Image reconstruction algorithm based on frequency-wavenumber decoupling for three-dimensional MIMO-SAR imaging

Hang Gao; Hang Gao; Hang Gao; Chao Li; Chao Li; Chao Li; Shiyou Wu; Shiyou Wu; Shen Zheng; Shen Zheng; Hongwei Li; Hongwei Li; Hongwei Li; Guangyou Fang; Guangyou Fang; Guangyou Fang

doi:10.1364/OE.382857

1. Introduction

Active millimeter wave (MMW) and terahertz (THz) imaging systems are receiving more and more attention recently because of their higher resolution with millimeter-scale compared with microwave and lower radio frequency waves, and better penetration for nonmetal materials such as clothes or wrappers compared with optical and infrared radiation. Due to such characteristics, they are found to be promising for plenty of applications [1–14], such as security and safety screening [2–10], non-destructive testing (NDT) [11–13] and ground penetrating radar [14].

By using a 2-D sampling aperture with wideband illumination, 3-D images with resolution approaching the diffraction limit can be obtained. MMW imaging systems in the early stage were usually developed based on single frequency holographic technique with the help of a stationary source and a scanned receiver system that employed optical (film-based) reconstruction [2]. This technique was dramatically improved by Collins et al. by utilizing a scanned 1-D monostatic holographic array and digital reconstruction [4]. A significant extension of the holographic imaging system from single-frequency operation to wide-band was then carried out in [5], which allows fully focused 3-D images and has been widely used in the security screening. Compressive holography was then extended to MMW imaging to minimize the data acquisition scan [15–16]. A system-level simulator was also developed in [17], which can support various monostatic MMW/THz SAR imaging systems studies in a convenient way.

As the ever growing of the security and NDT requirements, active imaging systems with higher data acquisition speed and lower cost become more and more necessary, especially for the real-time applications in high throughput security screening scenarios. An efficient way with the concept of MIMO array was introduced in MMW short-range imaging to achieve higher dynamic range with significantly reduced transmitters and receivers and higher data acquisition speed [18]. With the MIMO architecture, the total elements of the transceivers can be significantly reduced as compared with the conventional array topology [19]. The Quick Personnel Security Scanner (QPS) system developed by R&S Corporation is one of the famous representatives in this field. This kind of system employs a 2-D MIMO array without any scanning mechanism, so that backscattered waves can be obtained instantaneously. However, even the MIMO array is adopted, the number of transmitters and receivers of such system for a humanoid sized field of view (FOV) is still large because a 2-D real aperture is needed. A MIMO array based side-looking imaging scheme was studied in [20], which has a fast imaging speed and low cost with only 1-D linear array but lacks 3-D imaging capabilities.

In order to achieve fully focused 3-D images with high data acquisition speed and lower cost in the meanwhile, a compromise between the 1-D monostatic array and the 2-D MIMO array was made by combining a linear MIMO array with a rapid scanning quasi-optics in its perpendicular direction to cover a 2D FOV in THz band [21,22]. However, the fixed imaging distance is still expected in this imaging topology, since the real aperture focusing is used in the quasi-optical scanning dimension.

For the purpose to achieve a fully adaptive focused 3D images with high array efficiency, synthetic aperture thesis in the perpendicular direction of the linear MIMO array which is called MIMO-SAR imaging scheme is studied in this paper. More importantly, a novel high precision and efficient 3D imaging algorithm is proposed. As the schematic shown in Fig. 1, the 2-D FOV in x-y plane was covered by the scan of the 1-D MIMO array (x-direction) along the other dimension (y-direction). And the reconstruction of the image can be implemented by the combination of MIMO array signal processing and aperture synthetic in the corresponding dimensions. For the 3-D image reconstruction with the complex bistatic MIMO-SAR data, the methods based on the convolutions with the phase-corrected or time-compensated signals or coherent summations can be implemented, such as the conventional BP algorithm [23] and the Modified Kirchhoff algorithm [24]. However, such kind of methods are usually time consuming with heavily limited applications, especially for real-time imaging.

Fig. 1. Imaging system geometry of the MIMO-SAR configuration.

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So far, several works have been reported on the fast reconstruction methods for the 3-D MIMO-SAR imaging. A 3-D MIMO near-field RMA with high efficiency was proposed by Zhuge based on the 2-D MIMO array in [25] which reduced the computational complexity greatly as compared with the conventional BP algorithm. It was then extended to the MIMO-SAR regime by utilizing the spherical wave decomposition by Zhu in [26]. These wavenumber domain algorithms based on Stolt interpolation are usually efficient in dealing with the 3-D image reconstruction problems. However, the precision of the result largely relies on the interpolation approaches such as nearest neighbor interpolation, linear interpolation and cubic B-spline interpolation, and more accurate result usually means higher computational cost. More recently, two Modified Kirchhoff based frequency domain algorithms were also proposed in [27]. Similar to Zhu’s method, the first algorithm is also based on the Stolt interpolation. The second algorithm decomposes the MIMO imaging problem into several single-input-multiple-output (SIMO) problems, which allows a more flexible distribution of the transmitters and a better imaging accuracy, but on the cost of time-consuming problems.

