through-the-wall radar imaging using compressive sensing along ...
THROUGH-THE-WALL RADAR IMAGING USING COMPRESSIVE SENSING ALONG TEMPORAL FREQUENCY DOMAIN Yeo-Sun Yoon
Moeness G. Amin∗
R&D Division Samsung Thales Co., LTD. Yongin, Kyunggi, Korea
Center for Advanced Communications Villanova University Villanova, PA 19085, USA
ABSTRACT There are increasing demands on Through-the-Wall Radar Imaging (TWRI) systems to deliver high resolution images in both range and cross range. This requires using wideband signals and large array apertures, respectively. Wideband signals are typically implemented by transmitting a series of the narrowband signals. As such, the TWRI system operation involves transmitting and receiving multiple step-frequencies at each antenna location in either a physical or a synthetic aperture radar. Compressive sensing (CS) is an effective approach for decreasing the number of samples and, subsequently, reducing the data acquisition and post-processing time. In this paper, we propose a TWRI scheme based on CS in which a sizable reduction in the number of samples along the frequency axis is achieved without significant degradation in the image. The proposed approach applies the well-known Fourier-like measurement matrix and generates radar images of almost the same desirable quality as the image employing all data samples. Index Terms— Through-the-wall radar, radar imaging, compressive sensing 1. INTRODUCTION Through wall radar imaging (TWRI) has gained much attention in recent years due to its wide and broad applications in several civilian and military sectors [1, 2, 3]. Among many requirements for effective TWRI system is fast acquisition time. This requirement becomes more pronounced and challenging when deploying wideband large aperture systems for high resolution imaging. For the vast majority of systems which synthesize a large pulse by step frequencies, transmitting and receiving many narrowband signals at all antenna locations constitute a considerable percentage of the acquisition time. For example, for 100 narrowband signals and 50 element array, the total number of transmit-receive data samples becomes 5000. Once all the data is collected, the scene of interest can ∗ The
work by Prof. Moeness Amin is supported in part by the Office of Naval Research, Grant no. N00014-07-1-0043.
978-1-4244-4296-6/10/$25.00 ©2010 IEEE
2806
be constructed by image formation methods. In some cases, a long data collection time must be avoided, since all the objects in the scene should be stationary during the entire data collection operations to avoid image smearing and displacement of targets. There have been several approaches that apply compressive sensing (CS) to reduce the number of samples in radar imaging [4, 5]. In essence, CS aims at recovering the vector s from the vector y, where the two vectors are related by the matrix formula y = As, with A has more columns than rows. Sparse representations permit solutions where the required number of data samples (y) can be significantly reduced and yet allows recovery of the scene of interest (s). In reference [4], the authors introduced CS to high resolution radar imaging in urban sensing applications and tested the effectiveness of the CS approach with synthesized data. The work in [5] involves the application of CS to the ground penetrating radar (GPR) problems, which are similar to TWRI in the sense that image formation is performed using step-frequency radar signals. In this paper, we employ CS for TWRI in a new and different way. We apply CS in frequency domain only, leaving the spatial-domain sensing uncompressed. That is, we use all the antenna elements along the original system aperture, whether it is physical or synthesized. So, only the number of frequency bins is reduced by the CS. However, the corresponding CS optimization solution is not the final result, as is the case in [4, 5]. Rather, we use usual image formation methods, for example, delay and sum (DS) beamforming, following the CS. This proves advantageous, since residuals that might appear with CS can be reduced during the image formation process. Adopting a conventional image formation can also provide a desirable performance of the target detection and classification steps which immediately and typically follow the imaging step. 2. THROUGH-THE-WALL RADAR Electromagnetic wave can pass through a dielectric wall allowing scenes behind walls to be imaged by using radar imaging techniques [2]. In order to improve the down-range reso-
ICASSP 2010
lution, the signal bandwidth should be increased. High crossrange resolution requires a long array aperture. Assume that there are N antenna locations and a step-frequency signal of M narrowband signals are used. The radar signal received at the n-th antenna location with the m-th frequency bin is given by, P −1 σp exp{−j2πfm τn,p }, (1) y(n, m) = p=0
where σp is the reflection coefficient of the p-th target, and τn,p is the two-way signal propagation delay for the signal to travel between the n-th antenna and the p-th target. The frequency bins fm , for m = 0, 1, . . . , M − 1, are usually uniformly distributed over the frequency range. The signal propagation time depends on the propagation path between the antenna and the target, and the propagation speed. Unlike conventional radar, TWRI has a wall in between the system and the target. Since the wall induces refraction of the signal and a change in the propagation speed, τn,p should be properly computed according to factoring in the wall parameters such as thickness and the dielectric constant [2]. Throughout this paper, we will assume that the correct wall parameters are available and the wall reflections are taken into consideration [2, 6]. Once all the data samples are collected, delay-andsum (DS) beamforming can be applied to form an image. The gh-th pixel value using DS beamforming is, N −1 M −1 1 y(n, m) exp{j2πfm τn,(g,h) }. N M n=0 m=0 (2) In the above equation, τn,(g,h) is the two-way signal propagation time between the n-th antenna location and the gh-th pixel location. The total number of data samples that are used in (2) is M times N . In other words, it is the number of antenna locations times the number of frequency bins. Not all M N data samples are necessary to obtain an image. A high quality image can still be generated, although some of the data samples are missing. However, by merely using part of the frequency band, would provide an image quality that is degraded proportional to the number of missing data [7].
