Cross Validation Based Distributed Greedy Sparse Recovery for Multiview Through-the-Wall Radar Imaging

Multiview through-the-wall radar imaging (TWRI) can improve the imaging quality and target detection by exploiting the measurement data acquired from various views. Based on the established joint sparsity signal model for multiview TWRI, a cross validation (CV) baseddistributed greedy sparse recovery algorithmwhich combines the strengths of theCV technique and censored simultaneous orthogonalmatchingpursuit algorithm (CSOMP) is proposed in this paper.The developed imaging algorithm named byCV-CSOMPwhich separates the total measurements into reconstructionmeasurements andCVmeasurements is able to achieve the accurate imaging reconstruction and estimation of recovery error tolerance by the iterative CSOMP calculation.The proposed CV-CSOMP imaging algorithm not only can reduce the communication costs among radar units, but also can provide the desirable imaging performance without the prior information such as the sparsity or noise level. The experimental results have verified the validity and effectiveness of the proposed imaging algorithm.


Introduction
Through-the-wall radar imaging (TWRI) is a promising technique to detect, localize, and identify the objects behind the walls [1][2][3].The TWRI technology has a wide range of military and civilian applications, such as urban battle, law enforcement, and earthquake and avalanche rescue missions.Due to the shadowing effect and the limited scattering range of targets behind walls, the echoes of targets measured from single view have relatively weak energy, which may lead to inferior imaging quality.In contrast to single-view TWRI measurement configuration, multiview TWRI can provide more informative and high-quality imaging result by enhancing the target intensity and reducing the false alarms [4][5][6][7][8].
In order to achieve high imaging resolution in TWRI, both long observation aperture and ultrawideband illumination signal are demanded, which leads to a large amount of data to be acquired, stored, and processed.Recently, compressive sensing (CS) and sparse reconstruction techniques have been applied to TWRI to increase the speed of data acquisition and provide the enhanced imaging result [9][10][11][12][13][14][15].From the viewpoint of CS, the multiview images can be described by using the joint sparsity model.It is demonstrated that CS-based multiview image formation can enhance the quality of the reconstructed image compared to single-view TWRI using CS [16].A multiview TWRI formation algorithm based on joint Bayesian sparse recovery framework is proposed to enhance imaging quality in the strong wall clutter environment [17].A hybrid matching pursuit algorithm is developed to deal with the multiview TWRI in the distributed manner [18].The CS technique is applied to multiview TWRI reconstruction with the consideration of the multiple reflections off the targets in conjunction with the surrounding walls [19].
Nevertheless, the existing CS-based methods for multiview TWRI require data measurements from the radar unit at each view to be collected at a centralized location for processing.Due to the limitation of communication links and noncooperative environments, it may not be feasible to accumulate the measurement data at one centralized location.A distributed greedy signal recovery algorithm called censored simultaneous orthogonal matching pursuit (CSOMP) [20] without the centralized data accumulation for multiview TWRI is proposed, which has the advantage of reducing the communication costs among all radar units.However, this imaging approach requires prior information such as the sparsity or noise level for accurate imaging reconstruction.Such information is generally not available in practical multiview TWRI scenarios.To solve the above problem, this paper proposes a distributed signal recovery algorithm for multiview TWRI reconstruction which combines the advantages of CSOMP algorithm and cross validation (CV) technique [21,22].The proposed imaging approach called CV-CSOMP algorithm divides the total measurements into reconstruction measurements and CV measurements.The former are used to reconstruct the signal by the CSOMP algorithm and the latter are used to compute the CV residual.The process of accurate imaging reconstruction and estimation of recovery error tolerance are simultaneously achieved by the iterative CSOMP calculation in the proposed imaging algorithm.The proposed imaging approach can provide the accurate imaging reconstruction results without prior knowledge such as sparsity or noise level while reducing the communication costs among radar units from various views.
The remainder of this paper is organized as follows.In Section 2, the signal model of multiview TWRI is introduced.In Section 3, the CV-CSOMP algorithm is proposed in detail.The effectiveness of the proposed algorithm is evaluated by using simulated and measured multiview TWRI data.Finally, Section 5 gives concluding remarks.