In this paper, a frequency-wavenumber decoupling algorithm with high-efficiency and high-precise for 3-D MIMO-SAR imaging is proposed. By multi-dimensional Fourier transforming and Method of Stationary Phase (MSP), analytical expression of the object function in the frequency-wavenumber domain was derived. By further expanding the range Fourier transform factor to its Taylor series form, the range compression can be realized by a simple fast Fourier transform (FFT) without multi-dimensional interpolation. Then, a decoupling factor was multiplied to compensate the cross-range and range coupling. Finally, 2-D IFFT is carried out after rearrangement in the MIMO spatial frequency to get a fully focused 3-D image. Simulation and experimental results demonstrated that the algorithm can obtain the same high-precision image as BP, and has the same high efficiency as RMA while avoiding cumbersome multi-dimensional interpolation. The proposed algorithm was demonstrated by the proof-of-principle experiments in 0.1 THz.

This paper is organized as follows. The derivation of the proposed algorithm is presented in Section 2. Several implementation issues including sampling criteria, spatial resolution and computational complexity are discussed in Section 3. In Section 4 and 5, numerical simulation and experimental results are given in 0.1 THz band respectively, through which the effectiveness and the efficiency of the proposed algorithm are demonstrated. Finally, a conclusion is drawn in Section 6.

2. Formulation

As given in Fig. 1, the locations of the transmitters and receivers are represented as $({{x_t},y,{z_\textrm{0}}} )$ and $({{x_r},y,{z_\textrm{0}}} )$, respectively. The radiation aperture of linear MIMO array is assumed locating at a plane ${z_\textrm{0}}$ and scanned along the y axis forming a planar aperture to get the echoed field $s({{x_t},{x_r},y,{z_0},k} )$ where k is the usual wavenumber in free space. Assume that the imaging domain is denoted as D in the following content. The general target has a reflectivity distribution function $O({x^{\prime},y^{\prime},z^{\prime}} )$. Under such MIMO-SAR imaging regime, the corresponding scattered wave field can be expressed as

(1)$$s({{x_t},{x_r},y,{z_0},k} )= \mathop{\int\!\!\!\int\!\!\!\int}\limits_{D} {{G_t} \cdot O({x^{\prime},y^{\prime},z^{\prime}} )} \cdot {G_r}{\kern 1pt} {\kern 1pt} dx^{\prime}dy^{\prime}dz^{\prime}$$

where ${G_t}$ and ${G_r}$ are the Green’s functions of transmitting and receiving elements in the free-space, as follows:

(2)$${G_t} = \frac{{\exp ({ - jk{R_t}} )}}{{4\pi {R_t}}}$$

(3)$${G_r} = \frac{{\exp ({ - jk{R_r}} )}}{{4\pi {R_r}}}$$

${R_t}$ and ${R_r}$ are the distances from the target located at $(x^{\prime},y^{\prime},z^{\prime})$ to transmitters and receivers, respectively, as shown in the following:

(4)$${R_t} = \sqrt {{{({{x_t} - x^{\prime}} )}^2} + {{({y - y^{\prime}} )}^2} + {{({{z_\textrm{0}} - z^{\prime}} )}^2}}$$

(5)$${R_r} = \sqrt {{{({{x_r} - x^{\prime}} )}^2} + {{({y - y^{\prime}} )}^2} + {{({{z_\textrm{0}} - z^{\prime}} )}^2}}$$

By ignoring the amplitude attenuation term that has little effect on image reconstruction, the scattered wave field can be expressed as

(6)$$s({{x_t},{x_r},y,{z_0},k} )= \mathop{\int\!\!\!\int\!\!\!\int}\limits_{D} {O({x^{\prime},y^{\prime},z^{\prime}} )} {\kern 1pt} \cdot \exp ({ - jk({{R_t} + {R_r}} )} )dx^{\prime}dy^{\prime}dz^{\prime}$$

Based on (6), we further transform the scattered wave field into the spatial Fourier domain ${k_{xt}}$, ${k_{xr}}$ and ${k_y}$ over the three spatial dimensions ${x_t}$, ${x_r}$ and y, respectively, as follows:

(7)$$\tilde{s}({{k_{xt}},{k_{xr}},{k_y},{z_0},k} )= \mathop{\int\!\!\!\int\!\!\!\int}\limits_{D} {O({x^{\prime},y^{\prime},z^{\prime}} )} \cdot E({{k_{xt}},{k_{xr}},{k_y},k} )\cdot dx^{\prime}dy^{\prime}dz^{\prime}$$

where

(8)$$E({{k_{xt}},{k_{xr}},{k_y},k} )= \int {E({{k_{xt}},y,k} )\cdot E({{k_{xr}},y,k} )} \cdot \exp ({ - j{k_y}y} )dy$$

(8) can be computed as a Fourier transform along y dimension on the product of the two 1-D Fourier transforms over ${x_t}$ and ${x_r}$, respectively, as follows:

(9)$$E({{k_{xt}},y,k} )= \int {\exp ({ - jk{R_t}} )} \cdot \exp ({ - j{k_{xt}}{x_t}} )d{x_t}$$

(10)$$E({{k_{xr}},y,k} )= \int {\exp ({ - jk{R_r}} )} \cdot \exp ({ - j{k_{xr}}{x_r}} )d{x_r}$$

The integrations in (9) and (10) was further derived by applying the MSP as

(11)$$E({{k_{xt}},y,k} )= \exp \left( { - j{k_{xt}}x^{\prime} - j\sqrt {{k^2} - {k_{xt}}^2} \cdot \sqrt {{{({y - y^{\prime}} )}^2} + {{({{z_0} - z^{\prime}} )}^2}} } \right)$$

(12)$$E({{k_{xr}},y,k} )= \exp \left( { - j{k_{xr}}x^{\prime} - j\sqrt {{k^2} - {k_{xr}}^2} \cdot \sqrt {{{({y - y^{\prime}} )}^2} + {{({{z_0} - z^{\prime}} )}^2}} } \right)$$

