Entropy-Based Low-Rank Approximation for Contrast Dielectric Target Detection with Through Wall Imaging System

Abstract: In through wall imaging, clutter plays an important role in the detection of objects behind the wall. In the literature, extensive studies have been carried out to eliminate clutter in the case of targets with the same dielectric. Existing clutter reduction techniques, such as the sub-space approach, differential approach, entropy-based time gating, etc., are able to detect a single target or two targets with the same dielectric behind the wall. In a real-time scenario, it is not necessary that targets with the same dielectric will be present behind the wall. Very few studies are available for the detection of targets with different dielectrics; here we termed it “contrast target detection” in the same scene. Recently, low-rank approximation (LRA) was proposed to reduce random noise in the data. In this paper, a novel method based on entropy thresholding for low-rank approximation is introduced for contrast target detection. It was observed that our proposed method gives satisfactory results.


Introduction
Through wall imaging (TWI) is emerging as an important technology for surveillance, security and rescue missions. The main aim of TWI is seeing through a wall with the help of electromagnetic waves. In any radar system, the signal-to-clutter ratio (SCR) plays an important role in the improvement of the detection of the objects. The SCR can be improved either by classical or statistical methods [1]. Classical methods use different classical digital filters, while statistical methods exploit the statistical nature of the received signal to separate the clutter from the signal.
Digital filtering technique [2] uses frequency analysis of the clutter geometrical model and the signal geometrical model, while in [3], the coupled iterative procedure was used to reduce the ground reflections for the application of ground penetrating radar (GPR). The classical clutter reduction algorithm (CCRA) was proposed in [4,5], but this method does not optimize the coefficients for the representation of noise and the target. Kalman filtering uses the background component model in [6], but designing the Kalman filter is computationally intensive. Parametric clutter reduction was proposed in [3] by modeling the variations in the received signal, but it requires a reference signal, which cannot practically be made available.
In the literature, various statistical clutter reduction techniques have been proposed. Different statistical reduction techniques were compared in [7]. Verma et al. [1] applied various statistical techniques for clutter reduction such as singular value decomposition (SVD), principle component analysis (PCA), factor analysis (FA), and independent component analysis (ICA), and concluded that ICA performs better for the detection of low dielectric material behind a wall.

Data Collection
Data is collected by placing different dielectric materials such as metal or wood behind the wall at different distances in our experimental work. Synthetic aperture radar (SAR) in the multi-static mode in which an array of antennas are used to scan the whole wall at M different locations, and the reflection coefficient (S11) is measured for Ptargets in the scene using system parameters given in Table 1 and then set-up, which is shown in Figure 1. Received frequency domain data is converted to time domain [16] Step (1): Transformation from Time domain to spatial domain Time domain signal is converted to spatial domain to determine the range profile by using z = c * t/2 where c is the speed of light and t is a delay.
Step (2): External calibration The metallic plate is placed in front of the antenna [17] to find the delay due to the antenna system, which will be subtracted from the observed data. The range profile is corrected using the difference in the delay. Step (3): Velocity correction As the antenna is placed at the standoff distance from the wall, the signal propagates through the air, then the wall and again through the air up to the target. The presence of the wall scattered the signal, and shifting of the target position took place. This shifting was compensated using the method of velocity correction. The mathematical equation for this is given next in the paper. The metallic plate is placed in front of the antenna [17] to find the delay due to the antenna system, which will be subtracted from the observed data. The range profile is corrected using the difference in the delay.
Step (3): Velocity correction As the antenna is placed at the standoff distance from the wall, the signal propagates through the air, then the wall and again through the air up to the target. The presence of the wall scattered the signal, and shifting of the target position took place. This shifting was compensated using the method of velocity correction. The mathematical equation for this is given next in the paper.