b(g, h) =
The solution x in the above problem is not always equal to s. The probability that x = s changes depending on the K × L matrix A [8]. According to applications, the matrix A is defined differently. Even in the same application, the definition of s and y can be different. Therefore, the matrix A is not uniquely defined. In [4] and [5], they defined y as the whole measurement data and s as the scene of interest or the target space. In other words, the data sample y is defined by selecting K samples from the whole M × N samples and the recovered signal s is the final result (image). In this paper, CS is applied to the data collected at each antenna location separately and assume different forms. By doing so, we can recover all M frequency samples at each antenna location, allowing the application of the usual image formation method, such as DS beamformer. There are advantages of the proposed method over previous methods. First, the matrix A is similar to Fourier matrix which is well-known and widely studied. As such the minimum number of required sample K and the probability that the CS provides right solution can be analytically computed [9]. Second, since the output of CS is not the final solution, the residuals or errors from the CS can be suppressed during image formation process. Separate applications of the CS will result, if any, in independent residuals. The residuals will be averaged out during DS beamforming. Third, the frequency samples can be selected differently at each antenna location. This will decrease the chance of failure of the CS imaging, since the probability that the CS succeeds increases with increasing the randomness of the chosen data sets. 4. CS IN TWRI Let sn be the discrete range profile at the n-th antenna location and yn be the received signal at the n-th antenna location. Since we have M narrowband frequency signals, the dimension of yn will be M . The relation between y and the target location can be represented by linear equations (See eq.(1)). Suppose that the range of interest is divided into M equally spaced distances as ui for i = 0, 1, . . . , M − 1. Then, yn = Asn ,
(4)
where 3. COMPRESSIVE SENSING Compressive sensing has brought a new paradigm in recovering a signal from the limited number of data samples [8, 9]. CS is possible when the signal to recover is sparse and has a linear relationship with the sampled data [8]. Let s be the sparse signal to recover and y is the measured data sample vector. For y = As, the CS technique recovers the vector s from the vector y by solving the following optimization problem. (3) min x1 such that Ax = y. x
2807
yn [A]kl
= =
T
[y(n, 0) y(n, 1) · · · y(n, M − 1)] , exp{−2πfk τl },
and [sn ]k =
σp , τn,p = τk , 0, τn,p = τk ,
(5)
where τk is the two-way signal traveling time between the antenna location and uk ’s location. The dimension of the signal sn and the dimension of yn are the number of discrete ranges and the number of frequency bins, respectively. Note that
sn ’s are different for each antenna location due to the differences in the relative distances between antenna locations and the target locations. The A matrix is similar to the Fourier matrix which relates the frequency and time representations. ˜ n , which is a K(< M ) dimensional vector consisting Given y of elements randomly chosen from yn , we can recover sn by solving the following equation. ˜ =y ˜n, ˆsn = arg min x1 subject to Ax
(6)
x
where ˜n y ˜ A kl
T
= [y(n, i0 ) y(n, i1 ) . . . y(n, iK−1 )]
(7)
= [A]ik l .
(8)
In the above equation, ik ∈ [0, 1, · · · , M − 1] for k = 0, . . . , K − 1 are the indexes of the randomly chosen frequency bins. The selected frequency bins should be distributed over the entire frequency band. The set ik can be the same or different for each n.
