# Compressive sensing for through-the-wall radar imaging

**Moeness G. Amin**

Villanova University, Center for Advanced Communications, 800 E. Lancaster Avenue, Villanova, Pennsylvania 19085

**Fauzia Ahmad**

Villanova University, Center for Advanced Communications, Radar Imaging Lab, 800 E. Lancaster Avenue, Villanova, Pennsylvania 19085

*J. Electron. Imaging*. 22(3), 030901 (Jul 01, 2013). doi:10.1117/1.JEI.22.3.030901

#### Open Access

## Abstract

**Abstract.**
Through-the-wall radar imaging (TWRI) is emerging as a viable technology for providing high-quality imagery of enclosed structures. TWRI makes use of electromagnetic waves to penetrate through building wall materials. Due to the “see” through ability, TWRI has attracted much attention in the last decade and has found a variety of important civilian and military applications. Signal processing algorithms have been devised to allow proper imaging and image recovery in the presence of high clutter, which is caused by front walls and multipath due to reflections from internal walls. Recently, research efforts have shifted toward effective and reliable imaging under constraints on aperture size, frequency, and acquisition time. In this respect, scene reconstructions are being pursued with reduced data volume and within the emerging compressive sensing (CS) framework. We present a review of the CS-based scene reconstruction techniques that address the unique challenges associated with fast and efficient imaging in urban operations. Specifically, we focus on ground-based imaging systems for indoor targets. We discuss CS-based wall mitigation, multipath exploitation, and change detection for imaging of stationary and moving targets inside enclosed structures.

## Introduction

Through-the-wall radar imaging (TWRI) is an emerging technology that addresses the desire to see inside buildings using electromagnetic (EM) waves for various purposes, including determining the building layout, discerning the building intent and nature of activities, locating and tracking the occupants, and even identifying and classifying inanimate objects of interest within the building. TWRI is highly desirable for law enforcement, fire and rescue, and emergency relief, and military operations.^{1}^{–}^{6}

Applications primarily driving TWRI development can be divided based on whether information on motions within a structure or on imaging the structure and its stationary contents is sought out. The need to detect motion is highly desirable to discern about the building intent and in many fire and hostage situations. Discrimination of movements from background clutter can be achieved through change detection (CD) or exploitation of Doppler.^{7}^{–}^{24} One-dimensional (1-D) motion detection and localization systems employ a single transmitter and receiver and can only provide range-to-motion, whereas two- and three-dimensional (2-D and 3-D) multi-antenna systems can provide more accurate localization of moving targets. The 3-D systems have higher processing requirements compared with 2-D systems. However, the third dimension provides height information, which permits distinguishing people from animals, such as household pets. This is important since radar cross-section alone for behind-the-wall targets can be unreliable.

Imaging of structural features and stationary targets inside buildings requires at least 2-D and preferably 3-D systems.^{25}^{–}^{43} Because of the lack of any type of motion, these systems cannot rely on Doppler processing or CD for target detection and separation. Synthetic aperture radar (SAR) based approaches have been the most commonly used algorithms for this purpose. Most of the conventional SAR techniques usually neglect propagation distortions such as those encountered by signals passing through walls.^{44} Distortions degrade the performance and can lead to ambiguities in target and wall localizations. Free-space assumptions no longer apply after the EM waves propagate through the first wall. Without factoring in propagation effects, such as attenuation, reflection, refraction, diffraction, and dispersion, imaging of contents within buildings will be severely distorted. As such, image formation methods, array processing techniques, target detection, and image sharpening paradigms must work in concert and be reexamined in view of the nature and specificities of the underlying sensing problem.

