1. Introduction

Recent advances in microwave imaging (MWI) have made it an optimal imaging modality for many sensing applications. Being of low cost, providing higher safety due to the use of non-ionizing electromagnetic radiation, and the ability of penetration into the dielectric bodies, make MWI a promising tool for applications ranging from non-destructive testing of materials [1–4], security screening [5–9], and through-wall imaging [10–13] to medical diagnostics [14–20].

Although significant progress has been made on experimental implementations and hardware developments, effective processing techniques are still required in order to reconstruct the object of interest (OI) from the measurement data with the maximum possible image quality. Various approaches have been adopted to address this issue. Development of imaging algorithms with better performance is pursued in [14,17,21–24]. The use of prior information about the structure or the electrical properties of the OI is also proposed to enhance the quality of image reconstruction [25–28]. Furthermore, some pre/post-processing techniques are suggested to improve image quality by solving specific problems. For instance, a preprocessing filter is utilized in [29] to compensate for path-dependent attenuation and phase effects, or, a complex-valued blind de-convolution algorithm is employed in [30] as a post-processing technique to remove the blurring effect of non-point-wise antenna aperture.

Another more generally applicable approach to enhance the image quality is to reduce the side-lobe level (SLL) of the imaging system’s point spread function (PSF). PSF is known as an important mean of characterizing an imaging system by evaluating its capability to image a point scatterer. Thus, any method resulting in a more desired PSF, i.e. a sharp and focused one around the point target with minimized SLL, can enhance the image quality.

Since microwave imaging requires a wide dynamic range to capture the weak response of the target (e.g. a tumor), the high side-lobes of the imaging array can potentially leads to ambiguities and wrong interpretations of the OI. High side-lobe responses from strong scatterers or constructive accumulation of side-lobe responses from a few scatterers will cause appearance of false targets, i.e. artifacts. Furthermore, these strong side-lobe responses can easily obscure nearby weak targets and make their detection impossible. Another problem caused by high SLL is target break-up [31]. In imaging of a target distributed over space, strong side-lobe responses from the target break up the image of the target to some isolated pieces. Therefore, microwave imagery often requires side-lobe reduction.

In the literature, SLL reduction methods are studied mostly considering MWI systems with limited-view arrays. A post-processing technique using the prior knowledge of the point spread function of the sparse array is proposed in [31] to iteratively reduce the side-lobes from the initially formed imagery. In [32], a new array topology is proposed for a planar multistatic array that can reduce the SLL by 2-5 dB in comparison to other known topologies.

However, while in many MWI systems for civil, industrial and particularly medical applications a full-view array that surrounds the OI is commonly utilized, SLL suppression for full-view arrays is rarely discussed in the literature. On the other hand, the SLL reduction methods presented for limited-view arrays are not applicable to full-view arrays. In this paper, a novel SLL reduction technique is proposed for a multistatic full-view array in order to further enhance its imaging performance. We demonstrate that samples of low spatial frequencies collected by the array are responsible for a great part of array side-lobe response in space domain. The idea is to weaken the samples of zero and low spatial frequencies by means of a tapering function in order to enhance the image quality. Numerical simulations confirm about 5 dB SLL reduction. The experimental results also validate the efficacy of the proposed method in improving the image quality and avoiding appearance of artifacts and wrong interpretations of the target under imaging

This paper is organized as follows. Section II describes the imaging problem and discusses on three different implementations of full-view arrays, namely single-frequency multistatic, single-frequency monostatic and wideband monostatic in terms of K-space representations. PSF analysis of these three full-view arrays is also performed with numerical simulations. This detailed discussion illustrates the superiority of single-frequency multistatic imaging over the two others. Hence, the single-frequency multistatic full-view array is the focus of this study. Section III presents the idea behind our proposed SLL reduction method. To apply the proposed technique, a modified back-projection algorithm for image reconstruction is also suggested and then, numerical simulations compare the PSFs with and without SLL reduction. Experimental validation is reported in section IV to further verify the functionality of the proposed SLL reduction method. Finally, section V concludes the paper.

