Design of aperture radar auxiliary technology based on array mathematical model and statistical characteristic calculation

Abstract

In response to the problem of poor imaging quality caused by amplitude and phase errors in synthetic aperture imaging, this study develops an aperture radar-assisted technology based on an array of mathematical models and statistical characteristic calculations. Two error correction algorithms, active correction and iterative self-correction, are proposed by designing error correction algorithms based on the matrix space spectrum correction error concept. Simulation experiments showed that the mean square error of the two correction algorithms has decreased by an average of 36.23% compared to before correction, and the peak signal-to-noise ratio has increased by an average of 33.43% compared to before correction. Compared with other methods, the proposed two algorithms had an average increase of 139.51% in peak signal-to-noise ratio in two-dimensional imaging. The results indicate that it is feasible to use the traditional matrix space spectral correction method for comprehensive aperture imaging error correction. The designed error correction preprocessing algorithm based on an array of mathematical models and statistical characteristic calculations can improve imaging quality, reduce the impact of amplitude and phase errors, and has positive application value in synthetic aperture imaging technology.

Article highlights

Two new error correction algorithms are proposed to effectively reduce the amplitude and phase errors in SAR imaging;

A new signal processing framework is provided by introducing array mathematical models and statistical features;

The method is capable of significantly correcting imaging errors in 2D imaging with superior application potential.

Introduction

Synthetic Aperture Radar (SAR) is a technology that utilizes the transmission and reception of radar signals to obtain images of target areas. Through the phase information of the signal, it can maintain consistency during the movement of the radar platform, achieving higher directional resolution [1, 2–3]. SAR can transform the echo signal from the time domain to the frequency domain by performing Fourier Transform (FT), thereby obtaining the spatial frequency distribution of the target. It has been widely used in fields such as biological detection. However, in the context of practical SAR, the presence of non-ideal hardware can lead to the generation of amplitude and phase errors. These errors result in the appearance of additional beams in other directions within the weight matrix during the process of beamforming. Consequently, this can compromise the accuracy of the echo signal, as well as the contrast and resolution of the resulting image. To improve the accuracy and precision of SAR imaging, relevant scholars have conducted various studies.

SAR needs to solve the problem of systematic errors in practical work, including additive errors, multiplicative errors, etc. Therefore, industry scholars have conducted various explorations on how to correct SAR errors. Tang J et al. proposed a high plasticity error correction incremental learning method to address the issue of SAR automatic target recognition being unable to handle incremental recognition scenarios. The training of multiple optimal models for the correction of cumulative errors, in conjunction with the construction of class-balanced training batches, has been demonstrated to effectively address the issue of imbalanced data distribution. This approach has been shown to mitigate model degradation and plasticity decline in incremental scenarios [4]. Manzoni et al. proposed a motion estimation and compensation workflow aimed at conducting a comprehensive theoretical and experimental analysis of autofocus requirements in typical automotive scenes. This study analyzed and deduced the impact of trajectory estimation errors caused by navigation on SAR imaging, and used fixed ground control points in low resolution radar images to control the impact of position and focusing height on imaging. As a result, some common misconceptions and corresponding measures have been established [5]. Lagos et al. developed a single channel, fully dynamic pipeline SAR automatic target recognition system to reduce system errors and improve imaging quality. The use of ring amplification and background calibration has been demonstrated to enhance the compromise and looseness of the architecture, facilitating the implementation of new SAR dual helium and narrowband jitter injection techniques. This has led to the attainment of rapid and comprehensive background calibration for correlation recognition [6].

In theory, achieving urban and rural detection can be achieved by satisfying FT between the measured visibility function of SAR and the inverted brightness temperature. However, in practical applications, limited by the quality of SAR imaging equipment and the deployment properties of array combinations, the final imaging effect of SAR is difficult to meet the high-precision requirements of related fields [7]. Therefore, it is particularly necessary to carry out preprocessing of SAR images. Moskolaï et al. proposed a preprocessing technique that integrates the collection, preprocessing, and preparation of Sentinel-1 images. This technique was developed to address the complexity of satellite image preprocessing tasks, which leads to a lack of preprocessing datasets and time-consuming traditional preprocessing methods. By using scripts for collection and preprocessing, and establishing time-series Sentinel-1 images, an easily modifiable image collection and batch preprocessing program was provided for the community [8]. Xiang et al. proposed an efficient minimum entropy channel error estimation method based on finely focused SAR images to address the issue of reducing multi-channel signal reconstruction performance due to phase errors in SAR channels. By using the ranging Doppler imaging algorithm to preprocess each channel, the multi-channel fine focused SAR images were overlapped and iteratively calibrated for channel phase errors, thereby avoiding redundant reconstruction and imaging operations [9]. Xie et al. developed a super-pixel level constant false alarm rate preprocessing method to address the impact of speckle noise in SAR image propagation detection. The Johnsonsu function was employed to incorporate the superpixel clutter present in the background loop, and detection prefabrication was derived based on a specified false alarm rate. This approach enabled the optimization of candidate targets, facilitating the accurate and efficient extraction of the background superpixel region within each detected superpixel [10].