In addition to exterior walls, the presence of multipath and clutter can significantly contaminate the radar data leading to reduced system capabilities for imaging of building interiors and localization and tracking of targets behind walls. The multiple reflections within the wall result in wall residuals along the range dimension. These wall reverberations can be stronger than target reflections, leading to its masking and undetectability, especially for weak targets close to the wall.^{45} Multipath stemming from multiple reflections of EM waves off the targets in conjunction with the walls may result in the power being focused at pixels different than those corresponding to the target. This gives rise to ghosts, which can be confused with the real targets inside buildings.^{46}^{–}^{49} Further, uncompensated refraction through walls can lead to localization or focusing errors, causing offsets and blurring of imaged targets.^{26}^{,}^{39} SAR techniques and tomographic algorithms, specifically tailored for TWRI, are capable of making some of the adjustments for wave propagation through solid materials.^{26}^{–}^{30}^{,}^{36}^{–}^{41}^{,}^{50}^{–}^{57} While such approaches are well suited for shadowing, attenuation, and refraction effects, they do not account for multipath as well as strong reflections from the front wall.

The problems caused by the front wall reflections can be successfully tackled through wall clutter mitigation techniques. Several approaches have been devised, which can be categorized into those based on estimating the wall parameters and others incorporating either wall backscattering strength or invariance with antenna location.^{39}^{,}^{45}^{,}^{58}^{–}^{61} In Refs. ^{39} and ^{58}, a method to extract the dielectric constant and thickness of the nonfrequency dependent wall from the time-domain scattered field was presented. The time-domain response of the wall was then analytically modeled and removed from the data. In ^{45}, a spatial filtering method was applied to remove the DC component corresponding with the constant-type radar return, typically associated with the front wall. The third method, presented in Refs. ^{59}–^{61}, was based not only on the wall scattering invariance along the array but also on the fact that wall reflections are relatively stronger than target reflections. As a result, the wall subspace is usually captured in the most dominant singular values when applying singular value decomposition (SVD) to the measured data matrix. The wall contribution can then be removed by orthogonal subspace projection.

Several methods have also been devised for dealing with multipath ghosts in order to provide proper representation of the ground truth. Earlier work attempted to mitigate the adverse effects stemming from multipath propagation.^{27} Subsequently, research has been conducted to utilize the additional information carried by the multipath returns. The work in ^{49} considered multipath exploitation in TWRI, assuming prior knowledge of the building layout. A scheme taking advantage of the additional energy residing in the target ghosts was devised. An image was first formed, the ghost locations for each target were calculated, and then the ghosts were mapped back onto the corresponding target. In this way, the image became ghost-free with increased signal-to-clutter ratio (SCR).

More recently, the focus of the TWRI research has shifted toward addressing constraints on cost and acquisition time in order to achieve the ultimate objective of providing reliable situational awareness through high-resolution imaging in a fast and efficient manner. This goal is primarily challenged due to use of wideband signals and large array apertures. Most radar imaging systems acquire samples in frequency (or time) and space and then apply compression to reduce the amount of stored information. This approach has three inherent inefficiencies. First, as the demands for high resolution and more accurate information increase, so does the number of data samples to be recorded, stored, and subsequently processed. Second, there are significant data redundancies not exploited by the traditional sampling process. Third, it is wasteful to acquire and process data samples that will be discarded later. Further, producing an image of the indoor scene using few observations can be logistically important, as some of the measurements in space and frequency or time can be difficult, unavailable, or impossible to attain.

Toward the objective of providing timely actionable intelligence in urban environments, the emerging compressive sensing (CS) techniques have been shown to yield reduced cost and efficient sensing operations that allow super-resolution imaging of sparse behind-the-wall scenes.^{10}^{,}^{62}^{–}^{76} Compressive sensing is an effective technique for scene reconstruction from a relatively small number of data samples without compromising the imaging quality.^{77}^{–}^{89} In general, the minimum number of data samples or sampling rate that is required for scene image formation is governed by the Nyquist theorem. However, when the scene is sparse, CS provides very efficient sampling, thereby significantly decreasing the required volume of data collected.

