shifted upwards ascompared with the pixel in the first satellite image),the joint data vector si(i) is shown in Fig. 2, wherecircles represent SAR image pixels and where idenotes the desired pixel pair whose interferometricphase is to be estimated.

In this case the generalized correlation steeringvector ai(i) can be given by (8) and (9), where

r1 = [1,r21(i 2, i 1), r21(i 1, i 1), r21(i + 2, i 1),

r21(i + 3, i 1)]T

r2 = [1,r21(i, i), r21(i + 1, i), r21(i + 4, i), r21(i + 5, i)]T

r3 = [1,r21(i + 6, i +3),r21(i + 7, i + 3), (14)

r21(i +10, i +3),r21(i +11,i +3)]T

r4 = [1,r21(i + 8, i +4),r21(i + 9, i +4),r21(i +12, i +4),

r21(i +13, i +4)]T

LIAO & LI: ESTIMATION METHOD FOR INSAR INTERFEROMETRIC PHASE 1391


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Fig. 3. Joint data vector for coregistration error of pixels.

r21(i 2, i 1) = r21(i + 2, i 1) = r21(i + 3, i 1)

= 0

r21(i + 1, i) = r21(i + 4, i) = r21(i + 5, i) = 0

r21(i + 6, i + 3) = r21(i + 7, i + 3) = r21(i +10, i + 3)

= r21(i +11, i + 3) = 0

r21(i + 8, i + 4) = r21(i + 9, i + 4) = r21(i +12, i + 4)

= r21(i +13, i + 4) = 0

r21(i 1, i 1) = r21(i, i) = 1:

(15)

So the generalized correlation steering vector can begiven as

ai(i) = [1,0, eji ,0,0,1, eji ,0,0,0,1,0,0,0,0,1,0,0,0,0] T:

(16)

C. Coregistration Error of (0 < < 1) Pixels

When the azimuth coregistration error is (0 < < 1) pixels and when its direction is upwards (i.e.,the pixel of the image from the second satellite isshifted upwards as compared with the pixel in the firstsatellite image), the joint data vector si(i) is shown

in Fig. 3, where circles represent SAR image pixelsand where i denotes the desired pixel pair whoseinterferometric phase is to be estimated.

In this case the generalized correlation steeringvector ai(i) can be given by (8) and (9), where

r1 = [1, r21(A, i 1), r21(B, i 1), r21(E, i 1),

r21(F, i 1)]T

r2 = [1, r21(C, i), r21(D, i), r21(G, i), r21(H, i)]T

r3 = [1, r21(I, i +3), r21(J , i + 3), (17)

r21(M, i +3), r21(N, i +3)]T

r4 = [1, r21(K, i +4),r21(L, i +4),r21(O, i +4),

r21(P, i +4)]T

r21(A, i 1) = r21(E, i 1) = 0

r21(D, i) = r21(H, i) = 0

r21(I, i + 3) = r21(M, i + 3) = r21(N, i + 3) = 0

r21(L, i + 4) = r21(O, i + 4) = r21(P, i + 4) = 0:

(18)

So the generalized correlation steering vector canbe given as

ai(i) =

26664

1,0, r21(B, i 1)eji ,0, r21(F, i 1)e

ji ,1,

r21(C, i)eji ,0,r21(G, i)e

ji , 0,

1,0, r21(J , i + 3)eji ,0,0,1,

r21(K, i + 4)eji ,0,0,0

37775

T

:

(19)

From the results derived above, we can see that,benefiting from the changing of the generalizedcorrelation steering vector with the coregistrationerror, the proposed method can make full use of thecoherence information of the correlative pixel pairs inthe joint data vector to estimate the interferometricphase. In the following the correlative pixels ofthe second satellite image, with respect to the firstsatellite image pixel i for different coregistrationerrors, are given. As shown in Figs. 12, the secondsatellite image pixel i is the coherent pixel, withrespect to the first satellite image pixel i, for accurate

coregistration and coregistration errors of 1 pixel.The second satellite image pixels C and G are thecorrelative pixels, with respect to the first satelliteimage pixel i, for the coregistration error of pixelsas shown in Fig. 3. So the coherence information ofthe neighboring pixels can be employed to the greatestextent in order to provide an accurate estimation ofthe terrain interferometric phase in the presence oflarge coregistration errors.

