The problem of detecting and locating multiple scatterers in multibaseline Synthetic Aperture Radar (SAR) tomography, starting from compressive measurements and applying support detection techniques, is addressed. Different approaches based on the detection of the support set of the unknown sparse vector, that is, of the position of the nonzero elements in the unknown sparse vector, are analyzed. Support detection techniques have already proved to allow a reduction in the number of measurements required for obtaining a reliable solution. In this paper, a support detection method, based on a Generalized Likelihood Ratio Test (Sup-GLRT), is proposed and compared with the SequOMP method, in terms of probability of detection achievable with a given probability of false alarm and for different numbers of measurements.

Synthetic Aperture Radar (SAR) tomography exploits a stack of complex-valued SAR images, acquired with different view angle and at different times, for providing the fully 3D scene reflectivity profile along azimuth, range, and elevation directions [

In [

CS theory enables the reconstruction of sparse or compressible signals from a small set of linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist rate samples. Then, CS based techniques have been proved to be very effective for reducing the number of SAR images to be acquired and mitigating the effects due to nonuniform baseline spacing [

Nevertheless, several issues have still to be considered when dealing with 3D reflectivity profile reconstruction by means of CS based approaches. A first problem is the presence of outliers, produced by the presence of partially coherent clutter and noise and/or by possible solution instabilities. In addition, the so-called off-grid effect [

Another issue to be considered is that many tomographic applications do not require a full reconstruction of the signal. We are often interested only in the localization of multiple coherent scatterers and not in their intensity. This amounts in solving a sort of detection problem, dealing with the identification of only the position of the nonzero elements in the sparse unknown vector, whereas the full reconstruction of the sparse signal is not required.

In [

Recently, a Generalized Likelihood Ratio Test (Sup-GLRT), searching for the best support of the unknown signal matching the data, has been introduced [

In this paper, the performance of Sup-GLRT [

SAR tomography allows the reconstruction of the reflectivity profile of the observed scene along the coordinates of range

Multipass SAR geometry in the range-elevation plane (case

The recovery of the vector

A reduction of the number of measurements required for achieving a reliable solution can be obtained when a full reconstruction of the signal

In this section, we want to analyze some methods for estimating the position of multiple scatterers, by means of support detection techniques.

The detection of the support of a sparse signal can be addressed using different methods.

Since

A practical method to detect the position of the nonzero elements of the unknown sparse signal is LASSO [

In [

In contrast, optimal ML detection techniques can achieve scaling

Another common approach to support detection is the orthogonal matching pursuit (OMP) algorithm [

Then, also in this case, even as SNR scales to infinity, the minimum number of measurements does not scale as

The results summarized above suggest a performance gap between ML detection and algorithms like LASSO and OMP, especially when SNR is high. In particular, as SNR increases, the performance of these methods saturates at scaling in the number of measurements that can be significantly higher than that for ML.

A more practical method is a simplified version of OMP, called sequential OMP (SequOMP) [

SequOMP is a one-pass version of the OMP algorithm, since it is identical to the standard OMP algorithm of [

When the knowledge of conditional rank of signal components is not available, SequOMP has a performance worse than OMP and LASSO but exhibits a noticeably lower complexity.

The methods presented in the previous section are not Constant False Alarm Rate detection approaches. In this section, we present a CFAR approach, using a sequential GLRT for support detection, based on ML estimation. This approach is compared with the SequOMP, adapted in such a way to have a Constant False Alarm Rate.

The detection problem amounts to distinguish

The noise vector

When the scatterers are absent (hypothesis

Probability of detection

In 3D SAR tomography, the number of scatterers

Since the hypotheses

Note that the sequential test described by (

The thresholds

To analyze the compressive capability of the proposed detection scheme, we consider the problem of detecting one or two scatterers ^{−3} and for a scatterers separation distance

COSMO-SkyMed parameters | |
---|---|

Wavelength | 0.031 m |

View angle | 35° |

Range distance | 620 Km |

In this case, the proposed GRLT (

We compare the proposed method with the SequOMP approach. The SequOMP algorithm has been implemented using a threshold providing the desired

In order to compare Sup-GLRT with SequOMP, the probability of false alarm in both approaches is defined according to (

^{−3},

In Figure _{1} = SNR_{2} + 3 dB, where SNR_{2} corresponds to the weaker scatterer. In this case, it can be noted that the performance is equivalent for

_{1} = SNR_{2} + 3 dB) and at a separation distance ^{−3},

Eventually, we report in Figure _{1} = 8 dB and a weaker scatterer with SNR_{2} = 5 dB, and for a varying ratio

_{1} = 8 dB and SNR_{2} = 5 dB), at a separation distance ^{−3}.

In this paper we analyze the problem of identifying multiple scatterers lying in the same range azimuth resolution cell from a compressive number of multibaseline SAR images. We followed a support detection approach, which adapts well to the sparse unknown signal. Different support detection methods have been considered and the performance of two different schemes has been investigated: a GLRT based support detection and the SequOMP techniques. Preliminary results on simulated data show that the first one is more robust with respect to the reduction of the number of measurements, since it allows a higher probability of detection for a given probability of false alarm and a given number of measurements. It has been considered the detection and localization of two scatterers responding with the same intensities and of two scatterers responding with different intensities. Moreover, different ratios

The authors declare that there is no conflict of interests regarding the publication of this paper.