Signal Model
Consider  radar units, placed at known positions either along the front wall or surrounding the building being imaged.For the radar unit at each view, the transceiver moves in a fixed step along the direction parallel to the wall, resulting in  antenna positions.At each transceiver position, a stepped frequency signal of  frequencies equispaced over the desired bandwidth is used to illuminate the scene.Assuming the observed scene contains  point targets, the signal received by the th ( = 1, 2, ⋅ ⋅ ⋅, ) transceiver position and the th ( = 1, 2, ⋅ ⋅ ⋅, ) frequency from the th ( = 1, 2, ⋅ ⋅ ⋅, ) view can be expressed as where  , is the complex reflectivity of the th target corresponding to the th view,   =  1 + ( − 1)Δ is the th working frequency,  1 is the starting frequency, Δ is the uniform frequency step,  , represents the two-way traveling time between the th transceiver position and the th target corresponding to the th view, and V  (, ) is the measurement noise.Given the perfect knowledge of surrounding wall parameters such as thickness and permittivity is obtained,  , can be calculated using the Snell's law [23,24].The target reflectivities are assumed constant and independent of frequency.It is worth mentioning that the strong wall reflection signal and multipath propagation signal are omitted in the signal model of (1).The wall reflection signal can be suppressed by the suitable wall clutter mitigation technique such as background subtraction [25], spatial filtering [26], and subspace projection [27] when full data volume is available.The ghosts introduced by the multipath propagation signal can be eliminated by the methods developed in [28,29].Assume that the observed scene is divided into a finite number of pixels  ×  in the crossrange and downrange; an equivalent matrix-vector representation of the signal model in (1) can be obtained as where r  = [  (1,1),   (1, 2), ⋅⋅⋅,   (, )]  denotes the ×1 measurement vector obtained by stacking the signals from all the  transceiver locations,   is the scene reflectivity vector associated with the th view, and n  is the additive noise vector.Ψ  is the  ×  sensing matrix whose (, )th element is given by where = 1, 2, ⋅ ⋅ ⋅, ,  = 1, 2, ⋅ ⋅ ⋅, ,  =  mod ,  = ⌊/⌋, and  , is the round-trip propagation time between the th transceiver position and the th pixel associated with the th view.In many TWRI scenarios, the number of point targets is typically much less than the number of scene pixels and thus the scene reflectivity vector   is a sparse vector for each view.
To reduce the amount of data in multiview TWRI, consider an undersampled measurement vector y  , which is a vector of length ( ≪ ) composed of elements selected from y  as follows: where A  = Φ  Ψ  is the dictionary matrix; Φ  is the  ×  measurement matrix constructed by randomly selecting  rows of the  ×  identity matrix.It is noted that the wall reflection signal can be also removed by the technique proposed in [12,14] when we only have access to a reduced set of measurements.
Because the vectors   represent reflectivities of the same scene, the indices of the nonzero elements in sparse vectors   remain the same.On the other hand, because of the aspect angle dependence of target scattering characteristic, the values of these nonzero elements in sparse vectors   are different.As a consequence, all the scene reflectivity vectors share the same sparsity pattern for  = 1, 2, ⋅ ⋅ ⋅,  and the measurement data obtained by various views can be processed by the distributed sparse reconstruction methods.

Proposed Algorithm Description
In order to reduce the communication costs among the radar units from various views, the CSOMP algorithm can be utilized to reconstruct the scene reflectivity vectors associated with different views.In contrast to the traditional SOMP algorithm [30] which needs to share the complete observation vector among radar units, the CSOMP algorithm exploits the censoring procedure in the finding support set step, which means that only  largest elements and corresponding indices Input: The measurement vectors y  , the dictionary matrices Α  , for  = 1, 2, ⋅ ⋅ ⋅, , the measurement noise level  and the censored level .Initialization: Let the residual vectors r ,0 = y  , the support set Λ = ⌀, for  = 1, 2, ⋅ ⋅ ⋅, , and the iteration times  = 0; Iteration: ( Here, it should be emphasized that the preset measurement noise level  which is also known as the recovery error tolerance has a significant impact on the reconstruction results of CSOMP algorithm.The inaccurate selection of  will lead to the loss of genuine targets or the introducing of ghost targets.However, the noise level of multiview TWRI measurement data is usually unknown in practice.In order to estimate the measurement noise level, the CV-CSOMP algorithm which combines the CV technique and CSOMP algorithm is developed.It has been shown that CV strategy is the computationally efficient method to determine the recovery error tolerance [21,22].In the CV-CSOMP algorithm, the measurement vectors y  are separated into reconstruction measurement vectors y , and CV measurement vectors y , .
Accordingly, the dictionary matrices Α  are separated into the reconstruction matrices A , and CV matrices A , .The detailed steps of the CV-CSOMP algorithm are depicted in Algorithm 2. In each iteration of the CV-CSOMP algorithm, the scene reflectivity is reconstructed by the CSOMP algorithm and the outcome is evaluated by the CV technique, which is used to indicate that the iteration process starts to overfit the noise.The reconstructed imaging result is selected as the output on the criterion that its CV residual is the smallest one.

