An Efficient RMA with Chunked Nonlinear Normalized Weights and SNR-Based Multichannel Fusion for MIMO-SAR Imaging

Highlights

What are the main findings?

• The article proposes the chunked nonlinear normalized weights technique to suppress artifacts and noise.
• The article proposes the SNR-based multichannel fusion technique to solve the issue of missing target structures.

What is the implication of the main finding?

• The proposed chunked nonlinear normalized weights technique effectively suppresses artifacts and noise in reconstructed images.
• The proposed SNR-based multichannel fusion technique effectively preserves target structure while efficiently utilizing multichannel signals to enhance system robustness.

Abstract

Millimeter-wave multiple-input multiple-output synthetic aperture radar (MIMO-SAR) has been widely used in many scenarios such as geological exploration, post-disaster rescue, and security inspection. When faced with large complex scenes, the signal suffers from distortion problems due to amplitude-phase nonlinear aberrations, resulting in undesired artifacts. Many previous studies eliminate artifacts but result in missing target structures. In this paper, we propose to use chunked nonlinear normalized weights in conjunction with signal-to-noise ratio-based (SNR-based) multichannel fusion to address the above-mentioned problems. The chunked nonlinear normalized weights make use of the scene’s characteristics to separately perform the optimization of different regions of the scene. This approach significantly mitigates the effects of amplitude-phase distortion on signal quality, thereby facilitating the effective suppression of noise and artifacts. Applying SNR-based multichannel fusion solves the problem of missing target structures caused by the chunked weights. With the proposed techniques, we can effectively suppress artifacts and noise while maintaining the target structures to enhance the robustness of system. Based on practical experiments, the proposed techniques achieve the image entropy (IE) value, which reduces by approximately 1, and the image contrast (IC) value is increased by approximately 2~4. Furthermore, the computational time is only about 1.3 times that needed by the latest reported algorithm. Consequently, imaging resolution and system robustness are improved by implementing these techniques.

1. Introduction

Millimeter-wave synthetic aperture radar (SAR) has emerged as a prominent area of research due to its distinctive benefits in the detection and imaging of near-field objects. Millimeter wave is harmless to human body [1] and can work effectively in undesirable weather conditions such as rain and fog [2]. Therefore, millimeter-wave radar has great potential for application in various fields such as non-destructive testing [3,4], security monitoring [5], medical diagnosis [6,7], vehicle sensing, and ground probing [8,9]. Multiple-input multiple-output synthetic aperture radar (MIMO-SAR) extends the effective aperture through the spatial diversity of multiple antennas. It can significantly improve the imaging resolution and enhance the robustness. With its high resolution and wide coverage capability, MIMO-SAR shows promise for a wide range of applications in many fields [10].

Millimeter-wave MIMO-SAR is typically employed in applications that necessitate real-time imaging and high-precision target detection. It is necessary to choose the appropriate imaging algorithm. The back projection (BP) algorithm was initially introduced in the field of computerized tomography [11,12,13,14,15] as a non-geometrically approximated time-domain technique [16]. However, due to iterative computation, it requires high computational complexity when dealing with a huge amount of data. Additionally, it is less robust against noise and streak artifacts [17]. Large scene scanning is gradually gaining attention from researchers to deal with the problem of huge amount of data. Thereafter, compressed sensing (CS) emerged to break the Nyquist sampling limitation to a certain extent by using a small amount of sparse data for efficient imaging [18,19]. These methods greatly reduce the amount of data acquired and speed up the imaging process with degraded imaging resolution. In addition, structured prior is also helpful for sparse imaging. The distribution of targets often has certain structural patterns, and the prior information can guide the algorithm to better capture the true target structure [20,21]. Another heavily used imaging approach is the range migration algorithm (RMA), originating from seismic migration techniques [22]. RMA introduces the Fourier technique during the imaging process to greatly reduce the computational burden and maintain a high resolution. However, it introduces harmful artifacts in the image reconstruction due to the use of STOLT interpolation in the distance direction [23,24].