In this paper, we focus on CS for TWRI and present a review of $l1$ norm reconstruction techniques that address the unique challenges associated with fast and efficient imaging in urban operations. Sections 2ec3ec4–5 deal with imaging of stationary scenes, whereas moving target localization is discussed in Sec. 6 and 7. More specifically, Sec. 2 deals with CS based strategies for stepped-frequency based radar imaging of sparse stationary scenes with reduced data volume in spatial and frequency domains. Prior and complete removal of clutter is assumed, which renders the scene sparse. Section 3 presents CS solutions in the presence of front wall clutter. Wall mitigation in conjunction with application of CS is presented for the case when the same reduced frequency set is used from all of the employed antennas. Section 4 considers imaging of the building interior structures using a CS-based approach, which exploits prior information of building construction practices to form an appropriate sparse representation of the building interior layout. Section 5 presents CS based multipath exploitation technique to achieve good image reconstruction in rich multipath indoor environments from few spatial and frequency measurements. Section 6 deals with joint localization of stationary and moving targets using CS based approaches, provided that the indoor scene is sparse in both stationary and moving targets. Section 7 discusses a sparsity-based CD approach to moving target indication for TWRI applications, and deals with cases when the heavy clutter caused by strong reflections from exterior and interior walls reduces the sparsity of the scene. Concluding remarks are provided in Sec. 8. It is noted that for the sake of not overcomplicating the notation, some symbols are used to indicate different variables over different sections of the paper. However, for those cases, these variables are redefined to reflect the change.

The progress reported in this paper is substantial and noteworthy. However, many challenging scenarios and situations remain unresolved using the current techniques and, as such, further research and development are required. However, with the advent of technology that brings about better hardware and improved system architectures, opportunities for handling more complex building scenarios will definitely increase.

## CS Strategies in Frequency and Spatial Domains for TWRI

In this section, we apply CS to through-the-wall imaging of stationary scenes, assuming prior and complete removal of the front wall clutter.^{62}^{,}^{63} For example, if the reference scene is known, then background subtraction can be performed for removal of wall clutter, thereby improving the sparsity of the behind-the-wall stationary scene. We assume stepped-frequency-based SAR operation. We first present the through-the-wall signal model, followed by a description of the sparsity-based scene reconstruction, highlighting the key equations. It is noted that the problem formulation can be modified in a straightforward manner for pulsed operation and multistatic systems.

Consider a homogeneous wall of thickness $d$ and dielectric constant $\epsilon $ located along the $x$-axis, and the region to be imaged located beyond the wall along the positive $z$-axis. Assume that an $N$-element line array of transceivers is located parallel to the wall at a standoff distance $zoff$, as shown in Fig. 1. Let the $n$’th transceiver, located at $xn=(xn,\u2212zoff)$, illuminate the scene with a stepped-frequency signal of $M$ frequencies, which are equispaced over the desired bandwidth $\omega M\u22121\u2212\omega 0$,

^{27}

^{–}

^{28}

^{,}

^{40}

An equivalent matrix-vector representation of the received signals in Eq. (2) can be obtained as follows. Assume that the region of interest is divided into a finite number of pixels $Nx\xd7Nz$ in cross-range and downrange, and the point targets occupy no more than $P(\u226aNx\xd7Nz)$ pixels. Let $r(k,l)$, $k=0,1,\u2026,Nx\u22121$, $l=0,1,\u2026,Nz\u22121$, be a weighted indicator function, which takes the value $\sigma p$ if the $p$’th point target exists at the $(k,l)$’th pixel; otherwise, it is zero. With the values $r(k,l)$ lexicographically ordered into a column vector $r$ of length $NxNz$, the received signal corresponding to the $n$’th antenna can be expressed in matrix-vector form as

The expression in Eq. (7) involves the full set of measurements made at the $N$ array locations using the $M$ frequencies. For a sparse scene, it is possible to recover $r$ from a reduced set of measurements. Consider $y\u0306$, which is a vector of length $Q1Q2(\u226aMN)$ consisting of elements chosen from $y$ as follows:

^{80}Given $y\u0306$, we can recover $r$ by solving the following equation (ideally, minimization of the $l0$ norm would provide the sparsest solution. Unfortunately, it is NP-hard to solve the resulting minimization problem. The $l1$ norm has been shown to serve as a good surrogate for $l0$ norm.

