Abstract

Terahertz coded-aperture imaging relies on sequential coding and sampling. Its capturing speed is limited by the switching rate of the coded-aperture antenna; therefore, it is not suitable for capturing moving targets. In this paper, we focus on imaging for moving targets using the terahertz coded-aperture imaging technique. An incoherent detection array is adopted to replace sequential sampling by spatial sampling, therefore improving the capturing speed significantly. Then, a signal processing method based on phase retrieval and image enhancement is proposed to reconstruct targets with improved quality. Numerical experimental results verify the effectiveness of the method.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Terahertz coded-aperture imaging (TCAI) is a promising imaging technology with broad application prospects in autonomous driving, security checks, and terminal guidance [1]. TCAI uses a coded-aperture antenna to modulate the phase or amplitude information of the terahertz wave in the transmission process to generate a spatiotemporal independent radiation field [24]. Through mathematical modeling of the imaging system, the imaging equation can be obtained, and then the computational imaging algorithm [5,6] is used to reconstruct the target. TCAI has the advantages of both terahertz imaging [7] and optical coded-aperture imaging [8,9] technologies. For example, TCAI has a stronger penetrating ability than optical imaging and higher resolution than microwave imaging, and it is capable of forward looking and staring imaging without relying on any relative movement.

However, moving target reconstruction remains a great challenge for TCAI based on coherent detection [10,11]. When receiving echo signals with a coherent detector, the TCAI system needs a series of downconverting devices with high frequency stability, a local oscillator, and certain output power [12]. Therefore, the system structure can be quite complex, large, and expensive, making it difficult to miniaturize and integrate. Furthermore, high resolution imaging relies on solving a massive matrix equation with the support of a large number of echo sampling data. TCAI based on single channel coherent detection needs sequential sampling repeatedly, which is limited by the switching rate of the coded-aperture antenna. Research [13] shows that the response time of the electrically adjustable reflective terahertz phase shifter based on liquid crystal is 4 ms when working at 240 Hz. This will certainly lead to a long accumulation time in the sampling process, resulting in motion blur and even failure in imaging moving targets.

To address the issues of moving target reconstruction, the authors in [14] proposed an adaptive joint parametric estimation recovery algorithm. However, it imposes high computation requirements, as the reference signal matrix is too large and complex. Furthermore, TCAI is sensitive to phase errors; especially, it may have errors in estimating motion parameters, which may result in non-convergence in solving the imaging equation. According to the analysis above, a major obstacle for TCAI to reconstruct moving targets is the long sampling accumulation time. For TCAI, the entire imaging process can be divided into two main stages: echo detection and signal processing. The second stage can be accelerated through algorithm innovation. For example, the authors in [15] proposed an off-line end to end neural network for TCAI to reduce imaging time. The accumulation time of sequential coding and sampling takes up most of the time in the echo detection stage. Therefore, the detector can be designed as a detection array to realize spatial sampling instead of temporal sampling to save time. Thus, we can obtain enough samples with less coding and fewer sampling times. But the coherent detection array has a complex structure, and may introduce additional phase errors.

Fortunately, the development of incoherent detectors [16,17] can provide device support for TCAI to overcome problems introduced by imaging of moving targets. Incoherent detection technology can directly convert the intensity information of terahertz waves into current or voltage signals, without local oscillator signal sources or other equipment. Therefore, incoherent detection has a simple structure that can be miniaturized and large-scale integrated easily. An incoherent detection array is adopted to replace sequential sampling by spatial sampling, therefore improving the capturing speed significantly. The mathematic model for TCAI based on an incoherent detection array is a nonlinear nonconvex optimization problem. Luckily, in recent years, the development of the phase retrieval algorithm [18,19] has provided algorithmic support for the nonconvex optimization method.

In this paper, a novel imaging method is proposed for TCAI based on an incoherent detection array to reconstruct moving targets. There are two contributions in the proposed method. First, an incoherent detection array is used to sample echos for moving targets. Combining the simple structure of incoherent detection and the advantage of arrays replacing sequential sampling by spatial sampling, a single sample of an incoherent array can be used to reconstruct moving targets. Second, an enhancement algorithm based on phase recovery is proposed. Using this method, multiple initial target reconstructions are carried out to obtain better target reconstruction. The proposed method consists of two parts: initial reconstruction and image enhancement. Primarily, initial reconstruction can be obtained by incoherent detection array single sampling and phase retrieval algorithms. Afterwards multiple initial reconstructions obtained by multiple samplings are fused to improve image quality.

 figure: Fig. 1.

Fig. 1. Terahertz coded-aperture imaging system for moving targets based on incoherent detection array.

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This paper is organized as follows. We introduce the background of this technique and expound upon its principles and research state of the art. In Section 2, the mathematical model based on the incoherent detection array is verified. In Section 3, the phase retrieval algorithm and enhancement method based on registration and energy superposition are proposed. In Section 4, simulation experiments are carried out to verify the rationality of the design of incoherent detection elements. The results and analysis of image enhancement are performed. Finally, the main work and contributions of this paper are summarized.

2. SYSTEM CONFIGURATION AND MODELING

It is assumed that a TCAI system with one transmitter, one incoherent detection array, and one coded-aperture antenna at the receiving end works as the radar in this paper, as shown in Fig. 1. A brief introduction of the system is given below. The processor is used to control the signal generation module to generate the required signal and transmit the signal to the transmitter. The signal reflected by the target can be received by coded-aperture antenna. The coded-aperture antenna in this paper adopts a 1-bit digitally controlled coded-aperture antenna, which is convenient for repeatedly loading coding factors quickly. An incoherent detection array can detect only the intensity information of a modulated signal without phase information. Then the echo signal is sampled by the information acquisition module and sent to the processor to reconstruct target scatterers finally.

In the three-dimensional coordinate system, the coordinates of the transmitter antenna are ${{\textbf{r}}_t}$. Assume that the imaging plane is divided into $W$ grid cells, and the coordinates of each grid cell center are represented by ${{\textbf{r}}_w}$. The grid size represents the resolution, and the number of grid cells stands for the imaging area. The target is divided by the imaging grid cells evenly, and the strong scatterers of the target are exactly at the center of the grid cells. This paper makes a hypothesis that the moving target never exceeds the imaging plane. The coded-aperture antenna contains $M$ elements, and the coordinates of each element are represented by ${{\textbf{r}}_m}$. The incoherent detection array consists of $Q$ elements, and the coordinate of each unit is ${{\textbf{r}}_q}$. This paper uses the chirp signal as the transmitting signal, as shown in the following formula:

$${\rm St}(t) = A \cdot \exp \left[{j2\pi \left({{f_c}t + \frac{1}{2}\gamma {t^2}} \right)} \right],$$
where ${\rm St}(t)$ is the transmitting signal at time $t$, $A$ is the amplitude, ${f_c}$ is the center frequency of a terahertz wave, and $\gamma$ is the chirp rate of the signal. Through the signal transmitter, the terahertz wave reaches the target and is reflected by the target. At this time, the target signal ${\rm Str}({t_n})$ is expressed as
$${\rm Str}({t_n}) = \sum\limits_{w = 1}^W {\rm St}\left({{t_n} - \frac{{\left| {{{\textbf{r}}_t} - {{\textbf{r}}_w}} \right|}}{c}} \right) \cdot {\beta _w},$$
where $\frac{{| {{{\textbf{r}}_t} - {{\textbf{r}}_w}} |}}{c}$ is the time delay from the signal transmitter to the imaging plane, $c$ is the speed of light, and ${\beta _w}$ stands for the target corresponding to the $w$th imaging plane grid cell. Then the detection signal ${\rm Sc}({t_n},\varphi)$ arriving at the coded-aperture antenna at time ${t_n}$ is
$${\rm Sc}({t_n},{\boldsymbol \varphi}) = \sum\limits_{m = 1}^M {\rm Str}\left({t_n} - \frac{{| {{{\textbf{r}}_w} - {{\textbf{r}}_m}} |}}{c},{\boldsymbol \varphi} \right),$$
$$\begin{split}&{\rm Sc}({t_n},\varphi) = \sum\limits_{m = 1}^M \sum\limits_{w = 1}^W A \\&\cdot \exp \!\left\{\!{j\!\left[\!{2\pi \left(\!{\begin{array}{*{20}{l}}{{f_c}\left({{t_n} - \frac{{| {{{\textbf{r}}_t} - {{\textbf{r}}_w}} |}}{c} - \frac{{| {{{\textbf{r}}_w} - {{\textbf{r}}_m}} |}}{c}} \right) +}\\{\frac{1}{2}\gamma {{\left({{t_n} - \frac{{| {{{\textbf{r}}_t} - {{\textbf{r}}_w}} |}}{c} - \frac{{| {{{\textbf{r}}_w} - {{\textbf{r}}_m}} |}}{c}} \right)}^2}}\end{array}} \!\right) + {\varphi _m}\left({{t_n}} \right)} \!\right]}\! \right\} \cdot {\beta _w},\end{split}$$
where $\frac{{| {{{\textbf{r}}_w} - {{\textbf{r}}_m}} |}}{c}$ is the time delay from the imaging plane to the coded-aperture antenna, and ${\boldsymbol \varphi} = \{{{\varphi _m}({{t_n}})} \}_{m = 1}^{m = M}$ is the phase modulation factor loaded on the coded-aperture at time ${t_n}$. Hence, at time ${t_n}$, the echo signal strength received by the incoherent detection element is
$${{{\rm Sr}^{(q)}}({t_n},{\boldsymbol \varphi}) = {{\left| {{\rm Sc}\left({t_n} - \frac{{| {{{\textbf{r}}_m} - {{\textbf{r}}_q}}|}}{c},{\boldsymbol \varphi} \right)} \right|}^2}},$$
$${{\rm Sr}^{(q)}}({t_n},{\boldsymbol \varphi}) = \left| {\sum\limits_{m = 1}^M \sum\limits_{w = 1}^W A \cdot \exp \left\{{j\left[{2\pi \left({\begin{array}{*{20}{l}}{{f_c}\left({{t_n} - \frac{{| {{{\textbf{r}}_t} - {{\textbf{r}}_w}} |}}{c} - \frac{{| {{{\textbf{r}}_w} - {{\textbf{r}}_m}} |}}{c} - \frac{{| {{{\textbf{r}}_m} - {{\textbf{r}}_q}} |}}{c}} \right) +}\\{\frac{1}{2}\gamma {{\left({{t_n} - \frac{{| {{{\textbf{r}}_t} - {{\textbf{r}}_w}} |}}{c} - \frac{{| {{{\textbf{r}}_w} - {{\textbf{r}}_m}} |}}{c} - \frac{{| {{{\textbf{r}}_m} - {{\textbf{r}}_q}} |}}{c}} \right)}^2}}\end{array}} \right) + {{\varphi} _m}\left({{t_n}} \right)} \right]} \right\} \cdot {\beta _w}} \right|^2,$$
where $\frac{{| {{{\textbf{r}}_m} - {{\textbf{r}}_q}} |}}{c}$ is the time delay from the coded aperture antenna to the $q$th incoherent detection element. The signal will be interfered by noise during transmission. For the science of the experiment, noise ${\boldsymbol \omega} = [{{\omega _1},{\omega _2}, \cdots ,{\omega _Q}}]$ has been added in this paper artificially. By combining and simplifying Eq. (6), the imaging equation can be written as
$${\textbf{Sr}}({{t_n}}) = {| {{\textbf{S}}({{t_n}}) \cdot {\boldsymbol \beta}}|^2} + {\boldsymbol\omega} ,$$
where ${\textbf{Sr}}({{t_n}}) = \{{{{\rm Sr}^{(q)}}({t_n})} \}_{q = 1}^{q = Q}$ is the echo intensity detected by the entire incoherent detection array. ${\boldsymbol \beta} = \{{{\beta _w}} \}_{w = 1}^{w = W}$ is the vector representation of the target on the imaging plane. The reference signal matrix ${\textbf{S}}({{t_n}})$ can be expressed as
$$\begin{split}{\textbf{S}}\!\left({{t_n}} \right)&=\sum\limits_{m = 1}^M \sum\limits_{w = 1}^W A \\&\quad\cdot \exp \left\{{j\left[{2\pi \left({\begin{array}{*{20}{l}}{{f_c}\left({{t_n} - \frac{{| {{{\textbf{r}}_t} - {{\textbf{r}}_w}} |}}{c} - \frac{{| {{{\textbf{r}}_w} - {{\textbf{r}}_m}} |}}{c}} \right) +}\\{\frac{1}{2}\gamma {{\left({{t_n} - \frac{{| {{{\textbf{r}}_t} - {{\textbf{r}}_w}} |}}{c} - \frac{{| {{{\textbf{r}}_w} - {{\textbf{r}}_m}} |}}{c}} \right)}^2}}\end{array}} \right) + {\varphi _m}\left({{t_n}} \right)} \right]} \right\}.\end{split}$$

Equation (8) can be expressed as a matrix:

$${\boldsymbol{S}}({{t_n}}) = \left[{\begin{array}{*{20}{c}}{{S_{1,1}}({{t_n}})}&\quad{{S_{1,2}}({{t_n}})}& \quad\cdots &\quad{{S_{1,W}}({{t_n}})}\\{{S_{2,1}}({{t_n}})}&\quad{{S_{2,2}}({{t_n}})}& \quad\cdots &\quad{{S_{2,W}}({{t_n}})}\\ \vdots & \quad\vdots & \quad\ddots & \quad\vdots \\{{S_{Q,1}}({{t_n}})}&\quad{{S_{Q,2}}({{t_n}})}& \quad\cdots &\quad{{S_{Q,W}}({{t_n}})}\end{array}}\right].$$

3. ENHANCED PHASE RETRIEVAL ALGORITHM

The process of the proposed method in this paper includes two major steps as shown in Fig. 2. The first step aims at obtaining the initial reconstruction from single sampling based on the incoherent detection array with the phase retrieval algorithm. Sequently, based on the initial reconstructions of moving targets, registration and energy accumulation are performed to realize image enhancement to further improve imaging quality.

 figure: Fig. 2.

Fig. 2. Enhanced phase retrieval algorithm for moving targets based on incoherent detection array.

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A. Phase Retrieval Process

It is obvious that the imaging equation ${\textbf{Sr}}({{t_n}}) = {| {{\textbf{S}}({{t_n}}) \cdot {\boldsymbol \beta}} |^2} + {\boldsymbol \omega}$ is a non-convex quadratic program in which the traditional reconstruction algorithm of the TCAI system is no longer applicable. Fortunately, the phase retrieval algorithm is capable of solving such problems. Classic phase retrieval algorithms can be roughly divided into three types. The first one is the Gerchberg–Saxton (GS) algorithm [20], which is not suitable for the TCAI system, as it is highly dependent on the prior information of the target, and may fall into a local minimum during the solution process with non-convergence results frequently. Second, the PhaseLift algorithm [21] establishes a high-dimensional linear equation and transforms a non-convex problem into a convex problem. Although the PhaseLift algorithm can provide an accurate estimate, it imposes a high computational requirement as the size of the matrix and the dimension of $\beta$ increase. The last but crucial type is the Wirtinger flow (WF) algorithm [22], which directly uses the gradient descent method to iteratively update the estimated ${\boldsymbol {\tilde \beta}}$ and search for the global minimum. To estimate the true value accurately, the WF algorithm is often supported by a large amount of data samples. This brings us back to the problems caused by sampling accumulation during the reconstruction of moving targets. From the above analysis, it is determined that the employment of an incoherent array is capable of overcoming the issue of excessive time accumulation, which means a reasonable incoherent design scheme is necessary, so that the samples obtained by a single sampling can meet the requirements of the WF algorithm to the reconstruction target.

This paper adopts the WF with optimal stepsize (WFOS) algorithm [23], an improvement of the WF algorithm. Compared with the WF algorithm, the truncated WF algorithm [24], sparse WF algorithm [25], or sparse WF algorithm with optimal stepsize [26], the WFOS algorithm can not only reconstruct the extended target effectively, but also reduce the number of iterations and total calculation cost with the same solution accuracy and shorten the imaging time.

Using the least squares criterion, the solution to Eq. (7) can be transformed into

$$\min f ({\boldsymbol \beta} )\overset{\Delta}{=} \frac{1}{{2N}}\sum\limits_{n = 1}^N {(({\boldsymbol {S}}({t_n}){\boldsymbol \beta })^2 - {\textbf{Sr}}({t_n}))^2}.$$

The WFOS algorithm is used to perform gradient descent and optimized compensation operations on Eq. (7) to obtain the solution.