2. Imaging background

Designing any microwave imaging system deals with three important aspects, namely limited-view vs. full-view array geometry, monostatic vs. multistatic configuration, and single-frequency vs. wideband operation. The choices on these design aspects can be efficiently made considering: the specific application that the imaging system is to be designed for, the imaging requirements such as resolution and image quality, and also the acceptable cost and complexity of system implementation.

In medical applications such as breast cancer detection, brain, bone and heart imaging and also in various civil and industrial applications such as evaluation of concrete structures, wood and trees, a typical MWI array geometry is a circular (or generally any encircling) array that surrounds the target [33]. Thus, we focus on a 2D full-view array of transmitters and receivers’ antennas. Figure 1 schematically depicts this scenario.

 

Fig. 1. Schematic of imaging problem with a circular array and an object of area S

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A single-frequency monostatic array seems to be the simplest choice in terms of system cost and complexity. However, it is not known as a practical imaging system due to its weak imaging properties. The trade-off between imaging properties and system cost and complexity requires more efficient choices. Following this section, we briefly explain that why a wideband monostatic or a single-frequency multistatic array is generally considered as a better choice, by means of K-space representation and PSF analysis.

K-space or spatial frequency domain representation is utilized to provide an insight into the imaging properties of different imaging systems. Mathematically, the 2D spatial Fourier transform of any imaging target with an area S gives the target’s K-space spectrum which is an unlimited function over spatial frequency domain. However, it is physically not possible for the imaging system to sample the whole K-space spectrum of the target. “K-space support” demonstrates the limited sampling area of an imaging system over the target’s K-space spectrum [34].

Point spread function of an imaging system is known as an effective tool of evaluating the performance of the imaging system. PSF can be expressed as a 2D image of a point target or 1D cross-sections of the image, describing the system capability to image a point scatterer [35,36]. As the imaging system takes sample on a limited part of the target’s K-space, the point target would be reconstructed as a sinc-shaped spread function rather than a Dirac delta, being known as the PSF of the imaging system. The main features of PSF are the main-lobe width and the side-lobe level. The former determines the resolution of the imaging system and the latter plays the major role in image quality.

2.1 K-space representation

In [37] we thoroughly discussed the K-space representation of different imaging systems and the underlying theories. Here, we provide a brief discussion to compare the imaging properties of three imaging systems as follows.

2.1.1 Single-frequency multistatic full-view array

For a full-view array (Fig. 1) with multistatic configuration and single-frequency operation we know that

By defining ${k_x} = {k_{xt}} + {k_{xr}}$ and ${k_y} = {k_{yt}} + {k_{yr}}$, we have

Equation (3) represents the K-space sampling area of a single-frequency multistatic full-view array which is the locus of circles in K-space with a constant radius of k and variable centers of $({{k_{xt}},{k_{yt}}} )$. The center points of $({{k_{xt}},{k_{yt}}} )$ locate on a circle around the origin with the same radius k according to Eq. (1). Figure 2(a) shows this K-space support for a point target at the origin. Each transmitter corresponds a point on the central circle, shown in green, and then, receivers collect samples on a circle with that center, shown in blue.

 

Fig. 2. K-space support of (a) a multistatic full-view array with single-frequency operation, (b) and (c) a monostatic full-view array with single-frequency operation and wideband operation respectively [37].

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Note that adding a bandwidth to a multistatic full-view array imaging system, as long as ${k^{max}}$ in the wideband operation is not higher than the value of k in the single frequency operation, does not notably affect the K-space support, and consequently the imaging properties.

2.1.2 Single-frequency monostatic full-view array

To drive the K-space representation of a full-view array with monostatic configuration and single-frequency operation, we write ${k_x} = 2{k_{xt}}$ and ${k_y} = 2{k_{yt}}$ as the same antenna transmits and receives the signal. Hence, Eq. (3) turns to

This is the locus of a circle centered at the origin with a constant radius of $2k$, as shown in Fig. 2(b). For a point target at the origin, each transmitter corresponds a point on the central circle, shown in green, and then, the same antenna takes a sample on the outer circle in the K-space as receiver.