^{90}The $l1$ minimization problem is convex, which can be solved in polynomial time):

We note that the problem in Eq. (11) can be solved using convex relaxation, greedy pursuit, or combinatorial algorithms.^{91}^{–}^{96} In this section, we consider orthogonal matching pursuit (OMP), which is known to provide a fast and easy to implement solution. Moreover, OMP is better suited when frequency measurements are used.^{95} It is noted that the number of iterations of the OMP is usually associated with the level of sparsity of the scene. In practice, this piece of information is often unavailable *a priori*, and the stopping condition is heuristic. Underestimating the sparsity would result in the image not being completely reconstructed (underfitting), while overestimation would cause some of the noise being treated as signal (overfitting). Use of cross-validation (CV) has been also proposed to determine the stopping condition for the greedy algorithms.^{97}^{–}^{99} Cross-validation is a statistical technique that separates a data set into a training set and a CV set. The training set is used to detect the optimal stopping iteration. There is, however, a tradeoff between allocating the measurements for reconstruction or CV. More details can be found in Refs. ^{97} and ^{98}.

A through-the-wall wideband SAR system was set up in the Radar Imaging Lab at Villanova University. A 67-element line array with an inter-element spacing of 0.0187 m, located along the $x$-axis, was synthesized parallel to a 0.14-m-thick solid concrete wall of length 3.05 m and at a standoff distance equal to 1.24 m. A stepped-frequency signal covering the 1 to 3 GHz frequency band with a step size of 2.75 MHz was employed. Thus, at each scan position, the radar collects 728 frequency measurements. A vertical metal dihedral was used as the target and was placed at (0, 4.4) m on the other side of the front wall. The size of each face of the dihedral is $0.39\xd70.28\u2009\u2009m2$. The back and the side walls of the room were covered with RF absorbing material to reduce clutter. The empty scene without the dihedral target present was also measured to enable background subtraction for wall clutter removal.

The region to be imaged is chosen to be $4.9\xd75.4\u2009\u2009m2$ centered at (0, 3.7) m and divided into $33\xd773\u2009\u2009pixels$, respectively. For CS, 20% of the frequencies and 51% of the array locations were used, which collectively represent 10.2% of the total data volume. Figure 2(a) and 2(c) depict the images corresponding to the full dataset obtained with back-projection and $l1$ norm reconstruction, respectively. Figure 2(b) and 2(d) show the images corresponding to the measured scene obtained with back-projection and $l1$ norm reconstruction, respectively, applied to the reduced background subtracted dataset. In Fig. 2 and all subsequent figures in this paper, we plot the image intensity with the maximum intensity value in each image normalized to 0 dB. The true target position is indicated with a solid red rectangle. We observe that, with the availability of the empty scene measurements, background subtraction renders the scene sparse, and thus a CS-based approach generates an image using reduced data where the target can be easily identified. On the other hand, back-projection applied to reduced dataset results in performance degradation, indicated by the presence of many artifacts in the corresponding image. OMP was used to generate the CS images. For this particular example, the number of OMP iterations was set to five.

## Effects of Walls on Compressive Sensing Solutions

The application of CS for TWRI as presented in Sec. 2 assumed prior and complete removal of front wall EM returns. Without this assumption, strong wall clutter, which extends along the range dimension, reduces the sparsity of the scene and, as such, impedes the application of CS.^{71}^{–}^{73} Having access to the background scene is not always possible in practical applications. In this section, we apply joint CS and wall mitigation techniques using reduced data measurements. In essence, we address wall clutter mitigations in the context of CS.