By applying (11) and (12), (8) can be expressed as

(13)$$\begin{array}{l} E({{k_{xt}},{k_{xr}},{k_y},k} )= \exp ({ - j({{k_{xt}} + {k_{xr}}} )x^{\prime}} )\cdot \\ \int {\exp \left( { - j\left( {\sqrt {{k^2} - {k_{xt}}^2} } \right.} \right.} + \left. {\sqrt {{k^2} - {k_{xr}}^2} } \right)\left. { \cdot \sqrt {{{({y - y^{\prime}} )}^2} + {{({{z_0} - z^{\prime}} )}^2}} } \right) \cdot \exp ({ - j{k_y}y} )dy \end{array}$$

Similarly, after utilizing the MSP, (13) can be further expressed as

(14)$$\begin{array}{l} E({{k_{xt}},{k_{xr}},{k_y},k} )= \exp ({ - j({{k_{xt}} + {k_{xr}}} )x^{\prime} - j{k_y}y^{\prime}} )\\ \cdot \exp \left( { - j\left( {\sqrt {{{\left( {\sqrt {{k^2} - {k_{xt}}^2} + \sqrt {{k^2} - {k_{xr}}^2} } \right)}^2} - {k_y}^2} } \right)({z^{\prime} - {z_0}} )} \right) \end{array}$$

Then, substituting (14) into (7), we have

(15)$$\begin{array}{l} \tilde{s}({{k_{xt}},{k_{xr}},{k_y},{z_\textrm{0}},k} )= \mathop{\int\!\!\!\int\!\!\!\int}\limits_{D} {O({x^{\prime},y^{\prime},z^{\prime}} )\exp ({ - j({{k_{xt}} + {k_{xr}}} )x^{\prime} - j{k_y}y^{\prime}} )} \\ \cdot {\kern 1pt} \exp \left( { - j\left( {\sqrt {{{\left( {\sqrt {{k^2} - {k_{xt}}^2} + \sqrt {{k^2} - {k_{xr}}^2} } \right)}^2} - {k_y}^2} } \right)({z^{\prime} - {z_0}} )} \right)dx^{\prime}dy^{\prime}dz^{\prime} \end{array}$$

To express more compactly, 3-D dispersion relations are defined as

(16)$$\left\{ \begin{array}{l} {{\hat{k}}_x} = {k_{xt}} + {k_{xr}}\\ {{\hat{k}}_y}\textrm{ = }{k_y}\\ {{\hat{k}}_z} = \sqrt {{{\left( {\sqrt {{k^2} - {k_{xt}}^2} + \sqrt {{k^2} - {k_{xr}}^2} } \right)}^2} - {k_y}^2} \end{array} \right.$$

Thus, (15) can be further expressed as

(17)$$\tilde{s}^{\prime}({{k_{xt}},{k_{xr}},{k_y},k} )= F({{{\hat{k}}_x},{{\hat{k}}_y},{{\hat{k}}_z}} )$$

where

(18)$$F({{{\hat{k}}_x},{{\hat{k}}_y},{{\hat{k}}_z}} )\textrm{ = }\mathop{\int\!\!\!\int\!\!\!\int}\limits_{D} {O({x^{\prime},y^{\prime},z^{\prime}} )\exp ({ - j{{\hat{k}}_x}x^{\prime} - j{{\hat{k}}_y}y^{\prime} - j{{\hat{k}}_z}z^{\prime}} )} dx^{\prime}dy^{\prime}dz^{\prime}$$

is the spatial frequency form of $O({x^{\prime},y^{\prime},z^{\prime}} )$.

(19)$$\tilde{s}^{\prime}({{k_{xt}},{k_{xr}},{k_y},k} )\textrm{ = }\tilde{s}({{k_{xt}},{k_{xr}},{k_y},{z_\textrm{0}},k} )\cdot {\kern 1pt} \exp ({ - j{{\hat{k}}_z}{z_0}} )$$

is the spatial Fourier domain scattered wave field after matched filtering.

By applying 1-D inverse Fourier transform along the ${\hat{k}_z}$ dimension to (17) on both sides, we have

(20)$$F({{{\hat{k}}_x},{{\hat{k}}_y},\;{z^{\prime}}} )= \int {\tilde{s}^{\prime}({{k_{xt}},{k_{xr}},{k_y},k} )} \cdot \exp ({j{{\hat{k}}_z}z^{\prime}} )d{\hat{k}_z}$$

According to (16), $d{\hat{k}_z}/dk$ can be derived as

(21)$${a_1} = \frac{{2k + k\sqrt {\frac{{{k^2} - {k^2}_{xt}}}{{{k^2} - {k^2}_{xr}}}} + k\sqrt {\frac{{{k^2} - {k^2}_{xr}}}{{{k^2} - {k^2}_{xt}}}} }}{{\sqrt {{{\left( {\sqrt {{k^2} - {k_{xt}}^2} + \sqrt {{k^2} - {k_{xr}}^2} } \right)}^2} - {k_y}^2} }}$$

Thus, (20) can be further expressed as

(22)$$F({{{\hat{k}}_x},{{\hat{k}}_y},\;{z^{\prime}}} )= \int {\tilde{s}^{\prime\prime}({{k_{xt}},{k_{xr}},{k_y},k} )} \cdot \exp ({j{{\hat{k}}_z}z^{\prime}} )dk$$

where

(23)$$\tilde{s}^{\prime\prime}({{k_{xt}},{k_{xr}},{k_y},k} )= \tilde{s}^{\prime}({{k_{xt}},{k_{xr}},{k_y},k} )\cdot {a_1}$$

Since ${\hat{k}_z}$ represents the cross-range and range coupling relationship as shown in (16), Taylor series expanding here is used to obtain a more compact form with high precision.

