Compressive sensing and generalized likelihood ratio test in SAR tomography

Gianfranco Fornaro, Antonio Pauciullo, Diego Reale, Matthias Weis, Alessandra Budillon, Gilda Schirinzi
2016 2016 4th International Workshop on Compressed Sensing Theory and its Applications to Radar, Sonar and Remote Sensing (CoSeRa)  
The application of SAR tomography (TomoSAR) on the urban infrastructure and other man-made buildings has gained increasing popularity with the development of modern high-resolution spaceborne satellites. Urban tomography focuses on the separation of the overlaid targets within one azimuth-range resolution cell, and on the reconstruction of their reflectivity profiles. In this work, we build on the existing methods of compressive sensing (CS) and generalized likelihood ratio test (GLRT), and
more » ... est (GLRT), and develop a multiple scatterers detection method named CS-GLRT to automatically recognize the number of scatterers superimposed within a single pixel as well as to reconstruct the backscattered reflectivity profiles of the detected scatterers. The proposed CS-GLRT adopts a two-step strategy. In the first step, an L1-norm minimization is carried out to give a robust estimation of the candidate positions pixel by pixel with super-resolution. In the second step, a multiple hypothesis test is implemented in the GLRT to achieve model order selection, where the mapping matrix is constrained within the afore-selected columns, namely, within the candidate positions, and the parameters are estimated by least square (LS) method. Numerical experiments on simulated data were carried out, and the presented results show its capability of separating the closely located scatterers with a quasi-constant false alarm rate (QCFAR), as well as of obtaining an estimation accuracy approaching the Cramer-Rao Low Bound (CRLB). Experiments on real data of Spotlight TerraSAR-X show that CS-GLRT allows detecting single scatterers with high density, distinguishing a considerable number of double scatterers, and even detecting triple scatterers. The estimated results agree well with the ground truth and help interpret the true structure of the complex or buildings studied in the SAR images. It should be noted that this method is especially suitable for urban areas with very dense infrastructure and man-made buildings, and for datasets with tightly-controlled baseline distribution. The TomoSAR reconstruction can be regarded as an inverse problem, which was firstly solved by the nonparametric methods such as beamforming (BF) [1, 8] , singular value decomposition (SVD) [9,10], and adaptive Capon [11] [12] [13] . BF and SVD get their popularity from their high efficiency and robustness. Both methods can preserve the full resolution of SAR images, but suffer from low resolution and high side-lobe level problems. Capon allows super-resolution imaging at the expense of a spatial resolution loss as multi-look processing is required. Parametric methods, such as multiple signal classification (MUSIC) [14] , have also been introduced for TomoSAR. MUSIC can achieve super-resolution and side-lobe reduction but it requires a priori information and is sensitive to model errors. Based on the fact that target distribution along elevation is always sparse, especially in urban areas, compressive sensing (CS) [15] [16] [17] provides another solution for infrastructure tomographic reconstruction. It enhances the elevation resolution within the single-look archive and reduces the required number of measurements, while its main limitations stem from the off-grid effect so as to generate spurious outliers, the amplitude bias, and the inability to evaluate the probability of detection and the probability of false alarm. While these methods provide the foundations for significant advances, technical issues of the exact number of targets or scatterers remain to be decided when applied in urban scene study. Automatic methods for detection and reconstruction of infrastructure and other man-made structures in urban area have been widely studied. According to the different methods of model order selection (MOS), existing methods can be classified into three groups: the generalized likelihood ratio test (GLRT) based [18] [19] [20] [21] [22] [23] [24] ; information criterion based, such as Bayes information criterion (BIC) and Akaike information criterion (AIC) [25] [26] [27] ; and no additional MOS based or iterative reweighted method [28] [29] [30] . An approach based on GLRT for the detection of targets in the tomographic framework was proposed in [18] (BF based GLRT) for the first time. As an extension of the work in [18] , an approach focusing on the discrimination of single and double scatterers was introduced [19] . It could effectively estimate the positions and number of scatterers with constant false alarm rate (CFAR) efficiently. However, it suffered from the leakage effects related to side-lobe influence and from a low intrinsic elevation resolution, the so-called Rayleigh resolution. Rayleigh resolution is inversely proportional to the perpendicular baseline extension and is typically much worse than the resolution in azimuth and range, which makes it far from satisfactory when applied in the urban areas with high resolution and tightly-controlled TerraSAR/TanDEM-X. A support GLRT (sup-GLRT) method [20] was proposed to deal with the poor elevation resolution problem. It searches the best signal support to decide the number and positions of the significant scatterers based on nonlinear maximum for detecting at most K max scatterers. It is convenient when the scatterer distribution is very sparse (no more than two), while the computing complexity will increase dramatically with the increase of K max . To mitigate the computational burden, a fast sup-GLRT method [21] , referred to as M-sup-GLRT, was proposed. It transfers the multidimensional optimal searching problem into a K max 1D optimal searching one, so as to enjoy the computational efficiency as well a super-resolution capability comparable to that of sup-GLRT. Most recently, M-sup-GLRT has been extended to 5D application [22] and showed the ability to not only monitor temporal deformation but also to detect the thermal dilation. Another application of sup-GLRT is the investigation of polarimetric TomoSAR (Pol-TomoSAR), which allows the reduction of the number of acquisitions required. Multilook GLRT, referred to as M-GLRT has been proposed to improve the detection capability over man-made targets as well in areas characterized by low SNRs [23] . As an extension of the method in [23], the Capon filter has been introduced in M-GLRT [24] to get a super-resolution ability. SL1MMER [25, 26] has been proposed to eliminate the outliers and get the accurate parameter estimation by introducing two steps of BIC MOS and parameter estimation. It can effectively drop the outliers by penalizing the higher orders and estimating the parameters by scale down L2 method. SL1MMER is super-resolved and has been proven perfect for the urban reconstruction [25, 26] . Most recently, integrating with geographic information systems (GIS), its extended version, M-SL1MMER [31], has been proven to have an accurate tomographic reconstruction capability with
doi:10.1109/cosera.2016.7745706 fatcat:l67ttiprtvcpjnif36hrhcaiwm