B. Image Enhancement Process

Experiments show that the initial reconstruction results failed expectations, so some measures must be implemented to enhance the imaging results. From pulse accumulation, after accumulating the energy of $n$ echo signals, the energy of the target signal is $n$ times the original theoretically, whereas the energy of the noise will only slightly increase compared to the target signal since the noise is randomly distributed, thereby increasing the signal-to-noise ratio (SNR) [27]. Motivated by the pulse accumulation, the moving target can be sampled multiple times on the basis of the incoherent detection array to obtain multiple initial reconstruction targets whose energy can be superimposed to increase the SNR and improve imaging quality.

Imaging performance is measured by entropy whose expression is

$$\begin{split}{{p_{\textit{ij}}} = f({i,j})/{K^2},}\\{{\rm{H}} = \sum\limits_{i = 0}^{255} {p_{\textit{ij}}}\log {p_{\textit{ij}}},}\end{split}$$
where $i$ represents the gray value of a certain pixel. $j$ stands for the average gray value of the pixel neighborhood. $f({i,j})$ represents the probability of the feature binary group $(i,j)$. ${p_{\textit{ij}}}$ is the gray space feature vector, which can reflect the comprehensive characteristics of the gray value at a certain pixel position and the gray distribution of the surrounding pixels, and ${\textbf{H}}$ is the entropy of the image.

The prerequisite for image enhancement is that multiple initial reconstruction targets of moving targets have been obtained. The first step of image enhancement is to filter the reconstruction targets according to an objective filter criterion: average entropy. Initial reconstruction targets that are lower than the average entropy will be marked, and otherwise discarded. The main cause is that the amount of samples is insufficient, which makes the implementation of the WFOS algorithm to reconstruct the target not good or stable, so the initial reconstruction based on the incoherent detection array with the WFOS algorithm might fail. Under the present circumstances, the quality of fused images will be reduced with the failure of the initial reconstruction. Therefore, the step of the filter not only eliminates the damage caused by accidental reconstruction failure, but also enhances the accuracy of registration.

The second step is to perform target registration. The multiple initial reconstruction targets mentioned above are the results obtained by sampling at different times. Due to the movement of the target, the reconstructed target is located at different positions on the imaging plane. In different initial reconstruction targets, the different positions of the target in the coordinate system bring challenges to image enhancement. Luckily, the phase correlation method [28,29] used in translation, rotation, and zoom image registration addresses the problem in the initial reconstruction target. We use two of the initial reconstruction targets to illustrate the application of the phase correlation method in this paper. One of the initial reconstruction targets can be presented as ${f_1}(x,y)$, and the other initial reconstruction target has the following relationship with the first initial reconstruction target: ${g_2}(x,y) = {g_1}(x - {x_0},y - {y_0})$, where $g$ is an image representation function. $x$ and $y$ are the abscissa and ordinate of the initial reconstruction target, respectively, and ${x_0}$ and ${y_0}$ are the displacement between the two initial reconstruction targets. Through the Fourier transform, the relationship between the two initial reconstruction targets in the frequency domain can be obtained:

$${G_2}(u,v) = {G_1}(u,v) \cdot {e^{- i \cdot 2\pi \cdot (u \cdot {x_0} + v \cdot {y_0})}},$$
where $u$ and $v$ correspond to $x$ and $y$ in the frequency domain. The mutual energy spectrum presentation of Eq. (12) is
$$\begin{split}H(u,v) &= \frac{{{G_1} \cdot G_2^*}}{{| {{A_1}}| \cdot | {A_2^*}|}} \\ &= {e^{- i \cdot 2\pi \cdot ({u \cdot {x_0} + v \cdot {y_0}})}},\end{split}$$
where ${A_1}$ and ${A_2}$ are the maximum values of the spectra of the two initial reconstruction targets. Through the inverse Fourier transform of the mutual energy spectrum, it can be found that ${x_0}$ and ${y_0}$ are the initial values of the impulse function:
$${\rm IFFT}(H(u,v)) = \delta (u - {x_0},v - {y_0}).$$

Then we align the targets according to the displacement ${x_0}$ and ${y_0}$.

Finally, we superimpose the energy of the initial reconstruction target in the same position after registration, so as to obtain an enhancement image to realize the purpose of improving imaging quality.

4. NUMERICAL EXPERIMENTS AND DISCUSSION

In this section, the number of array elements and the distance between array elements are analyzed for the design of the incoherent detection array first. Then, the performance of the system is verified according to a reasonable array element design scheme. Next, the imaging results and other indicators are compared under the same imaging system and different receiving device conditions. The enhanced imaging results and imaging performance analysis are shown at the end. The main parameters of the experiment are shown in Table 1. The simulation setting is ${\rm SNR} = 20 \;{\rm dB}$. In this paper, the incoherent detection array is used to reconstruct targets through a single sampling. The sampling time is very short, which can be considered as an instantaneous completion. The position error caused by the movement of the target can be ignored. At this time, the moving target can be regarded as a static target. Therefore, the velocity of the target does not have a great impact on the method proposed in this paper.

Tables Icon

Table 1. Basic Parameter Settings Used in the Simulations

 figure: Fig. 3.

Fig. 3. Influence of L on WFOS algorithm with different k sparsity targets. (a) MSE. (b) RIE.

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A. Design of Incoherent Detection Array and System Performance Verification

To investigate the validity of the proposed system, the imaging performance can be measured by the mean squared error (MSE) and relative imaging error (RIE), whose expressions are ${\rm RIE} = 20 \,\mathop {\log}\nolimits_{10} \left({\frac{{{{\Vert {\boldsymbol{\tilde \beta }- {\boldsymbol \beta}} \Vert}_2}}}{{{{\Vert {\boldsymbol \beta}\Vert}_2}}}}\right)$ and ${\rm MSE} = \frac{1}{W}\sum\limits_{i = 1}^{{W_I}} \sum\limits_{j = 1}^{{W_J}} {({\beta _{i,j}} - {\hat \beta _{i,j}})^2}$, respectively, where $W = {W_I} \times {W_J}$. The ratio between the number of samples for imaging ($L$) and the number of imaging grids ($W$) can be written as $L/W$. Also, to investigate the validity of the proposed algorithms, numerical experiments are carried out on targets with different levels of sparsity. The sparseness criteria proposed in research [30] uses a measure based on the relationship between L1 and L2 norms of a given vector: $\textit{Sparseness}({\boldsymbol \beta}) = \frac{{\sqrt W - ({\sum | {{\beta _w}} |})/\sqrt {\sum \beta _w^2}}}{{\sqrt W - 1}}$. The Sparseness should be between zero and one. When the imaging plane is full of targets, the Sparseness is zero. When there are no scattering points in the imaging plane, the Sparseness is one. In this paper, $k = W \cdot \textit{Sparseness}({\boldsymbol \beta})$. The value of $k$ is between zero and $W$. The physical meaning of $k$ is the number of grids with scattering points on the imaging plane after the target is gridded by the imaging plane.

In this paper, the WFOS algorithm is used to reconstruct the initial reconstruction target using the single sampling data of the incoherent detection array. Logically, the demand of the WFOS algorithm for samples is embodied in the number of elements contained in the incoherent detection array. That is, $L = Q$. As the number of samples increases, the imaging effect using the WFOS algorithm gets better. However, more samples means more incoherent array elements are needed, and they must be integrated into an array, which puts forward higher requirements for the device technology. As a matter of fact, the initial reconstruction imaging effect tends to saturate with the WFOS algorithm under the support of a certain number of samples; in consequence, there is hope in finding a balance between the number of samples and the pressure device. In the following experiments, 20 Monte Carlo experiments were used to calculate the average values, avoiding accidental situations.

The system shown in Table 1 was adopted to reconstruct targets with different sparsity levels under different numbers of samples. Figures 3(a) and 3(b) show the changing trend of MSE and RIE for different sizes and shapes of targets, respectively. In Fig. 3, each curve represents the average results of 20 Monte Carlo trials, and the shape of the target changes randomly in each trial. It can been seen in Fig. 3(a) that MSE becomes smaller as the target sparsity $k$ decreases. In other words, the number of samples required by the WFOS algorithm increases as the target sparsity increases. But the sparsity $k$ of the target is not the main factor that affects the performance of the WFOS algorithm. Figure 3(b) shows that as the number of samples increases, RIE becomes smaller. When the number of samples becomes saturated, regardless of the sparsity $k$, the decline rates of MSE and RIE for all targets gradually slow down and tend to be consistent. This also confirms that the imaging effect will not be significantly improved with the increased number of samples, and will eventually tend to saturation. Moreover, the excessive amount of samples not only contains redundant information, but also causes excessive computational burden. Therefore, it is recommended that the element number of the incoherent detection array be greater than $4W$, but the specific value should be designed reasonably according to the system configuration and computing power.

 figure: Fig. 4.