RMA, as one of the most popular algorithms in near-field imaging, has been improved by numerous researchers from different perspectives. They mainly focused on three key aspects: minimizing computational complexity, enhancing image resolution, and reducing acquisition time. In terms of minimizing computational complexity, Zhu et al. [25] proposed three-dimensional (3D) RMA for MIMO-SAR imaging. It greatly reduces the computational effort by implementing the 3D fast Fourier transform (FFT) of the reflectivity through reduced dimensional superposition. In 2024, Lin et al. [26] proposed the range scaling algorithm (RSA) to address the challenges of real-time imaging. This algorithm eliminates the need for interpolation and accumulation operations by relying solely on matrix operations and FFT to achieve image reconstruction. However, most reported studies sacrifice image quality for computational speed. To improve the imaging resolution, Tan et al. [27] indicates that the linear components of range cell migration (RCM) and range offset (RO) within single-input multiple-output (SIMO) and MIMO data can be corrected using 3D stolt interpolation. Additionally, these components can be further compensated with phase multiplication and image domain interpolation to account for range errors. The researchers also attempted to improve microwave imaging resolution using deep learning (DL). In 2023, Rostami et al. [28] proposed the application of DL technique for image reconstruction. It is suitable for broadband imaging in diverse scenes. In this method, the network can autonomously learn to integrate the mutual information between these modalities. In 2022, Wang et al. [29] proposed iterative shrinkage thresholding network known as RMIST-Net by utilizing a range migration kernel. This network is designed to generate high-quality 3D images based on range-focused echoes while maintaining a rapid computational speed. DL techniques require huge amount of data for model trainings. Because collecting data in advance for training requires additional time, it is not suitable for scenarios whose statistical environments change frequently. Reducing acquisition time is also an essential aspect of real-time imaging. Array antenna is the most effective method for reducing acquisition time. In 2016, Zhu et al. [30] utilized transmitting and receiving units to construct virtual array units and proposed a novel near-field MIMO-SAR imaging algorithm based on distance compensation. It divides the large scene into manageable blocks and generates the final image by weighting the segmented scenes. This algorithm effectively reduces the computational load associated with distance compensation. In 2019, M.E.Yanik et al. [31] proposed a novel frequency domain imaging method for SIMO system that eliminates the need for interpolation in the frequency domain. This method generates frequency-specific sub-images through fast Fourier inversion and phase demodulation, ultimately coherently superimposing them to produce the final image. The use of array acquisition poses challenges to data reorganization and algorithm robustness. Especially for large complex scenes, erroneous data processing may lead to degradation of imaging accuracy. This problem might be solved by compensating signal-utilizing multichannel data.

The well-studied RMA still needs further improvement for more challenging applications. When facing large complex scenes, the multipath scattering effect and inter-target electromagnetic masking phenomenon are aggravated. Antenna-equivalent mechanisms based on the center-phase approximation have the potential to produce spurious scattering centers, which leads nonlinear amplitude-phase signal distortions to make the target scattering characteristics severely distorted. The imaging quality of the center and edge portions are not synchronized, which will result in artifacts. Conventional interpolation methods and compensation mechanisms are difficult to optimize imaging results based on the characteristics of the scene itself. This may result in artifacts not being completely removed. It is necessary to develop new technology to remove the artifacts efficiently.

MIMO arrays are the inevitable choice for large complex scenarios and can effectively reduce acquisition time. Array acquisition contains complex multichannel data. High level of data correction and spatial alignment is the prerequisite for high-quality images, and incorrect data processing may result in reduced imaging resolution. Therefore, some studies [32] use single channel for imaging, making it difficult to maximize the benefits of array acquisition. Furthermore, some studies [31,33] simply combine multichannel data, which can also have an impact on the quality of the data. In addition, some methods [34,35], while removing artifacts and noise, can also lead to the loss of the target structures. Therefore, it is necessary to develop new multichannel technology to combine data more reasonably and utilize the complementarity among data to maintain the integrity of the target structures, which can enhance the robustness of the algorithms.