6. SIMULATIONS 6.1. Synthesized Data First, the proposed method was tested with a synthesized data set. We consider there are three point targets located behind a wall. A concrete wall which is 0.14m thick is simulated in this scene. 1.2m-long linear array consist of 67 antenna locations and 2GHz bandwidth step-frequency signal (101 narrowband signals with 20MHz step) are used in the simulation. In the data, we assumed that the wall reflection has been properly removed [3, 6]. Figure 1 compares the three radar images: DS beamforming image with the whole data set, DS beamforming image with selected data samples, and DS beamforming image with selected data samples using the proposed method. The number of selected frequency bins is 24. Note that Fig. 1(c) is almost as good as Fig. 1(a). In this simulation, the selected frequency bins are different for each antenna. regular DS beamforming 0 2
í5
2.5
Δr , T = C/2
α α · δr = . C/2 B
í15 í20
3.5
í25
4
í30 í35 í40
5
í45
5.5 í2
í1
0 Crossírange, meter
1
2
í50
(a) thined DS beamforming
CS beamforming 0
2
0 2
í5
2.5 3
í20 3.5
í25
4
í30 í35
4.5
Downírange,meter
í10 í15
í5
2.5
í10 í15
3
í20 3.5
í25
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í30 í35
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í40 5 5.5 í2
í45 í1
0 Crossírange, meter
1
2
(b)
í50
í40 5 5.5 í2
í45 í1
0 Crossírange, meter
1
2
í50
(c)
(9)
where C is the speed of signal propagation. Let δr be the radar’s down-range resolution which is C/2B where B is the signal bandwidth. When Δr = α · δr, the time interval T is T =
í10
3
4.5
Downírange,meter
Once ˆsn , which is the recovered version of sn , is found for all n = 0, 1, . . . , N − 1, it is equivalent to have all the frequency samples at all the antenna locations. We can then obtain the radar image by applying conventional DS beamforming. The DS beamforming can be applied both in the frequency domain and the time domain. The frequency domain representation can be obtained via the discrete Fourier transform to sn . However, the frequency range and the frequency step in the Fourier transform of sn should be carefully redefined since these are different from those of the step-frequency signal used in the data collection. These parameters depend on the number of range bins and the distance between range bins. Let Δr be the distance between the range bins. Then, the time interval between range bins is given by
Downírange,meter
5. BEAMFORMING
(10)
Therefore, the frequency step used in the Fourier transform of ˆsn is B . (11) Δf = αM Note that α should be less than one to prevent missing targets between two consecutive range bins. The above Δf should be used in applying DS beamforming in the Frequency domain.
2808
Fig. 1. TWRI images of synthesized data. (a) regular DS image (101 frequency bins). (b) regular DS image using only 24 frequency bins. (c) DS image using the proposed method.
6.2. Experimental Data A real data has been collected at the Radar Imaging Lab at Villanova University, Villanova, PA. A single dihedral was located behind a 0.05m thick plywood wall. 67 antenna locations with 0.018m displacement are used for data collection. The step-frequency signal in this experiment has 67 frequency steps of size 30 MHz, covering the 1-3GHZ bandwidth. The wall is located at 1.0m from the antenna. The target (dihedral) was at 2.7m away from the wall. Figure 2 depicts the
aperture. Using CS, it is possible to recover all the frequency samples in the step-frequency signal from small subset. After recovery of the entire frequency samples, radar imaging can be obtained by conventional DS beamforming methods. We have tested the new method by computer simulations, using both synthesized data, and real experimental data generated from a an indoor two-dimensional radar imaging scanner. In both cases, the proposed method demonstrated that it can provide radar images that are very close to those obtained by processing the whole data set. The new approach can be used in other radar imaging system such as inverse synthetic aperture radars (ISAR) and GPRs.
0.018 m 1m 2.7 m 1.18 m
… 0.05 m
Fig. 2. Geometry of the experiment setup. 8. REFERENCES geometry of the experiment setup. Figure 3 shows similar results as the synthesized data case. While the regular DS image with small number of frequency bins suffers from many false alarms and low SNR (Fig. 3(b)), the DS image after CS recovers the original image (Fig. 3(a)) without significant degradation. The number of frequency bins used for Fig. 3(b) and (c) was 27. 0 í5
Downírange,meter
2.4
í10 í15
2.6
í20 2.8
í25 í30
3
[4] Y.-S. Yoon and M. Amin, “Compressed sensing technique for high-resolution radar imaging,” in Proc. of SPIE Signal Processing, Sensor Fusion, and Target Recognition XVII, vol. 6958, Orlando, FL, Mar. 2008.