Fig. 4. Influence of element spacing on reconstruction target with different numbers of samples.

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 figure: Fig. 5.

Fig. 5. Characteristics of the reference signal. (a) Instantaneous radiation field distribution. (b) Spatial autocorrelation function. (c) Temporal autocorrelation function.

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To analyze the effect of element spacing of the incoherent detection array on imaging quality, experiments were conducted to test the RIE with different $L/W$ under different element spacing conditions, as shown in Fig. 4. In this experiment, the target sparsity was set as $k = 100$. It can be seen in Fig. 4 that when the number of incoherent detection array elements is fixed, the RIE will gradually decrease and become stable as the distance between the array elements increases. If the distance between adjacent array elements is too small, the correlation of the signals received by each element of the detection array have a strong correlation, which reduces the utilization of data and affects the imaging results. Therefore, proper array element spacing should be arranged to ensure the difference in detecting signals between the array elements.

According to the aforementioned incoherent detection array design analysis, taking the demand of the WFOS algorithm and the pressure of device technology into consideration, we fix $k = 100$, and element spacing as ${10^{- 3}} \;{\rm m}$ and $L = 3.36\; {\rm W}$, i.e., in this system, when $Q = 3025$, the WFOS algorithm is capable of getting an initial reconstruction target. According to previous research [31], the system performance based on the abovementioned incoherent detection array design is evaluated. Figure 5(a) is a three-dimensional representation of the instantaneous radiation field distribution of the signal after modulation by a code-aperture antenna. It is a spatiotemporal independent radiation field and proves that the coded-aperture antenna has a good modulation effect on the transmission signal, making the target information carried by the echo signal more abundant to achieve high-resolution imaging. It can be seen in Fig. 5(b) that the TCAI system has good spatial distribution autocorrelation characteristics. The echo signal received by each detection array element has a large difference, which improves the utilization rate of the echo signal. Figure 5(c) is the autocorrelation function of 20 samples at different moments. It can be seen that the samples are uncorrelated at different times, which represents that the different coding sequences have great differences in echo modulation. In summary, the system is set up reasonably, which verifies the science and validity of the experiment.

B. Imaging Results of TCAI Systems with Different Receiving Devices

In the same TCAI system, a coherent detector, an incoherent detector, and incoherent detection array are used at the receiving end to receive a echo signal. The classic TVAL algorithm [32] is used to process the echo signal received by coherent detection. Experiments show that the TVAL algorithm demand for 0.8 W samples tends to be saturated. The TCAI system based on incoherent detection and an incoherent detection array both use the WFOS algorithm to calculate imaging with $L = 3.36\; {\rm W}$ samples. The incoherent detector needs to sample $Q$ times, i.e., the coded-aperture antenna needs to be modulated $Q$ times, and the sampling time is $Q \times \Delta t$, where $\Delta t$ represents the time required for the coding antenna to load the coding method in a single time. In contrast, the incoherent detection array can obtain $Q$ samples in a single sampling. Quoting the data of Ref. [13] as an example, at this time, $\Delta t = 4\; {\rm ms}$. According to the system parameter setting $Q = 3025$, the required sampling time of the coherent detector is $Q \times \Delta t = 12.1 \;{\rm s}$. If the system parameters shown in Table 1 are adopted, the data sampled are not usable for imaging, while the sampling of the non-coherent detection array needs only $\Delta t = 4\; {\rm ms}$. Therefore, we can see the effectiveness and importance of the method proposed in this paper. In Fig. 6, by comparing these three receiving methods, it is not difficult to find that only the TCAI system based on the incoherent detection array can reconstruct the moving target when the samples required by each algorithm are saturated. Table 2 and Fig. 6 are sufficient to reflect the imaging advantages of the TCAI system based on the incoherent detection array for moving targets. It can not only image moving targets in a short time, but also reduce the number of samplings to improve the imaging frame rate.

 figure: Fig. 6.

Fig. 6. Moving target imaging of TCAI based on different detections: (a) target; (b) coherent detection; (c) incoherent detection; (d) incoherent detection array.

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Table 2. Parameter Comparison Based on Different Receivers

 figure: Fig. 7.

Fig. 7. Entropy of the target for initial reconstruction of the $n$th sampling data.

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C. Enhancement Imaging Results and Entropy Analysis

It can be seen in Table 2 that the TCAI system based on the incoherent detection array can reconstruct moving targets, while the initial reconstruction still has room for improvement. Therefore, the image enhancement method has strong practicability, especially in the case of severe external interference. In this paper, the incoherent detection array is used to sample the moving target echo 20 times, and then 20 initial reconstruction targets are obtained with the WFOS algorithm. Then the 20 initial reconstruction target images are filtered and marked with a value lower than the average entropy. As shown in Fig. 7, the blue line represents the average value of entropy, and the red line stands for the entropy of 20 initial reconstruction targets. The initial reconstruction target that is lower than the average entropy is eliminated. On the contrary, the initial target that is lower than the average upper entropy is marked, and the corresponding sampling times are ${t_1},{t_3},{t_6},{t_{10}},{t_{11}},{t_{12}},{t_{13}},{t_{15}},{t_{20}}$.

 figure: Fig. 8.

Fig. 8. Reconstruction results for target: (a) original target; (b) initial reconstruction result at time ${t_1}$; (c) initial reconstruction result at time ${t_{20}}$; (d) enhancement result.

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 figure: Fig. 9.

Fig. 9. Entropy of the target for the $n$th enhancement image.

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We perform target registration on the marked initial reconstructions, and then superimpose the target signal energy after registration to obtain an enhancement image. Figures 8(a)–8(d) stand for the original target, the marked initial reconstruction targets at time ${t_1}$ and ${t_{20}}$, respectively, and the enhancement image, respectively. From the visual effect, it can be found that Fig. 8(d) is closer to the original target than Figs. 8(b) and 8(c). What is more, from the entropy standard, the entropy of the original target is ${\textbf{H}} = 2.521$. Figure 9 shows the change in entropy of the enhancement target during the enhancement process. It can be found that as enhancement time increases, the value of the enhancement target entropy gradually decreases, and the enhancement effect becomes better. It can be boldly predicted that as the amount of fusion processing gradually increases, the entropy value of the enhancement result will approach the entropy value of the target. It should be stated that the imaging resolution has not changed, but the image area has expanded during the image enhancement process.

5. CONCLUSION

In this paper, an enhanced imaging method based on the WFOS algorithm is presented for moving targets based on an incoherent detection array TCAI system. The incoherent detection array is used to receive the echo signal at the receiving end, which provides device support for the TCAI system to realize the reconstruction of moving targets. The WFOS algorithm provides algorithmic support to solve non-convex problems. The image enhancement method based on registration and energy superposition further improves imaging quality. Therefore, the method proposed in this paper can not only image moving targets, but also shorten imaging time, increase imaging frame rate, and improve imaging quality. Next, we will further promote experiments to verify the effectiveness of this method.

Funding

National Natural Science Foundation of China (61871386, 61921001, 61971427, 62035014).