We first expand ${\hat{k}_z}$ into a more concrete form as

(24)$${\hat{k}_z} = \sqrt {2{k^2} - {k_{xt}}^2 - {k_{xr}}^2 + 2{k^2}\sqrt {1 - \frac{{{k_{xt}}^2 + {k_{xr}}^2}}{{{k^2}}} + \frac{{{k_{xt}}^2{k_{xr}}^2}}{{{k^4}}}} - {k_y}^2}$$

The Taylor series expanding of $\sqrt {1 - \frac{{{k_{xt}}^2 + {k_{xr}}^2}}{{{k^2}}} + \frac{{{k_{xt}}^2{k_{xr}}^2}}{{{k^4}}}}$ are shown as

(25)$$\begin{array}{c} \sqrt {1 - \frac{{{k_{xt}}^2 + {k_{xr}}^2}}{{{k^2}}} + \frac{{{k_{xt}}^2{k_{xr}}^2}}{{{k^4}}}} = 1 - \frac{{{k_{xt}}^2 + {k_{xr}}^2}}{{2{k^2}}} - \frac{{{{({{k_{xt}}^2 - {k_{xr}}^2} )}^2}}}{{8{k^4}}} + \frac{{({{k_{xt}}^2 + {k_{xr}}^2} ){k_{xt}}^2{k_{xr}}^2}}{{4{k^6}}} - \frac{{{k_{xt}}^4{k_{xr}}^4}}{{8{k^8}}}\\ + o\left[ {{{\left( { - \frac{{{k_{xt}}^2 + {k_{xr}}^2}}{{{k^2}}} + \frac{{{k_{xt}}^2{k_{xr}}^2}}{{{k^4}}}} \right)}^3}} \right] \end{array}$$

For a common near-field imaging system, ${k^2}$ is usually much larger than ${k_{xt}}^2$, ${k_{xr}}^2$ and ${k_y}^2$[28]. In order to suit higher sampling rate systems and achieve a compact form, we keep the first three terms of (25). Then the second-order Taylor expansion of ${\hat{k}_z}$ can be similarly derived as

(26)$${\hat{k}_z} \approx 2k - {k_1}$$

where

(27)$${k_1} = \frac{{2({{k_{xt}}^2 + {k_{xr}}^2} )- {k^2}_y}}{{4{k_c}}} + \frac{{8({{k_{xt}}^4 + {k_{xr}}^4} )+ 4({{k_{xt}}^2 + {k_{xr}}^2} )\cdot {k^2}_y + {k^4}_y}}{{64{k_c}^3}}$$

where ${k_c}$ is the center wavenumber.

The effects of the approximation from (16) to (26) will be analyzed in detail combined with the actual array in the simulation section.

Thus, by applying (26), (22) can be further expressed as

(28)$$F({{{\hat{k}}_x},{{\hat{k}}_y},\;{z^{\prime}}} )= \int {\tilde{s}^{\prime\prime}({{k_{xt}},{k_{xr}},{k_y},k} )} \cdot \exp ({j2kz^{\prime}} )dk \cdot \exp ({ - j{k_1}z^{\prime}} )$$

From (28), we can find that $F({{{\hat{k}}_x},{{\hat{k}}_y},\;{z^{\prime}}} )$ can be derived from $\tilde{s}^{\prime\prime}({{k_{xt}},{k_{xr}},{k_y},{z_\textrm{0}},k} )$ with only one inverse Fourier transform along k to realize range compression without multi-dimensional interpolation. Then, the decoupling factor $\exp ({ - j{k_1}z^{\prime}} )$ was multiplied to compensate for the cross-range and range coupling in the spatial frequency domain.

Finally, 2-D Fourier transform is carried out as follows to obtain the final reconstructed image.

(29)$$O({x^{\prime},y^{\prime},z^{\prime}} )= \int\!\!\!\int {F({{{\hat{k}}_x},{{\hat{k}}_y},\;{z^{\prime}}} )\cdot } \exp ({j{{\hat{k}}_x}x^{\prime}} )\exp ({j{{\hat{k}}_y}y^{\prime}} )d{\hat{k}_x}d{\hat{k}_y}$$

The main steps can be summarized in Table 1.

Table 1. Overview of the proposed algorithm

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3. Implementation Issues

3.1 Sampling criteria and spatial resolution

The sampling criteria of the MIMO-SAR regime should both satisfy the Nyquist sampling criteria in the MIMO array direction [25] as well as the Nyquist sampling criteria in the synthetic aperture scan direction [5] to avoid aliasing

(30)$$\left\{ \begin{array}{c} {d_{xt}} \le {\lambda_{\min }}\; \; \frac{{\sqrt {{{({{L_{xt}} + {D_x}} )}^2}/4 + {z_0}^2} }}{{{L_{xt}} + {D_x}}}\; \; \\ {d_{xr}} \le {\lambda_{\min }}\; \; \frac{{\sqrt {{{({{L_{xr}} + {D_x}} )}^2}/4 + {z_0}^2} }}{{{L_{xr}} + {D_x}}}\\ {d_y} \le \frac{{{\lambda_{\min }}}}{{\min (4\sin ({{\theta_{HBW}}/2} ),2{D_y}{z_0})}} \end{array} \right.$$

where ${d_{xt}}$, ${d_{xr}}$ and ${d_y}$ are the intervals of the transmitters, receivers and the synthetic aperture scan path, and ${L_{xt}}$, ${L_{xr}}$ and ${L_y}$ are the length of transmitting array, receiving array and the synthetic aperture length respectively. ${\lambda _{\min }}$ is the smallest wavelength which corresponding to the maximum frequency within the data bandwidth, ${D_x}$ and ${D_y}$ are width for the FOV in x and y directions, ${z_0}$ is the minimum distance between the array plane and the FOV and ${\theta _{HBW}}$ is the half beam width (HBW) of antennas.