í35 3.2
í40 í45
3.4 í1.5
í1
í0.5 0 0.5 Crossírange, meter
í50
1
(a) Thinned DS beamforming
CS beamforming 0
2.2
0
2.2
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2.6
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í1
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1
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[5] A. Gurbuz, J. McClellan, and W. Scott, “A compressive sensing data acquisition and imaging method for stepped frequency GPRs,” IEEE Trans. Signal Processing, vol. 57, no. 7, pp. 2640–2650, July 2009.
í25 í30
3
í35 3.2
í40 í45
3.4 í1.5
[2] F. Ahmad and M. Amin, “Noncoherent approach to through-the-wall radar localization,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, no. 4, pp. 1405–1419, Oct. 2006. [3] M. Dehmollaian and K. Sarabandi, “Refocusing through building walls using synthetic aperture radar,” IEEE Trans. Geosci. Remote Sensing, vol. 46, no. 6, pp. 1589– 1599, June 2008.
Regular DS beamforming 2.2
[1] D. Ferris, Jr. and N. Currie, “A survey of current technologies for through-the-wall surveillance (TWS),” Proceedings of SPIE, no. 3577, pp. 62–72, Nov. 1998.
í1
í0.5 0 0.5 Crossírange, meter
(b)
1
í50
(c)
Fig. 3. TWRI images of synthesized data. (a) regular DS image (67 frequency bins). (b) regular DS image using only 27 frequency bins. (c) DS image using the proposed method.
7. CONCLUSIONS We have proposed a new method of reducing the data samples in TWRI applications. The new method is based on applying the CS technique independently to the data samples collected at each antenna location along the physical or synthesized
2809
[6] Y.-S. Yoon and M. Amin, “Spatial filtering for wall clutter mitigation in through-the-wall radar imaging,” IEEE Trans. Geosci. Remote Sensing, vol. 47, no. 9, pp. 3192– 3208, Sept. 2009. [7] L. He, S. Kassam, F. Ahmad, and M. Amin, “Sparse stepped-frequency waveform design for through-the-wall radar imaging,” in Proc. of the 2006 Waveform Diversity & Design Conference, Lihue, HI, Jan. 2006. [8] E. J. Cand`es, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math., vol. 59, pp. 1207–1223, Aug. 2006. [9] ——, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006.
Moeness G. Amin∗
R&D Division Samsung Thales Co., LTD. Yongin, Kyunggi, Korea
Center for Advanced Communications Villanova University Villanova, PA 19085, USA
ABSTRACT There are increasing demands on Through-the-Wall Radar Imaging (TWRI) systems to deliver high resolution images in both range and cross range. This requires using wideband signals and large array apertures, respectively. Wideband signals are typically implemented by transmitting a series of the narrowband signals. As such, the TWRI system operation involves transmitting and receiving multiple step-frequencies at each antenna location in either a physical or a synthetic aperture radar. Compressive sensing (CS) is an effective approach for decreasing the number of samples and, subsequently, reducing the data acquisition and post-processing time. In this paper, we propose a TWRI scheme based on CS in which a sizable reduction in the number of samples along the frequency axis is achieved without significant degradation in the image. The proposed approach applies the well-known Fourier-like measurement matrix and generates radar images of almost the same desirable quality as the image employing all data samples. Index Terms— Through-the-wall radar, radar imaging, compressive sensing 1. INTRODUCTION Through wall radar imaging (TWRI) has gained much attention in recent years due to its wide and broad applications in several civilian and military sectors [1, 2, 3]. Among many requirements for effective TWRI system is fast acquisition time. This requirement becomes more pronounced and challenging when deploying wideband large aperture systems for high resolution imaging. For the vast majority of systems which synthesize a large pulse by step frequencies, transmitting and receiving many narrowband signals at all antenna locations constitute a considerable percentage of the acquisition time. For example, for 100 narrowband signals and 50 element array, the total number of transmit-receive data samples becomes 5000. Once all the data is collected, the scene of interest can ∗ The
work by Prof. Moeness Amin is supported in part by the Office of Naval Research, Grant no. N00014-07-1-0043.