Based on the above discussion, we propose an improved RMA applicable to MIMO arrays. The targets are sampled with a MIMO array based on related studies in [36,37]. We built the millimeter-wave imaging system based on the commercial radar IWR1443BOOST from Texas Instruments. The slideway hardware and control program are designed independently and connected the mmWave studio with the control terminal. We can easily and freely adjust the experimental parameters. Then, the signal is processed by chunked nonlinear normalized weights from multichannel data. Compared to previous studies, the proposed imaging algorithms can significantly improve the image resolution in processing MIMO array data without much more time. At the same time, the algorithms maximize the utilization rate of multichannel data after array acquisition to enhance the robustness of the system.

The main contributions of this paper are as follows:

• We propose to add chunked nonlinear normalized weights to the echo signals. This technique uses the target scene’s properties to perform targeted enhancement of the received signals to suppress noise and artifacts in the reconstructed images.
• To cope with the possible problem of missing target structures due to chunked nonlinear normalized weights when imaging the weakly scattering and low-contrast targets, signal-to-noise ratio-based (SNR-based) multichannel fusion technique is proposed. This technique assigns weights to the data of each channel based on the SNR and selectively fuses the data of multiple channels. Such an arrangement effectively solves the problem of false attenuation and utilizes the data more efficiently to improve the robustness of the system.
• The proposed techniques are compared with the other methods including the original BP, the original RMA, the AHI-LFM, and the RSA in multiple scenes. The experimental results show that the proposed techniques can effectively suppress noise and artifacts to improve the imaging resolution without significantly increasing the computational complexity.

The rest of this article is organized as follows. Section 2 describes the signal model and the traditional RMA. Section 3 presents the theories related to the proposed algorithms. The improved algorithms include the determination of the chunked region, the construction of nonlinear weights, and multichannel fusion. Section 4 conducts experimental validation. Section 5 gives the discussion about the experiments. Section 6 summarizes the paper.

2. Signal Model and Traditional RMA

2.1. Signal Model

The signal transmitted by the transmitting antenna, upon interaction with the target, generates an echo to be subsequently received by the receiving antenna. Different types of signals require different perspectives to be considered in a necessary signal model. Figure 1 shows the schematic diagram of the MIMO antenna array and antenna center phase approximation.

The linear FMCW emitted by the radar can be expressed as

$$s_t\left(x,y,z_0\right)=a_1\cdot \exp [j2\pi \left(f_{c}t+{\displaystyle \frac{K}{2}}{t}^2\right)]$$

(1)

where $a_1$ represents the signal amplitude, $f_c$ refers to the carrier frequency, $t$ is the slow time, $K=B/T$ is the FM slope, $B$ is the bandwidth, and $T$ is the modulation period. Assuming a single-point target, the received echo signal can be expressed as

$$s_r\left(x,y,z^{\prime }\right)=a_2\cdot \sigma \cdot \exp \left[j2\pi \left(f_c\cdot \left(t-\tau \right)+{\displaystyle \frac{K}{2}}\left(t-\tau {)}^2\right)\right)\right]$$

(2)

where $\sigma =p/R^2$ is the reflectivity function of the target after distance attenuation [5,38], $p$ is the real reflectivity of the target, $R$ is the round-trip distance from the antenna $\left(x,y,z_0\right)$ to the target $\left(x^{\prime },y^{\prime },z^{\prime }\right)$, $\tau =\left(R_t+R_r\right)/c$ is the echo delay time. $c$ is the speed of light, and $R_t$ and $R_r$ are the round-trip distance between the transmitter antenna $\left(x_t,y_t,z_0\right)$ and receiver antenna $\left(x_r,y_r,z_0\right)$, respectively, to target $\left(x^{\prime },y^{\prime },z^{\prime }\right)$, expressed by

$$R_t=\sqrt{{(x_t-x^{\prime })}^2+{(y_t-y^{\prime })}^2+{(z_0-z^{\prime })}^2}$$

(3)

$$R_r=\sqrt{{(x_r-x^{\prime })}^2+{(y_r-y^{\prime })}^2+{(z_0-z^{\prime })}^2}$$

(4)