Acknowledgment

We thank the National Natural Science Foundation for help in identifying collaborators for this work.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES

1. K. B. Cooper, R. J. Dengler, N. Llombart, B. Thomas, G. Chattopadhyay, and P. H. Siegel, “THz imaging radar for standoff personnel screening,” IEEE Trans. Terahertz Sci. Technol. 1, 169–182 (2011). [CrossRef]  

2. C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014). [CrossRef]  

3. Y. B. Li, L. L. Li, B. B. Xu, W. Wu, R. Y. Wu, X. Wan, Q. Cheng, and T. J. Cui, “Transmission-type 2-bit programmable metasurface for single-sensor and single-frequency microwave imaging,” Sci. Rep. 6, 23731 (2016). [CrossRef]  

4. J. N. Gollub, O. Yurduseven, K. P. Trofatter, D. Arnitz, M. F. Imani, T. Sleasman, M. Boyarsky, A. Rose, A. Pedross-Engel, and H. Odabasi, “Large metasurface aperture for millimeter wave computational imaging at the human-scale,” Sci. Rep. 7, 42650 (2017). [CrossRef]  

5. J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013). [CrossRef]  

6. T. Zvolensky, J. N. Gollub, D. L. Marks, and D. R. Smith, “Design and analysis of a W-band metasurface-based computational imaging system,” IEEE Access 5, 9911–9918 (2017). [CrossRef]  

7. C. Jansen, S. Wietzke, O. Peters, M. Scheller, and M. Koch, “Terahertz imaging: applications and perspectives,” Appl. Opt. 49, E48–E57 (2010). [CrossRef]  

8. Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016). [CrossRef]  

9. L. Galvis, H. Arguello, and G. R. Arce, “Coded aperture design in mismatched compressive spectral imaging,” Appl. Opt. 54, 9875–9882 (2015). [CrossRef]  

10. A. Rogalski and F. Sizov, “Terahertz detectors and focal plane arrays,” Opto-Electron. Rev. 19, 346–404 (2011). [CrossRef]  

11. W. Shan, J. Yang, S. Shi, Q. Yao, Y. Zuo, Z. Lin, S. Chen, X. Zhang, W. Duan, and A. Cao, “Development of superconducting spectroscopic array receiver: a multibeam 2Sb SIS receiver for millimeter-wave radio astronomy,” IEEE Trans. Terahertz Sci. Technol. 2, 593–604 (2012). [CrossRef]  

12. J. Kawamura, C. Y. E. Tong, R. Blundell, D. C. Papa, T. R. Hunter, F. Patt, G. Gol’tsman, and E. Gershenzon, “Terahertz-frequency waveguide NBN hot-electron bolometer mixer,” IEEE Trans. Appl. Supercond. 11, 952–954 (2001). [CrossRef]  

13. Q. Yang, J. Zou, Y. Li, and S. T. Wu, “Fast-response liquid crystal phase modulators with an excellent photostability,” Crystals 10, 765 (2020). [CrossRef]  

14. G. Zha, H. Wang, Z. Yang, Y. Cheng, and Y. Qin, “Adaptive sparse reconstruction with joint parametric estimation for high-speed uniformly moving targets in coincidence imaging radar,” J. Appl. Remote Sens. 10, 025009 (2016). [CrossRef]  

15. F. Gan, C. Luo, X. Liu, H. Wang, and L. Peng, “Fast terahertz coded-aperture imaging based on convolutional neural network,” Appl. Sci. 10, 2661 (2020). [CrossRef]  

16. L. Liu, H. Xu, Y. Duan, A. W. Lichtenberger, and R. Ii, “A 200 Ghz Schottky diode quasi-optical detector based on folded dipole antenna,” in 20th International Symposium on Space Terahertz Technology (ISSTT) (2009), pp. 145–149.

17. J. Grzyb and U. Pfeiffer, “THz direct detector and heterodyne receiver arrays in silicon nanoscale technologies,” J. Infrared, Millimeter, Terahertz Waves 36, 998–1032 (2015). [CrossRef]  

18. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015). [CrossRef]  

19. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef]  

20. R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

21. E. J. Candès, T. Strohmer, and V. Voroninski, “PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013). [CrossRef]  

22. E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory. 61, 1985–2007 (2015). [CrossRef]  

23. J. Xue, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase retrieval,” IEEE Signal Process. Lett. 23, 1627–1631 (2016). [CrossRef]  

24. Y. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Commun. Pure Appl. Math. 70, 5 (2015).

25. Z. Yuan, H. Wang, and Q. Wang, “Phase retrieval via sparse Wirtinger flow,” J. Comput. Appl. Math. 355, 162–173 (2019). [CrossRef]  

26. L. Peng, C. Luo, B. Deng, H. Wang, S. Chen, and J. Dong, “Phaseless terahertz coded-aperture imaging for sparse target based on phase retrieval algorithm,” Sensors 19, 4617 (2019). [CrossRef]  

27. J. C. Toomay, Radar Principles (Xi’an Publishing House of Electronics Industry, 1995).

28. H. Stone, M. Orchard, E. Chang, and S. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39, 2235 (2001). [CrossRef]  

29. J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984). [CrossRef]  

30. P. O. Hoyer, “Nonnegative matrix factorization with sparseness constraints,” J. Mach. Learn. Res. 5, 1457–1469 (2004).

31. S. Chen, C. Luo, B. Deng, Y. Qin, and H. Wang, “Study on coding strategies for radar coded-aperture imaging in terahertz band,” J. Electron. Imaging 26, 053022 (2017). [CrossRef]  

32. C. Li, “An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing,” Ph.D. thesis (Rice University, 2011).

References

  • View by:
  • |
  • |
  • |

  1. K. B. Cooper, R. J. Dengler, N. Llombart, B. Thomas, G. Chattopadhyay, and P. H. Siegel, “THz imaging radar for standoff personnel screening,” IEEE Trans. Terahertz Sci. Technol. 1, 169–182 (2011).
    [Crossref]
  2. C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
    [Crossref]
  3. Y. B. Li, L. L. Li, B. B. Xu, W. Wu, R. Y. Wu, X. Wan, Q. Cheng, and T. J. Cui, “Transmission-type 2-bit programmable metasurface for single-sensor and single-frequency microwave imaging,” Sci. Rep. 6, 23731 (2016).
    [Crossref]
  4. J. N. Gollub, O. Yurduseven, K. P. Trofatter, D. Arnitz, M. F. Imani, T. Sleasman, M. Boyarsky, A. Rose, A. Pedross-Engel, and H. Odabasi, “Large metasurface aperture for millimeter wave computational imaging at the human-scale,” Sci. Rep. 7, 42650 (2017).
    [Crossref]
  5. J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
    [Crossref]
  6. T. Zvolensky, J. N. Gollub, D. L. Marks, and D. R. Smith, “Design and analysis of a W-band metasurface-based computational imaging system,” IEEE Access 5, 9911–9918 (2017).
    [Crossref]
  7. C. Jansen, S. Wietzke, O. Peters, M. Scheller, and M. Koch, “Terahertz imaging: applications and perspectives,” Appl. Opt. 49, E48–E57 (2010).
    [Crossref]
  8. Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
    [Crossref]
  9. L. Galvis, H. Arguello, and G. R. Arce, “Coded aperture design in mismatched compressive spectral imaging,” Appl. Opt. 54, 9875–9882 (2015).
    [Crossref]
  10. A. Rogalski and F. Sizov, “Terahertz detectors and focal plane arrays,” Opto-Electron. Rev. 19, 346–404 (2011).
    [Crossref]
  11. W. Shan, J. Yang, S. Shi, Q. Yao, Y. Zuo, Z. Lin, S. Chen, X. Zhang, W. Duan, and A. Cao, “Development of superconducting spectroscopic array receiver: a multibeam 2Sb SIS receiver for millimeter-wave radio astronomy,” IEEE Trans. Terahertz Sci. Technol. 2, 593–604 (2012).
    [Crossref]
  12. J. Kawamura, C. Y. E. Tong, R. Blundell, D. C. Papa, T. R. Hunter, F. Patt, G. Gol’tsman, and E. Gershenzon, “Terahertz-frequency waveguide NBN hot-electron bolometer mixer,” IEEE Trans. Appl. Supercond. 11, 952–954 (2001).
    [Crossref]
  13. Q. Yang, J. Zou, Y. Li, and S. T. Wu, “Fast-response liquid crystal phase modulators with an excellent photostability,” Crystals 10, 765 (2020).
    [Crossref]
  14. G. Zha, H. Wang, Z. Yang, Y. Cheng, and Y. Qin, “Adaptive sparse reconstruction with joint parametric estimation for high-speed uniformly moving targets in coincidence imaging radar,” J. Appl. Remote Sens. 10, 025009 (2016).
    [Crossref]
  15. F. Gan, C. Luo, X. Liu, H. Wang, and L. Peng, “Fast terahertz coded-aperture imaging based on convolutional neural network,” Appl. Sci. 10, 2661 (2020).
    [Crossref]
  16. L. Liu, H. Xu, Y. Duan, A. W. Lichtenberger, and R. Ii, “A 200 Ghz Schottky diode quasi-optical detector based on folded dipole antenna,” in 20th International Symposium on Space Terahertz Technology (ISSTT) (2009), pp. 145–149.
  17. J. Grzyb and U. Pfeiffer, “THz direct detector and heterodyne receiver arrays in silicon nanoscale technologies,” J. Infrared, Millimeter, Terahertz Waves 36, 998–1032 (2015).
    [Crossref]
  18. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
    [Crossref]
  19. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref]
  20. R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).
  21. E. J. Candès, T. Strohmer, and V. Voroninski, “PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
    [Crossref]
  22. E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory. 61, 1985–2007 (2015).
    [Crossref]
  23. J. Xue, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase retrieval,” IEEE Signal Process. Lett. 23, 1627–1631 (2016).
    [Crossref]
  24. Y. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Commun. Pure Appl. Math. 70, 5 (2015).
  25. Z. Yuan, H. Wang, and Q. Wang, “Phase retrieval via sparse Wirtinger flow,” J. Comput. Appl. Math. 355, 162–173 (2019).
    [Crossref]
  26. L. Peng, C. Luo, B. Deng, H. Wang, S. Chen, and J. Dong, “Phaseless terahertz coded-aperture imaging for sparse target based on phase retrieval algorithm,” Sensors 19, 4617 (2019).
    [Crossref]
  27. J. C. Toomay, Radar Principles (Xi’an Publishing House of Electronics Industry, 1995).
  28. H. Stone, M. Orchard, E. Chang, and S. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39, 2235 (2001).
    [Crossref]
  29. J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [Crossref]
  30. P. O. Hoyer, “Nonnegative matrix factorization with sparseness constraints,” J. Mach. Learn. Res. 5, 1457–1469 (2004).
  31. S. Chen, C. Luo, B. Deng, Y. Qin, and H. Wang, “Study on coding strategies for radar coded-aperture imaging in terahertz band,” J. Electron. Imaging 26, 053022 (2017).
    [Crossref]
  32. C. Li, “An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing,” Ph.D. thesis (Rice University, 2011).

