Robust Interferometric Phase Estimation in InSAR via Joint Subspace Projection

Synthetic aperture radar interferometry (InSAR) is an important remote sensing technique to retrieve the terrain digital elevation model (DEM)[1][2]. Image coregistration, InSAR interferometric phase estimation (or noise filtering) and interferometric phase unwrapping[3][4][5][6] are three key processing procedures of InSAR. It is well known that the performance of interferometric phase estimation suffers seriously from poor image coregistration.

and the problem formulation.In Section 3, we discuss the interferometric phase estimation method based on subspace projection.In Section 4, the modified interferometric phase estimation method via subspace projection is presented in details.Finally, numerical and experemental results are presented in Section 5. Section 6 concludes the whole chapter.

Data model and problem formulation
Assuming that the SAR images are accurately coregistered and the interferometric phases are flattened with a zero-height reference plane surface.The complex data vector, denoted as () i s , of a pixel pair i (corresponding to the same ground area) of the coregistered SAR images can be formulated as follows [19], 12 1 2 () [ () , () ] [ () , () ] ( ) () () where () [ 1 ,  ] is the spatial steering vector (i.e., the array steering vector) of the pixel i , superscript T denotes the vector transpose operation, i  is the terrain interferometric phase to be estimated,  denotes the Hadamard product, () i x is the complex magnitude vector (i.e., complex reflectivity vector of scene received by the satellites) of the pixel i , and () i n is the additive noise term.The complex data vector () i s can be modeled as a joint complex circular Gaussian random vector with zero-mean and the corresponding covariance matrix () s i C is given by () , () () () , () where () i s R is called the correlation coefficient matrix, I is a 22 identity matrix, () , and 1, 2 m  ) are the correlation coefficients between the satellites m and n , {} E denotes the statistical expectation, superscript H denotes vector conjugatetranspose, 2 () s i  is the echo power of the pixel i and 2 n  is the noise power.In order to simplify the mathematical expressions, the denotation i (denoting the pixel) in the right side of the following expressions is omitted.In practice, the statistical covariance matrix of (2) can be adaptively estimated using the sample covariance matrix.
If the SAR images are accurately coregistered and the cross-correlation coefficients (i.e., the nondiagonal elements) of () i s R are large enough, the number of the principal eigenvalues of the covariance matrix () s i C is one; i.e., the dimensions of the signal subspace and the noise subspace are both one (in the absence of the layover).In this case, the eigendecomposition of the covariance matrix () s i C is as follows: The minimization of 1 J can provide the optimum estimate of the interferometric phase i  .
The eigenvalues of C s (i) result in a low dispersion due to the increase of the coregistration error.Actually, it is induced by the increase of noise eigenvalue, i.e., the signal component spreads into the noise space.Moreover, the 2-dimensional space will be fully occupied by signal component with a worse coregistration error of one pixel.At this instant, the noise eigenvalue is almost equal to the bigger one.The noise subspace dimension becomes zero.
The degree of the signal component spreading to the noise subspace is smaller, the estimation of the InSAR interferometric phase is better(the subspace projection technique is used to estimate the InSAR interferometric phase ).On the contrary, the degree of the signal component spreading to the noise subspace is larger, the estimation of the InSAR interferometric phase is worse.The conclusions obtained from the preceding analysis are briefly summarized as follows: the degree of the signal component spreading to the noise subspace becomes larger and larger as the coregistration error increases when the example of formula ( 1) is used to build the data vector.In other words, the degree of dispersion is heavily impacted by the coregistration error.The simulation result shown in Fig. 1 demonstrates the above conclusions.The cost function given by ( 5) can be used to estimate the InSAR interferometric phase when the SAR images are accurately coregistered.However, in the presence of coregistration error, the cross-correlation coefficients are smaller than 1 and the noise eigenvalue becomes large.In the completely misregistered case, the rank of () i s R becomes 2, which means that the noise subspace vanishes in the eigenspace of () . So the cost function given by ( 5) can not be used to estimate the InSAR interferometric phase in the presence of coregistration error.

