# TomoSAR Imaging for the Study of Forested Areas: A Virtual Adaptive Beamforming Approach

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## Abstract

**:**

## 1. Introduction

## 2. Problem Phenomenology

**y**represents the set of L focused signals for a specified azimuth-range position, and is related to the complex random scene reflectivity vector

**s**via the linear equation of observation (EO) [3,7],

**s**,

**n**and

**y**in the EO (1), define the vectors composed of the decomposition coefficients ${\left\{{s}_{m}\right\}}_{m=1}^{M}$, ${\left\{{n}_{l}\right\}}_{l=1}^{L}$ and ${\left\{{y}_{l}\right\}}_{l=1}^{L}$, of the finite-dimensional approximations of the continuous signal S, noise N and observation Y fields, respectively; and matrix $A$ is the signal formation operator (SFO) that maps S → Y, the source Hilbert signal space S onto the observation Hilbert signal space Y [10].

**s**,

**n**and

**y**represent complex random Gaussian zero-mean vectors, which are characterized by their corresponding correlation matrices [7,10]

**b**at the principal diagonal of the diagonal matrix $D(b)$, composed of the averaged entries

**s**.

**y**. Recall that in this contribution, we intend to follow the DEDR-WDT optimization strategy and develop the new nonparametric TomoSAR WAVAB optimal solver.

## 3. Related State-of-the-Art Work

**b**.

**y**, in the EO (1), is customary modelled as a stochastic vector, which, in theory, represents an infinite number of different realizations of the data formation process. The pdf of such a complex-valued zero-mean Gaussian vector

**y**is given by [18]

**b**as the logarithm of the conditional pdf $p\left(y|b\right)$ [19],

**b**are ignored and the scene characteristics are retrieved from the measurement statistics presented by the data covariance matrix $Y$ [3,7],

**b,**related to ${R}_{y}={R}_{y}\left(b\right)=AD\left(b\right){A}^{+}+{N}_{0}I$, is equivalent to minimizing the covariance fitting Stein’s loss and yields the ML-inspired estimator (Equation (32) in Reference [19])

_{diag}returns the vector at the principal diagonal of the embraced matrix; in (16), $H={A}^{+}A$ represents the matrix-form ambiguity function (AF) operator of the MSF system that performs formation of the complex image $q={A}^{+}y$.

## 4. Proposed TomoSAR-Adapted WAVAB Approach

**I**is the identity matrix, $\mathsf{\Delta}={\mathsf{\Lambda}}^{(1)\mathrm{T}}{\mathsf{\Lambda}}^{(1)}$ defines the numerically approximated discrete-form Laplacian, computed applying the first-order finite difference operator ${\mathsf{\Lambda}}^{(1)}$ [21] over the original M-pixel formatted PLOS sensing direction z. The weight parameters ${\varsigma}_{1}$,${\varsigma}_{2}$,∈[0,1], scaled to the [0,1] interval, control the balance between the corresponding metrics measures in the image space ${\mathbb{B}}_{(M)}$. In the general case, $0<{\varsigma}_{1}$,${\varsigma}_{2}\le 1$, the Laplacian term in $U$ tends to contrast the image high spatial frequency information; thus, contrast preservation regularization is implicitly considered [22].

_{TL}is attained at some i = I (or the maximum admissible number of iterations is reached) [11,21]. In the experiments reported next, we followed the suggestions from Reference [11] for the equi-balanced adjustment, ${\varsigma}_{1}={\varsigma}_{2}=1$, which does not affect the overall convergence of (19) that is a direct sequence from the fundamental theorem of POCS.