The received and transmitted signals are mixed to obtain the intermediate frequency (IF) signal,

$$s_{\mathrm{IF}}\left(x,y,z\right)=s_t\left(x,y,z_0\right)\cdot {s_r}^{*}\left(x,y,z^{\prime }\right)$$

(5)

Putting (1) and (2) into (5), we obtain

$$s_{\mathrm{IF}}\left(x,y,z\right)=a\cdot \sigma \cdot \exp [j2\pi \left(f_c\tau +K\tau t-{\displaystyle \frac{K}{2}}{\tau }^2\right)]$$

(6)

where $K\tau t$ denotes the time delay relationship between target and radar, and $-K{\tau }^2/2$ is the phase offset, which can be neglected [39]. Since each distance corresponds to a delay time, (6) can be expressed as [40].

$$\begin{array}{cc}\hfill s_{\mathrm{IF}}\left(x,y,z\right)& =a\cdot \sigma \cdot \exp \left[j2\pi \left(f_c\tau +K\tau t\right)\right]\hfill \\ & =a\cdot \sigma \cdot \exp \left[j2\pi \tau \left(f_c+Kt\right)\right]\\ & =a\cdot \sigma \cdot \exp \left[j2\pi {\displaystyle \frac{R_t+R_r}{c}}\left(f_c+Kt\right)\right]\\ & =a\cdot \sigma \cdot \exp (j2\pi {\displaystyle \frac{R_t+R_r}{c}}{f}_i)\\ & =a\cdot \sigma \cdot \exp (j2Rk)\end{array}$$

(7)

where $k=2\pi f_i/c$ is the number of waves corresponding to a single frequency $f_i$ after delay.

2.2. Traditional RMA

In the previous section, we derived the form of the signal received when assuming a single target point. The antenna moves in the X-Y plane and the echo signal received at each observation position can be viewed as the scattering points acting together [1]. When the target is not a single point, the received signal can be expressed as

$$s\left(x,y,\omega \right)={\displaystyle \iiint a\cdot \sigma \left(x^{\prime },y^{\prime },z^{\prime }\right)\cdot \exp (-j2kR)\mathrm{d}{x}^{\prime }\mathrm{d}{y}^{\prime }\mathrm{d}{z}^{\prime }}$$

(8)

For small scene areas, the signal amplitudes $a$ can be regarded to be constant due to the small variation in the distance from the antenna to the target. It has trivial influence on the imaging and is ignored in the subsequent derivation process. According to Weyl’s related research, the spherical wave can be expressed as superposition of plane waves [41,42],

$$e^{-j2kR}=e^{-j2k\sqrt{{(x^{\prime }-x)}^2+{(y^{\prime }-y)}^2+{(z^{\prime }-z_0)}^2}}={\displaystyle \iint e^{j{k}_x\left(x-x^{\prime }\right)+j{k}_y\left(y-y^{\prime }\right)+j{k}_z\left(z_0-z^{\prime }\right)}\mathrm{d}{k}_x\mathrm{d}{k}_y}$$

(9)

$$k_x^2+k_y^2+k_z^2=4k^2=4{({\displaystyle \frac{\omega }{c}})}^2,k_z=\sqrt{{(2k)}^2-k_x^2-k_y^2}$$

(10)

where $k_x$, $k_y$, and $k_z$ correspond to spatial frequencies in the $x$, $y$, and $z$ directions. Respectively, $k_x$ and $k_y$ correspond to Fourier transforms in the $x$ and $y$ directions. Bringing (9) into (8) yields as follows:

$$s\left(x,y,\omega \right)={\displaystyle \iiint \sigma \left(x^{\prime },y^{\prime },z^{\prime }\right){\displaystyle \iint e^{j{k}_x\left(x-x^{\prime }\right)+j{k}_y\left(y-y^{\prime }\right)+j{k}_z\left(z_0-z^{\prime }\right)}\mathrm{d}{k}_x\mathrm{d}{k}_y}\mathrm{d}{x}^{\prime }\mathrm{d}{y}^{\prime }\mathrm{d}{z}^{\prime }}$$