Kang and Baek proposed an incremental SAR imaging method based on the perceptual dictionary matrix to solve the SAR imaging orientation ambiguity problem caused by irregular loss of received data and non-uniform undersampling. By optimizing the perceptual dictionary matrix estimation process, the SAR orientation blur suppression effect was improved [11]. Kang and Kim proposed a new framework for sparse aperture Inverse Synthetic Aperture Radar (ISAR) imaging and lateral scaling. This framework is based on compressed perception, a technique developed to address the problem of severe image blurring in ISAR imaging caused by targets with extreme manoeuvrability. The method performed ISAR image reconstruction by means of a perceptual matrix estimation technique, which used parametric signal model reconstruction to find a basis function that best represents the observed sparse aperture data [12]. To determine the best transformation in SAR image alignment to achieve the most accurate matching of two images, the two scholars further proposed a new method based on Tsallis entropy combined with sequential search strategy. The Tsallis entropy was utilized as a cost function of the similarity metric to assess the degree of focusing of the mean intensity projection contour of SAR images. This cost function was processed in the distance direction and azimuthal direction in the coarse and fine alignment steps, respectively [13].

Based on the above, many scholars have designed the research results of SAR image preprocessing through concrete error elimination methods. However, the method of handling the quality degradation of SAR visibility function cannot completely eliminate the non-ideality of the measured visibility function. Therefore, this study proposes a SAR aperture radar auxiliary technology based on Array Mathematical Model (AMM) and Statistical Characteristic Calculation (SCC), with the optimization of visibility function as the research approach, to solve the problem of large errors in SAR mapping preprocessing. Firstly, a comprehensive Aperture Signal Processing Transfer Method (ASPTM) is explored based on AMM and SCC, and a correction preprocessing method suitable for SAR aperture matrix is further proposed. The novelty of the research lies in the exploration of SAR imaging in signal processing based on AMM and SCC, thereby establishing a multi-dimensional SAR ASPTM.

The study introduces the mathematical model and SCC method in ASPT to SAR imaging and proposes a SAR-assisted technique based on AMM and SCC. The image quality problems caused by amplitude error and phase error in imaging are solved by optimizing the SAR visibility function. The proposed method offers a novel approach to error correction in SAR imaging, with the potential to enhance the quality of the imaging results and mitigate the impact of hardware errors on the imaging process. In addition, two error correction algorithms, Active Calibration (AC) and Iterative Self-Correction (ISC), are proposed, which provide new solutions to improve imaging quality in high-precision correction scenarios and complex scenarios. The proposed methods have important application value in bio-detection, military reconnaissance, environmental monitoring, etc., and can meet the needs of high-precision imaging.

Methods and materials

To solve the problem of poor imaging quality caused by non-ideal visibility functions in SAR imaging processing, this study first proposes an ASPTM based on AMM and SCC. Secondly, based on ASPTM, this study further designs an aperture imaging preprocessing algorithm that combines error correction ideas.

ASPTM based on AMM and SCC

Array Signal Processing Technology (ASPT) analyzes the signals received by various receivers in the array and extracts information about the position, velocity, intensity, and other aspects of the signal source, which is similar to the signal processing concept of SAR [14, 15]. Therefore, this study explores ASPTM based on ASPT. The core of ASPT lies in utilizing the spatial specificity of the antenna matrix for signal processing and extraction [16, 17]. Among them, Spatial Spectral Estimation (SSE) is mainly used to determine the location and quantity information of signal sources. Assuming that there are multiple narrowband signals distributed at long distances in space, and the receiving matrix has multiple matrix elements, this study first constructs a mathematical model for the SSE matrix. Equation (1) can be obtained for sources with narrowband form and narrowband long-distance signal sources.

1

$$\left\{\begin{array}{c}s(t)=z(t)e^{j(w_{0}t+\varphi (t))}\hfill \\ z(t-\upsilon )\approx z(t)\hfill \\ \varphi (t-\upsilon )\approx \varphi (t)\hfill \end{array}\right)$$

In Eq. (1), $s(t)$ is a source with narrowband form. $z(t)$ is the amplitude of the signal. $e$ is logarithm. $w_0$ is the angular frequency of the signal. $\varphi (t)$ is the phase of the signal. $\upsilon$ is latency. The expression of the signal received by the matrix at a certain moment is shown in Eq. (2).

2

$$\left\{\begin{array}{c}{x}_k(t)=\sum _{i=1}^N{h}_{\mathit{ik}}{s}_i(t-{\upsilon }_{\mathit{ik}})+n_k(t)\hfill \\ \mathrm{X}(t)=\mathrm{A}S(t)+\mathrm{N}(t)\hfill \\ \mathrm{A}={[a_1(w_0),a_2(w_1),...,a_N(w_0)]}^{\mathrm{T}}\hfill \\ a_i(w_0)={[e^{-j{w}_0{\upsilon }_{1i}},e^{-j{w}_0{\upsilon }_{2i}},...,e^{-j{w}_0{\upsilon }_{\mathit{Mi}}}]}^{\mathrm{T}}\hfill \end{array}\right)$$

In Eq. (2), $n_k(t)$ and $x_k(t)$ are the noise of the $k$-th matrix element and the received signal. $\mathrm{T}$ is the transpose matrix. $N$ is the number of narrowband signals over long distances. $h_{\mathit{ik}}$ is the gain coefficient applied to the $i$-th signal by the $k$-th matrix element. $\mathrm{X}(t)$ and $\mathrm{N}(t)$ are data vectors for matrices and noise. The spatial matrix of $\mathrm{A}$ is a popular matrix. $a_i(w_0)$ is the guiding vector. $M$ is the number of matrix elements. The popular matrix is defined by the time it takes for the elements in each matrix to reach the receiving point during signal transmission or processing, in accordance with the basic theory of ASPT. Therefore, this study further constructs two dimensional matrix models. The specific 1D model is shown in Fig. 1.