III. SEVERAL ESTIMATION METHODS FOR INSARINTERFEROMETRIC PHASE

In this section we show how beamforming andCapon methods can be applied to estimate the InSARinterferometric phase by using the generalizedcorrelation steering vector in the presence of largecoregistration errors.

A. Beamforming

The beamforming technique with the generalizedcorrelation steering vector can be used to estimatethe InSAR interferometric phase. In this case the costfunction can be formulated as follows

J2 = aiH(i)Csi(i)ai(i) (20)

where

ai(i) = [T(i),

T(i),T(i),

T(i)]T [rT1 , r

T2 , r

T3 ,r

T4 ]

T

(21)

(i) = [1, eji , eji , eji , eji ]T: (22)

The maximum J2 corresponds to the estimate of

the InSAR interferometric phase i, i.e., i = i.



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B. Capon

Let wcap be a Capon filter designed to minimize

cost function wHcapCsi(i)wcap under the constraint

wHcapai(i) = 1.By assuming that the inverse of Csi(i) exists, it is

known [19] that

wcap =C1si (i)ai(i)

aiH(i

)C1

si

(i)ai(i

)(23)

which yields the estimated power of the Capon filteroutput

2capon(i) =1

aiH(i)C1si (i)ai(i)

: (24)

The maximum (24) corresponds to the estimate of the

InSAR interferometric phase i, i.e., i = i.

IV. PROCESSING PROCEDURES

In this section we give the detailed steps for the

proposed method.Step 1 Coregister SAR images. The SAR images

are coarsely coregistered by using the crosscorrelationinformation of the SAR image intensity or otherstrategies [1, 2] after SAR imaging of the echoesacquired by each satellite.

REMARK 1 The required image coregistrationaccuracy for the proposed method can be one pixel,which is much lower than the required accuracy (from1/10 to 1/100 pixel) for conventional methods. Thelow coregistration accuracy requirement can greatlyfacilitate coregistration processing.

Step 2 Estimate the covariance matrix. Thecovariance matrix Csi(i) can be estimated by using

joint data vector si(i), shown in Fig. 1. Underthe assumption that the neighboring pixels havethe identical terrain height and that the complexreflectivity is independent from pixel to pixel [15, 21],the covariance matrix Csi(i) can be estimated by its

sample covariance matrix Csi(i), i.e.,

Csi(i) =1

2K+ 1

KXk=K

si(i + k)siH(i + k) (25)

where 2K+ 1 is the number of independentand identically distributed (IID) samples fromthe neighboring pixel pairs. According to theReed-Mallett-Brennan rule [22], the number ofIID samples that 2K+ 1 2M 1 would make theestimation loss within 3 dB if the dimensions of thecovariance matrix Csi(i) are MM [15, 21].

REMARK 2 It is easy to obtain enough IID samplesfor locally flat terrains. However an imaging terrainin practice cannot be relied upon to be so flat that

the adjacent pixels have identical terrain height. Ifthe local terrain slope is available in advance orif it can be estimated [16, 23], the steering vector(i.e., the interferometric phase) variation due to thedifferent terrain heights from pixel to pixel can becompensated, which greatly enlarges the size of thesample window.

Step 3 Estimate the generalized correlationsteering vector. The generalized correlation steering

vector can be estimated by

aicorrelation(i) = [T(i),

T(i),T(i),

T(i)]T

[rT1 , rT2 , r

T3 , r

T4 ]

T (26)

where

r1 = [1, r21(i 6, i 1), r21(i 5, i 1), r21(i 2, i 1),

r21(i 1, i 1)]T

r2 = [1, r21(i 4, i), r21(i 3, i), r21(i, i), r21(i + 1, i)]T

r3 = [1, r21(i + 2, i +3), r21(i + 3, i + 3), (27)

r21(i + 6, i +3), r21(i + 7, i +3)]T

r4 = [1, r21(i + 4, i +4), r21(i + 5, i +4), r21(i + 8, i +4),

r21(i + 9, i +4)]T

r21(m, n) =

PLk=L s2(m + k)s

1(n + k)

qPL

k=L js2(m + k)j2PL

k=L js1(n + k)j2

,

m = i 6, : : : , i + 9, n = i 1, i, i + 3, i + 4: (28)

Here 2L + 1 is the number of IID samples from theneighboring pixel pairs.