978-1-4244-4296-6/10/$25.00 ©2010 IEEE
2806
be constructed by image formation methods. In some cases, a long data collection time must be avoided, since all the objects in the scene should be stationary during the entire data collection operations to avoid image smearing and displacement of targets. There have been several approaches that apply compressive sensing (CS) to reduce the number of samples in radar imaging [4, 5]. In essence, CS aims at recovering the vector s from the vector y, where the two vectors are related by the matrix formula y = As, with A has more columns than rows. Sparse representations permit solutions where the required number of data samples (y) can be significantly reduced and yet allows recovery of the scene of interest (s). In reference [4], the authors introduced CS to high resolution radar imaging in urban sensing applications and tested the effectiveness of the CS approach with synthesized data. The work in [5] involves the application of CS to the ground penetrating radar (GPR) problems, which are similar to TWRI in the sense that image formation is performed using step-frequency radar signals. In this paper, we employ CS for TWRI in a new and different way. We apply CS in frequency domain only, leaving the spatial-domain sensing uncompressed. That is, we use all the antenna elements along the original system aperture, whether it is physical or synthesized. So, only the number of frequency bins is reduced by the CS. However, the corresponding CS optimization solution is not the final result, as is the case in [4, 5]. Rather, we use usual image formation methods, for example, delay and sum (DS) beamforming, following the CS. This proves advantageous, since residuals that might appear with CS can be reduced during the image formation process. Adopting a conventional image formation can also provide a desirable performance of the target detection and classification steps which immediately and typically follow the imaging step. 2. THROUGH-THE-WALL RADAR Electromagnetic wave can pass through a dielectric wall allowing scenes behind walls to be imaged by using radar imaging techniques [2]. In order to improve the down-range reso-
ICASSP 2010
lution, the signal bandwidth should be increased. High crossrange resolution requires a long array aperture. Assume that there are N antenna locations and a step-frequency signal of M narrowband signals are used. The radar signal received at the n-th antenna location with the m-th frequency bin is given by, P −1 σp exp{−j2πfm τn,p }, (1) y(n, m) = p=0
where σp is the reflection coefficient of the p-th target, and τn,p is the two-way signal propagation delay for the signal to travel between the n-th antenna and the p-th target. The frequency bins fm , for m = 0, 1, . . . , M − 1, are usually uniformly distributed over the frequency range. The signal propagation time depends on the propagation path between the antenna and the target, and the propagation speed. Unlike conventional radar, TWRI has a wall in between the system and the target. Since the wall induces refraction of the signal and a change in the propagation speed, τn,p should be properly computed according to factoring in the wall parameters such as thickness and the dielectric constant [2]. Throughout this paper, we will assume that the correct wall parameters are available and the wall reflections are taken into consideration [2, 6]. Once all the data samples are collected, delay-andsum (DS) beamforming can be applied to form an image. The gh-th pixel value using DS beamforming is, N −1 M −1 1 y(n, m) exp{j2πfm τn,(g,h) }. N M n=0 m=0 (2) In the above equation, τn,(g,h) is the two-way signal propagation time between the n-th antenna location and the gh-th pixel location. The total number of data samples that are used in (2) is M times N . In other words, it is the number of antenna locations times the number of frequency bins. Not all M N data samples are necessary to obtain an image. A high quality image can still be generated, although some of the data samples are missing. However, by merely using part of the frequency band, would provide an image quality that is degraded proportional to the number of missing data [7].