2.1.3 Wideband monostatic full-view array

By adding a bandwidth to the above-mentioned monostatic system, a set of circles with different radius corresponding to different values of k can be sampled on the K-space. Figure 2(c) depicts the K-space support of a wideband monostatic full-view array with ${k_{min}} \le k \le {k_{max}}$.

Note that the three imaging systems have no superiority over each other in terms of resolution. For a fixed maximum frequency of operation, the three imaging systems provide the same resolution as their K-space support limit equals to $4{k^{max}}$ [37]. However, Fig. 2 clearly explains the reason of superiority of a single-frequency multistatic or a wideband monostatic imaging system over a single-frequency monostatic one. The significant difference of Figs. 2(a) and (c) with Fig. 2(b) is sampling on an area of K-space instead of a ring. In [37] it is shown that how sampling on an area of K-space makes the PSF softer; i.e., lowers the SLL and broadens the main-lobe of the PSF. In summary, an area of K-space can be expressed as the sum of a few rings with different radii. Therefore, the inverse Fourier transform of this area can be also expressed as a summation of the PSFs of rings. As these PSFs are sinc-shaped functions focused at the point target position [38], the maxima add constructively, while the side-lobes tend to add destructively. Consequently, the final PSF is softer than the PSF of a single ring of K-space.

2.2 PSF analysis

To complete our discussion on comparison of three above-mentioned imaging systems, numerical simulations are conducted in MATLAB to produce the scattered data of a point target received by the antenna array of the imaging systems. A point target is located at the origin inside the full-view array (Fig. 1) with an arbitrary diameter (20 cm here), and the propagation medium is assumed to be free space. The single-frequency multistatic and single-frequency monostatic systems are simulated at 10 GHz, and the wideband monostatic system is simulated with different bandwidth, all with maximum frequency of 10 GHz. To evaluate the PSF of imaging systems, back-projection (BP) reconstruction method is applied on the received data [6]. For an object $O({x,y} )$ to be imaged, back-projection can be expressed as

Figure 3 shows the 1D PSFs for three imaging systems. The high SLL of a single-frequency monostatic imaging system (Fig. 3(b)) makes it an undesirable tool for imaging. That is why imaging systems with monostatic configuration usually rely on an ultra-wide frequency band to have an acceptable performance.

 

Fig. 3. 1D PSF of the (a) single-frequency multistatic at 10 GHz, (b) single-frequency monostatic at 10 GHz, and (c) wideband monostatic with three different bandwidth (BW) arrays with a point target. The target is located at (0,0) and PSF is calculated a at $y = 0$.

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According to Fig. 3(c), among three different simulated bandwidths, the wideband monostatic system has its lowest SLL with the ultra-wide bandwidth of 1-10 GHz which is about -22 dB. Nevertheless, the single-frequency multistatic system (Fig. 3(a)) also provides a reasonably comparable imaging properties, as reported in Table 1.

Table 1. Imaging properties of full-view arrays

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Interestingly, it is concluded that a full-view imaging system can work at a single frequency instead of demanding an ultra-wide bandwidth, and still presenting quite the same performance, only by collecting data in a multistatic configuration. Single-frequency operation is preferable as it avoids many of the problems of UWB operation such as cost and complexity of hardware (specially the RF components), interference with other electronic devices, difficulties in calibration and also challenges of imaging dispersive objects [39,40]. Furthermore, frequency-domain measurement of scattering (S) parameters in a single-frequency imaging system takes less time in comparison to an UWB operation while measurement time is critical in many real-time imaging applications [41]. Thus, in this study we focus on the single-frequency multistatic system for full-view microwave imaging platforms in order to enhance its performance. In the following section, we present our technique to further improve the PSF of single-frequency multistatic imaging with about 5 dB SLL reduction.