Fig. 1 [Images not available. See PDF.]

Array modelling of 1D line arrays

Fig. 1a and b are schematic diagrams of 1D linear arrays for long-range and near-field waves. Among them, $d$ is the distance between matrix elements. $\theta$ is the angle between the direction of the incoming wave and the normal direction of the matrix. The combination of 1(a) and 1(b) in a 1D linear array with long-range waves suggests that electromagnetic waves from a narrowband emission source can be regarded as plane waves under specific conditions. These conditions include the presence of both the narrowband emission source and a matrix with a finite frequency band and clearly defined band length, all meeting the long-range conditions. At this point, there is a wave path difference between the narrowband emission source and the reference matrix element and a certain signal source. On the contrary, when the conditions do not hold, the assumption of plane waves propagating electromagnetic waves does not hold, and they are under near-field conditions. Among them, the delay expression formula corresponding to the wave path difference in the 1D linear array's long-distance incoming waves is shown in Eq. (3)

3

$${\upsilon }_{\mathit{ij}}=\frac{2\pi m·dsin\theta }{\gamma }$$

In Eq. (3), $m$ is the number of sources. $\gamma$ is the wavelength of the radiated electromagnetic wave. The expression ${\upsilon }_m^{\mathit{Near}}$ for the wave path difference delay of the 1D linear array near-field incoming wave is shown in Eq. (4).

4

$${\upsilon }_m^{\mathit{Near}}\approx \frac{2\pi r_k}{\gamma }\left(\frac{m^2·d^{2}cos\theta }{2r_k^2}-\frac{m·dsin\theta }{r_k}\right)$$

In Eq. (4), $r_k$ is the distance from the narrowband emission source to $k$ matrix elements. The matrix model of the 2D array is shown in Fig. 2.

Fig. 2 [Images not available. See PDF.]

Array models for 2D matrix planar arrays

In Fig. 2, in the incoming wave of the 2D array, the unit vector of the narrowband emission source signal in the incident direction and the corresponding delay are expressed as Eq. (5).

5

$$\left\{\begin{array}{c}b=-{[sin\theta cos\varphi ,sin\theta sin\varphi ,cos\varphi ]}^{\mathrm{T}}\hfill \\ {\upsilon }_m^e=\frac{2\pi (m·d_{x}sin\theta cos\varphi +n{d}_{y}sin\theta sin\varphi )}{\gamma }\hfill \\ {\overline{\upsilon }}_m^{\mathit{Near}}=\frac{2\pi r_{m,n}}{\gamma }(\frac{X_m^2+Y_n^2}{2r_{m,n}^2}-\frac{X_{m}sin\theta cos\varphi +Y_{n}sin\theta sin\varphi }{r_{m,n}})\hfill \end{array}\right)$$

In Eq. (5), $b$ is the unit vector of the narrowband transmission source signal intake direction. $X_m$ and $Y_n$ are the coordinates of matrix elements. $\varphi$ is the phase angle. ${\upsilon }_m^e$ represents the delay of the matrix element with coordinates $(X_m,Y_n,0)$ relative to the reference matrix element (0,0,0). ${\overline{\upsilon }}_m^{\mathit{Near}}$ is the near-field delay of the 2D array. $r_{m,n}$ represents the distance between the matrix elements with coordinates $(X_m,Y_n,0)$ and the narrowband emission source. In the process of spatial spectrum signal processing, it is necessary to calculate the statistical characteristics of the matrix based on matrix [18]. Therefore, the present study posits the hypothesis that the transmitted signal originates from a distant area, belongs to the narrowband category, and exhibits uniformity in its values or characteristics across various directions. This hypothesis further assumes consistent channel performance and the absence of interference or coupling among elements. Assuming that the noise of all matrix signals is normal noise and has an independent relationship and mutual independence with the input signal of the narrowband transmission source. Based on the above assumptions, this study conducts SCC. Among them, the variable correlation matrix of the signal vector and matrix receiving the signal is shown in Eq. (6).

6

$$\left\{\begin{array}{c}\Re =\mathrm{E}[(\mathrm{X}(t)-m_x(t){(\mathrm{X}(t)-m_x(t))}^{\mathrm{H}}]\hfill \\ {\Re }^{\prime }=\mathrm{E}[\mathrm{X}(t){\mathrm{X}}^{\mathrm{H}}(t)]\hfill \end{array}\right)$$

In Eq. (6), $\Re$ and ${\Re }^{\prime }$ are the variable correlation matrices of the received signal vector and matrix. $m_x(t)$ represents the mathematical expectation of $\mathrm{X}(t)$. $\mathrm{H}$ is the conjugate transpose of a matrix. $\mathrm{E}$ is the expectation. Due to the independence between various narrowband emission sources and the independence between noise and narrowband emission sources, the matrix variable correlation matrix is transformed into a similar diagonalization equation through unitary transformation, as shown in Eq. (7).