b(g, h) =
The solution x in the above problem is not always equal to s. The probability that x = s changes depending on the K × L matrix A [8]. According to applications, the matrix A is defined differently. Even in the same application, the definition of s and y can be different. Therefore, the matrix A is not uniquely defined. In [4] and [5], they defined y as the whole measurement data and s as the scene of interest or the target space. In other words, the data sample y is defined by selecting K samples from the whole M × N samples and the recovered signal s is the final result (image). In this paper, CS is applied to the data collected at each antenna location separately and assume different forms. By doing so, we can recover all M frequency samples at each antenna location, allowing the application of the usual image formation method, such as DS beamformer. There are advantages of the proposed method over previous methods. First, the matrix A is similar to Fourier matrix which is well-known and widely studied. As such the minimum number of required sample K and the probability that the CS provides right solution can be analytically computed [9]. Second, since the output of CS is not the final solution, the residuals or errors from the CS can be suppressed during image formation process. Separate applications of the CS will result, if any, in independent residuals. The residuals will be averaged out during DS beamforming. Third, the frequency samples can be selected differently at each antenna location. This will decrease the chance of failure of the CS imaging, since the probability that the CS succeeds increases with increasing the randomness of the chosen data sets. 4. CS IN TWRI Let sn be the discrete range profile at the n-th antenna location and yn be the received signal at the n-th antenna location. Since we have M narrowband frequency signals, the dimension of yn will be M . The relation between y and the target location can be represented by linear equations (See eq.(1)). Suppose that the range of interest is divided into M equally spaced distances as ui for i = 0, 1, . . . , M − 1. Then, yn = Asn ,
(4)
where 3. COMPRESSIVE SENSING Compressive sensing has brought a new paradigm in recovering a signal from the limited number of data samples [8, 9]. CS is possible when the signal to recover is sparse and has a linear relationship with the sampled data [8]. Let s be the sparse signal to recover and y is the measured data sample vector. For y = As, the CS technique recovers the vector s from the vector y by solving the following optimization problem. (3) min x1 such that Ax = y. x
2807
yn [A]kl
= =
T
[y(n, 0) y(n, 1) · · · y(n, M − 1)] , exp{−2πfk τl },
and [sn ]k =
σp , τn,p = τk , 0, τn,p = τk ,
(5)
where τk is the two-way signal traveling time between the antenna location and uk ’s location. The dimension of the signal sn and the dimension of yn are the number of discrete ranges and the number of frequency bins, respectively. Note that
sn ’s are different for each antenna location due to the differences in the relative distances between antenna locations and the target locations. The A matrix is similar to the Fourier matrix which relates the frequency and time representations. ˜ n , which is a K(< M ) dimensional vector consisting Given y of elements randomly chosen from yn , we can recover sn by solving the following equation. ˜ =y ˜n, ˆsn = arg min x1 subject to Ax
(6)
x
where ˜n y ˜ A kl
T
= [y(n, i0 ) y(n, i1 ) . . . y(n, iK−1 )]
(7)
= [A]ik l .
(8)
In the above equation, ik ∈ [0, 1, · · · , M − 1] for k = 0, . . . , K − 1 are the indexes of the randomly chosen frequency bins. The selected frequency bins should be distributed over the entire frequency band. The set ik can be the same or different for each n.
6. SIMULATIONS 6.1. Synthesized Data First, the proposed method was tested with a synthesized data set. We consider there are three point targets located behind a wall. A concrete wall which is 0.14m thick is simulated in this scene. 1.2m-long linear array consist of 67 antenna locations and 2GHz bandwidth step-frequency signal (101 narrowband signals with 20MHz step) are used in the simulation. In the data, we assumed that the wall reflection has been properly removed [3, 6]. Figure 1 compares the three radar images: DS beamforming image with the whole data set, DS beamforming image with selected data samples, and DS beamforming image with selected data samples using the proposed method. The number of selected frequency bins is 24. Note that Fig. 1(c) is almost as good as Fig. 1(a). In this simulation, the selected frequency bins are different for each antenna. regular DS beamforming 0 2
í5
2.5
Δr , T = C/2
α α · δr = . C/2 B
í15 í20
3.5
í25
4
í30 í35 í40
5
í45
5.5 í2
í1
0 Crossírange, meter
1
2
í50
(a) thined DS beamforming
CS beamforming 0
2
0 2
í5
2.5 3
í20 3.5
í25
4
í30 í35
4.5
Downírange,meter
í10 í15
í5
2.5
í10 í15
3
í20 3.5
í25
4
í30 í35
4.5
í40 5 5.5 í2
í45 í1
0 Crossírange, meter
1
2
(b)
í50
í40 5 5.5 í2
í45 í1
0 Crossírange, meter
1
2
í50
(c)
(9)
where C is the speed of signal propagation. Let δr be the radar’s down-range resolution which is C/2B where B is the signal bandwidth. When Δr = α · δr, the time interval T is T =
í10
3
4.5
Downírange,meter
Once ˆsn , which is the recovered version of sn , is found for all n = 0, 1, . . . , N − 1, it is equivalent to have all the frequency samples at all the antenna locations. We can then obtain the radar image by applying conventional DS beamforming. The DS beamforming can be applied both in the frequency domain and the time domain. The frequency domain representation can be obtained via the discrete Fourier transform to sn . However, the frequency range and the frequency step in the Fourier transform of sn should be carefully redefined since these are different from those of the step-frequency signal used in the data collection. These parameters depend on the number of range bins and the distance between range bins. Let Δr be the distance between the range bins. Then, the time interval between range bins is given by
Downírange,meter
5. BEAMFORMING
(10)
Therefore, the frequency step used in the Fourier transform of ˆsn is B . (11) Δf = αM Note that α should be less than one to prevent missing targets between two consecutive range bins. The above Δf should be used in applying DS beamforming in the Frequency domain.