The spatial resolution of this algorithm is shown in (31) which is a combined result of previous work [9,29,30] due to the combination feature of this regime.

(31)$$\left\{ \begin{array}{l} {\delta_x} = \textrm{0}\textrm{.886}\frac{{{\lambda_c}z}}{{{L_{tx}} + {L_{rx}}}}\\ {\delta_y} = \textrm{0}\textrm{.443}\frac{{{\lambda_c}z}}{{{L_y}}}\\ {\delta_z} = \textrm{0}\textrm{.44}\frac{c}{B} \end{array} \right.$$

3.2 Computational cost

Table 2 illustrated the detailed computational cost of the proposed algorithm. In practical processing the simulation or experimental data, we have empirically found that the third step in Table 1 has actually no effect on final imaging result, thus this step can be ignored during reconstruction. Therefore, the computational cost of the third step is not included here. To measure more compactly, the final computational cost is evaluated by the number of floating-point operation (FLOP), which can be either a real multiplication or a real addition [26]. The final computation cost of the proposed algorithm is shown as

(32)$$\begin{array}{l} {C_{proposed}} = 5{N_f}{N_{xt}}{N_{xr}}{N_y}{\log _2}({N_f}{N_{xt}}{N_{xr}}{N_y})\textrm{ + 4}{N_{xt}}{N_{xr}}{N_y}({\textrm{1}\textrm{.5}{N_f}\textrm{ + 2}{N_z}} )\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + 5{N_z}{N_x}{N_y}{\log_2}({N_x}{N_y})\textrm{ - 2}{N_z}{N_x}{N_y}_{}^{}FLOP \end{array}$$

where ${N_f}$ is the number of frequency steps of the SFCW signal, ${N_x}$ is the number of monostatic spatial frequency in the x-direction, which has an expression as ${N_x} = {N_{xt}} + {N_{xr}} - 1$, ${N_y}$ is the number of scanning points in the y-direction, and ${N_z}$ is the number of the resulted planes in the ROI.

Table 2. Computational cost of the proposed algorithm

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For comparison, the computational cost of an improved BP algorithm with FFT integrated in range domain for acceleration, as given in [27], is shown as

(33)$$\begin{aligned}{C_{BP}} &= 5{N_{xt}}{N_{xr}}{N_y}{N_f}{\log _2}{N_f}\\ &+ 12{N_{xt}}{N_{xr}}{N_y}{N_x}^\prime {N_y}^\prime {N_z}{^\prime _{}}FLOP \end{aligned}$$

where ${N^{\prime}_x}$, ${N^{\prime}_y}$, ${N^{\prime}_z}$ are the reconstructed points of the ROI.

The total computation cost of the RMA proposed in [26] is

(34)$$\begin{array}{l} {C_{RMA}} = 5{N_{xt}}{N_{xr}}{N_y}[{\textrm{N}_f}{\log_2}({N_{xt}}{N_{xr}}{N_y}) + 1.2{\textrm{N}_f} + 1.6{N_z}]\\ \begin{array}{ccc} {}&{}&{} \end{array} + 5{N_x}{N_z}{N_y}{[\log({N_x}{N_z}{N_y}) - 0.4]_{}}\textrm{FLOP} \end{array}$$

To further simplify the comparison, we suppose that ${N_y}$, ${N_f}$, ${N_z}$, ${N^{\prime}_x}$, ${N^{\prime}_y}$, ${N^{\prime}_z}$ are in the same order with a given number $N$. For ${N_{xt}}$, ${N_{xr}}$,since the number of EPC is determined by ${N_{xt}}{N_{xr}}$, it is more reasonable to assume that ${N_{xt}}$, ${N_{xr}}$ are in the order of $\sqrt N$ respectively. Therefore, we obtained more intuitive expressions of (32), (33) and (34), which are $O({N^3}\log N)$, $O({N^\textrm{5}})$ and $O({N^3}\log N)$ respectively. Compared with the improved BP, the efficiency of the proposed algorithm is greatly improved and has the same order as the RMA. The efficiency of the proposed algorithm will be further validated by simulation and experiments in the following sections.

4. Simulation

4.1 Simulation setup

The 1-D MIMO array structure shown in Fig. 2 was used in the simulation below, with 6 transmitters and 39 receivers. The total length of the MIMO array is 0.3 m, the spacing of the transmitting antenna is 2.5 mm, and the spacing of the receiving antenna is 7.5 mm, and the scanning length of the synthetic aperture direction is 0.3 m with a spatial interval of 5 mm, which is in accordance with the Nyquist sampling criteria shown in (30). Since the main point of this paper is on the improvement of image reconstruction methods, we didn’t adopt optimization method for array design here. The signal frequency is centered at 0.1 THz with a bandwidth of 15.75 GHz and is sampled at 31 points in steps of 525 MHz.