Interferometric phase estimation via subspace projection
Considering the difficulties in accurate coregistration, we use not only the corresponding pixel pair i of the coarsely coregistered SAR image pair (as given in ( 1)) but also the neighboring pixel pairs centered on the pixel pair i to jointly construct the data vector.An example of the construction method for the data vector is shown in Fig. 2, where a circle represents a SAR image pixel and i denotes the centric pixel pair (i.e., the desired pixel pair whose interferometric phase is to be estimated).We call this extended data vector () i si the joint data vector.The number of the neighboring pixel pairs to construct the joint data vector is 8 as shown in Fig. 2. Fig. 2. A construction method for the joint data vector.
The joint data vector () i si as shown in Fig. 2 can be written as: () [( 4) , ( 3 ) , , () , , ( 4 ) ] The corresponding joint covariance matrix is given by where are referred to as the joint steering vector and the joint correlation function matrix of the pixel pair i , respectively.In fact, most of the nature terrain can be approximated by a local plane.After we estimate the local slopes and correct the neighboring pixels, we can assume that the neighboring pixels have the almost identical terrain height [20].That is, the spatial steering vectors of the pixel pairs in () i si are assumed to be identical, i.e.,    Comparing Fig. 3 with (4), the conventional estimation approaches based on the single pixel pair will be not feasible if the coregistration error is very large, for example, larger than 0.5 pixel, while our approach can still achieve the optimum estimation due to the use of multiple neighboring pixel pairs.
From the literature [21], we can know that: for the accurate coregistration, the dimensions of the signal subspace and the noise subspace of the joint covariance matrix () i si C (18 18  ) are both 9.For the coregistration error of 1 pixel, the dimensions of the signal subspace and the noise subspace of the joint covariance matrix () i si C are changed to 12 and 6, respectively.The method can also provide accurate estimation of the terrain interferometric phase even if the coregistration error reaches one pixel only by changing the dimension of the noise subspace from 9 to 6.
As mentioned above, the signal subspace () () The minimization of 2 J can provide the optimum estimate of the interferometric phase i  .Comparing these figures, we can observe that the large coregistration error has almost no effect on the interferograms obtained by the proposed method.We can see that the subspace projection method in this chapter is robust to large coregistration errors (up to one pixel).

Modified interferometric phase estimation via subspace projection
The joint subspace projection method mentioned above employs the projection of the joint signal subspace onto the corresponding joint noise subspace which is obtained from the eigendecomposition of the joint covariance matrix to estimate the terrain interferometric phase, and takes advantage of the coherence information of the neighboring pixel pairs to auto-coregister the SAR images, where the phase noise is reduced simultaneously.However, the noise subspace dimension of the covariance matrix changes with the coregistration error.For accurate estimating the InSAR interferometric phase, the noise subspace dimension of the covariance matrix must be known, and the performance of the method (i.e.subspace projection method) degrades when the noise subspace dimension is not estimated correctly.In this chapter, an modified joint subspace projection method for InSAR interferometric phase estimation is proposed.In this method, the benefit from the new formulation of joint data vector is that the noise subspace dimension of the covariance matrix is not affected by the coregistration error (i.e., the noise subspace dimension of the corresponding covariance matrix with the coregistration error  (0 1)    pixel is the same as that of the covariance matrix with accurate coregistration).So the method does not need to calculate the noise subspace dimension before estimating the InSAR interferometric phase.

Data modeling of the modified method
For avoiding the trouble of calculating the noise subspace dimension before estimating the InSAR interferometric phase, the new formulation of joint data vector is used in the proposed method.An example to construct the new joint data vector () i si is shown in Fig. 5.


  pixel and its direction is upwards (i.e., the pixel of the image from the second satellite is shifted upwards compared to the in the first satellite image), the joint data vector, () i si ,is shown in Fig. 8(a).
From the results derived above, we can see that: the new formulation of joint data vector proposed in this chapter has the advantage that the noise subspace dimension of the corresponding covariance matrix is independent of the coregistration error.That is to say, the noise subspace dimension of the corresponding covariance matrix with the coregistration error  (0 1)    pixel is the same as that of the accurate covariance matrix.Therefore, it is not required to calculate the noise subspace dimension, thus avoiding the trouble of calculating the noise subspace dimension before estimating the InSAR interferometric phase.

Summary of the modified method
In this section, we give the detailed steps for the modified interferometric phase estimation method based on subspace projection.
Step 1. Coregister SAR images.The SAR images are coarsely coregistered using the crosscorrelation information of the SAR image intensity or other strategies [1][2] after SAR imaging of the echoes acquired by each satellite.However, the allowable coregistration error of the proposed method can be very large (such as one pixel), which is useful in practice.The low coregistration accuracy requirement can greatly mitigate complexity in image coregistration processing.
Step 2. Estimate the covariance matrix.The covariance matrix () i si C can be estimated by using joint data vector () i si shown in Fig. 5.Under the assumption that the neighboring pixels have the identical terrain height and the complex reflectivity is independent from pixel to pixel [20][21], the covariance matrix () i si C can be estimated by its sample covariance matrix ˆ() where 21 L  is the number of i.i.d.samples from the neighboring pixel pairs.
Remark 1: It is easy to obtain enough i.i.d.samples for locally flat terrains.However, an imaging terrain in practice can not be relied upon to be so flat that the adjacent pixels have the identical terrain height.If the local terrain slope is available in advance or can be estimated [20], the steering vector (i.e., the interferometric phase) variation due to the different terrain height from pixel to pixel can be compensated, which greatly enlarges the size of the sample window.
Step 3. Subspace estimation by Eigendecomposing.The estimated covariance matrix ˆ() where , (1)   (2) ˆˆˆˆK ) corresponding to the smaller eigenvalues () ) span the noise subspace, i.e,   ˆˆ, , , whereas the larger eigenvectors () ˆm csi (1 , 2 , , mK   ) corresponding to the principal The noise power is often estimated by 20 () 2 1 1 20 The joint correlation function matrix ˆ() i si R can be approximated as the amplitude (i.e., the absolute value) of the estimated covariance matrix ˆ() i si C [20], i.e., Step 4. Projection of signal subspace onto noise subspace.The projection of the signal subspace onto the corresponding noise subspace is performed as follows: The cost function given by ( 29) is used to estimate the terrain interferometric phase i  .And the minimization of 3 J provide the optimum estimate of the interferometric phase i Remark 2: The computational burden will be high if the minimization of 3 J is obtained via search of i  in the principal phase interval [ , ]     .To reduce the computational burden, a fast algorithm to compute the minimization of 3 J is developed in Appendix B, where the closed-form solution to the estimate of i  is directly obtained by using the fast algorithm.
Using the above four steps, the terrain interferogram can be recovered after the pixel pairs of the SAR images are processed separately.