(11)

Reordering this quintuple integral and applying a Fourier inverse transform yields the final expression of the reflectance function,

$$\sigma \left(x^{\prime },y^{\prime },z^{\prime }\right)={\mathrm{IFT}}_{3\mathrm{D}}\left\{{\mathrm{FT}}_{2\mathrm{D}}\left[s\left(x,y,\omega \right)\right]e^{-j{k}_z{z}_0}\right\}{=\mathrm{IFT}}_{3\mathrm{D}}\left[S\left(k_x,k_y,\omega \right)e^{-j\sqrt{{(2k)}^2-k_x^2-k_y^2}{z}_0}\right]$$

(12)

which is the RMA based on matched filtering [41] used to reconstruct the image for uniform data. In the actual imaging process, the data is uniformly sampled in $x$ and $y$ directions. After Fourier transform, it is uniformly arranged in $k_x$ and $k_y$ directions; however, the data in $k_z$ direction is non-uniformly arranged. To be able to carry out the 3D inverse Fourier transform in (12), the stolt interpolation is often used for $k_z$ to reorganize the data so that the data is uniformly distributed. Based on the above analysis, (12) can be finally expressed as

$$\sigma \left(x^{\prime },y^{\prime },z^{\prime }\right)={\mathrm{IFT}}_{3\mathrm{D}}\left\{\mathrm{Stolt}\left[{\mathrm{FT}}_{2\mathrm{D}}\left[s\left(x,y,\omega \right)\right]e^{-j{k}_z{z}_0}\right]\right\}{=\mathrm{IFT}}_{3\mathrm{D}}\left\{\mathrm{Stolt}\left[S\left(k_x,k_y,\omega \right)e^{-j\sqrt{{(2k)}^2-k_x^2-k_y^2}{z}_0}\right]\right\}$$

(13)

The method based on the center-phase approximation applied in the traditional RMA is not suitable for all imaging scenarios [43]. When the antenna array is close to the target or the target size is larger than the SAR resolution, the distances from the equivalent antenna to different locations on the target have large variations. The distance variation leads to nonlinear aberrations that inevitably introduce artifacts to degrade the imaging resolution. The effects of artifacts and noise can be mitigated by adding external compensation to the signal. This requires choosing the right compensation methods with additional computational resources. There have not been any studies in the literature using the signal’s characteristics for targeted adjustment. By using the proposed chunked nonlinear normalized weighting and SNR-based multichannel fusion technique, the researchers can effectively remove artifacts while maintaining target structures. The relevant theories will be explained in Section 3 below.

3. Improved Algorithm

The chunked nonlinear normalized weights and SNR-based multichannel fusion is shown in Figure 2. The yellow block performs the regular processing steps. The green area presents the chunked nonlinear normalized weights. The method calculates weights for different regions by chunking the raw data in a specific way. The method determines the type of the chunked region to remove signal noise and image artifacts. The blue area performs the SNR-based multichannel fusion. As mentioned above, chunked weight is applied to each single-channel data within the multichannel. The multichannel data fusion is implemented based on the SNR of individual channels. Due to the full consideration and reasonable fusion of data from multiple channels, this method can effectively solve the problem of missing target structures and improve the system robustness.