Fig. 2. MIMO-SAR set up used in the numerical simulation, with the 1-D MIMO array in front of nine distributed point scatters at a distance of 1 m.

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The target shown in Fig. 2 consists of 9-point scatters distributed in the region of 0.15 m × 0.15 m × 0.15 m. The center point of the target is located at 1 m away from the planar aperture and the azimuth position is at the center of the planar array.

In order to simulate the backscattered data as realistically as possible in a convenient way, the simulation is based on the field distribution acquired by the commercial electromagnetic computation software FEKO of the horn we used in the following experiment part. The horn of the transmitters and receivers are the same, as shown in Fig. 3. With the simulated field distribution of each transmitter and receiver, the MIMO-SAR echoed data of each transceiver pair can be obtained by multiplying the target distribution function under the first-order Born approximation.

Fig. 3. Horn antenna for simulation and experiment with detailed dimensions.

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4.2 Approximation analysis

Before the reconstruction, we analyzed the effect of the approximation from (16) to (27) for the current simulation setup. The relative error between them are as follows:

(35)$$RE({{k_{xt}},{k_{xr}},{k_y},k} )= \left|{\frac{{{{\hat{k}}_z} - ({2k - {k_1}} )}}{{{{\hat{k}}_z}}}} \right|$$

It is easy to find that the minimum relative error $RE({{k_{xt}},{k_{xr}},{k_y},k} )$ occurs at ${k_{xt}}\textrm{ = 0}$, ${k_{xr}}\textrm{ = 0}$ and ${k_y}\textrm{ = 0}$ according to (16) and (27). To show more intuitively, $RE({{k_{xt}},{k_{xr}},{k_y},k} )$ along ${k_y}$ at ${k_{xt}}\textrm{ = 0}$ and ${k_{xr}}\textrm{ = 0}$ for different k is shown in Fig. 4(a). Similarly, Fig. 4(b) shows the maximum $RE({{k_{xt}},{k_{xr}},{k_y},k} )$ along ${k_y}$ for different k which appears at $\left|k_{x r}\right|=k_{x r_{m}}$ and $\left|k_{x v}\right|=k_{x v_{m}}$ where $k_{x t_{m}}$ and $k_{x r_{m}}$ represent the maximum value of ${k_{xt}}$ and ${k_{xr}}$ respectively. Through observing Fig. 4(b), we can find that the maximum relative error is less than 0.02 which can be totally neglected. Moreover, for general short-range imaging systems, the wavenumber energy is mainly concentrated in areas with low spatial frequency which means that it is usually close to area around ${k_{xt}}\textrm{ = 0}$, ${k_{xr}}\textrm{ = 0}$ and ${k_y}\textrm{ = 0}$, in that way, the relative error could be even smaller. Therefore, the approximation from (16) to (27) is reasonable, and the algorithm processing can be simplified to only FFT and multiplication operation while avoiding cumbersome multidimensional interpolation. In the simulation and experimental sections below, comparing the imaging results with the golden-rule BP algorithm also proves the rationality of this approximation.

Fig. 4. Approximation error analysis from (16) to (26). (a) Minimum relative error along ${k_{y}}$ at ${k_{xt}}$ = 0 and ${k_{xr}}$ = 0 for different k. (b) Maximum relative error along ${k_{y}}$ at $\left|k_{x t}\right|=k_{x t_{m}}$ and $\left|k_{x r}\right|=k_{x r_{m}}$ for different k.

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4.3 Simulation results

In this part, the superiority of the proposed algorithm is verified by comparing the computational complexity of image quality and consumption with the improved BP algorithm and RMA for a more comprehensive comparison.

Figure 5 shows the reconstructed images of these three algorithms for the 9-point targets as setup in Fig. 2. The 3-D images have been reconstructed within a 0.2 m ×0.2 m ×0.2 m cube. The 3-D images and its front/side views projecting onto the x-y and z-y planes of the 9-point targets are all shown. Through the reconstructed images in Fig. 5, we can find that results of the three algorithms are all fully focused. In order to illustrate the difference more clearly, Fig. 6 shows the PSFs of the three algorithms for a single point target at (0,0,1). The well consistency between them can be clearly observed, however, at the position of about 0.025 m we can find a sidelobe of the MIMO-SAR RMA is about 3-dB higher than the MIMO-PSM, and 6-dB higher than BP.

Fig. 5. Imaging results of the simulation shown in Fig. 2 with the MIMO-SAR configuration. (a) 3-D image by BP. (b) 3-D image by MIMO-SAR RMA. (c) 3-D image by the proposed algorithm. (d) Front view of the image obtained by BP.(e) Front view of the image obtained by MIMO-SAR RMA. (f) Front view of the image obtained by the proposed algorithm. (g) Side view by BP. (h) Side view by MIMO-SAR RMA. (i) Side view by the proposed algorithm. The front/side views are obtained by maximum projection of the 3-D image on to the x-y and z-y planes, respectively.

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Fig. 6. PSFs of different algorithms.

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Table 3 lists the amount of FLOP of these three algorithms as well as the actual time executed on a Lenovo PC with 2 Intel core i3-8100 CPUs. These algorithms are all implemented in the MATLAB language. From Table 3, we can find that the calculation amount of the proposed algorithm is the same as the RMA, which is much smaller than the calculation amount of BP. However, as shown in the third column in Table 3, since the code is implemented in MATLAB, which is relatively friendly to matrix calculations, the proposed algorithm consisting of only multiplication and FFT processing is much more efficient than RMA with multi-dimensional interpolation.