There are several approaches, which successfully mitigate the front wall contribution to the received signal.^{39}^{,}^{45}^{,}^{58}^{–}^{61} These approaches were originally introduced to work on the full data volume and did not account for reduced data measurements especially randomly. We examine the performance of the subspace projection wall mitigation technique^{60} in conjunction with sparse image reconstruction. Only a small subset of measurements is employed for both wall clutter reduction and image formation. We consider the case where the same subset of frequencies is used for each employed antenna. Wall clutter mitigation under use of different frequencies across the employed antennas is discussed in Refs. ^{68} and ^{73}. It is noted that, although not reported in this paper, the spatial filtering based wall mitigation scheme^{45} in conjunction with CS provides a similar performance to the subspace projection scheme.^{73}

We first extend the through-the-wall signal model of Eq. (2) to include the front wall return. Without the assumption of prior wall return removal, the output of the $n$’th transceiver corresponding to the $m$’th frequency for a scene of $P$ point targets is given by

From Eq. (12), we note that $\tau w$ does not vary with the antenna location since the array is parallel to the wall. Furthermore, as the wall is homogeneous and assumed to be much larger than the beamwidth of the antenna, the first term in Eq. (12) assumes the same value across the array aperture. Unlike $\tau w$, the time delay $\tau p,n$, given by Eq. (3), is different for each antenna location, since the signal path from the antenna to the target is different from one antenna to the other.

The signals received by the $N$ antennas at the $M$ frequencies are arranged into an $M\xd7N$ matrix, $Y$,

^{59}and

^{60}. The subspace orthogonal to the wall subspace is

^{60}

Subspace projection method for wall clutter reduction relies on the fact that the wall reflections are strong and assume very close values at the different antenna locations. When the same set of frequencies is employed for all employed antennas, the condition of spatial invariance of the wall reflections is maintained.^{72}^{,}^{73} This permits direct application of the subspace projection method as a preprocessing step to the $l1$ norm based scene reconstruction of Eq. (11).

We consider the same experimental setup as in Sec. 2.3. Figure 3(a) shows the result obtained with $l1$ norm reconstruction using 10.2% of the raw data volume without background subtraction. The number of OMP iterations was set to 100. Comparing Fig. 3(a) and the corresponding background subtracted image of Fig. 2(d), it is evident that in the absence of access to the background scene, the wall mitigation techniques must be applied, as a preprocessing step, prior to CS in order to detect the targets behind the wall.

First, we consider the case when the same set of reduced frequencies is used for a reduced set of antenna locations. We employ only 10.2% of the data volume, i.e., 20% of the available frequencies and 51% of the antenna locations. The subspace projection method is applied to a $Y$ matrix of reduced dimension $146\xd734$. The corresponding $l1$ norm reconstructed image obtained with OMP is depicted in Fig. 3(b). It is clear that, even when both spatial and frequency observations are reduced, the joint application of wall clutter mitigation and CS techniques successfully provides front wall clutter suppression and unmasking of the target.

## Designated Dictionary for Wall Detection

In this section, we address the problem of imaging building interior structures using a reduced set of measurements. We consider interior walls as targets of interest and attempt to reveal the building interior layout based on CS techniques. We note that construction practices suggest the exterior and interior walls to be parallel or perpendicular to each other. This enables sparse scene representations using a dictionary of possible wall orientations and locations.^{76} Conventional CS recovery algorithms can then be applied to reduced number of observations to recover the positions of various walls, which is a primary goal in TWRI.

Considering a monostatic stepped-frequency SAR system with $N$ antenna positions located parallel to the front wall, as shown in Fig. 1, we extend the signal model in Eq. (12) to include reflections from multiple parallel interior walls, in addition to the returns from the front wall and the $P$ point targets. That is, the received signal at the $n$’th antenna location corresponding to the $m$’th frequency can be expressed as

Note that the above model contains contributions only from interior walls parallel to the front wall and the antenna array. This is because, due to the specular nature of the wall reflections, a SAR system located parallel to the front wall will only be able to receive direct returns from walls, which are parallel to the front wall. The detection of perpendicular walls is possible by concurrently detecting and locating the canonical scattering mechanism of corner features created by the junction of walls of a room or by having access to another side of the building. Extension of the signal model to incorporate corner returns is reported in ^{76}.

