Super-Resolution Persistent Scatterer Interferometry (SR-PSI) – PS densiﬁcation through Capon based SAR reprocessing

The Capon algorithm can be applied to reprocess SAR images, resulting in super-resolution super-resolution recon-structed scenes with lower sidelobe levels. Base on this idea, we have proposed a processing chain of Super-Resolution Persistent Scatterer Interferometry (SR-PSI), to increase PS density. In this paper, we review the main aspect of SR-PSI. We also propose a revised robust Capon algorithm. The result of real-life Sentinel-1 (S-1) data is shown.


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Super-Resolution Persistent Scatterer Interferometry (SR-PSI) The SAR imaging can be cast as a spectral estimation problem [1,2,3]. Let {s n1,n2 }, n 1 = 0, 1, . . . , N 1 − 1, n 2 = 0, 1, . . . , N 2 − 1 denote a 2-D data matrix and (ω 1 , ω 2 ) be the interested frequency pair. We can model s n1,n2 as s(n 1 , n 2 ) = α ω1,ω2 · e j(n1·ω1+n2·ω2) + e(n 1 , n 2 ), (1) where α ω1,ω2 denotes the complex amplitude of a sinusoid with frequency (ω 1 , ω 2 ), and e(n 1 , n 2 ) represents the unmodeled noise and interference. The focusing process is to estimate α(ω 1 , ω 2 ) from the 2-D data s n1,n2 . The most efficient method is the 2-D Fourier transform, or matchedfilter based focusing algorithm. Single Look Complex (SLC) images generated through this approach suffer from the limited resolution and relatively high sidelobe levels [1]. We can apply modern spectral-estimation methods to calculate α(ω 1 , ω 2 ), resulting in improved resolution and suppressed sidelobe levels [1,3]. By integrating these algorithms into Persistent Scatterer Interferometry, we can increase the PS density [4]. Since the Capon algorithm seems promising in the context of PSI, here we only consider the Capon algorithm. A detailed discussion of the proposed Super-Resolution PSI (SR-PSI) can be found in [4]. Compared to the traditional PSI, the main change of SR-PSI is the Super-Resolution reprocessing and PSC selection method. The step of Super-Resolution reprocessing is to refocus the SAR images using the Capon algorithm. It includes spectral equalization, image chipping, Discrete Fourier Transform (DFT), the Capon processing, and chip-images mosaicking [3] We can apply a peak-detection based Persistent Scatterer Candidate (PSC) selection approach to the reprocessed images rather than the regular normalized amplitude dispersion approach [4].

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Revised robust Capon algorithm

Revised robust Capon algorithm
The Capon estimator is designed so that the power of the filtered signal is minimized with the constraint that the gain of the filter remains one at the selected frequency [5]: where R is the covariance matrix, a is the Fourier matrix, h is the filter to be constructed and (·) H denotes conjugate transpose operator. The solution of (2) is given by In the application of standard Capon algorithm, R is usually replaced by the sample covariance matrixR, where N is the number of snapshots and z(n) is the n-th snapshot (snapshots are defined as the sub-matrices of the data matrix). To make the calculation more robust, a fully diagonal loading (DL) approach is proposed by [6]: is used instead ofR. The α and β coefficients are determined by minimizing the Mean Square Error (MSE) ofR with the constraints that α ≥ 0 and β ≥ 0, and guaranteeing thatR is a positive semidefinite matrix. The solution of α and β is expressed as: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 12 October 2020 doi:10.20944/preprints202010.0225.v1 andυ = tr(R)/M .
Since α and β can be calculated from the data, the method is parameter-free. The covariance matrixR can be estimated by combining (5) and (6). We found that DL approaches tend to cause discontinuities between the adjacent chip images. To avoid this as far as possible, we should apply the DL approach only when necessary. We set the required condition as when the condition number is above a given threshold. Thus the covariance matrix is estimated bȳ

Its effect on PSC selection
The benefits of the proposed robust approach are illustrated in Fig. 1. We can observe edge effect on the upper subplot while the lower image is seamless. The edge effect results in spurious peaks that are identified as additional PSCs, as for example within the rectangle near the uppler left corner. We also observe that some new PSCs are present in the middle of the plot, which may indicate the figure reprocessed by regular Capon algorithm is resolved better than the one reprocessed by the standard automatic DL-Capon algorithm.

Experimental Results
We applied the robust-Capon-based SR-PSI to a stack of interferometric Sentinel-1 images. Fig. 2 shows a detailed comparison of the deformation velocities of the PSs. As indicated by the optical image, the two lines of PSs in the right subplot can be interpreted as dihedrals formed by the piers and the roof. We can observe the PSs are separated on the Capon-based reprocessed image while they are mixed on the original image, which shows that the resolution is improved. Correspondingly, the PS density is increased, which may confirms the effectiveness of SR-PSI.

Figure 1
The upper subplot and the lower subplot present PSC selected from the images reprocessed by the automatic DL Capon algorithm and by our proposed algorithm, respectively.The background shows the squared mean intensity image.

Figure 2
The left subplot shows the optical image of the area from the Google Earth. The middle subplot and the right subplot show the PSs selected from the original stack and from the Capon-based reprocessed images, respectively.