2020 (2)

Q. Yang, J. Zou, Y. Li, and S. T. Wu, “Fast-response liquid crystal phase modulators with an excellent photostability,” Crystals 10, 765 (2020).
[Crossref]

F. Gan, C. Luo, X. Liu, H. Wang, and L. Peng, “Fast terahertz coded-aperture imaging based on convolutional neural network,” Appl. Sci. 10, 2661 (2020).
[Crossref]

2019 (2)

Z. Yuan, H. Wang, and Q. Wang, “Phase retrieval via sparse Wirtinger flow,” J. Comput. Appl. Math. 355, 162–173 (2019).
[Crossref]

L. Peng, C. Luo, B. Deng, H. Wang, S. Chen, and J. Dong, “Phaseless terahertz coded-aperture imaging for sparse target based on phase retrieval algorithm,” Sensors 19, 4617 (2019).
[Crossref]

2017 (3)

S. Chen, C. Luo, B. Deng, Y. Qin, and H. Wang, “Study on coding strategies for radar coded-aperture imaging in terahertz band,” J. Electron. Imaging 26, 053022 (2017).
[Crossref]

J. N. Gollub, O. Yurduseven, K. P. Trofatter, D. Arnitz, M. F. Imani, T. Sleasman, M. Boyarsky, A. Rose, A. Pedross-Engel, and H. Odabasi, “Large metasurface aperture for millimeter wave computational imaging at the human-scale,” Sci. Rep. 7, 42650 (2017).
[Crossref]

T. Zvolensky, J. N. Gollub, D. L. Marks, and D. R. Smith, “Design and analysis of a W-band metasurface-based computational imaging system,” IEEE Access 5, 9911–9918 (2017).
[Crossref]

2016 (4)

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

G. Zha, H. Wang, Z. Yang, Y. Cheng, and Y. Qin, “Adaptive sparse reconstruction with joint parametric estimation for high-speed uniformly moving targets in coincidence imaging radar,” J. Appl. Remote Sens. 10, 025009 (2016).
[Crossref]

Y. B. Li, L. L. Li, B. B. Xu, W. Wu, R. Y. Wu, X. Wan, Q. Cheng, and T. J. Cui, “Transmission-type 2-bit programmable metasurface for single-sensor and single-frequency microwave imaging,” Sci. Rep. 6, 23731 (2016).
[Crossref]

J. Xue, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase retrieval,” IEEE Signal Process. Lett. 23, 1627–1631 (2016).
[Crossref]

2015 (5)

Y. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Commun. Pure Appl. Math. 70, 5 (2015).

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory. 61, 1985–2007 (2015).
[Crossref]

J. Grzyb and U. Pfeiffer, “THz direct detector and heterodyne receiver arrays in silicon nanoscale technologies,” J. Infrared, Millimeter, Terahertz Waves 36, 998–1032 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

L. Galvis, H. Arguello, and G. R. Arce, “Coded aperture design in mismatched compressive spectral imaging,” Appl. Opt. 54, 9875–9882 (2015).
[Crossref]

2014 (1)

C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
[Crossref]

2013 (2)

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref]

E. J. Candès, T. Strohmer, and V. Voroninski, “PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

2012 (1)

W. Shan, J. Yang, S. Shi, Q. Yao, Y. Zuo, Z. Lin, S. Chen, X. Zhang, W. Duan, and A. Cao, “Development of superconducting spectroscopic array receiver: a multibeam 2Sb SIS receiver for millimeter-wave radio astronomy,” IEEE Trans. Terahertz Sci. Technol. 2, 593–604 (2012).
[Crossref]

2011 (2)

K. B. Cooper, R. J. Dengler, N. Llombart, B. Thomas, G. Chattopadhyay, and P. H. Siegel, “THz imaging radar for standoff personnel screening,” IEEE Trans. Terahertz Sci. Technol. 1, 169–182 (2011).
[Crossref]

A. Rogalski and F. Sizov, “Terahertz detectors and focal plane arrays,” Opto-Electron. Rev. 19, 346–404 (2011).
[Crossref]

2010 (1)

2004 (1)

P. O. Hoyer, “Nonnegative matrix factorization with sparseness constraints,” J. Mach. Learn. Res. 5, 1457–1469 (2004).

2001 (2)

H. Stone, M. Orchard, E. Chang, and S. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39, 2235 (2001).
[Crossref]

J. Kawamura, C. Y. E. Tong, R. Blundell, D. C. Papa, T. R. Hunter, F. Patt, G. Gol’tsman, and E. Gershenzon, “Terahertz-frequency waveguide NBN hot-electron bolometer mixer,” IEEE Trans. Appl. Supercond. 11, 952–954 (2001).
[Crossref]

1984 (1)

1982 (1)

1972 (1)

R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).

Arce, G. R.

Arguello, H.

Arnitz, D.

J. N. Gollub, O. Yurduseven, K. P. Trofatter, D. Arnitz, M. F. Imani, T. Sleasman, M. Boyarsky, A. Rose, A. Pedross-Engel, and H. Odabasi, “Large metasurface aperture for millimeter wave computational imaging at the human-scale,” Sci. Rep. 7, 42650 (2017).
[Crossref]

Blundell, R.

J. Kawamura, C. Y. E. Tong, R. Blundell, D. C. Papa, T. R. Hunter, F. Patt, G. Gol’tsman, and E. Gershenzon, “Terahertz-frequency waveguide NBN hot-electron bolometer mixer,” IEEE Trans. Appl. Supercond. 11, 952–954 (2001).
[Crossref]

Boyarsky, M.

J. N. Gollub, O. Yurduseven, K. P. Trofatter, D. Arnitz, M. F. Imani, T. Sleasman, M. Boyarsky, A. Rose, A. Pedross-Engel, and H. Odabasi, “Large metasurface aperture for millimeter wave computational imaging at the human-scale,” Sci. Rep. 7, 42650 (2017).
[Crossref]

Brady, D.

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref]

Candes, E. J.

Y. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Commun. Pure Appl. Math. 70, 5 (2015).

Candès, E. J.

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory. 61, 1985–2007 (2015).
[Crossref]

E. J. Candès, T. Strohmer, and V. Voroninski, “PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

Cao, A.

W. Shan, J. Yang, S. Shi, Q. Yao, Y. Zuo, Z. Lin, S. Chen, X. Zhang, W. Duan, and A. Cao, “Development of superconducting spectroscopic array receiver: a multibeam 2Sb SIS receiver for millimeter-wave radio astronomy,” IEEE Trans. Terahertz Sci. Technol. 2, 593–604 (2012).
[Crossref]

Chang, E.