Step 4 Estimate the InSAR interferometricphase. The cost functions of estimating the InSARinterferometric phase in the presence of largecoregistration errors can be formulated as

JBF = aiHcorrelation(i)Csi(i)aicorrelation(i) (29)

JCAPON =1

aiHcorrelation(i)C1si (i)aicorrelation(i)

:

(30)

The InSAR interferometric phase can be estimatedby maximizing any one of (29) and (30).

REMARK 3 The computational burden is high if thecost functions are computed via the search of i inthe principal phase interval [, +]. To reduce thecomputational burden, a fast algorithm for computingthe cost functions is developed in the Appendix,where the closed-form solution to the estimate of iis directly obtained by using the fast algorithm.

With the use of the above four steps, the terraininterferogram can be recovered after the pixel pairs ofthe SAR images are processed separately.



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Fig. 4. Accurate coregistration. Interferograms obtained by (a) pivoting mean filtering, (b) pivoting median filtering, (c) adaptive phase

noise filtering, (d) adaptive contoured window filtering, (e) beamforming using the generalized correlation steering vector and

(f) Capon using the generalized correlation steering vector.

Fig. 5. Image coregistration error of 0.5 pixels. Interferograms obtained by (a) pivoting mean filtering, (b) pivoting median filtering,

(c) adaptive phase noise filtering, (d) adaptive contoured window filtering, (e) beamforming using the generalized correlation steering

vector and (f) Capon using the generalized correlation steering vector.



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Fig. 6. Image coregistration error of 1 pixel. Interferograms obtained by (a) pivoting mean filtering, (b) pivoting median filtering,

(c) adaptive phase noise filtering, (d) adaptive contoured window filtering, (e) beamforming using the generalized correlation steering

vector and (f) Capon using the generalized correlation steering vector.

V. PERFORMANCE INVESTIGATION

In this section we demonstrate the robustness ofthe method to coregistration errors by using two setsof simulated data and a real data set. Some practicalconsiderations and the computational efficiency of the

proposed method are also discussed in the following.We assume that there are two formation-flying

satellites in the cartwheel formation, and we selectone orbit position for simulation, with an effectivecross-track baseline of 127.60 m, an orbit height of750 km, and an incidence angle of 45. We use atwo-dimensional Hann window to simulate the terrain,and we use the statistical model to generate thecomplex SAR image pairs [24]. The signal-to-noiseratio (SNR) of the SAR images is 18 dB.

Here the sample window for pivoting meanfiltering and pivoting median filtering is 7 (in range)7 (in azimuth), the number of the samples for theadaptive contoured window filtering to estimate theterrain interferometric phases is 49, and the numberof the samples used to estimate the covariance matrixis 7 (in range) 7 (in azimuth) = 49. The variationof the standard deviation of the interferometric phasewith the increasing coregistration error is computed bymeans of 1000 Monte-Carlo simulations.

Figs. 46 compare the simulation results forvarious techniques and coregistration errors. Bycomparing Figs. 4, 5, and 6, we can observe that

the large coregistration error heavily affects theinterferograms obtained by pivoting mean filtering,pivoting median filtering, adaptive phase noisefiltering, and adaptive contoured window filtering.On the contrary the large coregistration error hasalmost no effect on the interferograms obtained by the

proposed methods by using the generalized correlationsteering vector.

Figs. 78 compare the differential interferogramsand their histograms of the proposed methods forvarious coregistration errors. From the simulationresults we observe that the performance of theproposed methods does not degrade with increasingcoregistration error. That is to say that the proposedmethods can accurately estimate the correspondingterrain interferometric phase in the presence of largecoregistration errors.

By comparing Figs. 48, we can conclude that theproposed methods that use the generalized correlationsteering vector in this paper are robust to largecoregistration errors (up to one pixel).

Fig. 9 shows the variation of the standarddeviation of the interferometric phase with theincreasing coregistration error. We can see that theperformance of our method is as good as that of the

joint subspace projection method [15]. However ourmethod avoids the trouble of calculating the noisesubspace dimension before estimating the InSARinterferometric phase.



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Fig. 7. Differential interferograms (difference between interferograms obtained by beamforming and Capon using generalized

correlation steering vector for coregistration error of 0.5 pixels and accurate coregistration) and their histograms. (a) Differential

interferogram obtained by beamforming approach. (b) Differential interferogram obtained by Capon approach. (c) Histogram of (a).

