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Motivated by the use of unmanned aerial vehicles (UAVs) for buried landmine detection, we consider the spectral classification of dispersive point targets below a rough air-soil interface. The target location can be estimated using a previously developed method for ground-penetrating synthetic aperture radar involving principal component analysis for ground bounce removal and Kirchhoff migration. For the classification problem, we use the approximate location determined from this imaging method to recover the spectral characteristics of the target over the system bandwidth. For the dispersive point target we use here, this spectrum corresponds to its radar cross section (RCS). For a more general target, this recovered spectrum is a proxy for the frequency dependence of the RCS averaged over angles spanning the synthetic aperture. The recovered spectrum is noisy and exhibits an overall scaling error due to modeling errors. Nonetheless, by smoothing and normalizing this recovered spectrum, we compare it with a library of precomputed normalized spectra in a simple multiclass classification scheme. Numerical simulations in two dimensions validate this method and show that this spectral estimation method is effective for target classification. 

Motivated by applications in unmanned aerial based ground penetrating radar for detecting buried landmines, we consider the problem of imaging small point like scatterers situated in a lossy medium below a random rough surface. Both the random rough surface and the absorption in the lossy medium significantly impede the target detection and imaging process. Using principal component analysis we effectively remove the reflection from the air‐soil interface. We then use a modification of the classical synthetic aperture radar imaging functional to image the targets. This imaging method introduces a user‐defined parameter,δ, which scales the resolution by allowing for target localization with sub wavelength accuracy. Numerical results in two dimensions illustrate the robustness of the approach for imaging multiple targets. However, the depth at which targets are detectable is limited due to the absorption in the lossy medium. 

Abstract We study the effects of absorption in the medium on synthetic aperture imaging. We model absorption using the loss tangent, which is the imaginary part of the relative dielectric permittivity, and study two cases: (i) the loss tangent is known and (ii) the loss tangent is unknown. When the loss tangent is known and used in Kirchhoff migration (KM), we find that images of targets are range-shifted by approximately a central wavelength so that their predicted locations are closer to the synthetic aperture than they actually are. In contrast, we find that when the medium is unknown, the KM image does not exhibit this range-shift. Hence, we determine that it is better to not make use of any knowledge of the absorption when imaging. Using a recently developed transformation of KM images, which we call reciprocal-KM (rKM), we achieve tunably high-resolution images of targets through adjusting the value of a user-defined parameterε. When rKM is applied to an imaging region containing two targets, we find that their predicted locations shift, especially in range, but within a fraction of central wavelength of their true locations. 

We introduce a dispersive point target model based on scattering by a particle in the far-field. The synthetic aperture imaging problem is then expanded to identify these targets and recover their locations and frequency dependent reflectivities. We show that Kirchhoff migration (KM) is able to identify dispersive point targets in an imaging region. However, KM predicts target locations that are shifted in range from their true locations. We derive an estimate for this range shift for a single target. We also show that because of this range shift we cannot recover the complex-valued frequency dependent reflectivity, but we can recover its absolute value and hence the radar cross-section (RCS) of the target. Simulation results show that we can detect, recover the approximate location, and recover the RCS for dispersive point targets thereby opening opportunities to classifying important differences between multiple targets such as their sizes or material compositions. 

Abstract We have recently introduced a modification of the multiple signal classification method for synthetic aperture radar. This method incorporates a user‐defined parameter,ϵ, that allows for tunable quantitative high‐resolution imaging. However, this method requires relatively large single‐to‐noise ratios (SNR) to work effectively. Here, we first identify the fundamental mechanism in that method that produces high‐resolution images. Then we introduce a modification to Kirchhoff Migration (KM) that uses the same mechanism to produce tunable, high‐resolution images. This modified KM method can be applied to low SNR measurements. We show simulation results that demonstrate the features of this method. 

Abstract We develop and analyze a quantitative signal subspace imaging method for single-frequency array imaging. This method is an extension to multiple signal classification which uses (i) the noise subspace to determine the location and support of targets, and (ii) the signal subspace to recover quantitative information about the targets. For point targets, we are able to recover the complex reflectivity and for an extended target under the Born approximation, we are able to recover a scalar quantity that is related to the product of the volume and relative dielectric permittivity of the target. Our resolution analysis for a point target demonstrates this method is capable of achieving exact recovery of the complex reflectivity at subwavelength resolution. Additionally, this resolution analysis shows that noise in the data effectively acts as a regularization to the imaging functional resulting in a method that is surprisingly more robust and effective with noise than without noise. 

The ability to detect sparse signals from noisy, high-dimensional data is a top priority in modern science and engineering. It is well known that a sparse solution of the linear system A ρ = b 0 can be found efficiently with an ℓ 1 -norm minimization approach if the data are noiseless. However, detection of the signal from data corrupted by noise is still a challenging problem as the solution depends, in general, on a regularization parameter with optimal value that is not easy to choose. We propose an efficient approach that does not require any parameter estimation. We introduce a no-phantom weight τ and the Noise Collector matrix C and solve an augmented system A ρ + C η = b 0 + e , where e is the noise. We show that the ℓ 1 -norm minimal solution of this system has zero false discovery rate for any level of noise, with probability that tends to one as the dimension of b 0 increases to infinity. We obtain exact support recovery if the noise is not too large and develop a fast Noise Collector algorithm, which makes the computational cost of solving the augmented system comparable with that of the original one. We demonstrate the effectiveness of the method in applications to passive array imaging. 

In this paper, we consider imaging problems that can be cast in the form of an underdetermined linear system of equations. When a single measurement vector is available, a sparsity promoting ℓ1-minimization-based algorithm may be used to solve the imaging problem efficiently. A suitable algorithm in the case of multiple measurement vectors would be the MUltiple SIgnal Classification (MUSIC) which is a subspace projection method. We provide in this work a theoretical framework in an abstract linear algebra setting that allows us to examine under what conditions the ℓ1-minimization problem and the MUSIC method admit an exact solution. We also examine the performance of these two approaches when the data are noisy. Several imaging configurations that fall under the assumptions of the theory are discussed such as active imaging with single- or multiple-frequency data. We also show that the phase-retrieval problem can be re-cast under the same linear system formalism using the polarization identity and relying on diversity of illuminations. The relevance of our theoretical analysis in imaging is illustrated with numerical simulations and robustness to noise is examined by allowing the background medium to be weakly inhomogeneous.