H. Stone, M. Orchard, E. Chang, and S. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39, 2235 (2001).
[Crossref]

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Chattopadhyay, G.

K. B. Cooper, R. J. Dengler, N. Llombart, B. Thomas, G. Chattopadhyay, and P. H. Siegel, “THz imaging radar for standoff personnel screening,” IEEE Trans. Terahertz Sci. Technol. 1, 169–182 (2011).
[Crossref]

Chen, S.

L. Peng, C. Luo, B. Deng, H. Wang, S. Chen, and J. Dong, “Phaseless terahertz coded-aperture imaging for sparse target based on phase retrieval algorithm,” Sensors 19, 4617 (2019).
[Crossref]

S. Chen, C. Luo, B. Deng, Y. Qin, and H. Wang, “Study on coding strategies for radar coded-aperture imaging in terahertz band,” J. Electron. Imaging 26, 053022 (2017).
[Crossref]

W. Shan, J. Yang, S. Shi, Q. Yao, Y. Zuo, Z. Lin, S. Chen, X. Zhang, W. Duan, and A. Cao, “Development of superconducting spectroscopic array receiver: a multibeam 2Sb SIS receiver for millimeter-wave radio astronomy,” IEEE Trans. Terahertz Sci. Technol. 2, 593–604 (2012).
[Crossref]

Chen, Y.

Y. Chen and E. J. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Commun. Pure Appl. Math. 70, 5 (2015).

Cheng, Q.

Y. B. Li, L. L. Li, B. B. Xu, W. Wu, R. Y. Wu, X. Wan, Q. Cheng, and T. J. Cui, “Transmission-type 2-bit programmable metasurface for single-sensor and single-frequency microwave imaging,” Sci. Rep. 6, 23731 (2016).
[Crossref]

Cheng, Y.

G. Zha, H. Wang, Z. Yang, Y. Cheng, and Y. Qin, “Adaptive sparse reconstruction with joint parametric estimation for high-speed uniformly moving targets in coincidence imaging radar,” J. Appl. Remote Sens. 10, 025009 (2016).
[Crossref]

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Cooper, K. B.

K. B. Cooper, R. J. Dengler, N. Llombart, B. Thomas, G. Chattopadhyay, and P. H. Siegel, “THz imaging radar for standoff personnel screening,” IEEE Trans. Terahertz Sci. Technol. 1, 169–182 (2011).
[Crossref]

Cui, T. J.

Y. B. Li, L. L. Li, B. B. Xu, W. Wu, R. Y. Wu, X. Wan, Q. Cheng, and T. J. Cui, “Transmission-type 2-bit programmable metasurface for single-sensor and single-frequency microwave imaging,” Sci. Rep. 6, 23731 (2016).
[Crossref]

Deng, B.

L. Peng, C. Luo, B. Deng, H. Wang, S. Chen, and J. Dong, “Phaseless terahertz coded-aperture imaging for sparse target based on phase retrieval algorithm,” Sensors 19, 4617 (2019).
[Crossref]

S. Chen, C. Luo, B. Deng, Y. Qin, and H. Wang, “Study on coding strategies for radar coded-aperture imaging in terahertz band,” J. Electron. Imaging 26, 053022 (2017).
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W. Shan, J. Yang, S. Shi, Q. Yao, Y. Zuo, Z. Lin, S. Chen, X. Zhang, W. Duan, and A. Cao, “Development of superconducting spectroscopic array receiver: a multibeam 2Sb SIS receiver for millimeter-wave radio astronomy,” IEEE Trans. Terahertz Sci. Technol. 2, 593–604 (2012).
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F. Gan, C. Luo, X. Liu, H. Wang, and L. Peng, “Fast terahertz coded-aperture imaging based on convolutional neural network,” Appl. Sci. 10, 2661 (2020).
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J. Xue, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase retrieval,” IEEE Signal Process. Lett. 23, 1627–1631 (2016).
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Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
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K. B. Cooper, R. J. Dengler, N. Llombart, B. Thomas, G. Chattopadhyay, and P. H. Siegel, “THz imaging radar for standoff personnel screening,” IEEE Trans. Terahertz Sci. Technol. 1, 169–182 (2011).
[Crossref]

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F. Gan, C. Luo, X. Liu, H. Wang, and L. Peng, “Fast terahertz coded-aperture imaging based on convolutional neural network,” Appl. Sci. 10, 2661 (2020).
[Crossref]

L. Peng, C. Luo, B. Deng, H. Wang, S. Chen, and J. Dong, “Phaseless terahertz coded-aperture imaging for sparse target based on phase retrieval algorithm,” Sensors 19, 4617 (2019).
[Crossref]

S. Chen, C. Luo, B. Deng, Y. Qin, and H. Wang, “Study on coding strategies for radar coded-aperture imaging in terahertz band,” J. Electron. Imaging 26, 053022 (2017).
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C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
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J. N. Gollub, O. Yurduseven, K. P. Trofatter, D. Arnitz, M. F. Imani, T. Sleasman, M. Boyarsky, A. Rose, A. Pedross-Engel, and H. Odabasi, “Large metasurface aperture for millimeter wave computational imaging at the human-scale,” Sci. Rep. 7, 42650 (2017).
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C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
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J. Kawamura, C. Y. E. Tong, R. Blundell, D. C. Papa, T. R. Hunter, F. Patt, G. Gol’tsman, and E. Gershenzon, “Terahertz-frequency waveguide NBN hot-electron bolometer mixer,” IEEE Trans. Appl. Supercond. 11, 952–954 (2001).
[Crossref]

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J. Kawamura, C. Y. E. Tong, R. Blundell, D. C. Papa, T. R. Hunter, F. Patt, G. Gol’tsman, and E. Gershenzon, “Terahertz-frequency waveguide NBN hot-electron bolometer mixer,” IEEE Trans. Appl. Supercond. 11, 952–954 (2001).
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J. N. Gollub, O. Yurduseven, K. P. Trofatter, D. Arnitz, M. F. Imani, T. Sleasman, M. Boyarsky, A. Rose, A. Pedross-Engel, and H. Odabasi, “Large metasurface aperture for millimeter wave computational imaging at the human-scale,” Sci. Rep. 7, 42650 (2017).
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F. Gan, C. Luo, X. Liu, H. Wang, and L. Peng, “Fast terahertz coded-aperture imaging based on convolutional neural network,” Appl. Sci. 10, 2661 (2020).
[Crossref]

L. Peng, C. Luo, B. Deng, H. Wang, S. Chen, and J. Dong, “Phaseless terahertz coded-aperture imaging for sparse target based on phase retrieval algorithm,” Sensors 19, 4617 (2019).
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Pfeiffer, U.

J. Grzyb and U. Pfeiffer, “THz direct detector and heterodyne receiver arrays in silicon nanoscale technologies,” J. Infrared, Millimeter, Terahertz Waves 36, 998–1032 (2015).
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S. Chen, C. Luo, B. Deng, Y. Qin, and H. Wang, “Study on coding strategies for radar coded-aperture imaging in terahertz band,” J. Electron. Imaging 26, 053022 (2017).
[Crossref]

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J. Xue, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase retrieval,” IEEE Signal Process. Lett. 23, 1627–1631 (2016).
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J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
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Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
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Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
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Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
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W. Shan, J. Yang, S. Shi, Q. Yao, Y. Zuo, Z. Lin, S. Chen, X. Zhang, W. Duan, and A. Cao, “Development of superconducting spectroscopic array receiver: a multibeam 2Sb SIS receiver for millimeter-wave radio astronomy,” IEEE Trans. Terahertz Sci. Technol. 2, 593–604 (2012).
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C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
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K. B. Cooper, R. J. Dengler, N. Llombart, B. Thomas, G. Chattopadhyay, and P. H. Siegel, “THz imaging radar for standoff personnel screening,” IEEE Trans. Terahertz Sci. Technol. 1, 169–182 (2011).
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A. Rogalski and F. Sizov, “Terahertz detectors and focal plane arrays,” Opto-Electron. Rev. 19, 346–404 (2011).
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J. N. Gollub, O. Yurduseven, K. P. Trofatter, D. Arnitz, M. F. Imani, T. Sleasman, M. Boyarsky, A. Rose, A. Pedross-Engel, and H. Odabasi, “Large metasurface aperture for millimeter wave computational imaging at the human-scale,” Sci. Rep. 7, 42650 (2017).
[Crossref]