Beamforming
The antenna is placed in front of the wall at a fixed standoff distance and data is collected for M different locations that received the signal, represented as (1).
where ( ) s t is the transmitted signal convolved with the transfer function of the wall [14] for the SFCW radar. p σ is the reflection coefficient, and , n p τ is the two-way delay-the time between the th n antenna position and the target P. When the signal propagates between the th n antenna positions and th p target, the two-way delay-time is given as (2)

Beamforming
The antenna is placed in front of the wall at a fixed standoff distance and data is collected for M different locations that received the signal, represented as Equation (1).
where s(t) is the transmitted signal convolved with the transfer function of the wall [14] for the SFCW radar. σ p is the reflection coefficient, and τ n,p is the two-way delay-the time between the n th antenna position and the target P. When the signal propagates between the n th antenna positions and p th target, the two-way delay-time is given as Equation (2) where c is the speed of light. In this paper, the focused image has been developed using DS (delay-and-sum) beamforming for collected data. The i th pixel value in the DS image is given by Equation where τ n (i, j) is the propagation delay on both sides of the wall. The SFCW radar, for which stepped size depends upon the selection of frequency bins, requires trade-off between a number of frequency bins and scanning time. The SFCW radar waveform consists of Q-narrowband signals defined as Equation (4) b where x(n, f q ) is the signal received at the n th antenna position for the frequency q. S11 data is collected either in the time domain or frequency domain [18]. We collected data for three targets with different dielectrics by arranging them at different positions behind the wall (refer to Appendix A). Our method was tested on collected data for illustration purposes, with a few results given in Sections 3 and 4. The geometry for the TWI is shown in Figure 2, if we consider the point target at X p , then developing the image transformation is required from the time to the spatial domain.
where c is the speed of light.
In this paper, the focused image has been developed using DS (delay-and-sum) beamforming for collected data. The th i pixel value in the DS image is given by (3) where ( , j) n i τ is the propagation delay on both sides of the wall. The SFCW radar, for which stepped size depends upon the selection of frequency bins, requires trade-off between a number of frequency bins and scanning time. The SFCW radar waveform consists of Q-narrowband signals defined as (4) where ( , ) q x n f is the signal received at the th n antenna position for the frequency q . S11 data is collected either in the time domain or frequency domain [18]. We collected data for three targets with different dielectrics by arranging them at different positions behind the wall (refer to Appendix A). Our method was tested on collected data for illustration purposes, with a few results given in Sections 3 and 4. The geometry for the TWI is shown in Figure 2, if we consider the point target at p X , then developing the image transformation is required from the time to the spatial domain. Delay-sum-beamforming is the most popular and least complex imaging algorithm, interested readers may refer to [19] where m and q represents the spatial index and frequency index, respectively. In our experimental set-up, we considered a homogenous wall of thickness d = 15 cm and relative permittivity of the wall of

5.3
w ε = . The dielectric constant of the wall is measured as described in [20]. The distance from the antenna to the wall is ( ) a z and from a wall to the target is ( t z ). The velocity correction [21] for geometry shown in Figure 2 is given by (6) Delay-sum-beamforming is the most popular and least complex imaging algorithm, interested readers may refer to [19] for more details. The signal received at the antenna location is z[m, q] of the frequency f n with a delay τ p,m , then z[m, q] can be represented as Equation (5) where m and q represents the spatial index and frequency index, respectively. In our experimental set-up, we considered a homogenous wall of thickness d = 15 cm and relative permittivity of the wall of ε w = 5.3. The dielectric constant of the wall is measured as described in [20]. The distance from the antenna to the wall is (z a ) and from a wall to the target is (z t ). The velocity correction [21] for geometry shown in Figure 2 is given by Equation (6) where d v is the actual distance between the antenna and the target after velocity correction. τ n,p can be estimated by putting Equation (6) into Equation (2) τ n,p = (x tr 0 − x tr n ) 2 + (d v + X p ) 2 (7) where k and l are the number of pixels in the image.