7

$$\left\{\begin{array}{c}\mathrm{E}[S(t)\mathrm{N}(t)]=0\hfill \\ {\Re }_S=\mathrm{E}[S(t)S^{\mathrm{H}}(t)]\hfill \\ \mathrm{E}[\mathrm{N}(t){\mathrm{N}}^{\mathrm{H}}(t)]={\sigma }^2\mathrm{I}\hfill \\ {\Re }^{\prime }=\sum _{i=1}^M\gamma {\mathrm{Z}}_i{\mathrm{Z}}_i^{\mathrm{H}}\hfill \end{array}\right)$$

In Eq. (7), ${\Re }_S$ is the narrowband emission source correlation matrix. ${\sigma }^2$ is the variance. $\mathrm{I}$ is the noise vector. ${\mathrm{Z}}_i$ is the feature vector. Based on the above derivation, this study conducts ASPTM design for 2D matrix long-range and near-field. This study assumes that there is directional independence between each element in the 2D SAR matrix, good separation conditions between each radiation channel, and the existence of several unrelated long-range narrowband emission sources. The emission source is independent of noise. Therefore, the expression of long-distance conditions and the formula for the received signal of this matrix can be obtained, as shown in Eq. (8).

8

$$\left\{\begin{array}{c}r\gg \frac{L_{max}^2}{\gamma }\hfill \\ \mathrm{X}(t)=\mathrm{A}S(t)+\mathrm{N}(t)\hfill \\ \mathrm{A}=[a({\alpha }_1,{\beta }_1),a({\alpha }_2,{\beta }_2),...,a({\alpha }_N,{\beta }_N)]\hfill \end{array}\right)$$

In Eq. (8), $L$ is the maximum baseline length of the comprehensive aperture matrix. $a({\alpha }_i,{\beta }_i)$ is the directional vector of the radiation signal. ${\alpha }_i$ is the azimuth angle. ${\beta }_i$ is the pitch angle. Assuming that the noise follows a zero mean Gaussian process with variance, and that a random element in the variable correlation matrix has a discretized approximate FT format [19, 20]. This study solves the radiation temperature and brightness of narrowband emission sources using Rayleigh formula, and processes the variable correlation matrix using classical array signal processing theory. On this basis, the variable correlation matrix of the received signal and the calculation of the comparative synthetic aperture imaging are shown in Eq. (9).

9

$$\left\{\begin{array}{c}{\Re }^{\prime }(j,k)=\frac{2\kappa }{{\gamma }^2}\underset{{\epsilon }^2+{\beta }^2\le 1}{\iint }Q({\alpha }_i,{\beta }_i)e^{-j2\pi (p_{\mathit{jk}}\alpha +q_{\mathit{jk}}\beta )}d\alpha d\beta \hfill \\ T(p,q)=\underset{{\epsilon }^2+{\beta }^2\le 1}{\iint }Q({\alpha }_i,{\beta }_i)e^{-j2\pi (p\alpha +q\beta )}d\alpha d\beta \hfill \end{array}\right)$$

In Eq. (9), ${\Re }^{\prime }(j,k)$ is any element in the variable correlation matrix. $p$ and $q$ are baselines for the X and Y axes. $\kappa$ is the Boltzmann constant. $T(p,q)$ is the basic formula for synthetic aperture imaging. $Q({\alpha }_i,{\beta }_i)$ is the brightness temperature distribution of narrowband emission sources. $\epsilon$ is the threshold. $d\alpha$ and $d\beta$ are the integrals of two directional angles. The comparison diagram of the aperture matrix and matrix element arrangement, as well as the baseline distribution, under the condition of long-distance wave arrival in the 2D matrix is shown in Fig. 3.

Fig. 3 [Images not available. See PDF.]

Array and array element alignments and baseline distributions for 2D array far-field conditions

Figure 3 shows the arrangement and baseline distribution of the T-shaped aperture matrix and matrix elements under the condition of a 2D matrix with 10 matrix elements at a distance. In instances where the matrix between the narrowband emission source and the comprehensive aperture does not satisfy the condition of long distance, it undergoes a transformation into a 2D matrix near-field. The delay expression formula and variable correlation matrix are shown in Eq. (10).

10

$$\left\{\begin{array}{c}{\overline{\upsilon }}_m^{\mathit{Near}}=\frac{2\pi r_{m,n}}{\gamma }(\frac{X_m^2+Y_n^2}{2r_{\sigma }^2}-\frac{X_{m}sin\theta cos\varphi +Y_{n}sin\theta sin\varphi }{r_{\sigma }})\hfill \\ {\Re }^{\prime }(j,k)=\sum _{i=1}^N{\sigma }_s^2({\alpha }_i,{\beta }_i)e^{-j2\pi (p_{\mathit{jk}}{\alpha }_i+q_{\mathit{jk}}{\beta }_i-\frac{x_j^2-x_k^2+y_j^2-y_k^2}{2\gamma r_{\sigma }})}\hfill \end{array}\right)$$

In Eq. (10), $r_{\sigma }$ is the distance from the center of the narrowband transmission source to the middle of two antennas. In SAR imaging, there is a situation where multiple antennas have the same baseline. The visibility function for additional baselines is calculated using the average distance between antennas. Therefore, prior to calculating the variable correlation matrix, it is necessary to ascertain the phase difference of antennas that meet the same baseline situation. This will allow for the correction of the near-field variable correlation matrix and the elimination of errors in the variable correlation matrix elements. The Rayleigh-Kings formula is introduced to update the matrix expression, as shown in Eq. (11).