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[Crossref]

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T. Zvolensky, J. N. Gollub, D. L. Marks, and D. R. Smith, “Design and analysis of a W-band metasurface-based computational imaging system,” IEEE Access 5, 9911–9918 (2017).
[Crossref]

C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
[Crossref]

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref]

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E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory. 61, 1985–2007 (2015).
[Crossref]

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H. Stone, M. Orchard, E. Chang, and S. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39, 2235 (2001).
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Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
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K. B. Cooper, R. J. Dengler, N. Llombart, B. Thomas, G. Chattopadhyay, and P. H. Siegel, “THz imaging radar for standoff personnel screening,” IEEE Trans. Terahertz Sci. Technol. 1, 169–182 (2011).
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J. Kawamura, C. Y. E. Tong, R. Blundell, D. C. Papa, T. R. Hunter, F. Patt, G. Gol’tsman, and E. Gershenzon, “Terahertz-frequency waveguide NBN hot-electron bolometer mixer,” IEEE Trans. Appl. Supercond. 11, 952–954 (2001).
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F. Gan, C. Luo, X. Liu, H. Wang, and L. Peng, “Fast terahertz coded-aperture imaging based on convolutional neural network,” Appl. Sci. 10, 2661 (2020).
[Crossref]

L. Peng, C. Luo, B. Deng, H. Wang, S. Chen, and J. Dong, “Phaseless terahertz coded-aperture imaging for sparse target based on phase retrieval algorithm,” Sensors 19, 4617 (2019).
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S. Chen, C. Luo, B. Deng, Y. Qin, and H. Wang, “Study on coding strategies for radar coded-aperture imaging in terahertz band,” J. Electron. Imaging 26, 053022 (2017).
[Crossref]

G. Zha, H. Wang, Z. Yang, Y. Cheng, and Y. Qin, “Adaptive sparse reconstruction with joint parametric estimation for high-speed uniformly moving targets in coincidence imaging radar,” J. Appl. Remote Sens. 10, 025009 (2016).
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Wu, J.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

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Y. B. Li, L. L. Li, B. B. Xu, W. Wu, R. Y. Wu, X. Wan, Q. Cheng, and T. J. Cui, “Transmission-type 2-bit programmable metasurface for single-sensor and single-frequency microwave imaging,” Sci. Rep. 6, 23731 (2016).
[Crossref]

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Q. Yang, J. Zou, Y. Li, and S. T. Wu, “Fast-response liquid crystal phase modulators with an excellent photostability,” Crystals 10, 765 (2020).
[Crossref]

Wu, W.

Y. B. Li, L. L. Li, B. B. Xu, W. Wu, R. Y. Wu, X. Wan, Q. Cheng, and T. J. Cui, “Transmission-type 2-bit programmable metasurface for single-sensor and single-frequency microwave imaging,” Sci. Rep. 6, 23731 (2016).
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Y. B. Li, L. L. Li, B. B. Xu, W. Wu, R. Y. Wu, X. Wan, Q. Cheng, and T. J. Cui, “Transmission-type 2-bit programmable metasurface for single-sensor and single-frequency microwave imaging,” Sci. Rep. 6, 23731 (2016).
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L. Liu, H. Xu, Y. Duan, A. W. Lichtenberger, and R. Ii, “A 200 Ghz Schottky diode quasi-optical detector based on folded dipole antenna,” in 20th International Symposium on Space Terahertz Technology (ISSTT) (2009), pp. 145–149.

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J. Xue, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase retrieval,” IEEE Signal Process. Lett. 23, 1627–1631 (2016).
[Crossref]

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W. Shan, J. Yang, S. Shi, Q. Yao, Y. Zuo, Z. Lin, S. Chen, X. Zhang, W. Duan, and A. Cao, “Development of superconducting spectroscopic array receiver: a multibeam 2Sb SIS receiver for millimeter-wave radio astronomy,” IEEE Trans. Terahertz Sci. Technol. 2, 593–604 (2012).
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Q. Yang, J. Zou, Y. Li, and S. T. Wu, “Fast-response liquid crystal phase modulators with an excellent photostability,” Crystals 10, 765 (2020).
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G. Zha, H. Wang, Z. Yang, Y. Cheng, and Y. Qin, “Adaptive sparse reconstruction with joint parametric estimation for high-speed uniformly moving targets in coincidence imaging radar,” J. Appl. Remote Sens. 10, 025009 (2016).
[Crossref]

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W. Shan, J. Yang, S. Shi, Q. Yao, Y. Zuo, Z. Lin, S. Chen, X. Zhang, W. Duan, and A. Cao, “Development of superconducting spectroscopic array receiver: a multibeam 2Sb SIS receiver for millimeter-wave radio astronomy,” IEEE Trans. Terahertz Sci. Technol. 2, 593–604 (2012).
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Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)

Fig. 1.
Fig. 1. Terahertz coded-aperture imaging system for moving targets based on incoherent detection array.
Fig. 2.
Fig. 2. Enhanced phase retrieval algorithm for moving targets based on incoherent detection array.
Fig. 3.
Fig. 3. Influence of L on WFOS algorithm with different k sparsity targets. (a) MSE. (b) RIE.
Fig. 4.
Fig. 4. Influence of element spacing on reconstruction target with different numbers of samples.
Fig. 5.
Fig. 5. Characteristics of the reference signal. (a) Instantaneous radiation field distribution. (b) Spatial autocorrelation function. (c) Temporal autocorrelation function.
Fig. 6.
Fig. 6. Moving target imaging of TCAI based on different detections: (a) target; (b) coherent detection; (c) incoherent detection; (d) incoherent detection array.
Fig. 7.
Fig. 7. Entropy of the target for initial reconstruction of the $n$th sampling data.
Fig. 8.
Fig. 8. Reconstruction results for target: (a) original target; (b) initial reconstruction result at time ${t_1}$; (c) initial reconstruction result at time ${t_{20}}$; (d) enhancement result.
Fig. 9.
Fig. 9. Entropy of the target for the $n$th enhancement image.

Tables (2)

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Table 1. Basic Parameter Settings Used in the Simulations

Tables Icon

Table 2. Parameter Comparison Based on Different Receivers

Equations (14)

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S t ( t ) = A exp [ j 2 π ( f c t + 1 2 γ t 2 ) ] ,
S t r ( t n ) = w = 1 W S t ( t n | r t r w | c ) β w ,
S c ( t n , φ ) = m = 1 M S t r ( t n | r w r m | c , φ ) ,
S c ( t n , φ ) = m = 1 M w = 1 W A exp { j [ 2 π ( f c ( t n | r t r w | c | r w r m | c ) + 1 2 γ ( t n | r t r w | c | r w r m | c ) 2 ) + φ m ( t n ) ] } β w ,
S r ( q ) ( t n , φ ) = | S c ( t n | r m r q | c , φ ) | 2 ,
S r ( q ) ( t n , φ ) = | m = 1 M w = 1 W A exp { j [ 2 π ( f c ( t n | r t r w | c | r w r m | c | r m r q | c ) + 1 2 γ ( t n | r t r w | c | r w r m | c | r m r q | c ) 2 ) + φ m ( t n ) ] } β w | 2 ,
Sr ( t n ) = | S ( t n ) β | 2 + ω ,
S ( t n ) = m = 1 M w = 1 W A exp { j [ 2 π ( f c ( t n | r t r w | c | r w r m | c ) + 1 2 γ ( t n | r t r w | c | r w r m | c ) 2 ) + φ m ( t n ) ] } .
S ( t n ) = [ S 1 , 1 ( t n ) S 1 , 2 ( t n ) S 1 , W ( t n ) S 2 , 1 ( t n ) S 2 , 2 ( t n ) S 2 , W ( t n ) S Q , 1 ( t n ) S Q , 2 ( t n ) S Q , W ( t n ) ] .
min f ( β ) = Δ 1 2 N n = 1 N ( ( S ( t n ) β ) 2 Sr ( t n ) ) 2 .
p ij = f ( i , j ) / K 2 , H = i = 0 255 p ij log p ij ,
G 2 ( u , v ) = G 1 ( u , v ) e i 2 π ( u x 0 + v y 0 ) ,
H ( u , v ) = G 1 G 2 | A 1 | | A 2 | = e i 2 π ( u x 0 + v y 0 ) ,
I F F T ( H ( u , v ) ) = δ ( u x 0 , v y 0 ) .

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