Instead of the point-target based sensing matrix described in Eq. (7), where each antenna accumulates the contributions of all the pixels, we use an alternate sensing matrix, proposed in ^{68}, to relate the scene vector, $r$, and the observation vector, $y$. This matrix underlines the specular reflections produced by the walls. Due to wall specular reflections, and since the array is assumed parallel to the front wall and, thus, parallel to interior walls, the rays collected at the $n$’th antenna will be produced by portions of the walls that are only in front of this antenna [see Fig. 4(a)]. The alternate matrix, therefore, only considers the contributions of the pixels that are located in front of each antenna. In so doing, the returns of the walls located parallel to the array axis are emphasized. As such, it is most suited to the specific building structure imaging problem, wherein the signal returns are mainly caused by EM reflections of exterior and interior walls. The alternate linear model can be expressed as

Since the number of parallel walls is typically much smaller compared with the downrange extent of the building, the decomposition of the image into parallel walls can be considered as sparse. Note that although other indoor targets, such as furniture and humans, may be present, their projections onto the horizontal lines are expected to be negligible compared to those of the walls.

In order to obtain a linear matrix-vector relation between the scene and the horizontal projections, we define a sparsifying matrix $R$ composed of possible wall locations. Specifically, each column of the dictionary $R$ represents an image containing a single wall of length $lx$ pixels, located at a specific cross-range and at a specific downrange in the image. Consider the cross-range to be divided into $Nc$ nonoverlapping blocks of $lx$ pixels each [see Fig. 5(a)], and the downrange division defined by the pixel grid. The number of blocks $Nc$ is determined by the value of $lx$, which is the minimum expected wall length in the scene. Therefore, the dimension of $R$ is $NxNz\xd7NcNz$,where the product $NcNz$ denotes the number of possible wall locations. Figure 5(b) shows a simplified scheme of the sparsifying dictionary generation. The projection associated with each wall location is given by

It is noted that we are implicitly assuming that the extents of the walls in the scene are integer multiples of the block of $lx$ pixels. In case this condition is not satisfied, the maximum error in determining the wall extent will be at most equal to the chosen block size. Note that incorporation of the corner effects will help resolve this issue, since the localization of corners will identify the wall extent.^{76}

A through-the-wall SAR system was set up in the Radar Imaging Lab, Villanova University. A stepped-frequency signal consisting of 335 frequencies covering the 1 to 2 GHz frequency band was used for interrogating the scene. A monostatic synthetic aperture array, consisting of 71-element locations with an inter-element spacing of 2.2 cm, was employed. The scene consisted of two parallel plywood walls, each 2.25 cm thick, 1.83 m wide, and 2.43 m high. Both walls were centered at 0 m in cross-range. The first and the second walls were located at respective distances of 3.25 and 5.1 m from the antenna baseline. Figure 6(a) depicts the geometry of the experimental scene.

The region to be imaged is chosen to be $5.65(cross-range)\xd74.45\u2009\u2009m(down range)$, centered at (0, 4.23) m, and is divided into $128\xd7128\u2009\u2009pixels$. For the CS approach, we use a uniform subset of only 84 frequencies at each of the 18 uniformly spaced antenna locations, which represent 6.4% of the full data volume. The CS reconstructed image is shown in Fig. 6(b). We note that the proposed algorithm was able to reconstruct both walls. However, it can be observed in Fig. 6(b) that ghost walls appear immediately behind each true wall position. These ghosts are attributed to the dihedral-type reflections from the wall-floor junctions.

## CS and Multipath Exploitation

In this section, we consider the problem of multipath in view of the requirements of fast data acquisition and reduced measurements. Multipath ghosts may cast a sparse scene as a populated scene, and at minimum will render the scene less sparse, degrading the performance of CS-based reconstruction. A CS method that directly incorporates multipath exploitation into sparse signal reconstruction for imaging of stationary scenes with a stepped-frequency monostatic SAR is presented. Assuming prior knowledge of the building layout, the propagation delays corresponding to different multipath returns for each assumed target position are calculated, and the multipath returns associated with reflections from the same wall are grouped together and represented by one measurement matrix. This allows CS solutions to focus the returns on the true target positions without ghosting. Although not considered in this section, it is noted that the clutter due to front wall reverberations can be mitigated by adapting a similar multipath formulation, which maps back multiple reflections within the wall after separating wall and target returns.^{100}