Results and Discussion
4.1.Simulation Results.In this section, simulation experiments are conducted using the synthetic data.In the simulation, the surrounding homogeneous walls composed of solid concrete blocks have the thickness of 0.2 m and the relative permittivity of 7.66.The scene containing nine targets is illuminated at the down, left, and right views, respectively.At each view, a total of 41 transceiver positions parallel to the wall with a spacing of 0.08 m have a standoff distance of 1 m from the corresponding wall.The stepped frequency signal ranging from 2 GHz to 3 GHz with a frequency step of 5 MHz is used for imaging.Thus, the TWRI system transmits and receives 8241 (41×201) monochromatic signals.
The actual geometry of multiview TWRI under consideration is depicted in Figure 1.The 4 m× 4 m square region to be imaged is partitioned into 33×33 pixels along the crossrange and downrange directions.The reflectivities of nines targets drawn from the complex Gaussian distribution CN(0, 1) are assumed to remain unchanged for each view, but vary with different views due to the aspect-dependent scattering characteristic of the targets.The complex white Gaussian noise is added to the measurements to account for the measurement noise.
The value of  is set to 4100 where one-half of frequencies and total antenna locations are used for scene reconstruction.The censored levels of CSOMP and CV-CSOMP algorithms are both set to 50.For the CV-CSOMP algorithm, 60% of total reduced measurements are used to reconstruct the targets via the CSOMP algorithm and 40% of remaining total reduced measurements are utilized to compute the CV residual, while for the CSOMP algorithm, the measurement noise level as the input parameter is unknown and empirically set to  = 0.1 ∑  =1 ‖y  ‖ 2 2 .The imaging reconstruction results of CSOMP and CV-CSOMP algorithms are depicted in Figure 2. All the images are normalized to their own maxima and shown on same 20 dB scale.Figures 2(a) and 2(b) show that both the CSOMP and CV-CSOMP algorithms are able to achieve the accurate reconstruction of the targets when the signalto-noise ratio (SNR) is equal to 10 dB.It is observed from Figure 2(c) that, because of the improper selection of the measurement noise level, the CSOMP algorithm misses two targets when SNR is set to 20 dB.However, the CV-SOMP algorithm can accurately localize the nine targets in the case that SNR is equal to 20 dB.As seen in Figures 2(e) and 2(f), the CV residual of the CV-CSOMP algorithm decreases to a constant value after about 7 iterations in the iteration process, which means that the CV-CSOMP algorithm is able to determine the measurement noise level and obtain the promising imaging reconstruction results with a small number of iterations.
In order to provide the quantitative comparison of imaging performance, the normalized mean square error (NMSE) is applied to evaluate the imaging quality.The definition of the NMSE is given by where  is the original signal vector, σ is the reconstructed signal vector, and  is the number of Monte Carlo experiment.
For the analysis of imaging performance, several parameters are varied.First, we vary the SNR from 0 dB to 20 dB with 5 dB increment.For a fixed value of SNR, the experiment is repeated 100 times and the NMSE of the formed images is computed.Figure 3(a) shows the NMSE of the reconstruction results obtained by CSOMP and CV-CSOMP algorithms as a function of different SNR values.It is observed that the NMSE of CV-CSOMP algorithm is slightly higher than that of CSOMP algorithm when the SNR is less than 10 dB.When the SNR is greater than 10 dB, due to the improper selection of the measurement noise level, the imaging performance of CSOMP algorithm starts to degrade severely.However, the CV-CSOMP algorithm can still maintain high imaging accuracy because it can adaptively determine the proper measurement noise level through the CV procedure.
To further assess the effectiveness of the CV-CSOMP imaging algorithm, the relationship of NMSE and the percentage of full measurement set when the SNR is equal to 20 dB is demonstrated in Figure 3(b).As can be seen, the CV-CSOMP algorithm achieves consistently lower NMSE than the CSOMP algorithm.The imaging quality of CSOMP algorithm is nearly unchanged while the imaging quality of CV-CSOMP algorithm improves with the increase of the percentage of measurements.It is also observed from Figure 3(b) that the CV-CSOMP algorithm can exploit only 20% of full measurement data to successfully detect and localize the targets.In practical scenarios, the number of measurements critically impacts the time required for data collection time for the synthetic aperture system [31].It   can be concluded that the proposed CV-CSOMP algorithm allows the significant reduction of data acquisition time and provides the desirable image quality.
The selection of the number of reconstruction measurements and CV measurements is very important for the CV-CSOMP algorithm.For the one-half reduced data set, Figure 4 presents the relationship of the NMSE and the percentage of reconstruction measurements chosen from the total reduced measurement set.It can be observed that the NMSE decreases when the number of reconstruction measurements increases.When the percentage of reconstruction measurements exceeds 60%, the value of NMSE starts to remain almost unchanged.To obtain a reliable imaging reconstruction, 60% of total measurements are used as reconstruction measurements and 40% of remaining total measurements are used as CV measurements in the simulated and experimental scenarios.
To analyze the influence of different censoring levels on the reconstruction performance of the CV-CSOMP algorithm, Figure 5 depicts the NMSE versus the different censoring levels.It is obviously observed that the CV-CSOMP algorithm has the nearly same NMSE when the censoring level varies from 10 to 100.A censoring level of  = 50 is selected for the considered scenario.