(d) Histogram of (b).

The second simulated data is the interferometricdata pair of Mount Etna (the data are produced based

on the SIR-C/X-SAR data acquired at X-band).2

Fig. 10 shows the interferograms generated from

the Mount Etna data. Fig. 10(a) is the interferogramobtained by conventional processing [15] (i.e., directly

computing the interferometric phase pixel by pixel),and Fig. 10(b) is the interferogram obtained by theapproach proposed in this paper.

In the following we verify the validity of ourmethod with the ERS1/ERS2 (European Remote

Sensing 1 and 2 tandem satellites, ERS1 orbit =32585, ERS2 orbit = 12912, frame = 2781,

1997-10-08/09, Zhangbei, China) real data.Fig. 11 shows the interferograms generated

from the ERS1/ERS2 real data. Fig. 11(a) is theinterferogram obtained by conventional processing

[15] (i.e., directly computing the interferometric phasepixel by pixel), and Fig. 11(b) is the interferogram

obtained by the approach proposed in this paper.

2Epsilon Nought, Radar Remote Sensing: http://epsilon.nought.de/.

As mentioned above in practice we use theadjacent pixels as the IID samples to estimate the

covariance matrix Csi(i). The IID assumption isgenerally satisfied for a locally flat terrain [21].The number of IID samples is one of the importantfactors that affect the estimation accuracy of theinterferometric phase. The phase estimation accuracyas a function of the number of IID samples is shownin Fig. 12. It can be seen from Fig. 12 that theestimation accuracy increases asymptotically with theincrease of the number of samples.

However for most natural terrains, the assumptionof identical height of neighboring pixels is not strictlysatisfied because of the height variation betweenneighboring pixels. It is important, in practicalapplications, to investigate the impact of the terrainphase variation due to different terrain height betweensamples on the phase estimation accuracy. Fig. 13shows the estimated phase accuracy with the terrainphase variation of samples, and the number ofsamples used to estimate the covariance matrix isfixed to 49. In fact if a priori knowledge about thelocal terrain slope (such as a coarse DEM) is availableor if the local slope can be coarsely estimated, the



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Fig. 8. Differential interferograms (difference between interferograms obtained by beamforming and Capon using generalized

correlation steering vector for coregistration error of 1 pixel and accurate coregistration) and their histograms. (a) Differential

interferogram obtained by beamforming approach. (b) Differential interferogram obtained by Capon approach. (c) Histogram of (a).

(d) Histogram of (b).

Fig. 9. (a) Standard deviation of interferometric phase versus coregistration error. (b) Magnified part of left picture.

terrain phase variation between samples can becompensated in order to significantly mitigate itsimpact on phase estimation accuracy.

To evaluate the computational complexity ofthe proposed method, we follow a strategy similarto that presented in [25]. The approach consistsof estimating the number of multiplications and

additions for each major step of the algorithm. Foreasy discussion we only evaluate the computationalcomplexity of the beamforming technique by usingthe generalized correlation steering vector. Table Ishows the total number of multiplications andadditions of the beamforming technique using thegeneralized correlation steering vector for the pixel



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Fig. 10. Interferograms obtained by (a) conventional processing, and (b) proposed method for Mount Etna data.

Fig. 11. Interferograms obtained by (a) conventional processing, and (b) proposed method for ERS1/ERS2 real data.

Fig. 12. Phase estimation accuracy versus numbers of IID samples using (a) beamforming technique and (b) Capon.

pair i whose interferometric phase is to be estimated(the computational complexity of image coregistrationis not included).

Table II shows the computational time of theproposed method and the conventional techniquefor the first simulated data set, which is a 701 261pixel simulated interferogram. The computational

time for this table is obtained based on a computerwith a 3.00 GHz Intel Pentium 4 processor equippedwith 1 GB access memory. The searching step size ofthe proposed method is 1. The relevant simulationparameters have been mentioned before.

REMARK 4 Similar to Table I the computationaltime of image coregistration for various techniques



11/15

Fig. 13. Phase estimation accuracy versus terrain phase variation between sample pixels using (a) beamforming technique and

(b) Capon.

