A rank-deficient and sparse penalized optimization model for compressive indoor radar target localization

h Real Data
computed and listed in Table II. As we expect from the 4.2.1 Experimental setup: Real radar datasets were
visual interpretation, the proposed joint nuclear-norm collected from a scene, shown in Figure 5, that consists
and `1-norm approach significantly enhances the image of one dihedral target placed behind a wooden wall.
quality in terms of TCR; it yields the target image with This behind-wall scene is sensed by a 81-element linear
8 REV Journal on Electronics and Communications, Vol. 10, No. 1–2, January–June, 2020
 Table III
 Computational Results of the Proposed Low-rank and Sparse Regularized Model for Different Noise Levels using 50% of the
 Total Measurements
 w w
 f (Z , st+1) f (Z , st)
 SNR (dB) Rank(Zw) | t+1 − t | Iterations TCR (dB)
 f (Zw, s )
 | t t |
 15 1 9.63 10 7 155 19.45
 − × −
 10 1 9.31 10 7 143 30.96
 − × −
 5 1 8.86 10 7 135 40.60
 − × −
 0 1 8.54 10 7 134 43.49
 × −
 5 1 8.34 10 7 134 43.58
 × −
 10 1 8.56 10 7 132 44.75
 × −
 15 1 8.31 10 7 130 45.86
 × −
 Table IV
 Computational Results of the Low-rank and Sparse Model for Different Reduced Datasets
 w w
 f (Z , st+1) f (Z , st)
 Dataset Rank(Zw) | t+1 − t | Iterations TCR (dB)
 f (Zw, s )
 | t t |
 20% 1 6.90 10 7 567 25.07
 × −
 25% 1 7.97 10 7 368 32.26
 × −
 30% 1 7.59 10 7 253 34.07
 × −
 35% 1 8.01 10 7 203 37.64
 × −
 40% 1 7.71 10 7 168 38.87
 × −
 45% 1 8.05 10 7 144 42.59
 × −
 50% 1 7.84 10 7 134 44.79
 × −
 full data volume, by randomly selecting only half of the
 available antennas (41 out of 81). Each selected antenna
 uses only 20% (160 out of 801) frequencies, which are
 !"#$%&' randomly sampled. Thus, the used dataset constitutes
 only 10% of the full data volume. For comparison, we
 also implement the existing direct CS and multistage
 CS-based approaches for wall clutter mitigation and
 *), + target image reconstruction using the same reduced
 dataset.
 Figure 6 depicts the target images reconstructed
 ()*- + using the different imaging methods. The direct CS
 method yields a degraded image shown in Figure 6(a)
 *)( +
 due to the presence of strong wall interferences, hin-
 ()(* + dering target identification. The multistage CS-based
 methods with the spatial filtering and subspace projec-
Figure 5. Layout of the imaged scene containing a wooden wall tion produce the target images in Figures 6(b) and (c),
and one dihedral used as target for through-the-wall radar data respectively. In these formed target images, strong
acquisition. wall clutter is suppressed, which enables the target
 localization. The proposed joint low-rank and sparse
 model gives the image shown in Figure 6(d), where the
antenna array synthesized by a monostatic radar system target pixels are highlighted and background clutter are
with the inter-element spacing of 0.01 m. The radar substantially reduced.
system transceives a 801-step frequency signal covering Table V presents the performance measure of TCRs
the bandwidth of [1–3 GHz]. This stepped-frequency by the different CS-based techniques. It can be observed
SAR system was positioned in front of the 0.16 m-thick from the table that the proposed joint nuclear-norm and
wooden wall, at a standoff distance of 1.0 m. `1-norm approach outperforms the exisiting CS-based
 4.2.2 Clutter mitigation & image reconstruction: This imaging techniques in terms of TCR; it has the highest
experiment evaluates the proposed rank-deficient and TCR value of 29.97 dB, followed by the multistage
sparse approach under generic CS operations where CS with subspace projection of 16.74 dB, multistage
both antennas and frequencies are highly compressed. CS with spatial filtering of 6.05 dB, and direct CS of
CS radar imaging acquires a reduced dataset, instead of 1.13 dB.
V. H. Tang & V.-G. Nguyen: A Rank-Deficient and Sparse Model for Indoor Radar Target Localization 9
 !" #"
 !" #"
 $" %"
 $" %"
Figure 6. Images produced by different CS-based wall clutter reduc-
tion and target image reconstruction methods with 50% antennas and Figure 7. Detection of target by thresholding the images produced
20% frequencies (collectively represent only 10% of full measure- by different CS-based wall clutter reduction and target image recon-
ments): (a) direct `1-norm minimization without clutter mitigation, struction methods with 50% antennas and 20% frequencies (collec-
(b) multistage CS approach with signal estimation, spatial filtering, tively represent only 10% of full measurements): (a) direct `1-norm
and `1-norm minimization, (c) multistage CS approach with signal minimization without clutter mitigation, (b) multistage CS approach
 `
estimation, subspace projection, and `1-norm minimization, and (d) with signal estimation, spatial filtering, and 1-norm minimization, (c)
proposed joint nuclear-norm and `1-norm method. multistage CS approach with signal estimation, subspace projection,
 and `1-norm minimization, and (d) proposed joint nuclear-norm and
 `1-norm method.