3. SLL reduction method

As explained in the previous section, the K-space representation of a single-frequency multistatic full-view array is a circular area with the radius of $2k$ (Fig. 2(a)) that can be expressed as the sum of rings with different radii centered at the origin. From the Fourier transform properties we know that the radius of a ring in the spatial frequency domain and the main-lobe width of its corresponding PSF in space domain have an inverse relation to each other, i.e. rings with smaller radii result in PSFs with wider main-lobe. In the extreme case of zero spatial frequency, the main-lobe width of its PSF approaches infinity.

Figure 4 illustrates the PSFs of a point target located at the center of a single-frequency multistatic full-view array, reconstructed by samples of three different rings selected from its total K-space samples (Fig. 2(a)). The numerical simulation setting is the same as before. As the radius of K-space ring decreases from $2k$ to k and to$\; 0.2k$, the PSF main-lobe gets wider.

 

Fig. 4. 1D PSFs of the full-view multistatic system at 10 GHz arrays with a point target at (0,0), for three K-space rings with different radii (ρ=2k,k,0.2k). PSF is calculated at y=0.

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The total PSF of the system shown in Fig. 3(a) is the summation over all the PSFs corresponding to different rings. As mentioned before, the main-lobes add constructively, while the side-lobes tend to add destructively and thus, the total SLL is lower than the SLL of each ring’s PSF - the total SLL in Fig. 3(a) is -15.7 dB while the SLL of each single ring’s PSF in Fig. 4 is -7.9 dB. However, it can be concluded from our discussion on Fig. 4 that rings with small radii, i.e. low spatial frequency samples, seem to be the major determinant of the total SLL. Their wide main-lobes cause to add a significant amount to the total SLL. Thus, it gives a clue to reduce the total SLL by omitting or reducing the effect of low spatial frequency samples in image reconstruction. Indeed, instead of giving equal weights to all samples of K-space representation shown in Fig. 3(a), samples of low spatial frequencies located in a small disk with the center of the origin are weakened. This results in losing data and leading to lower Signal-to-Noise Ratio (SNR). However, note that the effect of this discarded part of samples on SNR is not significant, as it is a small part comparing to the whole collected data, whereas it makes a considerable impact on reducing SLL as discussed.

In practice, the antenna array collects data in space domain. But to apply the above-mentioned idea we need to gain access to the low spatial frequency samples of the target K-space spectrum. Although it is possible to transform the collected data to the K-space spectrum of the target with the Fourier imaging methods [34,38], it gives rise to particular difficulties. The Fourier transformation of the signal received by the array and the corresponding dispersion relations vary with the shape and geometry of the antenna array and hence, must be re-extracted for different arbitrary geometries. Dispersion relations for simple geometries such as linear/planar limited-view arrays are available in the literature [6,34,42], whereas extracting these relations for more complicated geometries with arbitrary antenna array shape might be nor available in the literature neither easy to express in closed-form. Furthermore, the transformed data is irregular in K-space domain and must be mapped to a regular grid by interpolation before applying IFT. The interpolation process is a critical step in Fourier imaging methods that significantly affects the imaging quality of the system and requires high efficiency and accuracy [6,43]. We introduce a simple method in order to weaken the low spatial frequencies of the data without dealing with these difficulties.

Theoretically, for the zero spatial frequency of the K-space representation in Fig. 3(a) we have$\; {k_x} = {k_y} = 0$. According to the definitions of ${k_x}$ and$\; {k_y}$, it requires that ${k_{xt}} ={-} {k_{xr}}$ and $ {k_{yt}} ={-} {k_{yr}}$, i.e. ${\varphi _t} = {\varphi _r} \pm \pi $. In other words, for each pixel of the imaging domain S inside the multistatic full-view array, any pair of transmitter and receiver antennas that are in front of each other with respect to the selected pixel, takes a sample on the zero spatial frequency. Figure 5 pictorially explains this notion. For the selected pixel in Fig. 5, and considering one of the transmitters, the depicted in front receiver takes a sample on the zero spatial frequency of the target K-space. Clearly, other receivers in close vicinity of the in front receiver also collect the low spatial frequency samples. Therefore, when reconstructing this pixel of the imaging domain, it suffices to weaken the signal received by these in-front receiver antennas to reduce the SLL of its PSF.