Numerical and experemental results
In this section we demonstrate the robustness of the modified method via subspace projection to coregistration errors by using simulated data and real data.
The simulated data are described as follows.We assume there are two formation-flying satellites in the cartwheel formation, and we select one orbit position for simulation, with an effective cross-track baseline of 281.46 m, an orbit height of 750 kilometers and an incidence angle of 45.We use a two-dimensional window to simulate the terrain and use the statistical model to generate the complex SAR image pairs [23].The signal-to-noise ratio (SNR) of the SAR images is 18 dB.
Here, the number of the samples to estimate the covariance matrix is 7 (in range)  7 (in azimuth)=49.In the following, we will verify the validity of the modified method with the ERS1/ERS2 (European Remote Sensing 1 and 2 tandem satellites) real data.Fig. 13 shows the interferograms generated from the ERS1/ERS2 real data.Fig. 13(a) is the interferogram obtained by the conventional processing, and Fig. 13(b) is that obtained by the modified method proposed in this chapter.

Conclusions
In this chapter, the interferometric phase estimation method based on subspace projection and its modified method were presented.The interferometric phase estimation methods based on joint subspace projection can provide accurate estimate of the terrain interferometric phase (interferogram) even if the coregistration error reaches one pixel.
Benefiting from the new formulation of joint data vector, the modified method does not need to calculate the noise subspace dimension, thus avoiding the trouble of calculating the noise subspace dimension before estimating the InSAR interferometric phase.A fast algorithm is developed to implement the modified method, which can significantly reduce the computational burden.Theoretical analysis and simulations demonstrate the efficiency of the proposed new algorithms.

Acknowledgement
This work is supported in part by the National Natural Science Foundations of China under grant 60736009, 61071194 and 60979002, by the Fund of Civil Aviation University of China under grant ZXH2009D018 and 2011kyE06.

Appendix B 9.1 Fast algorithm for optimal interferometric phase estimation
If U , V and W are arbitrary complex column vectors, then [20] www  So the closed-form solution to the estimate of i  is directly obtained by using the fast algorithm, which can significantly reduce the computational burden.
. From Fig.3we can see that when the SAR images are not accurately coregistered, the same imaged ground area can be coregistered to different pixel positions; thus its correlation information appears in other non-diagonal element position of ()i si R. The gray level denotes the magnitude of each element with the white strongest and black zero.

Fig.s 4
Fig.s 4 shows the simulation results for various coregistration errors by the subspace projection method.Comparing these figures, we can observe that the large coregistration error has almost no effect on the interferograms obtained by the proposed method.We can see that the subspace projection method in this chapter is robust to large coregistration errors (up to one pixel).

Fig. 4 .
Fig. 4. The interferograms obtained by the subspace projection method for the accurate coregistration (a), the coregistration error of 0.5 pixels (b) and the coregistration error of one pixel (c).

Fig. 5 .
Fig. 5. Formulation of the joint data vector.In the following the noise subspace dimension of the joint covariance matrix () i si C for

Fig. 7 .
Fig. 7. Joint data vector for the coregistration error of one pixel.The corresponding covariance matrix () i si' C is given by (b)[21], which does not change the eigenvalues of the corresponding covariance matrix.The joint data vector () i js , shown in Fig.8(b), can be written as

Fig. 8 .
Fig. 8. Joint data vector for the coregistration error of  pixels.

Fig.s 9 -Fig. 9 .Fig. 10 .Fig. 11 . 12 .
Fig.s 9-12 compare the simulation results for various techniques and coregistration errors.Comparing Fig. 9,10 ,11and 12, we can observe that the large coregistration error heavily affects the interferograms obtained by pivoting median filtering, pivoting mean filtering and adaptive contoured window filtering.On the contrary, the large coregistration error has

Fig. 13 .
Fig. 13.The interferograms obtained by (a) the conventional processing, and (b) the modified method for the ERS1/ ERS2 real data.

Fig. A. 1 .
Fig. A.1.Formulation of the joint data vector () i ss . , 1 .intechopen.comRecent Interferometry Applications in Topography and Astronomy