We refer to the signal that propagates from the antenna through the front wall to the target and back to the antenna as the direct target return. Multipath propagation corresponds with indirect paths, which involve reflections at one or more interior walls by which the signal may reach the target. Multipath can also occur due to reflections from the floor and ceiling and interactions among different targets. In considering wall reflections and assuming diffuse target scattering, there are two typical cases for multipath. In the first case, the wave traverses a path that consists of two parts—one part is the propagation path to the target and back to the receiver, and the other part is a round trip path from the target to an interior wall. As the signal weakens at each secondary wall reflection, this case can usually be neglected. Furthermore, except when the target is close to an interior wall, the corresponding propagation delay is high and, most likely, would be equivalent to the direct-path delay of a target that lies outside the perimeter of the room being imaged. Thus, if necessary, this type of multipath can be gated out. The second case is a bistatic scattering scenario, where the signal propagation on transmit and receive takes place along different paths. This is the dominant case of multipath with one of the paths being the direct propagation, to or from the target, and the other involving a secondary reflection at an interior wall.

Other higher-order multipath returns are possible as well. Signals reaching the target can undergo multiple reflections within the front wall. We refer to such signals as wall ringing multipaths. Also the reflection at the interior wall can occur at the outer wall-air interface. This will result, however, in additional attenuation and, therefore, can be neglected. In order to derive the multipath signal model, we assume perfect knowledge of the front wall, i.e., location, thickness, and dielectric constant, as well as the location of the interior walls.

Consider the antenna-target geometry illustrated in Fig. 7(a), where the front wall has been ignored for simplicity. The $p$’th target is located at $xp=(xp,zp)$, and the interior wall is parallel to the $z$-axis and located at $x=xw$. Multipath propagation consists of the forward propagation from the $n$’th antenna to the target along the path $P\u2032\u2032$ and the return from the target via a reflection at the interior wall along the path $P\u2032$. Assuming specular reflection at the wall interface, we observe from Fig. 7(a) that reflecting the return path about the interior wall yields an alternative antenna-target geometry. We obtain a virtual target located at $x\u2032p=(2xw\u2212xp,zp)$, and the delay associated with path $P\u2032$ is the same as that of the path $P\u02dc\u2032$ from the virtual target to the antenna. This correspondence simplifies the calculation of the one-way propagation delay $\tau p,n(P\u2032)$ associated with path $P\u2032$. It is noted that this principle can be used for multipath via any interior wall.

From the position of the virtual target of an assumed target location, we can calculate the propagation delay along path $P\u2032$ as follows. Under the assumption of free space propagation, the delay can be simply calculated as the Euclidean distance from the virtual target to the receiver divided by the propagation speed of the wave. In the TWRI scenario, however, the wave has to pass through the front wall on its way from the virtual target to the receiver. As the front wall parameters are assumed to be known, the delay can be readily calculated from geometric considerations using Snell’s law.^{28}

The effect of wall ringing on the target image can be delineated through Fig. 7(b), which depicts the wall and the incident, reflected, and refracted waves. The distance between the target and the array element in cross-range direction, $\Delta x$, can be expressed as

^{28}. From Snell’s law,

^{101}

Having described the two principal multipath mechanisms in TWRI, namely the interior wall and wall ringing types of multipath, we are now in a position to develop a multipath model for the received signal. We assume that the front wall returns have been suppressed and the measured data contains only the target returns. The case with the wall returns present in the measurements is discussed in ^{100}.