Clutter Reduction Techniques
Clutter is the unwanted reflections due to other objects in the room. Clutter overwhelms the target, and so clutter reduction techniques can be used to separate clutter from the target. The data collected for the n th observation can be denoted as where s a (t) are the reflections due to an antenna mismatch, s w (t) are the reflections due to the wall and s p (t) are the contribution due to p (the number of targets behind the wall). The discrete form of collected data for M antenna locations at N different instances can be arranged in an M × N data matrix.
We try to separate the s p (t) signal from s a (t) and s w (t) using clutter reduction methods [22]. Imaging for raw data was done with the help of DS beamforming as discussed in Section 2. Raw normalized images for different targets after pre-processing are shown in Figure 3. It can be seen from the DS images for different targets, along with the target, clutter due to the wall and other objects also dominate, which may obscure the target. The reflections from the low dielectric targets are generally weak compared to clutter from the interior and exterior of the wall, making detection of such targets difficult. Efficient clutter reduction technique is required to remove the clutter from the B-scan image to detect low dielectric targets [23]. After pre-processing, different commonly used clutter reduction techniques are implemented and the results of this are discussed here. Their performance is given in terms of the PSNR (peak signal-to-noise ratio) [24].
where v d is the actual distance between the antenna and the target after velocity correction. , n p τ can be estimated by putting (6) into (2) We can recover the image [ , ] s k l by DS-beamforming using (8) where k and l are the number of pixels in the image.

Clutter Reduction Techniques
Clutter is the unwanted reflections due to other objects in the room. Clutter overwhelms the target, and so clutter reduction techniques can be used to separate clutter from the target. The data collected for the th n observation can be denoted as s t s t s t s t = + +  Imaging for raw data was done with the help of DS beamforming as discussed in Section 2. Raw normalized images for different targets after pre-processing are shown in Figure 3. It can be seen from the DS images for different targets, along with the target, clutter due to the wall and other objects also dominate, which may obscure the target. The reflections from the low dielectric targets are generally weak compared to clutter from the interior and exterior of the wall, making detection of such targets difficult. Efficient clutter reduction technique is required to remove the clutter from the B-scan image to detect low dielectric targets [23]. After pre-processing, different commonly used clutter reduction techniques are implemented and the results of this are discussed here. Their performance is given in terms of the PSNR (peak signal-to-noise ratio) [24].

Average Trace Subtraction
In general, clutter remains constant with respect to target reflections for data collected [25], hence we can consider that clutter will be constant for a homogenous wall. We can separate a constant signal from the non-constant signal using spatial filtering [26] in the time domain, which can be represented as  (11) can be given as This condition cover [− 0 k , 0 k ] and no filtering is introduced by the grating lobe. Due to the Dirichlet condition appearing in Equation (12), the low-frequency spatial spectrum of target signal n s will also get rejected, and due to this target which is placed near the wall, cannot be detected.

Differential Approach
In this approach the clutter is removed by subtracting the adjacent two traces [27], for the time domain data this is represented as [28] ( , ) ( The corresponding spatial spectrum can be given as

Average Trace Subtraction
In general, clutter remains constant with respect to target reflections for data collected [25], hence we can consider that clutter will be constant for a homogenous wall. We can separate a constant signal from the non-constant signal using spatial filtering [26] in the time domain, which can be represented as where s(n, m) is the data matrix element and s(m) is the average of the data matrix. The Fourier transform for (11) can be given as where k x represents spatial frequency (∆x ≤ 2π/k 0 ), and k 0 is the frequency wave number. This condition cover [−k 0 ,k 0 ] and no filtering is introduced by the grating lobe. Due to the Dirichlet condition appearing in Equation (12), the low-frequency spatial spectrum of target signal s n will also get rejected, and due to this target which is placed near the wall, cannot be detected.

Differential Approach
In this approach the clutter is removed by subtracting the adjacent two traces [27], for the time domain data this is represented as [28] The corresponding spatial spectrum can be given as The observation point corresponds to x n = (n + 1/2)∆x n = 0, . . . , N − 1 and n = 0, . . . , N−2 in the first and second term. It can be seen from Equation (14) that de-cluttering can be achieved locally and that not all traces are exploited simultaneously. If ∆x is very small, then the sin term rises very slowly, and if it is very high, then artifacts corrupt the reconstruction.