2808
Fig. 1. TWRI images of synthesized data. (a) regular DS image (101 frequency bins). (b) regular DS image using only 24 frequency bins. (c) DS image using the proposed method.
6.2. Experimental Data A real data has been collected at the Radar Imaging Lab at Villanova University, Villanova, PA. A single dihedral was located behind a 0.05m thick plywood wall. 67 antenna locations with 0.018m displacement are used for data collection. The step-frequency signal in this experiment has 67 frequency steps of size 30 MHz, covering the 1-3GHZ bandwidth. The wall is located at 1.0m from the antenna. The target (dihedral) was at 2.7m away from the wall. Figure 2 depicts the
aperture. Using CS, it is possible to recover all the frequency samples in the step-frequency signal from small subset. After recovery of the entire frequency samples, radar imaging can be obtained by conventional DS beamforming methods. We have tested the new method by computer simulations, using both synthesized data, and real experimental data generated from a an indoor two-dimensional radar imaging scanner. In both cases, the proposed method demonstrated that it can provide radar images that are very close to those obtained by processing the whole data set. The new approach can be used in other radar imaging system such as inverse synthetic aperture radars (ISAR) and GPRs.
0.018 m 1m 2.7 m 1.18 m
… 0.05 m
Fig. 2. Geometry of the experiment setup. 8. REFERENCES geometry of the experiment setup. Figure 3 shows similar results as the synthesized data case. While the regular DS image with small number of frequency bins suffers from many false alarms and low SNR (Fig. 3(b)), the DS image after CS recovers the original image (Fig. 3(a)) without significant degradation. The number of frequency bins used for Fig. 3(b) and (c) was 27. 0 í5
Downírange,meter
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3
[4] Y.-S. Yoon and M. Amin, “Compressed sensing technique for high-resolution radar imaging,” in Proc. of SPIE Signal Processing, Sensor Fusion, and Target Recognition XVII, vol. 6958, Orlando, FL, Mar. 2008.
í35 3.2
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(a) Thinned DS beamforming
CS beamforming 0
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[5] A. Gurbuz, J. McClellan, and W. Scott, “A compressive sensing data acquisition and imaging method for stepped frequency GPRs,” IEEE Trans. Signal Processing, vol. 57, no. 7, pp. 2640–2650, July 2009.
í25 í30
3
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[2] F. Ahmad and M. Amin, “Noncoherent approach to through-the-wall radar localization,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, no. 4, pp. 1405–1419, Oct. 2006. [3] M. Dehmollaian and K. Sarabandi, “Refocusing through building walls using synthetic aperture radar,” IEEE Trans. Geosci. Remote Sensing, vol. 46, no. 6, pp. 1589– 1599, June 2008.
Regular DS beamforming 2.2
[1] D. Ferris, Jr. and N. Currie, “A survey of current technologies for through-the-wall surveillance (TWS),” Proceedings of SPIE, no. 3577, pp. 62–72, Nov. 1998.
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Fig. 3. TWRI images of synthesized data. (a) regular DS image (67 frequency bins). (b) regular DS image using only 27 frequency bins. (c) DS image using the proposed method.
7. CONCLUSIONS We have proposed a new method of reducing the data samples in TWRI applications. The new method is based on applying the CS technique independently to the data samples collected at each antenna location along the physical or synthesized
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[6] Y.-S. Yoon and M. Amin, “Spatial filtering for wall clutter mitigation in through-the-wall radar imaging,” IEEE Trans. Geosci. Remote Sensing, vol. 47, no. 9, pp. 3192– 3208, Sept. 2009. [7] L. He, S. Kassam, F. Ahmad, and M. Amin, “Sparse stepped-frequency waveform design for through-the-wall radar imaging,” in Proc. of the 2006 Waveform Diversity & Design Conference, Lihue, HI, Jan. 2006. [8] E. J. Cand`es, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math., vol. 59, pp. 1207–1223, Aug. 2006. [9] ——, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006.