Each path $P$ from the transmitter to a target and back to receiver can be divided into two parts, $P\u2032$ and $P\u2032\u2032,$ where $P\u2032\u2032$ denotes the partial path from the transmitter to the scattering target and $P\u2032$ is the return path back to the receiver. For each target-transceiver combination, there exist a number of partial paths due to the interior wall and wall ringing multipath phenomena. Let $Pi1\u2032$, $i1=0,1,\u2026,R1\u22121$, and $Pi2\u2032\u2032$, $i2=0,1,\u2026,R2\u22121$, denote the feasible partial paths. Any combination of $Pi1\u2032$ and $Pi2\u2032\u2032$ results in a round-trip path $Pi$, $i=0,1,\u2026,R\u22121$. We can establish a function that maps the index $i$ of the round-trip path to a pair of indices of the partial paths, $i\u21a6(i1,i2)$. Hence we can determine the maximum number $R\u2264R1R2$ of possible paths for each target-transceiver pair. Note that, in practice, $R\u226aR1R2$, as some round-trip paths may be equal due to symmetry while some others could be strongly attenuated and thereby can be neglected. We follow the convention that $P0$ refers to the direct round-trip path.

The round-trip delay of the signal along path $Pi$, consisting of the partial parts $Pi1\u2032$ and $Pi2\u2032\u2032$,can be calculated as

Without loss of generality, we assume the same number of propagation paths for each target. The unavailability of a path for a particular target is reflected by a corresponding path amplitude of zero. The received signal at the $n$’th antenna due to the $m$’th frequency can, therefore, be expressed as

The matrix-vector form for the received signal under multipath propagation is given by

Within the CS framework, we aim at undoing the ghosts, i.e., inverting the multipath measurement model and achieving a reconstruction, wherein only the true targets remain.

In practice, any prior knowledge about the exact relationship between the various subimages $r(i)$ of the sparse scene is either limited or nonexistent. However, we know with certainty that the sub-images $r(0),r(1),\u2026r(R\u22121)$ describe the same underlying scene. That is, the support of the $R$ images is the same, or at least approximately the same. The common structure property of the sparse scene suggests the application of a group sparse reconstruction.

All unknown vectors in Eq. (35) can be stacked to form a tall vector of length $NxNzR$

We proceed to reconstruct the images $r\u2192$ from $y\u0306$ under measurement model in Eq. (38). It has been shown that a group sparse reconstruction can be obtained by a mixed $l1\u2212l2$ norm regularization.^{102}^{–}^{105} Thus we solve

^{106}The convex optimization problem in Eq. (39) can be solved using SparSA,

^{102}YALL group,

^{103}or other available schemes.

^{105}

^{,}

^{107}

Once a solution $r\u2192^$ is obtained, the subimages can be noncoherently combined to form an overall image with an improved signal-to-noise-and-clutter ratio (SCNR), with the elements of the composite image $r^GS$ defined as

An experiment was conducted in a semi-controlled environment at the Radar Imaging Lab, Villanova University. A single aluminum pipe (61 cm long, 7.6 cm diameter) was placed upright on a 1.2-m-high foam pedestal at 3.67 m downrange and 0.31 m cross-range, as shown in Fig. 8. A 77-element uniform linear monostatic array with an inter-element spacing of 1.9 cm was used for imaging. The origin of the coordinate system is chosen to be at the center of the array. The 0.2-m-thick concrete front wall was located parallel to the array at 2.44 m downrange. The left sidewall was at a cross-range of $\u22121.83\u2009\u2009m$, whereas the back wall was at 6.37 m downrange (see Fig. 8). Also there was a protruding corner on the right at 3.4 m cross-range and 4.57 m downrange. A stepped-frequency signal, consisting of 801 equally spaced frequency steps covering the 1 to 3 GHz band was employed. The left and right side walls were covered with RF absorbing material, but the protruding right corner and the back wall were left uncovered.

We consider background-subtracted data to focus only on target multipath. Figure 9(a) depicts the backprojection image using all available data. Apparently, only the multipath ghosts due to the back wall, and the protruding corner in the back right are visible. Hence we only consider these two multipath propagation cases for the group sparse CS scheme. We use 25% of the array elements and 50% of the frequencies. The corresponding CS reconstruction is shown in Fig. 9(b). The multipath ghosts have been clearly suppressed.