Subspace Projection Approach
Subspace approaches are used to separate out complementary subspaces, called the target and clutter, in order to increase SCR. In the literature, many methods for clutter reduction using the subspace projection approach are given. Here we will restrict our discussion to SVD and ICA.
where u i v i T are the Eigen-component and w i is the Eigen-value for the i th component. The first Eigen-value represents the strong reflections, which are generally from the wall in the case of TWI, the remaining values represent other reflections from the target and noise. We can categorize Eigen-space into target sub-space and noise sub-space.
where E 1→k represents strong reflections, E k+1→p represents reflections from the target and E p+1→N represents noise.

Independent Component Analysis (ICA)
ICA divides the data into statistically independent components. Statistical independence considers higher order moments for data matrix S. ICA takes a linear combination of S x such that where A is the matrix holding N independent source components. The output signal matrix Y is for the input matrix, matrix I can be determined with the help of the full rank matrix (W), such that where W is the matrix which makes I as independent as possible for dependent sensor signals S.

Entropy-based Time Gating
Recently in [25], entropy-based time gating was proposed for clutter reduction, and it is shown that this method is efficient for clutter reduction compared to earlier methods in the literature. In this method behavior of the clutter, which is similar over each time trace, is used to exploit the entropy. Clutter signal gives higher entropy compared to the target. The threshold set in this paper is as Equation (20) W where α < 1 is the tolerance for the threshold and N is the number of scanning points The time trace after incorporating the threshold is given as To illustrate the results using the above methods, we used the data collected with a single metal target and two targets of metal and wood for contrast imaging as described in Section 2. The PSNR for each method in both cases is given in Tables 2 and 3. It can be seen from Figure 4 that both targets are visible along with some clutter by using average trace subtraction, but they are not visible by other methods. It is necessary to develop an efficient clutter removal technique to detect a weak target in the presence of the strong clutter.
PSNR is the ratio to analyze the distortion in the final image with respect to the input low-resolution image is given by PSNR = 10 log 10 1  The time trace after incorporating the threshold is given as To illustrate the results using the above methods, we used the data collected with a single metal target and two targets of metal and wood for contrast imaging as described in Section 2. The PSNR for each method in both cases is given in Table 2 and Table 3.   It can be seen from Figure 4 that both targets are visible along with some clutter by using average trace subtraction, but they are not visible by other methods. It is necessary to develop an efficient clutter removal technique to detect a weak target in the presence of the strong clutter.

A Proposed Novel Method for Contrast Imaging
Low-rank approximation (LRA) has been used recently for seismic data [12]. LRA is efficient compared to SVD and ICA as it exploits the noise space by considering the large rank, this motivates us to use LRA for the detection of contrast targets. LRA is a rank reduction technique, the principle requirement of the LRA is that the data should be low rank. In TWI imaging, the number of targets are less than the number of scanning points (P < MN), hence the collected data is inherently sparse and low rank. Steps for traditional LRA are Step (1): Calculate the SVD for data matrix S.
While selecting the n largest singular or Eigen-values from matrix D, we ignore the first Eigen-value, which represents the strongest reflections from the wall [1]. The modified LRA for TWI can be given as Step (4): Ignore first Eigen-value corresponding to wall reflections, hence the matrix D is given as To satisfy the principle of the algorithm, LRA works in the local windows, where deciding the optimum rank is difficult. Since LRA cannot estimate the optimum rank in the local window, it will consider the large rank to preserve the useful energy in the signal. In the attempt to preserve the useful energy using the large rank, unwanted clutter is also added to the signal. This problem can be solved by using an optimum threshold while selecting Eigen-values during the reconstruction of the useful signal.
Step (5): Select the optimum threshold for Eigen-values from LRA using the entropy-based criterion.
We used the entropy-based criterion [25] for selecting the threshold for the Eigen-values in the LRA. The idea for entropy-based thresholding was adapted to discriminate between target and clutter signals. To select the optimum threshold, we consider the criteria that entropy is maximum for clutter and minimum for the target. First, we construct the normalized time traces as At each instant, normalized data is considered to be a probability density function (PDF) [29]. Introducing PDF allows us to adopt the entropy-based criterion to determine the threshold and entropy measure, which is given as The entropy of the clutter signals gives large values and the clutter in the observations is generally constant, hence the average value for the Eigen-values can be the optimum threshold. The flow chart for the proposed method is given in Figure 5.  For subspace approaches, such as SVD and ICA, only a few dominant Eigen-values are considered as the target and lower Eigen-values are considered as the noise, as a result, weak targets are considered as noise. In the proposed developed method, we consider all the Eigen-values and set the optimum threshold to eliminate the noise-space, hence we are able to detect weak targets such as wood along with a strong target such as metal.
The data is processed with different targets to check the capability of the method. For contrast imaging, two targets with different dielectrics are chosen behind the wall, i.e., metal and wood. The results for which are shown in Figure 6 For subspace approaches, such as SVD and ICA, only a few dominant Eigen-values are considered as the target and lower Eigen-values are considered as the noise, as a result, weak targets are considered as noise. In the proposed developed method, we consider all the Eigen-values and set the optimum threshold to eliminate the noise-space, hence we are able to detect weak targets such as wood along with a strong target such as metal.
The data is processed with different targets to check the capability of the method. For contrast imaging, two targets with different dielectrics are chosen behind the wall, i.e., metal and wood. The results for which are shown in Figure 6.
targets are considered as noise. In the proposed developed method, we consider all the Eigen-values and set the optimum threshold to eliminate the noise-space, hence we are able to detect weak targets such as wood along with a strong target such as metal.
The data is processed with different targets to check the capability of the method. For contrast imaging, two targets with different dielectrics are chosen behind the wall, i.e., metal and wood. The results for which are shown in Figure 6.

