Compressive sensing (CS) has been shown to be a useful tool for subsurface or through-the-wall imaging (TWI) using ground penetrating radar (GPR). It has been used to decrease both time/frequency or spatial measurements and generate high-resolution images. Although current works apply CS directly to TWI, questions on the required number of measurements for a sparsity level, measurement strategy to subsample in frequency and space, or imaging performance in varying noise levels and limits on CS range resolution performance still needs to be answered. In addition current CS-based imaging methods are based on two basic assumptions; targets are point like and positioned at only discrete grid locations and wall thickness and its dielectric constant are perfectly known. However, these assumptions are not usually valid in most TWI applications. This work extends the theory of CS-based radar imaging developed for subsurface imaging to TWI and outlines the performance of the proposed imaging for the above-mentioned questions using numerical simulations. The effect of unknown parameters on the imaging performance is analyzed, and it is observed that off-the-grid point targets and big modeling errors decreases the performance of CS imaging.

Through-the-wall imaging (TWI) radar [

Compressive sensing (CS) [

Compressive sensing was introduced in the general context of radar imaging in [

In all previous CS-TWI or CS-GPR literature, the imaging theory depends on several assumptions. First, the targets are point-like reflectors located only at discrete grid positions. Second, the wave velocities or medium parameters, such as permittivity or wall thickness, are assumed to be perfectly known. These assumptions were used to build a forward model between the target space and measurements, but actual targets might not fall on grid points and wave velocities or wall thickness might only be known approximately. Hence the created forward model will definitely include modeling errors. It is very important to understand the effect of such parameters on the robustness of CS-based TWI. Additionally, most CS-TWI papers show simulation results for imaging of randomly placed several point targets with using less space/frequency measurements as compared to observing all space/frequency domain. However, no specific result for subsurface or TWI has been shown detailing the required number of measurements, a function of the sparsity level of the imaged region. The general result of CS, which briefly says

Another important question is what should be the random sampling strategy for CS. This question turns out to be a design of random measurement matrix, which controls how the space and frequency domain is sampled. In [

Range resolution performance of CS-TWI is another important issue. CS has shown to resolve targets spaced closer than the Rayleigh range resolution limit in radar applications [

The organization of the paper is as follows. Section

Through-the-wall imaging (TWI) algorithms like delay and sum beamforming generate images by mainly applying a coherent matched filter of the measured data with the impulse response of the data acquisition process [

In this development a stepped frequency (SF) radar system is considered. A detailed explanation on CS-based imaging theory for subsurface imaging for a 2-layer geometry is given in [

Our goal is to linearly relate the sensor measurements to the target space. To do so, a target model for which the expected target return can be calculated should be used. Under the Born approximation, neglecting the mutual interactions between targets, the scattering field for a continuous target space

Finally when the radar is at the

In generating the data dictionary, the crucial point is to calculate the time delay value

Standard stepped frequency systems measure a regularly spaced set of

In this section the effects of the parameters in CS imaging system are tested through numerical simulations. A preliminary understanding on the required number of measurements for correct reconstruction of the target space at varying sparsity levels, the effect of acquiring these measurements from frequency or space, the level of noise where CS imaging still works, the effect of system bandwidth and resolution limits, the effect of discritization of target space and off-the-grid targets, and the estimation errors in parameters like wave velocity or wall thickness in TWI are aimed in the simulations presented in this section. For these tests a TWI geometry with a monostatic antenna with a 20 cm offset from a 30 cm thick wall of dielectric constant 4 is considered. The SF system collects frequency domain measurements from 500 MHz to 5.5 GHz with 40 MHz frequency steps. Thus, at each scan position, GPR acquires 126 frequency measurements if all frequency steps are measured. A target space of size 60 cm by 60 cm is considered with a 3 cm discretization on both axes generating an

In CS-based subsurface imaging or TWI literature, it is shown that the total number of measurements acquired can be decreased if the target space is sparse, but it is important to know the relation between the required number of measurements for correct imaging and varying levels of sparsity specifically for GPR applications. To understand this relation, target spaces with sparsity levels changing from 2 to 12 are tested. For each case the target space is imaged with 10 to 500 compressed measurements using proposed CS technique. This procedure is repeated 50 times for each sparsity level and measurement number. For each case a random target space and random measurement selections are done and correct reconstructions are counted. The correct reconstruction ratio is obtained as the ratio of total number of correct reconstructed cases to the total number of tests which is 50 in this case. Two measurement strategies as random and uniform are compared. In the random measurement strategy acquiring

Figures

Correct reconstruction ratio versus measurement number for varying levels of target sparsity for (a) random measurement strategy, (b) uniform measurement strategy. The required number of measurements to achieve a 95% correct reconstruction for random measurement strategy.