11

$$\left\{\begin{array}{c}C(j,k)=e^{j2\pi \frac{x_j^2-x_k^2+y_j^2-y_k^2}{2\gamma r_{\sigma }}}\hfill \\ \widehat{\Re }(j,k)=\frac{2\kappa }{{\gamma }^2}\underset{{\epsilon }^2+{\beta }^2\le 1}{\iint }Q(\alpha ,\beta )e^{-j2\pi (p_{\mathit{jk}}\alpha +q_{\mathit{jk}}\beta )}d\alpha d\beta \hfill \end{array}\right)$$

In Eq. (11), $C(j,k)$ represents the phase error generated by the antenna on acquisition points $(x_j,y_j)$ and $(x_k,y_k)$. $\widehat{\Re }(j,k)$ is the corrected variable correlation matrix. In the near-field scenario, the detection image of a binary interferometer placed in the same plane in the near-field region of the narrowband emission source is shown in Fig. 4.

Fig. 4 [Images not available. See PDF.]

Near-field binary interferometer probing of two-dimensional arrays

In Fig. 4, the distance between the micro element center point $d_{\sigma }$ of the narrowband transmission source and the two antennas $r_i$ and $r_j$, as well as $r_{\sigma }$, have the relationship shown in Eq. (12).

12

$$\left\{\begin{array}{c}{r}_{\sigma }=z_{\sigma }-0\hfill \\ r_i-r_j=\frac{x_i^2-x_j^2+y_i^2-y_j^2}{2r_{\sigma }}-\frac{x_{\sigma }(x_i-x_j)}{r_{\sigma }}-\frac{y_{\sigma }(y_i-y_j)}{r_{\sigma }}\hfill \end{array}\right)$$

In Eq. (12), $z_{\sigma }$ is the Z-axis coordinate of $d_{\sigma }$. The relevant output expression of the binary interferometer is shown in Eq. (13).

13

$$\begin{array}{c}T(p,q)=\underset{S}{\iint }\frac{{\sigma }_{\mathrm{E}}(x_{\sigma },y_{\sigma })e^{j\frac{2\pi }{\gamma }(r_i-r_j)}}{r_i{r}_j}d\sigma \\ =e^{-j\frac{2\pi (x_i^2-x_j^2+y_i^2-y_j^2)}{2\gamma r_{\sigma }}}{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }{\sigma }_{\mathrm{E}}(x_{\sigma },y_{\sigma })e^{-j2\pi (p\alpha +q\beta )}d\alpha d\beta \\ =e^{j\frac{2\pi (x_i^2-x_j^2+y_i^2-y_j^2)}{2\gamma r_{\sigma }}\frac{2\kappa }{{\gamma }^2}}{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }Q(\alpha ,\beta )e^{-j2\pi (p\alpha +q\beta )}d\alpha d\beta \\ \end{array}$$

In Eq. (13), ${\sigma }_{\mathrm{E}}(x_{\sigma },y_{\sigma })$ represents the spatial power distribution of the radiation source. Based on the above, the ASPTM designed on the basis of AMM and SCC, regardless of whether it is under long-distance or near-field conditions, the derived variable correlation matrix is still in the same form as the visibility function matrix of SAR imaging. It is feasible to transfer the ASPT algorithm and signal processing concepts to SAR imaging.

Aperture imaging preprocessing algorithm based on ASPTM

Based on the ASPTM established earlier, this study further designs the aperture imaging preprocessing algorithm. SAR achieves high-resolution imaging through the complexity of system and signal processing, so hardware errors are divided into three types: antenna, channel, and analog digital errors [21]. Among them, channel error mainly refers to the amplitude and phase errors generated by the imperfect channel itself during data transmission or signal processing. Therefore, this study mainly designs and solves algorithms based on amplitude and phase errors. Firstly, based on the correction ideas of amplitude and phase errors in ASPTM and traditional matrix signal processing, an AC algorithm is proposed. In the AC algorithm, the matrix $\mathbf{R}$ of SAR measurement visibility affected by amplitude and phase errors is expressed as Eq. (14).

14

$$\mathbf{R}=\mathrm{\Gamma }{\mathbf{R}}_s{\mathrm{\Gamma }}^{\mathrm{H}}$$

In Eq. (14), $\mathrm{\Gamma }$ is a matrix composed of all amplitude and phase errors. ${\mathbf{R}}_s$ is the ideal visibility matrix. The formula for the autocorrelation function $F(i,j)$ received between two channels $i$ and $j$ is shown in Eq. (15).

15

$$F(i,j)={\sigma }_s^2{e}^{j({\vartheta }_{\mathit{is}}-{\vartheta }_{\mathit{js}})}$$

In Eq. (15), ${\vartheta }_{\mathit{is}}-{\vartheta }_{\mathit{js}}$ is the phase difference of the antenna corresponding to the $i$ and $j$ channels when the auxiliary signal source $s$ is incident. The autocorrelation matrix of the source $s$ affected by the amplitude and phase errors of the matrix elements is shown in Fig. 5.