Conclusions
In this paper, the problem of contrast imaging is addressed and it is shown that our developed method is able to detect a weak target in the presence of a strong target. The inherent problem of considering the large rank in LRA is solved by setting the optimum threshold using the entropy-based criterion. The entropy-based LRA method was compared to other methods, such as average trace subtraction, entropy-based time gating, SVD, ICA and DA, and found to be very effective for different types and arrangements of the target.
Another advantage of the proposed method is that it also avoids the filtering of low spatial targets and therefore this method allows a better reconstruction of low as well as high dielectric targets in the scene. In our future work, we are going to work on rank optimization problem of LRA.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
We collected 45 datum points for different arrangements of a target position behind the wall, target size and target thickness. The distance of the target is measured from the antenna mouth. Table 3 gives target types while Figure A1, Figure A2 and Figure A3 gives A-scan, B-scan and C-scan images using the proposed method as described in Section 4, which are given for validation.

Conclusions
In this paper, the problem of contrast imaging is addressed and it is shown that our developed method is able to detect a weak target in the presence of a strong target. The inherent problem of considering the large rank in LRA is solved by setting the optimum threshold using the entropy-based criterion. The entropy-based LRA method was compared to other methods, such as average trace subtraction, entropy-based time gating, SVD, ICA and DA, and found to be very effective for different types and arrangements of the target.
Another advantage of the proposed method is that it also avoids the filtering of low spatial targets and therefore this method allows a better reconstruction of low as well as high dielectric targets in the scene. In our future work, we are going to work on rank optimization problem of LRA.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
We collected 45 datum points for different arrangements of a target position behind the wall, target size and target thickness. The distance of the target is measured from the antenna mouth. Table 3 Electronics 2019, 8, 634 13 of 17 gives target types while Figures A1-A3 gives A-scan, B-scan and C-scan images using the proposed method as described in Section 4, which are given for validation. TWI scanning methods-In TWI, three types of scanning are done for target detection and shape identification. The A-Scan or range profile is a dimensional plot, which provides information about the presence of a target along with the approximate location. The B-Scan gives information about a number of targets present in the down-range and the C-scan gives information about shape, height, and width. A-Scan plots for different target ID are shown in Figure A1. B-Scan and C-Scan images are developed using 30 horizontal scans and 15 vertical scans.  Figure A2. B-scan images for different targets. Target ID 06  Target ID 11 Target ID 12 Target ID 13