Another important point in a measurement strategy is to decide on a good division in observing measurement from frequency or space. It is important to understand the tradeoff between subsampling from space or frequency. As an example, if a total of 50 measurement would be taken from the whole space-frequency domain how would you distribute the measurements to space and frequency; that is, would you take random 10 scan points and measure 5 frequencies at each scan point or measure 5 scan points and take 10 frequency measurements at each point. Which one would be more effective in reconstructing the target space image? For this test the target sparsity level is fixed to 5 targets, and varying numbers of randomly selected frequency measurements from 5 to 125 at varying spatial scan points from 2 to 20 are observed. At each case the independent target space generations and random measurement selections are repeated 50 times and the number of correct reconstructions are counted. Figure

Correct reconstruction ratio versus the random frequency measurements per each scan point. The term “sp” in legend represents “scan point.”

It can be initially observed that imaging performance is very low if only one scan point is used regardless of the measurement number, but even increasing it to two scan points increases the performance. For the same total space-frequency measurement numbers, distributing measurements across scan points increases correct reconstruction performance. For example, observing 90 measurements at a single scan point generates a 0.1 reconstruction ratio where observing 40 measurements at two scan points or 30 measurements at 3 scan points increases reconstruction performance dramatically. It can also be observed that the total number of measurements becomes the important factor if at least 4 out of 20 scan points are used. As a conclusion, random subsampling in both frequency and space can be done but for successful reconstructions spatial scan points should not be too few. Further and more general tests are needed to be done to obtain more general results including whether this tradeoff between space and frequency division is effected by the sparsity level of the scene.

To analyze the impact of noise level on the imaging performance, a simulation is performed. First, with frequency domain data for a single point target SNRs from −5 to 25 dB are tested with 50 different trials using additive complex white gaussian noise (CWGN) at each SNR level using varying number of total of space-frequency measurements from 50 to 300 observed randomly at 20 scan points and the target space is reconstructed with (

Correct reconstruction ratio versus SNR for different total measurement number

It can be seen in simulation result that with increasing measurement numbers, the same reconstruction performance could be obtained at lower SNR values. At lower SNR values than 0 dB, correct reconstructions were not obtained for the tested measurement numbers. Above 15 dB SNR, even very low measurement numbers generate highly correct reconstruction ratios.

One of the important properties of the CS method is its ability to resolve targets spaced closer than the conventional range resolution of an SFCW GPR [

Imaging performance of backprojection when the range separation between two targets is (a) 10 cm, (b) 6 cm, and (c) performance of CS-based imaging when two targets are separated by 3 cm.

In the GPR imaging applications, the assumed signal sparsity is in the continuous target space and the sparsity basis

In the literature the simplistic and the general approach to the problem is to increase the grid size and use a reconstruction algorithm that can handle additive noise in measurements. Although this could provide close to satisfactory results for some applications, it is not the solution to the actual problem. No matter how fine the discritization is, the actual signal parameters might not be on the grid. In addition increasing grid size increases the coherence between dictionary columns making restricted isometry property (RIP) [

In this part, off-the-grid targets are simulated, and effect of the discritization or grid size on the TWI performance is observed. To observe only the effect of grid size, parameters like wall thickness or wave velocity in all mediums are assumed to be known perfectly. A target space of size ^{2} area is simulated. Two point reflectors at off-the-grid positions as (−0.192, 0, −0.143) and (0.218,0, −0.382) are placed, and the target space are discritized with 1 cm, 2 cm, and 5 cm grid sizes. An SNR of 10 dB is used. The obtained compressive sensing and back-projection reconstructions using only the 20% of the total frequency-space data randomly are shown in Figure

Effects of off-the-grid targets to the CS-based TWI: (a) backprojection, (b) CS image for discritization of 1 cm, (c) backprojection, (d) CS image for discritization of 2 cm, (e) backprojection, (f) CS image for discritization of 5 cm.

It can be observed from Figure

To solve the off-grid target problem, we are working on new techniques that utilize gradients with respect to the parameters at each grid location and apply greedy or joint perturbations to basis columns. This solution for the off-grid problem is combined with the orthogonal matching pursuit (OMP) algorithm for a new iterative off-the-grid OMP method. Our initial results are given in [

Another fundamental problem in CS-based TWI is that the parameters like wall thickness and the velocity of the wave in the wall that are used in creating the data dictionary might be different than the actual parameters. These problems are also studied in classical TWI in [

Effects of unknown wall characteristics to CS imaging: wall permittivity is (a) 4.1 and (b) 4.5 and wall thickness (c) 30.2 cm and (d) 31 cm. Effects of unknown wall characteristics to back-projection imaging: wall permittivity is (e) 4.7 and Wall thickness (f) 33 cm.

In this paper, a compressive-sensing-based through-the-wall imaging algorithm is presented. Initial results through numerical simulations are obtained for questions on the required number of measurements for a sparsity level, measurement strategy to subsample in frequency and space, imaging performance in varying noise levels, and enhanced CS range resolution performance. Simulations with off-the-grid targets and unknown parameters are performed, and it is observed that if the grid size or the error in the unknown parameters is not too big, the imaging performance is not severely affected; however, big grid sizes or errors degrade the reconstructed image. Although obtained results are not too general, we believe that they are preliminary to a much deeper study.

This work was supported by the Scientific and technical Research Council of Turkey (TUBITAK) under Contract Agreement 109E280 and within the Marie Curie IRG Grant with Grant Agreement PIRG04-GA-2008-239506.