3.1. Chunked Nonlinear Normalized Weights

Assume that the imaging matrix obtained after preliminary signal processing can be expressed as $R(x,y),(x,y)\in \mathrm{\Omega }$, where $\mathrm{\Omega }$ denotes the imaging domain. First, it is necessary to discriminate the target region, the transition region, and the background region based on mathematical statistics. Processing $R(x,y)$ with a fixed-size window $W$, the size can be adjusted according to the size of the data matrix. The corresponding mean ${\mu }_{x,y}$ and standard deviation ${\sigma }_{x,y}$ are computed at the center of each window,

$${\mu }_{x,y}={\displaystyle \frac{1}{\left|W\right|}}{\displaystyle \sum _{(i,j)\in W(x,y)}R(i,j)}$$

(14)

$${\sigma }_{x,y}=\sqrt{{\displaystyle \frac{1}{\left|W\right|}}{\displaystyle \sum _{(i,j)\in W(x,y)}{\left[R(i,j)-{\mu }_{x,y}\right]}^2}}$$

(15)

where $W(x,y)$ is the window centered on $(x,y)$, and $\left|W\right|$ is the number of the data in the window. Next, we define threshold function $u$ and $v$ to construct two thresholds,

$$T_{\mathrm{h}}(x,y)={\mu }_{x,y}+u{\sigma }_{x,y}$$

(16)

$$T_{\mathrm{l}}(x,y)={\mu }_{x,y}-v{\sigma }_{x,y}$$

(17)

The regions are defined as follows based on the constructed thresholds:

• Target region (T):

$$W(x,y)\ge T_{\mathrm{h}}(x,y).$$

(18)

• Background region (B):

$$W(x,y)\le T_{\mathrm{l}}(x,y).$$

(19)

• Transition region (M):

$$T_{\mathrm{l}}(x,y)<W(x,y)<T_{\mathrm{h}}(x,y).$$

(20)

These regions are to be differentially weighted. The aim is to achieve better signal retention in the target region, noise suppression in the background region, and smooth processing in the transition region. Then, the echo signal to obtain the original normalization factor is calculated by

$$\alpha (x,y)={\displaystyle \frac{R(x,y)}{\max \left|R(x,y)\right|}}$$

(21)

which has the same dimensions as $R(x,y)$. Next, different weighting factors are assigned to three corresponding regions,

$$\begin{array}{ll}{A}_{\mathrm{T}}(x,y)={\displaystyle \frac{T_{\mathrm{h}}(x,y)}{T_{\mathrm{l}}(x,y)}},\hfill & R(x,y)\ge T_{\mathrm{h}}(x,y)\hfill \\ A_{\mathrm{B}}(x,y)={\displaystyle \frac{T_{\mathrm{l}}(x,y)}{T_{\mathrm{h}}(x,y)}},\hfill & R(x,y)\le T_{\mathrm{l}}(x,y)\hfill \\ A_{\mathrm{M}}(x,y)=\lambda A_{\mathrm{T}}(x,y)+(1-\lambda )A_{\mathrm{B}}(x,y),\hfill & T_{\mathrm{l}}(x,y)<R(x,y)<T_{\mathrm{h}}(x,y)\hfill \end{array}$$

(22)

where $A_{\mathrm{T}}(x,y)$ is the nonlinear normalized weight factor of the target region, $A_{\mathrm{B}}(x,y)$ is the nonlinear normalized weight factor of the background region, and $A_{\mathrm{M}}(x,y)$ is the nonlinear normalized weight factor of the transition region. Based on the position of $W(x,y)$ within $R(x,y)$, the factor $\lambda$ is calculated by

$$\lambda ={\displaystyle \frac{W(x,y)-T_{\mathrm{l}}(x,y)}{T_{\mathrm{h}}(x,y)-T_{\mathrm{l}}(x,y)}}$$

(23)

As $W(x,y)$ converges to $T_{\mathrm{h}}$, i.e., close to the target region, $\lambda \to 1$, and as $W(x,y)$ converges to $T_{\mathrm{l}}$, i.e., close to the background region, $\lambda \to 0$. After the computation of the chunked nonlinear normalized weights is obtained, the echo signal matrix can be expressed as

$$R_{\mathrm{re}}(x,y)=R(x,y)\odot A(x,y)\odot \alpha (x,y)$$

(24)

where $\odot$ is the Hadamard product that multiplies the matrix element-by-element. The reflectivity function is obtained by putting $R_{\mathrm{re}}(x,y)$ into (13). With this technique, the more energetic echo signal is enhanced and the less energetic echo signal is suppressed.