TABLE I

Number of Multiplications and Additions for Beamforming using the Generalized Correlation Steering Vector

Step # Number of Multiplications Number of Additions

2. Estimate the covariance matrix 400 (2K+ 1) + 2 2K 400

3. Estimate the generalized correlation steering vector (3 (2L + 1 ) + 2 ) 16 + 20 3 2L 16

4. Estimate the InSAR interferometric phase (20 21)(2= + 1) (19 21)(2= + 1)

Total 800K+ 96L +840= + 922 800K+ 96L +798= +399

Note: In this table is the searching step size for estimating the InSAR interferometric phase.

is not included in this table. It has been foundthat the drawback of the proposed method is ahigh computational load due to the fact that thesearching technique is adopted. In order to enhancethe computational efficiency of the approach, a fastalgorithm has been developed in Appendix A.

VI. CONCLUSIONS

We have proposed a new method to estimate theterrain interferometric phases from the InSAR imagepair. Benefiting from the generalized correlationsteering vector in the presence of large coregistrationerrors, the method can auto-coregister the SARimages and reduce the interferometric phase noisesimultaneously. A fast algorithm is developed toimplement the method, which significantly reduce thecomputational burden. The effectiveness of the methodis verified via simulated data and real data.

APPENDIX A. FAST ALGORITHM TO COMPUTE THEOPTIMUM INTERFEROMETRIC PHASE ESTIMATE

The cost function of (29) can be rewritten as

JBF = aiHcorrelation(i)Csi(i)aicorrelation(i) (31)

aicorrelation(i) = (i) rst (32)

(i) = [1, eji ,eji ,eji ,eji ,1,eji ,eji ,eji ,eji ,1,

eji ,eji , eji , eji , 1,eji ,eji ,eji ,eji ]T (33)

TABLE II

Computational Time of the Proposed Method and the

Conventional Technique

Different Filtering Techniques Computational Time

Pivoting mean filtering 59.687 s

Pivoting median filtering 59.828 sAdaptive phase noise filtering 232.844 s

Adaptive contoured window filtering 1607.094 s

Beamforming using the generalized

correlation steering vector

3234.359 s

Capon using the generalized correlation

steering vector

3380.063 s

rst = [1,r2, r3,r4,r5,1,r7, r8, r9, r10,1, r12, r13,r14,r15,1, r17,

r18,r19, r20]T: (34)

Let p = [2,3, 4,5,7,8,9, 10,12,13,14,15,17,18,

19,20] and

Csi(i) =

266666666664

c1,1 c1,2 c1,3 c1,18 c1,19 c1,20

c2,1 c2,2 c2,3 c2,18 c2,19 c2,20

.... . .

...

.... . .

...

c19,1 c19,2 c19,3 c19,18 c19,19 c19,20

c20,1 c20,2 c20,3 c20,18 c20,19 c20,20

377777777775

:



12/15

It can be easily proved that Csi(i) is a Hermitianmatrix, i.e., cnm = cmn (n = 2,3, : : : ,19,20 m =2,3, : : : ,19,20 m 6= n), so we get0

@ Xn=1,6,11,16

Xm=p

cnmrm

1A

=X

n=1,6,11,16

Xm=p

cmnrm:

Let

Xn=1,6,11,16

Xm=p

cnmrm =

Xn=1,6,11,16

Xm=p

cnmrmej

where < . Then

JBF = aiHcorrelation(i)Csi(i)aicorrelation(i)

= aiHcorrelation(i)

266666666664

c1,1 c1,2 c1,3 c1,18 c1,19 c1,20

c2,1 c2,2 c2,3 c2,18 c2,19 c2,20

.... . .

...... . . .

.

..

c19,1 c19,2 c19,3 c19,18 c19,19 c19,20

c20,1 c20,2 c20,3 c20,18 c20,19 c20,20

377777777775

aicorrelation(i)

=X

m=1,6,11,16

Xn=1,6,11,16

cmn +Xm=p

Xn=p

cmnrmrn +

Xn=1,6,11,16

Xm=p

cmnrme

ji +X

n=1,6,11,16

Xm=p

cnmrmeji

=X

m=1,6,11,16

Xn=1,6,11,16

cmn +Xm=p

Xn=p

cmnrmrn +

0@ X

n=1,6,11,16

Xm=p

cnmrmeji

1A

+X

n=1,6,11,16

Xm=p

cnmrmeji

=X

m=1,6,11,16

Xn=1,6,11,16

cmn +Xm=p

Xn=p

cmnrmrn +

0@ X

n=1,6,11,16

Xm=p

cnmrm

ej eji

1A +

X

n=1,6,11,16

Xm=p

cnmrm

ej eji

=X

m=1,6,11,16

Xn=1,6,11,16

cmn +Xm=p

Xn=p

cmnrmrn + 2

Xn=1,6,11,16

Xm=p

cnmrm

cos( + i): (35)