Table 3. Comparison on computational times of different algorithm of the simulation

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5. Experiments

5.1 Experiment setup

To illustrate the performance of the algorithm in practice, as shown in Fig. 7, a prototype imaging system in 0.1 THz band was developed. The system is based on a microwave vector network analyzer (VNA) that combines two transmitters to multiply the signal into the 0.1 THz band, and a receiver to convert the echo signal into an intermediate frequency (IF) for the VNA to extract the amplitude and phase information of the echo data. Three identical horns with 3 dB beam width of 19° as shown in Fig. 3 are used as the transmit and receive antennas. The 1-D MIMO array can be equivalently implemented by scanning the two transmitters and one receiver in the horizontal direction (x direction) by means of three motor drive platforms. The movement of the entire equivalent MIMO array in the vertical direction (y direction) is controlled by a fourth motor driven platform to cover the FOV in the SAR direction.

Fig. 7. Photograph of the 2-D-scanner of the experiment setup.

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In the experimental setup, the same virtual MIMO array as the simulation shown in Fig. 2 is equivalently implemented by controlling two transmitters and receivers moving in a special manner. The experimental system has an entire frequency band of 75 to 110 GHz and a sampling step size of 175 MHz. To avoid data redundancy, we chose 15.75 GHz of experimental data, with a center frequency of 0.1 THz and a sampling step size of 525 MHz, which is also consistent with the simulation settings.

5.2 Experimental results

To compare more comparatively and be consistent with the simulation, the proposed algorithm was verified through the comparison with the improved BP algorithm and the RMA in image quality and consumed imaging computational complexity.

Two experimental results are shown here. In each experimental scenario, the target is made up of three weapons that are prohibited from being carried on the flight. Three weapons are placed in different ranges as shown in Fig. 8(a) and Fig. 9(a). The center of the targets is located at 1 m away from the sampling aperture.

Fig. 8. Adaptive focusing experiment 1 for three targets distributed in different ranges. (a) Photo of front view of three targets located in different ranges. (b)Adaptive focusing result reconstructed by BP algorithm. (c) Adaptive focusing result reconstructed RMA algorithm. (d) Adaptive focusing result reconstructed by the proposed algorithm.

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Fig. 9. Adaptive focusing experiment 2 for three targets distributed in different ranges. (a) Photo of front view of three targets located in different ranges. (b)Adaptive focusing result reconstructed by BP algorithm. (c) Adaptive focusing result reconstructed RMA algorithm. (d) Adaptive focusing result reconstructed by the proposed algorithm.

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As can be seen in the first experiment shown in Fig. 8, the projection of the fully focused 3-D imaging results of these three algorithms on the XY plane can be seen in Fig. 8(b)-(d). However, a better imaging quality is obtained by the proposed algorithm than the RMA through comparing the knife part, where a strong interference can be found in the result of the RMA labeled by arrows as shown in Fig. 8(c). In the second experiment shown in Fig. 9, it was again demonstrated by observing the trigger portion of the gun that the proposed method can achieve accurate imaging results consistent with BP algorithm, and more accurate imaging results than the RMA. Cautiously, we think this is mainly due to the non-interpolation feature of the proposed algorithm which avoids the numerical error resulted from the Stolt interpolation, which is also consistent with the simulation results in Fig. 6.

The computational complexity of the three algorithms of these two experiments are shown in Table 4. The computational times of the 3-D reconstruction of the RMA and the proposed algorithm are far less than that of the improved BP algorithm. And the computational complexity of the proposed algorithm is the same as the RMA. However, since the actual code implementation is done in MATLAB, which is relatively friendly to matrix calculations, the proposed algorithm consisting of multiplication and FFT processing is much more efficient than RMA with multi-dimensional interpolation. The calculations of these three algorithms in the experimental scene are also consistent with the simulation part.

Table 4. Comparison on computational times of different algorithm of the experiments

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6. Conclusion

In this paper, an imaging reconstruction algorithm with high efficiency and high precise based on a frequency-wavenumber decoupling was proposed for 3-D MIMO-SAR imaging. By multidimensional Fourier transforming and MSP, analytical expression of the object function in the wavenumber domain is derived. By further expanding the range Fourier transform factor to its Taylor series form, the range compression can be realized by a simple FFT without multi-dimensional interpolation. Then, a decoupling factor was multiplied to compensate the cross-range and range coupling. Finally, 2-D IFFT is carried out after rearrangement in the MIMO spatial frequency to get a fully focused 3-D image. With the proposed algorithm, the total reconstruction process can be conveniently implemented by only several FFTs and two multiplications with little approximation. Analysis on the approximation is also given which almost has no effect on the final accurate image reconstruction. Besides, comprehensively comparison with the most representative algorithms is also presented. By effectively employing the fast Fourier transform in the algorithm, a great reduction on the time consuming was achieved as compared with the improved BP algorithm, without loss of the imaging qualities. Compared with the RMA, the proposed algorithm avoids cumbersome multi-dimensional interpolation, and has higher precision with the same high efficiency, even faster in the MATLAB implementation. A bistatic prototype imager was designed for the proof-of-principle experiments in 0.1 THz band. The imaging results of different targets were given to demonstrate the theoretical results and effectiveness of the proposed algorithm for MIMO-SAR imaging.

Funding

National Key Research and Development Program of China (2017YFA0701004, 2018YFF01013004); National Natural Science Foundation of China (61671432, 61731020, 61988102); Key Project of Equipment Preresearch Fund (6140413010401); Key Program of Scientific and Technological Innovation from Chinese Academy of Sciences(KGFZD-135-18-029); Science and Technology Key Project of Guangdong Province, China (2019B010157001).