- As first input (zero-step iteration), retrieve an initial estimate of the PSP using the MSF beamforming technique, ${v}_{\mathrm{MSF}}={\widehat{b}}_{\mathrm{MSF}}={\left\{{A}^{+}YA\right\}}_{\mathrm{diag}}$.
- Specify the balancing factors (degrees-of-freedom) ${\varsigma}_{1}$,${\varsigma}_{2}$,∈[0,1] for each particular case; as suggested in Reference [11], we recommend the equi-balanced adjustment ${\varsigma}_{1}={\varsigma}_{2}=1$. Next, perform the DEDR-inspired reconstructive processing in (19) until convergence or when the maximum admissible number of iterations is reached, ${\widehat{v}}^{\left[0\right]}={\widehat{b}}^{\left[I\right]}={\widehat{b}}_{\text{DEDR-VAB}}$.
- Last, using the previously reconstructed image ${\widehat{v}}^{\left[0\right]}$ as first input (zero-step iteration), execute the WDT-based iterative recovery stage in (20) until convergence, ${\widehat{v}}^{\left[T\right]}={\widehat{b}}_{\mathrm{WAVAB}}$. The WT basis and the Donoho’s soft thresholding shrinkage parameter $\phi $ are specified for the specific cases. As recommended in the previous related studies, we suggest the use of the Daubechies Symlet WT basis with four vanishing moments and three decomposition levels [8,9], and the median estimator for $\phi $ [13]. The second iterative process performs suppression of artifacts and provides additional image enhancement to the last (I-th) iteration retrieval of (19) ${\widehat{b}}_{\text{DEDR-VAB}}$.

## 5. Separation of Ground and Canopy

## 6. Numerical Examples

## 7. Experimental Results

## 8. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## List of Acronyms

BMR | Bayes Minimum Risk |

CS | Compressed Sensing |

DCRCB | Doubly Constrained Robust Capon Beamforming |

DEDR | Descriptive Experiment Design Regularization |

DOA | Direction of Arrival |

MB | Multi-Baseline |

ML | Maximum Likelihood |

MP | Multi-Polarimetric |

MSE | Mean Squared Error |

MSF | Matched Spatial Filter |

MUSIC | Multiple Signal Classification |

Probability Density Function | |

PLOS | Perpendicular to the Line of Sight |

POCS | Projections onto Convex Sets |

PSF | Point Spread Function |

PSP | Power Spectrum Pattern |

RS | Remote Sensing |

SAR | Synthetic Aperture Radar |

SFO | Signal Formation Operator |

SKP | Sum of Kronecker Products |

SM | Scattering Mechanism |

SSA | Spatial Spectral Analysis |

TomoSAR | SAR Tomography |

VAB | Virtual Adaptive Beamforming |

WAVAB | WDT-refined VAB |

WDT | Wavelet Domain Thresholding |

WT | Wavelet Transform |

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**Figure 2.**Distribution of two scatterers among $J=250$ independent looks, following two Gaussian distributions with phase-centers (means) located at ${z}_{1}=0$ m and ${z}_{2}=6\text{}$m, respectively, and spread (standard deviation) $\sigma =0.05$ m.

**Figure 4.**Height estimation of two uncorrelated scatterers with equal reflectivity, versus number of looks. The scatterers are located at ${z}_{1}=0$ m and ${z}_{2}=6$ m, respectively. The presented plot is the average of 100 Monte-Carlo trials.

**Figure 5.**Height estimation of two uncorrelated scatterers with equal reflectivity, versus height difference between scatterers $\Delta z$. The considered number of looks equals $J=250$. The presented plot is the average of 100 Monte-Carlo trials.

**Figure 6.**Fully polarimetric SAR image of the test site in northern Sweden (colours correspond to the channels: red: HH; green: HV; blue: VV). The region of interest is located within the yellow rectangle along azimuth.

**Figure 7.**Tomographic acquisition constellation. The location of each slave, with respect to the master track at position (0, 0), is indicated in meters

**Figure 8.**Tomograms recovered after applying MSF beamforming [3] for three different polarizations: (

**a**) HH, (

**b**) HV and (

**c**) VV. (

**d**) The color-coded tomogram, calibrated to the same [0,1] scales, for the three corresponding polarimetric channels (red HH, green HV and blue VV).