The proposed chunked nonlinear normalized weights technique offers several advantages. By tailoring weighting strategies according to local statistical characteristics of the image, this technique achieves more precise enhancement and suppression effects compared to a single global operator. Furthermore, the technique exhibits universality, as its performance does not rely on specific assumptions about artifact morphology but leverages intrinsic image features. Therefore, it can maintain robustness across diverse scenarios.

Figure 3 demonstrates how the chunked nonlinear normalized weights are used. The artifacts around the image after chunked nonlinear normalized weights are effectively suppressed. However, some target areas are attenuated to make the target structure missing, as shown in the yellow dashed box.

3.2. SNR-Based Multichannel Fusion

Similarly to the previously reported studies [34,35], the direct application of the chunked nonlinear normalization weights potentially has the problem of attenuating weak scattering and low-contrast targets resulting in missing target structures. Some previous studies [31,33] also suffered from inefficient utilization of multichannel data. The problem of false attenuation can be remedied by multichannel data fusion when data are collected using MIMO arrays.

Firstly, the data from each receiving channel is preprocessed and phase-compensated separately to ensure the spatial alignment of the data across different channels. We use the conventional calibration method. Data is collected using a corner reflector or a smooth iron plate as an ideal scatterer. The data is used to construct a calibration matrix to calibrate the different channels. Then, the weights of each channel are calculated, and the comprehensive utilization of multichannel data is realized through weighted fusion. SNR can be calculated without reference image [44], and variance is sensitive to noise fluctuations. The variance method [45,46] is selected to calculate the SNR which is expressed as the inverse of the coefficient of variation (CV), or equivalently, it is defined as the ratio of the standard deviation to the mean. By calculating the mean ${\mu }_k$ and standard deviation ${\sigma }_k$ from the different channels of data $R_k(x,y), k=1,2,\dots ,L$, $SN{R}_k$ can be expressed as

$$SN{R}_k={\displaystyle \frac{{\mu }_k}{{\sigma }_k}}$$

(25)

$${\mu }_k={\displaystyle \frac{1}{N}}{\displaystyle \sum _{(x,y)\in \mathrm{\Omega }}{R}_k(x,y)}$$

(26)

$${\sigma }_k=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{(x,y)\in \mathrm{\Omega }}{\left[R_k(x,y)-{\mu }_k\right]}^2}}$$

(27)

where $N$ is the total number of pixels of the echo signal matrix and $L$ is the total number of channels. Weights are computed for each channel based on the SNR,

$${\omega }_k={\displaystyle \frac{SN{R}_k}{{{\displaystyle \sum }}_{j=1}^{L}SN{R}_j}}$$

(28)

Then, the signal after multichannel fusion can be expressed as

$$\overline{R}(x,y)={\omega }_k{R}_k(x,y), k=1,2,\dots ,L$$

(29)

According to (29), the signals and normalization factors under SNR-based multichannel fusion can be combined into the following matrix form,

$$R_{\mathrm{MC}}(x,y)=\left[{\overline{R}}_1,{\overline{R}}_2,\dots ,{\overline{R}}_L\right]$$

(30)

$$\mathrm{A}=\left[A_1,A_2,\dots ,A_L\right]$$

(31)

where $R_{\mathrm{MC}}(x,y)$ is the signal after sorting and combining data from multiple channels, and $\mathsf{A}$ is the set of normalization factors after SNR-based multichannel fusion. Then, the signal of each channel after adding the chunked nonlinear normalization factor is

$$R_{\mathrm{A}}(x,y)=\mathrm{A}\odot R_{\mathrm{MC}}(x,y)$$

(32)

The final signal can be expressed as

$$R(x,y)={\displaystyle \sum _{i=1}^L{R}_{\mathrm{A}}(x,y)}={\displaystyle \sum _{i=1}^L\mathrm{A}\odot R_{\mathrm{MC}}(x,y)}$$

(33)

With (33), more fine-tuning is also possible. If the number of channels is not large, manual screening can be performed to eliminate channels with small SNR and retain only those with high SNR. Thresholds can also be set to automatically select channels within the target range. The final reconstruction is performed for $R(x,y)$ by regular RMA processing.

