Section: New Results

Underfoliage object imaging using SAR tomography and wavelet-based sparse estimation methods

Participants : Yue Huang, Jacques Lévy-Vehel

Hybrid environments refer to a scenario of deterministic objects embedded in a host natural random environment and their scattering patterns consist of a complex mixture of diverse mechanisms, like, in the case of this study, volume scattering from the canopy, double bounce reflection between the ground and under-foliage objects as well as between objects and trunks, surface scattering from the underlying ground, etc. The resulting SAR information is characterized by a strong complexity, and its analysis using 2-D images or even data acquired in InSAR configuration remains difficult. Using Multi-baseline(MB) InSAR data, SAR tomography can be applied to reconstruct in 3-D the measured scattering responses and polarimetric patterns. Natural volumes, such as forest canopies, being composed of a large number of scatterers whose responses cannot be discriminated at the resolution of analysis, their scattering patterns are generally considered as a vertical density of random or speckle-affected reflectivity. On the other hand, localized objects, such as artificial targets on the ground are associated to point-like contributions, that may be separable in the vertical direction. The global response of under-foliage objects with a deterministic scattering response embedded in surrounding distributed environments, can be described by a mixed spectrum. Conventional tomographic techniques like the Capon and Beamforming methods, estimate continuous Power Spectral Density (PSD) and hence are well adapted to the characterization of continuous volumetric media, but cannot discriminate closely-spaced scatterers, e.g. scattering responses from trucks, due to limited spatial resolution. Conventional high-resolution methods like MUSIC and subspace fitting estimators as well as sparse estimation techniques such as LASSO [52] and FOCUSS [40], are well adapted to the characterization of discrete scatterers like truck top, truck-ground interaction and calibrators over bare soils, or buildings over urban areas [53], but cannot properly handle the high dimensionality of the scattering responses of natural volumes. Usual tomographic techniques cannot simultaneously cope with both types of spectrum, and not able to deal with mixed spectral estimation problems, characteristic of underfoliage object imaging scenario.

Wavelet-based techniques present a high potential for such applications, as they permit to parameterize in a sparse way continuous functions, i.e. canopy PSDs in the present case. Wavelet-based tomographic techniques have been used for tomographic imaging of forested areas [27], and for such regular signals, large wavelet coefficients being often concentrated in the approximation space, scale thresholding may be implemented to extract the most significant wavelet coefficients for an accurate volume signal recovery [27]. In the underfoliage object scenario, discrete scatterers embedded in a continuous medium, result in a mixed vertical PSD that may be associated to an irregular signal with wavelet coefficients distributed both in the approximation and detail spaces, and a simple scale cut-off is hence not adapted to separate the wavelet coefficients of discrete scatterers from those of continuous media. Therefore, we propose a new wavelet-based method to extract underfoliage objects from their speckle-affected distributed environment and characterize them with a high resolution.

For an MB-InSAR configuration with $M$ acquisition positions, considering an azimuth-range resolution cell containing a mixture of backscattering contributions from object ($o$) and volume ($v$) scatterers located at different heights $z$, the observed data vector at $l$th realization can be represented by:

where the steering matrix, ${𝐀}_x\left({𝐳}_x\right)$, contains the interferometric phase information associated to the InSAR responses of the scatterers located at the unknown elevation positions ${𝐳}_x=[z_{x_1},\cdots ,z_{x_{N_x}}]$ above the reference focusing plane, and the source signal vector, ${𝐬}_x={[s_{x_1}\cdots s_{x_{N_x}}]}^T\in {ℂ}^{N_x\times 1}$, contains the unknown complex backscattering coefficients of the $N_x$ source scatterers. The vertical reflectivity function can be represented as ${𝐩}_x=E\left(\right|{𝐬}_x{|}^2)$ ($x=o,v$).

Over speckle-affected environments, unknown reflectivity and elevation parameters are generally estimated from second-order statistics, i.e. from the covariance matrix $\widehat{𝐑}\in {ℂ}^{M\times M}$ of the observed MB-InSAR data $𝐲\in {ℂ}^{M\times 1}$. The proposed tomographic processing technique is based on the minimization of the Least-Square (LS) fitting between the observed and modeled data covariance $\left|\right|𝐑-\widehat{𝐑}{\left|\right|}_F$. The modeled covariance matrix is composed by the covariances of object and volume contributions $𝐑={𝐑}_o+{𝐑}_v$, each of them being simply related to its discretized vertical density of reflectivity ${𝐩}_x$ through ${𝐑}_x=𝐀\left({𝐳}_x\right)diag\left({𝐩}_x\right){𝐀}^H\left({𝐳}_x\right)\in {ℂ}^{M\times M}$. The proposed method can be represented by a $l_1$ norm minimization in a transformed space subject to quadratic constraints between the observed and modeled data covariance:

where

• $𝐩={\left[{𝐩}_o^T{𝐩}_v^T\right]}^T\in {ℝ}^{+N_s\times 1}$ stands for vertical backscattering power distribution for the resolution cell under analysis,

• $𝐁=\left[\begin{array}{cc}{𝐈}_{(N_o\times N_o)}& \mathbf{0}\\ \mathbf{0}& {\Psi }_{(N_v\times N_v)}\end{array}\right]\in {ℝ}^{(N_s\times N_s)}$ represents the hybrid sparsifying basis with the wavelet basis $\Psi$

This tomographic technique is suitable for the mixed-spectrum estimation problem, because it maintains the spectral continuity for the backscattering power of forest canopies and the high-resolution for the vertical reflectivity of objects. The effectiveness of this new approach is demonstrated using L-band airborne tomographic SAR data aquired by the DLR over Dornstetten, Germany. The undeniable performance can be shown by the results in [21] and [20].

This work has been presented in European SAR conference 2016 . Some refined results have been presented in IGARSS conference 2016 as an invited talk. By extending this work in details, a journal paper [24] has been submitted to IEEE Geoscience and Remote Sensing Letters (GRSL) and is currently under reviewing.