Fig. 5 [Images not available. See PDF.]

Auxiliary source correlation matrix for AC algorithm

In Fig. 5, the main diagonal part of the autocorrelation matrix of the source is a real number and is related to the amplitude error of the matrix elements. Therefore, the formula for further solving the amplitude error and phase error of the signal source is shown in Eq. (16).

16

$$\left\{\begin{array}{c}{h}_i=\left|{\mathrm{\Gamma }}_i\right|=\sqrt{\frac{\left|\widehat{\mathbf{R}}(i,i)\right|}{\left|F(i,i)\right|}}=\frac{\sqrt{\left|\widehat{\mathbf{R}}(i,i)\right|}}{{\sigma }_s}\hfill \\ {[{\varphi }_1,{\varphi }_2,...,{\varphi }_K]}^{\mathrm{T}}={\mathrm{A}}^{-1}\ast {[{\varphi }_1-{\varphi }_2,{\varphi }_2-{\varphi }_3,...,{\varphi }_{K-1}-{\varphi }_K]}^{\mathrm{T}}\hfill \end{array}\right)$$

In Eq. (16), ${\mathrm{\Gamma }}_i$ represents the amplitude and phase errors of the matrix elements. Based on the above, the proposed AC algorithm solves the estimation values of amplitude and phase errors by using an auxiliary signal source with known precise azimuth and power in space, thereby achieving amplitude and phase error correction of the visibility function. This algorithm requires high azimuth accuracy from external auxiliary signal sources and can only perform offline calculations. Therefore, this study further proposes an ISC algorithm that does not require additional auxiliary signal sources. The specific calculation process is shown in Fig. 6.

Fig. 6 [Images not available. See PDF.]

ISC algorithm computation flow

In Fig. 6, the expression of the visibility matrix ${\mathit{T}}_k$ for the current iteration round solved by the ISC algorithm is shown in Eq. (17).

17

$${\mathit{T}}_k=\sum _{\alpha ,\beta }{\widehat{Q}}_k(\alpha ,\beta )e^{-j2\pi (p\alpha +q\beta )}$$

In Eq. (17), ${\widehat{Q}}_k(\alpha ,\beta )$ is the inverted brightness temperature for the current iteration. The amplitude error $h_{k+1}(i)$ of the SAR aperture matrix in the ISC algorithm is calculated as Eq. (18).

18

$$h_{k+1}(i)=\left|{\mathrm{\Gamma }}_{k+1}(i)\right|=\sqrt{\frac{\left|\mathbf{R}(i,i)\right|}{\left|{\mathit{T}}_k(0,0)\right|}}$$

The mathematical equation expression value of phase error is shown in Eq. (19).

19

$${\left[\begin{array}{c}{\vartheta }_{k+1}(1)-{\vartheta }_{k+1}(2)\\ {\vartheta }_{k+1}(2)-{\vartheta }_{k+1}(3)\\ ⋮\\ {\vartheta }_{k+1}(N-1)-{\vartheta }_{k+1}(N)\end{array}\right]}_{(N-1)\times 1}={\left[\begin{array}{cccc}1& -1& \cdots & 0\\ 0& 1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & -1\end{array}\right]}_{(N-1)\times N}\ast {\left[\begin{array}{c}{\vartheta }_{k+1}(1)\\ {\vartheta }_{k+1}(2)\\ ⋮\\ {\vartheta }_{k+1}(N)\end{array}\right]}_1$$

By solving the phase error equation, the estimated phase error value for the current round can be obtained, thereby generating the amplitude and phase error estimates as well as the cost function for the current iteration round. The specific calculation is shown in Eq. (20).

20

$$\left\{\begin{array}{c}{\mathrm{\Gamma }}_{k+1}(i)=h_{k+1}(i){\vartheta }_{k+1}(i)\hfill \\ O_{k+1}=\sum _{\alpha }\sum _{\beta }{∥{\widehat{Q}}_k(\alpha ,\beta )∥}^2\hfill \end{array}\right)$$

In Eq. (20), ${\vartheta }_{k+1}(i)$ is the estimated phase error value for the current iteration. $O_{k+1}$ is the cost function for the current round.

Results

To verify the effectiveness of the aperture imaging preprocessing algorithms AC and ISC based on ASPTM, this study conducted simulation experiments using Matlab R2016b. Firstly, the simulation performance of AC and ISC algorithms in 1D uniform line matrix was compared, and secondly, the performance of the two algorithms in 2D uniform T-matrix was compared. Finally, existing SAR imaging preprocessing methods were introduced for comparison.

Experimental verification of 1D uniform linear matrix

To verify the preprocessing ability of AC and ISC for inversion imaging, this study conducted algorithm simulation analysis using point source targets and rectangular spread source targets. This study selected point sources (1, 2, 3) with wave directions (0°, 45°, 60°) and brightness temperatures (200 K, 400 K, 300 K) for point source target inversion imaging experiments. The incoming wave direction of rectangular source 1 was − 10°–10°, and the brightness temperature was 300kK. The wave direction of rectangular source 2 was 20°–40°, and the brightness temperature was 400 K. The specific results are shown in Fig. 7.