**Figure 11.**Tomograms recovered after applying the wavelet-based CS technique [8,9] for three different polarizations: (

**a**) HH, (

**b**) HV and (

**c**) VV. A sparsifying basis based on the Daubechies Symlet 4 wavelet family, with three decomposition levels, is considered. (

**d**) The color-coded tomogram, calibrated to the same [0,1] scales, for the three corresponding polarimetric channels (red HH, green HV and blue VV).

**Figure 12.**Tomograms recovered after applying the first stage of the WAVAB technique in (19), with convergence at I = 8 iterations, for three different polarizations: (

**a**) HH, (

**b**) HV and (

**c**) VV. (

**d**) The color-coded tomogram, calibrated to the same [0,1] scales, for the three corresponding polarimetric channels (red HH, green HV and blue VV).

**Figure 13.**Tomograms recovered after applying the overall TomoSAR WAVAB method (19), (20), with convergence at I = 8 iterations at the DEDR-VAB stage (19), followed by T = 3 iterations at the WDT stage (20); in total 11 iterations, for three different polarizations: (

**a**) HH, (

**b**) HV and (

**c**) VV. (

**d**) The color-coded tomogram, calibrated to the same [0,1] scales, for the three corresponding polarimetric channels (red HH, green HV and blue VV).

**Figure 14.**Tomograms obtained after applying SKP separation of ground and canopy. Ground (red) separation is done using MUSIC [3,7] with a single source model. Canopy (green) separation is followed by: (

**a**) MSF beamforming [3]; (

**b**) Capon beamforming [3,7]; (

**c**) the DCRCB technique [24,25]; (

**d**) wavelet-based CS [8,9], with a sparsifying basis based on the Symlets 4 wavelet family with three decomposition levels; (

**e**) the introduced TomoSAR WAVAB technique, with convergence at I = 8 iterations performed at the DEDR-VAB stage (19) followed by T = 3 iterations at the WDT stage (20); in total, 11 iterations.

**Figure 15.**Contour of the top of the canopy (red), retrieved from the DCRCB phase-center estimates at the highest position, superimposed on a tomogram obtained using WAVAB.

Frequency band | P |

Wavelength | 0.85631 m |

Chirp bandwidth | 94 MHz |

Altitude above ground | 3891.30 m |

Pulse repetition frequency | 500 Hz |

Look angle (near-, mid-, far-range) | 25°, 40°, 53° |

Azimuth resolution | 4 m |

Range resolution | 1.6 m |

TomoSAR Focusing Technique | Processing Time in Seconds |
---|---|

MSF | 123.518 |

Capon beamforming | 198.435 |

DCRCB | 343.373 |

Wavelet-based CS | 2159.678 |

WAVAB | 429.654 |

^{1}A set of 50 experimental realizations (range lines) is considered. The experiments were performed using Python(x,y)-2.7.10.0 in an Intel© Core i7 at 2.20 GHz PC with 8GB in RAM. The reported results include the processing time due to the SKP decomposition stage.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Martín del Campo, G.D.; Shkvarko, Y.V.; Reigber, A.; Nannini, M. TomoSAR Imaging for the Study of Forested Areas: A Virtual Adaptive Beamforming Approach. *Remote Sens.* **2018**, *10*, 1822.
https://doi.org/10.3390/rs10111822

**AMA Style**

Martín del Campo GD, Shkvarko YV, Reigber A, Nannini M. TomoSAR Imaging for the Study of Forested Areas: A Virtual Adaptive Beamforming Approach. *Remote Sensing*. 2018; 10(11):1822.
https://doi.org/10.3390/rs10111822

**Chicago/Turabian Style**

Martín del Campo, Gustavo D., Yuriy V. Shkvarko, Andreas Reigber, and Matteo Nannini. 2018. "TomoSAR Imaging for the Study of Forested Areas: A Virtual Adaptive Beamforming Approach" *Remote Sensing* 10, no. 11: 1822.
https://doi.org/10.3390/rs10111822