 

Fig. 5. The in front receiver for a selected pixel and transmitter.

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To apply the proposed SLL reduction method, we suggest a modified back-projection (BP) method in which a tapering function is utilized to reduce the effect of low spatial frequencies. Let us define ${\rho _{lf}}$ as the radius of the K-space area to be considered as low frequencies, i.e. samples of this area need to be weakened, leading to the minimum possible SLL. ${\rho _{max}}$ is also defined as the maximum radius of K-space sampled area that is generally$\; 2k$. In this modified BP algorithm, for reconstruction of each pixel $({x,y} )$ of the target, first the radial location of K-space sample collected by each pair of transmitter and receiver is calculated as follows

Thus, Eq. (5) can be rewritten as

It is worth mentioning that determining$\; {\rho _{lf}}$ is straightforward. It just needs to be determined once for an imaging system using any simple search algorithm, and then can be used for any target under imaging with that system. For a step tapering function it is about 0.1 to 0.3 of$\; {\rho _{max}}$, depending on the properties of the imaging system such as array geometry and topology and for a smooth function e.g. a linear one it is approximately twice the ${\rho _{lf}}$ of the step function.

 Figure 6 compares the total PSF of a full-view multistatic array working at 10 GHz reconstructed with and without SLL reduction. The simulation setting is as before and ${\rho _{lf}} = 0.3\; {\rho _{max}}$ is used in the modified BP. The SLL is reduced from -15.7 dB to -20.2 dB. It is notable that SLL is reduced without widening the main-lobe width or degrading the resolution.

 

Fig. 6. 1D PSFs of the full-view multistatic system at 10 GHz arrays with a point target at (0,0), before and after SLL reduction. PSF is calculated at y=0.

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4. Experimental validation

The proposed SLL reduction method is also verified using an experimental dataset provided by institute Fresnel, Marseille, France [44]. The experimental setup consists of a large anechoic chamber with $14.50\; m$ long, $6.50\; m$ wide and $6.50\; m$ high, and a set of three positioner to adjust the antennas or targets positions. Two targets under imaging are presented in Fig. 7(a) and (b), including two identical dielectric circular cylinders with radius $15mm$ and ${\varepsilon _r} = 3 \pm 0.3$ and a U-shaped metallic target respectively. The transmitting source is fixed at ${0^\circ }$ and by rotation of targets, a full-view scan of targets is enabled for the transmitter. For each position of target’s rotation, the receiver antenna moves around the target to collect data. However, due to the technical restrictions of the system design, the receiver antenna cannot be closer to the source more than$\; {60^\circ }$, i.e. it scans from ${60^\circ }$ to ${300^\circ }$. Thus, according to Eq. (6), one can conclude that k-space representation of this system covers an area of radius$\; {\rho _{max}} = k\sqrt 3 $. The measured multistatic data of only one single frequency is used to reconstruct both targets; the two cylinders at 8GHz and the U-shaped target at 16GHz.

 

Fig. 7. Cross sectional dimensions of targets under measurement at the institute Fresnel, Marseille, France [44], (a) two identical dielectric cylinders and (b) metallic target with ‘U-shaped’ cross section.

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To apply our method, first we need to find the optimum value of$\; {\rho _{lf}}$, corresponding to the minimum SLL, for this system. Since we do not have access to the measured PSF of this system i.e. measured data for a sufficiently small target, we simulate the imaging system in MATLAB as described above and reconstruct the PSF of this system. Figure 8 shows the PSF of the Fresnel imaging system before and after SLL reduction at 8 and 16 GHz. For both frequencies, a linear tapering function with an optimum $\; {\rho _{lf}} = 0.4\; {\rho _{max}}$ leads to a $8\; dB$ SLL reduction from $- 16.6\; dB$ to$\; - 24.5\; dB$. Note that frequency of operation affects the resolution or main-lobe width of PSF, however it has no effect on the SLL of PSF, the optimum value of$\; {\rho _{lf}}/\; {\rho _{max}}\; $ and the amount of reduction in SLL.