 Table V
 Target-to-Clutter Ratio Performances for the Images
 Produced by Several CS-based Techniques with 50% Antennas
and 20% Frequencies (Collectively Represent only 10% of Full 5 Conclusion
 Measurements)
 This paper introduced a rank-deficient and sparsity
 Clutter mitigation & image formation TCR (dB) regularized optimization model to address two im-
 methods portant problems of wall clutter mitigation and target
 Direct ` min. without clutter
 1 1.13 image formation in compressive indoor radar imaging.
 mitigation [8, 9] The task of wall clutter suppression and target image
 Multistage signal esti. & spatial filt. reconstruction is formulated as a composite nuclear and
 6.05
 & `1 min. [11, 15] `1-penalized minimization problem and an iterative
 Multistage signal esti. & sub. proj. algorithm based on the proximal forward-backward
 16.74
 & `1 min. [11, 16] splitting technique is developed, which captures wall
 Proposed rank-deficient and sparsity clutter and yields an indoor target image simultane-
 29.97
 approach ously. Through extensive experimental evaluations on
 simulated and real radar data, we found that exploiting
 both rank-deficient and sparsity structures improves
 the accuracy of clutter suppression and enhances target
 To evaluate the capability of target detection by dif-
 detection even with highly compressed measurements.
ferent wall clutter mitigation and image reconstruction
 Furthermore, the proposed model enables the process-
methods, we apply a thresholding technique to the
 ing of wall and target signals jointly, which results in a
target images formed by the different methods shown
 significant better performance than those of multistage
in Figure 6. Here, the input image is partitioned into
 approaches where the wall clutter removal and target
two classes: object and background. The threshold used
 image reconstruction are performed separately.
for classification is found by maximizing the between-
class variance. Figure 7 presents the detection results
after applying the same threshold value to the form Acknowledgments
images. As expected, without wall clutter mitigation,
the direct CS fails to localize the target as demonstrated The authors would like to thank the Editor-in-Chief,
in Figure 7(a). By contrast, it can be observed from Associate Editor, and three anonymous reviewers for
Figures 7(b) and (c) that the multistage CS approach the constructive and detailed comments that helped us
can detect the target with a high level of false alarms. improve the paper.
The detection result in Figure 7(d) shows that the pro- This research is funded by Vietnam National Founda-
posed model is able to localize the target well, with no tion for Science and Technology Development (NAFOS-
appearance of false alarms. TED) under grant number 102.01-2017.307.
10 REV Journal on Electronics and Communications, Vol. 10, No. 1–2, January–June, 2020
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 Van Ha Tang received the B.Eng. degree in
 Remote Sensing, vol. 51, no. 2, pp. 891–906, 2012. 2005 and M.Eng. degree in 2008, all in com-
[12] V. H. Tang, A. Bouzerdoum, S. L. Phung, and F. H. C. puter engineering, from Le Quy Don Technical
 Tivive, “Enhanced wall clutter mitigation for compressed University, Hanoi, Vietnam. In 2016, he ob-
 through-the-wall radar imaging using joint Bayesian tained the Ph.D. degree in computer engineer-
 sparse signal recovery,” in Proceedings of the IEEE Interna- ing from the University of Wollongong, Aus-
 tional Conference on Acoustics, Speech and Signal Processing tralia. Since 2016, he has been working as a
 (ICASSP). IEEE, 2014, pp. 7804–7808. Postdoctoral Research Fellow in the School of
[13] F. Ahmad, J. Qian, and M. G. Amin, “Wall clutter mit- Electrical Computer and Telecommunications
 igation using discrete prolate spheroidal sequences for Engineering, University of Wollongong. Dr.
 Tang is now with the Faculty of Information
 sparse reconstruction of indoor stationary scenes,” IEEE Technology, Le Quy Don Technical University, Hanoi, Vietnam. His
 Transactions on Geoscience and Remote Sensing, vol. 53, research interests include radar imaging and through-wall radar
 no. 3, pp. 1549–1557, 2014. signal processing, statistical and probabilistic modeling, and inverse
[14] A. Bouzerdoum, F. H. C. Tivive, and V. H. Tang, “Multi- problems in image and signal processing.
 polarization through-the-wall radar imaging using joint
 Bayesian compressed sensing,” in Proceedings of the 19th
 International Conference on Digital Signal Processing. IEEE,
 2014, pp. 783–788.
[15] Y.-S. Yoon and M. G. Amin, “Spatial filtering for wall- Van-Giang Nguyen received his B.S. degree
 in computer science from Le Quy Don Tech-
 clutter mitigation in through-the-wall radar imaging,” nical University , Hanoi, Vietnam, in 2005,
 IEEE Transactions on Geoscience and Remote Sensing, and his M.Sc. and Ph.D. degrees in electronic
 vol. 47, no. 9, pp. 3192–3208, 2009. engineering from Paichai University, Daejeon,
[16] F. H. C. Tivive, A. Bouzerdoum, and M. G. Amin, “A Korea, in 2009 and 2012, respectively. Since
 subspace projection approach for wall clutter mitigation 2013, he has been with the Department of
 in through-the-wall radar imaging,” IEEE Transactions on Information Systems, Le Quy Don Techni-
 Geoscience and Remote Sensing, vol. 53, no. 4, pp. 2108– cal University, Hanoi, Vietnam. His current
 2122, 2014. research interests include image processing,
[17] H. Van Tang and V.-G. Nguyen, “A rank-deficient and computer vision, machine learning and their
 applications to medical physics and medical imaging.
 sparse penalized optimization model for compressive
 indoor radar imaging,” in Proceedings of the 3rd Interna-