Obviously the maximum of JBF can be obtainedfor +

i

= 2k (k is an integer, and =angle(

Pn=1,6,11,16

Pm=p cnmrm)). Since < and

< i < , thus i = : (36)

The cost function of (30) can be rewritten as

JCAPON =1

aiHcorrelation(i)C1si (i)aicorrelation(i)

=1

J(37)

where

J = aiHcorrelation(i)C1si (i)aicorrelation(i): (38)

Let

B = C1si (i)

=

26666666666664

b1,1 b1,2 b1,3 b1,18 b1,19 b1,20

b2,1 b2,2 b2,3 b2,18 b2,19 b2,20

.... . .

...

.... . .

...

b19,1 b19,2 b19,3 b19,18 b19,19 b19,20

b20,1 b20,2 b20,3 b20,18 b20,19 b20,20

37777777777775

:

It can be easily proved that B is a Hermitian matrix,i.e., bnm = bmn (n = 2,3, : : : ,19,20 m = 2,3, : : : ,19,20

m 6= n), so we get0@ X

n=1,6,11,16

Xm=p

bnmrm

1A

=X

n=1,6,11,16

Xm=p

bmnrm:

Let

Xn=1,6,11,16

Xm=p

bnmrm =

Xn=1,6,11,16

Xm=p

bnmrm

e

j



13/15

where < . Then

J = aiHcorrelation(i)C1si (i)aicorrelation(i)

= aiHcorrelation

(i)

266666666664

b1,1 b1,2 b1,3 b1,18 b1,19 b1,20

b2,1 b2,2 b2,3 b2,18 b2,19 b2,20

.... . .

...

... . . . ...

b19,1 b19,2 b19,3 b19,18 b19,19 b19,20

b20,1 b20,2 b20,3 b20,18 b20,19 b20,20

377777777775

aicorrelation

(i)

=X

m=1,6,11,16

Xn=1,6,11,16

bmn +Xm=p

Xn=p

bmnrmrn +

Xn=1,6,11,16

Xm=p

bmnrme

ji +X

n=1,6,11,16

Xm=p

bnmrmeji

=X

m=1,6,11,16

Xn=1,6,11,16

bmn +Xm=p

Xn=p

bmnrmrn +

0@ X

n=1,6,11,16

Xm=p

bnmrmeji

1A

+X

n=1,6,11,16

Xm=p

bnmrmeji

= Xm=1,6,11,16

Xn=1,6,11,16

bmn + Xm=p

Xn=p

bmnrmrn +

0@ X

n=1,6,11,16Xm=p

bnmrm

e

j eji

1A

+

X

n=1,6,11,16Xm=p

bnmrm

e

j eji

=X

m=1,6,11,16

Xn=1,6,11,16

bmn +Xm=p

Xn=p

bmnrmrn + 2

X

n=1,6,11,16

Xm=p

bnmrm

cos( + i): (39)

Obviously the minimization of J can be obtainedfor + i = + 2k (k is an integer, and =angle(

Pn=1,6,11,16

Pm=p bnmrm)). Since < and

< i < thus

i = ( 0)

( > 0): (40)

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Guisheng Liao (M96) received his B.S. degree from Guangxi University,Guangxi, China and his M.S. and Ph.D. degrees from Xidian University, Xian,China, in 1985, 1990, and 1992, respectively.

His research interests are mainly in signal processing for radar andcommunication and in smart antenna for wireless communication.

Hai Li was born in Tianjin, China, on August 12, 1976. He received his B.S.degree and his M.S. degree in measurement and control engineering andinstrumentation and his Ph.D. degree in electronic information engineering fromXidian University, Xian, China, in 1999, 2002, and 2008, respectively.

He currently works in the Tianjin Key Lab for Advanced Signal Processing,at the Civil Aviation University of China, Tianjin, China. His current researchis in InSAR signal processing, space-time adaptive processing and SARground-moving target indication.


estimation method for insar

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