 

Fig. 8. Simulated 1D PSFs of the Fresnel system at (a) 8 GHz and (b) 16 GHz with a point target at (0,0), before and after SLL reduction. PSF is calculated at y=0.

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Now we can reconstruct the images of the targets using Eq. (8) and the estimated value of the optimum$\; \; {\rho _{lf}}$. To better evaluate the performance of the proposed method and quantify the image quality, we utilize a commonly used metric namely signal-to-mean ratio (SMR). SMR compares the mean intensity of the region of interest (ROI) of the target (${I_{ROI}}$) and the mean intensity of background clutter (${I_c}$) outside the ROI

ROI can be determined according to the physical boundaries of the targets (Fig. 7(a) and (b)), and extended to where the backscattered target intensity drops by one half or -3dB, i.e. the full-width half maximum (FWHM) region. SMR is usually reported in dB.

Figures 9 and 10 compare reconstruction of the targets with and without SLL reduction. A threshold of 0.5 (-3 dB) on intensity is also applied on reconstructed images to better expose the effect of SLL reduction on images. Clearly, much improved image quality is achieved after SLL reduction in modified BP algorithm, especially for the more complicated U-shaped target. Notice that in complicated targets, high SLL creates non-negligible artifacts that can possibly be considered as target and this leads to incorrect imaging.

 

Fig. 9. Reconstruction of measured data at 8GHz for two identical circular cylinders, (a) and (b) reconstructed images before and after SLL reduction, (c) and (d) reconstructed images before and after SLL reduction with threshold of 0.5.

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Fig. 10. Reconstruction of measured data at 16GHz for metallic U-shaped target, (a) and (b) reconstructed images before and after SLL reduction, (c) and (d) reconstructed images before and after SLL reduction with threshold of 0.5.

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Table 2 reports the SMR achieved in image reconstruction of Fresnel targets with and without SLL reduction. It is worth to mention that in the modified BP algorithm used for SLL reduction, image reconstruction of Fresnel targets with different values of$\; {\rho _{lf}}$ shows that the reported SMRs in Table II are the best amounts of possible achievable SMR. It confirms the validity of the estimated value obtained for the optimum$\; {\rho _{lf}}$. According to Table II, considerable enhancement to the image quality is witnessed by applying the proposed SLL reduction method.

Table 2. Quantitative Comparison of Imaging Performance in terms of SMR^a

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5. Conclusion

In this paper, first we emphasized on both the possibility and the superiority of full-view microwave imaging utilizing single-frequency multistatic array instead of using single-frequency or wideband monostatic arrays. Single-frequency operation is reassuringly advantageous by avoiding many of the problems of UWB operation such as cost and complexity of hardware, difficulties in calibration and challenges of imaging dispersive objects. On the other hand, multistatic configuration with full-view geometry allows for sampling on a wide area of spatial frequency domain and hence, guarantees a desired PSF with focused main-lobe and low side-lobes.

Then, an innovative idea for SLL reduction is proposed to further enhance the image quality of a full-view multistatic array. The idea comes from the understanding of the effect of low spatial frequency samples collected by the array on SLL. These samples lead to very wide main-lobes in space domain according to inverse Fourier transform and thus, add a considerable amount to the side-lobe response of the array. By weakening the cooperation of these samples in image reconstruction, much quality improvement can be achieved for images. We have suggested a modified BP algorithm to apply the proposed SLL reduction method. Numerical simulations have shown a SLL reduction of about 5dB. The proposed method has been also experimentally verified using Fresnel measurement dataset for two different targets. The conventional BP algorithm and the modified BP equipped with the proposed SLL reduction method have been applied to the measured multistatic data at a single frequency. The results demonstrate a considerable improvement in image quality, particularly for the complicated U-shaped target.

Noise, dispersion, clutter and losses are also important factors to be considered when assessing the performance of a single-frequency multistatic imaging system and the proposed SLL reduction method. Although the effectiveness of the proposed method on enhancing the image quality has been proved with experimental results, these further considerations will be discussed in future contributions.