Experimental Results.
The proposed CV-CSOMP algorithm is evaluated using the real multiview TWRI data.As depicted in Figure 6(a), the slider controller and vector network analyzer (VNA) are under the control of a desktop computer, which achieves the mechanical scan and data collection.The scanning frequency band of VNA is from 1 GHz to 3 GHz with the step size of 10 MHz.A dual-polarized horn antenna is used as the transceiver and mounted on  Then the spatial filtering method [26] can be directly applied to the reduced data set to remove the strong wall reflection signal as well as the background clutter from the room.
Figure 7 demonstrates the imaging results obtained by the CSOMP and CV-CSOMP algorithms, respectively.The censoring level is set to  = 50 for both two algorithms.The CSOMP algorithm fails to recover the trihedral target at the downrange of 162 cm.In comparison, the proposed CV-CSOMP algorithm can accurately detect and locate the two trihedral targets.

Conclusion
In this paper, we have proposed a distributed greedy sparse recovery algorithm for multiview TWRI.The proposed CV-CSOMP algorithm exploits the CV technique to determine the proper measurement noise level through the iterative CSOMP calculation.The simulation and experimental results demonstrate that the proposed CV-CSOMP algorithm can provide the desirable multiview TWRI imaging results without the prior information such as the sparsity or noise level while the communication load among the radar units is significantly reduced.It is worth mentioning that although the proposed algorithm can localize the target correctly, it cannot retain the edge and shape of the target.The development of advanced imaging approaches which can preserve the edge and shape of extended target will be pursued in the future research.

Figure 1 :
Figure 1: The actual geometry of multiview TWRI under consideration.

Figure 2 :
Figure 2: (a) and (b) are, respectively, the imaging results of CSOMP and CV-CSOMP algorithms when SNR equals to 10 dB.(c) and (d) are, respectively, the imaging results of CSOMP and CV-CSOMP algorithms when SNR equals to 20 dB.(e) and (f) are, respectively, CV residual curves of CV-CSOMP algorithm when SNR equals 10 dB and 20 dB.

Figure 4 :
Figure 4: NMSE versus the percentage of reconstruction measurements.