Fig. 7 [Images not available. See PDF.]

Rectangular spread-source target validation in point-source target domains

Compared to Fig. 7a, the inversion imaging obtained by the AC after correcting amplitude and phase errors had a direction and brightness temperature that was closer to the point source set by the simulation observation. The ISC algorithm obtained a point source with higher brightness temperature. Both algorithms could obtain clear and distinguishable point source targets after processing. The rectangular spread source detection results in Fig. 7b showed that the brightness temperature of the rectangular spread source target obtained by the ISC algorithm after correcting amplitude and phase errors was closer to the spread source set by the simulation observation. However, the imaging quality of the two algorithms preprocessed could not be directly observed from the inversion imaging curve alone. Therefore, this study further compared the Root Mean Square Error (RMSE) and Peak Signal-to-Noise Ratio (PSNR) of the two algorithms, as shown in Table 1.

Table 1. Comparison of inversion imaging quality before and after error correction for amplitude and phase of 1D correction algorithm

Methods	Point source		Rectangular show source
	RMSE	PSNR (dB)	RMSE	PSNR (dB)
Pre-correction	43.30	22.35	163.79	13.01
AC	34.77	24.21	65.98	20.29
ISC	36.56	19.27	81.85	20.25

The comparison of RMSE and PSNR of the two algorithms in point source targets showed that after correction for amplitude and phase errors, the imaging quality was significantly improved. Among them, the RMSE of the AC decreased by 19.70% compared to before correction and by 4.90% compared to the ISC. The PSNR of AC increased by 8.33% compared to before calibration and 25.64% compared to the ISC. Compared to before calibration, the PSC algorithm had a worse PSNR, which might be due to its more complex calculation and the lack of additional auxiliary information sources, resulting in a lower PSNR. Based on the comparison results of rectangular expansion sources, in terms of RMSE, the AC algorithm had more advantages, while in terms of PSNR, the two algorithms had similar effects. The comparison with the uncorrected imaging could further confirm the effectiveness of the proposed AC and ISC. After comparing the amplitude and phase errors corrected by two algorithms, the RMSE of inversion imaging was significantly reduced, while the PSNR was significantly improved, resulting in improved imaging quality.

Experimental verification of 2D uniform T-matrix

To verify the processing performance of the proposed algorithm in 2D matrices, this study sets the simulation environment for inversion imaging as a Multi-Matrix Spread Source (MMSS) and complex contour subject for verification. All matrices use a uniform T-matrix with a SAR center frequency of 0.22 THz and 48 matrix elements. Among them, the imaging results of multi-matrix source inversion are shown in Fig. 8.

Fig. 8 [Images not available. See PDF.]

MMSS inversion imaging experiments with a comprehensive aperture 2D correction algorithm

A comparison of the inversion imaging results depicted in Fig.8b, c, and d with the simulated observation imaging quality presented in Fig. 8a was conducted. It was determined that the inversion imaging quality in the unprocessed Fig. 8b was the least optimal, as it lacked the capacity to discern multiple matrix sources within the image. After processing, the AC algorithm could clearly see images with multiple matrix sources, and the processing effect was more ideal than the ISC algorithm. In the inversion imaging obtained by the ISC algorithm in Fig. 8d, the contour of the MMSS was relatively blurry. This might be because it was mainly imaged by internal auxiliary information sources, and the calculation process was complex and cumbersome, resulting in too much impact information. The performance comparison of preprocessing algorithms in MMSS and subject scenarios is shown in Fig. 9.

Fig. 9 [Images not available. See PDF.]

Performance comparison of preprocessing algorithms in MMSS and examinee scenarios

In Fig. 9a, the research algorithm significantly reduced the RMSE value of inversion imaging in MMSS and subject scenarios. Among them, the RMSE of AC in MMSSs was reduced by 23.56% compared to ISC, but in the subject scenario, the RMSE increased by 5.24% compared to ISC. Comparing the PSNR corrected by the two algorithms, the AC algorithm was superior. This might be because AC calculation was relatively simple, and additional auxiliary sources had good performance in inverting imaging of MMSSs with simple contours. In the subject scenario, the contour information source was relatively complex, making it difficult for AC to perform effective calculations. Overall, research algorithms can effectively correct amplitude and phase errors, improve the quality of SAR inversion imaging, among which the AC algorithm performs better and the inversion imaging effect is more ideal.

Performance comparison of different preprocessing algorithms

To further confirm the effectiveness of the research algorithm, other error correction algorithms are introduced for performance comparison. The validation scenario employed in this study is consistent with the examinee inspection scenario, wherein the examinee is observed transporting hazardous materials, such as knives and simulated firearms. The parameter settings utilized in this study are analogous to those employed in the preceding study. The experimental platform is a personal computer end, with CPU model Intel Xeon Gold 6240R and main frequency 2.4 GHz. The specific inversion imaging results are shown in Fig. 10.

Fig. 10 [Images not available. See PDF.]