Disclosures

The authors declare no conflicts of interest.

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Operation	Real multiplications	Real additions
Spatial 3-D FFT	$2 N_{f} N_{x t} N_{x r} N_{y} \log_{2} N_{x t} N_{x r} N_{y}$	$3 N_{f} N_{x t} N_{x r} N_{y} \log_{2} N_{x t} N_{x r} N_{y}$
Apply $\exp [- j {\hat{k}}_{z} z_{0}]$	$4 N_{f} N_{x t} N_{x r} N_{y}$	$2 N_{f} N_{x t} N_{x r} N_{y}$
IFFT along $k$	$2 N_{x t} N_{x r} N_{y} N_{f} \log_{2} N_{f}$	$3 N_{x t} N_{x r} N_{y} N_{f} \log_{2} N_{f}$
Apply $\exp (- j k_{1} z^{'})$	$4 N_{z} N_{x t} N_{x r} N_{y}$	$2 N_{z} N_{x t} N_{x r} N_{y}$
Reorganize	0	$2 N_{y} N_{z} (N_{x t} N_{x r} - N_{x})$
Spatial 2-D IFFT	$2 N_{z} N_{x} N_{y} \log_{2} N_{x} N_{y}$	$3 N_{z} N_{x} N_{y} \log_{2} N_{x} N_{y}$
Total	$\begin{array}{l} 2 N_{f} N_{x t} N_{x r} N_{y} \log_{2} N_{f} N_{x t} N_{x r} N_{y} \\ + 2 N_{z} N_{x} N_{y} \log_{2} N_{x} N_{y} \\ + 4 N_{x t} N_{x r} N_{y} (N_{f} + N_{z}) \end{array}$	$\begin{array}{l} 3 N_{f} N_{x t} N_{x r} N_{y} \log_{2} N_{f} N_{x t} N_{x r} N_{y} \\ + 3 N_{z} N_{x} N_{y} \log_{2} N_{x} N_{y} \\ + 2 N_{y} N_{x t} N_{x r} (2 N_{z} + N_{f}) - 2 N_{y} N_{z} N_{x} \end{array}$

Operation	Real multiplications	Real additions
Spatial 3-D FFT	$2 N_{f} N_{x t} N_{x r} N_{y} \log_{2} N_{x t} N_{x r} N_{y}$	$3 N_{f} N_{x t} N_{x r} N_{y} \log_{2} N_{x t} N_{x r} N_{y}$
Apply $\exp [- j {\hat{k}}_{z} z_{0}]$	$4 N_{f} N_{x t} N_{x r} N_{y}$	$2 N_{f} N_{x t} N_{x r} N_{y}$
IFFT along $k$	$2 N_{x t} N_{x r} N_{y} N_{f} \log_{2} N_{f}$	$3 N_{x t} N_{x r} N_{y} N_{f} \log_{2} N_{f}$
Apply $\exp (- j k_{1} z^{'})$	$4 N_{z} N_{x t} N_{x r} N_{y}$	$2 N_{z} N_{x t} N_{x r} N_{y}$
Reorganize	0	$2 N_{y} N_{z} (N_{x t} N_{x r} - N_{x})$
Spatial 2-D IFFT	$2 N_{z} N_{x} N_{y} \log_{2} N_{x} N_{y}$	$3 N_{z} N_{x} N_{y} \log_{2} N_{x} N_{y}$
Total	$\begin{array}{l} 2 N_{f} N_{x t} N_{x r} N_{y} \log_{2} N_{f} N_{x t} N_{x r} N_{y} \\ + 2 N_{z} N_{x} N_{y} \log_{2} N_{x} N_{y} \\ + 4 N_{x t} N_{x r} N_{y} (N_{f} + N_{z}) \end{array}$	$\begin{array}{l} 3 N_{f} N_{x t} N_{x r} N_{y} \log_{2} N_{f} N_{x t} N_{x r} N_{y} \\ + 3 N_{z} N_{x} N_{y} \log_{2} N_{x} N_{y} \\ + 2 N_{y} N_{x t} N_{x r} (2 N_{z} + N_{f}) - 2 N_{y} N_{z} N_{x} \end{array}$

Image reconstruction algorithm based on frequency-wavenumber decoupling for three-dimensional MIMO-SAR imaging

Abstract

1. Introduction

2. Formulation

3. Implementation Issues

3.1 Sampling criteria and spatial resolution

3.2 Computational cost

4. Simulation

4.1 Simulation setup

4.2 Approximation analysis

4.3 Simulation results

5. Experiments

5.1 Experiment setup

5.2 Experimental results

6. Conclusion

Funding

Disclosures

References

Cited By

Figures (9)

Tables (4)

Equations (35)

Optics Express

Algorithms	FLOP ( $10^{10}$ )	Actual execution time (10 s)
BP	431.8	5085.8
RMA	0.3	3.6
Proposed	0.3	0.3

Algorithms	FLOP ( $10^{10}$ )	Actual execution time (10 s)
BP	431.8	5098.2
RMA	0.3	3.7
Proposed	0.3	0.3

Algorithms	FLOP ( $10^{10}$ )	Actual execution time (10 s)
BP	431.8	5085.8
RMA	0.3	3.6
Proposed	0.3	0.3

Algorithms	FLOP ( $10^{10}$ )	Actual execution time (10 s)
BP	431.8	5098.2
RMA	0.3	3.7
Proposed	0.3	0.3