Comparison of inverse imaging results of different algorithms for subject scenes

Figure 10a shows the ideal inversion results of simulated observations, and Fig. 10b shows the uncorrected inversion imaging. Compared with the preprocessed inversion imaging in Fig.10b and c–f, the inversion imaging quality before correction is poor and the resolution is extremely low. Comparing the inversion imaging results of different preprocessing algorithms, AC and ISC are significantly better than other methods. Table 2 shows the performance comparison of different preprocessing algorithms.

Table 2. Comparison of inverse imaging results of different algorithms for subject scenes

Methods	RMSE	PSNR (dB)
Pre-correction	297.63	− 8.79
AC	38.46	33.25
ISC	42.18	29.98
Reference [22]	59.44	20.05
Reference [23]	205.60	6.35

In Table 2, AC and ISC are significantly superior to the other two methods in terms of inversion imaging RMSE and PSNR. The RMSE of AC is the lowest among all methods, only 38.46, and its PSNR value has increased by an average of 35.11% compared to other methods. The RMSE value of ISC decreases by an average of 68.42% compared to the other two methods. Overall, the proposed error correction preprocessing algorithm has significant advantages in amplitude and phase error processing, and the inversion imaging quality after correction is significantly improved.

Computational cost analysis of the proposed method

Finally, the study further carries out the computational cost analysis of the proposed method. The AC algorithm relies on an external auxiliary signal source, which may increase the complexity and cost of the system in practical applications. While the ISC algorithm does not require an external auxiliary signal source, but its computational process is more complex. Consequently, the study validates the computational time comparison between the two. The experimental environment is still Intel Xeon Gold 6240R processor with 2.4 GHz and 128 GB of memory. The experimental data are SAR imaging datasets of different sizes, and the dataset sizes are 100 × 100, 500 × 500, and 1000 × 1000 pixels, respectively. The experimental results are shown in Table 3.

Table 3. Comparison of computation time for different algorithms

Data set size	AC calculation time (s)	ISC calculation time (s)
100 × 100	0.45	0.78
500 × 500	12.34	23.56
1000 × 1000	45.66	89.23

In Table 3, the computation time of both the AC and ISC algorithms increases significantly as the size of the dataset increases. The computation time of the ISC algorithm is close to twice that of the AC algorithm on a large-scale dataset of 1000 × 1000 pixels. This indicates that the ISC algorithm is computationally more expensive when dealing with large-scale data. This may limit its application in scenarios with high real-time requirements. However, relatively speaking, the proposed method still has excellent ability in improving the quality of SAR imaging.

Conclusion

To improve the quality of SAR inversion imaging, this study proposes an aperture radar-assisted technology based on AMM and SCC. By exploring and analyzing AMM and SCC, two error correction algorithms AC and ISC are proposed based on the traditional amplitude and phase error correction ideas. The AC algorithm is suitable for scenarios with an external auxiliary signal source and requires high-precision correction, especially when dealing with simple contour targets. The primary advantage of this approach is that the calculation process is relatively straightforward and the correction accuracy is high. However, the efficacy of this method is contingent upon the availability of an external auxiliary signal source, and it is particularly well-suited for offline calculation. The ISC algorithm is well-suited for scenarios lacking an external auxiliary signal source and necessitates real-time processing. It demonstrates particular efficacy in the context of complex contour targets. A notable advantage of this approach is that it does not necessitate an external auxiliary signal source, is adaptable, and is well-suited for real-time correction in complex environments. However, it should be noted that the calculation process is more intricate.

The results showed that the RMSE and PSNR of the AC algorithm in 1D imaging decreased by 19.70–59.72% and increased by 8.33–55.96% compared to before correction. The RMSE of ISC decreased by 15.57–50.03% compared to before correction, and the PSNR increased by 13.78–55.65%. Compared with other methods, the imaging quality of AC and ISC was significantly improved, with an RMSE value of only 38.46 for AC, which was an average reduction of 68.42% compared to other methods. The PSNR value of AC was as high as 33.25 dB, which was an average increase of 35.11% compared to other methods. The imaging quality of AC was the best among all methods, followed by the ISC. Research has shown that the design of amplitude and phase correction methods based on AMM and SCC is reasonable and effective. It is feasible to perform error correction in SAR imaging based on traditional amplitude and phase error correction methods. The proposed AC and ISC algorithms have positive application significance in the field of SAR synthetic aperture imaging.

However, this study only explored the correction of amplitude and phase errors. Future research will consider combining intelligent algorithms such as deep learning to meet the high-precision requirements of SAR detection. Furthermore, the AC and ISC algorithms proposed in the study are computationally expensive, particularly when dealing with large-scale data, and the computation time increases considerably. This may impose limitations on their practical application in scenarios characterized by stringent real-time requirements. In the future, parallel computing or distributed computing techniques will be introduced to reduce the computational cost of AC and ISC algorithms and improve their real-time performance in large-scale data processing.

Author contributions

JY provided the concept and wrote the draft manuscript; YF collected and analyzed the data; SQY validated the research and revised the paper critically. All authors approved this submission.

Funding

The research is supported by Hubei Province education science planning project: Research on the joint training path of "school, village and enterprise" for family farm management talents in Hubei Province under the perspective of gray association and symbiosis (2022GB207).

Data availability

All data generated or analyzed during this study are included in this article.

Declarations

Ethics approval and consent to participate

An ethics statement was not required for this study type, no human or animal subjects or materials were used.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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