Figure 4 shows the schematic diagram of the SNR-based multichannel fusion. The left image is mis-attenuated, and the right one is obtained after the SNR-based multichannel fusion. This technology automatically assigns higher credibility to channels with superior SNR, ensuring fusion results remain unaffected by low-quality data contamination. It effectively mitigates signal attenuation in weakly scattering regions, thereby preserving target structural integrity.

4. Experiments and Results

4.1. Experimental Indicators and Parameters

In this section, we verify the effectiveness of the proposed algorithms compared to other reported ones through simulation and practical experiments. Comparisons are made among the original BP, the original RMA, the AHI-LFM [47], the RSA [26], and the two algorithms proposed in this paper. It should be noted that the above algorithms use single-channel data except for the final multichannel fusion algorithm. The practical experimental parameters are shown in

Table 1. Experiment parameters setting.

Parameters	Value
Frequency	77~81 GHz
Antenna number	2T4R
Sample point	256
Sample rate	5000 ksps
Slope	70.295 MHz/μs
Horizontal step (Y)	8 mm
Vertical step (X)	1 mm

.

In addition to qualitatively analyzing the target reconstructed images, we also quantitatively analyze the imaging results using image entropy (IE) and image contrast (IC). IE characterizes the uncertainty or complexity of the distribution of image pixel values. In general, the larger the IE value is, the more even the distribution of the pixel values is and the more information is available. However, larger IE may also mean the presence of noise or artifacts in the image. IC characterizes the difference between high and low brightness in the target region. The larger the value, the more obvious the distinction and the better the imaging results. In addition, considering the acquisition of reference images, we also chose to use image correlation and PSNR as metrics for the simulation experiment. The IE, IC, CORR, and PSNR are defined by

$$\mathrm{IE}=-{{\displaystyle \sum }}_{i=1}^m{{\displaystyle \sum }}_{j=1}^n{\displaystyle \frac{{\left|g_{i,j}\right|}^2}{{‖G‖}_F^2}}\ln {\displaystyle \frac{{\left|g_{i,j}\right|}^2}{{‖G‖}_F^2}}$$

(34)

$$\mathrm{IC}=\sqrt{{\displaystyle \frac{mn}{{‖G‖}_F^2}}{{\displaystyle \sum }}_{i=1}^m{{\displaystyle \sum }}_{j=1}^n{\left|g_{i,j}\right|}^4-1}$$

(35)

$$\mathrm{CORR}={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n\left|\frac{g_{i.j}}{\sqrt{{\displaystyle {\sum }_{i=1}^m{\displaystyle {\sum }_{j=1}^n{g}_{i,j}^2}}}}·\frac{\overline{g_{i.j}}}{\sqrt{{\displaystyle {\sum }_{i=1}^m{\displaystyle {\sum }_{j=1}^n{\overline{g_{i.j}}}^2}}}}\right|}}$$

(36)

$$\mathrm{PSNR}=10{\log }_{10}\left[{\displaystyle \frac{\max {(g_{i,j})}^2}{\mathrm{MSE}(g_{i,j},\overline{g_{i,j}})}}\right]$$

(37)

4.2. Simulation Experiment

To verify the effectiveness of the proposed algorithms, we designed a simulation experiment with a simulated “lemon” target, which consists of randomly distributed point targets. We use the toolbox [48] developed by Smith team for the simulation experiment. We have adjusted the parameters of the toolbox accordingly. The frequency is 77–81 GHz, and the number of antennas is 2T4R. The sample points are 256, the sample rate is 5000 ksps, and the slope is 70.295 MHz/μs. The difference from the practical experiments lies in the vertical antenna arrangement direction in the simulation experiment. The distance from the target to the antennas is 0.1 m. The antenna setup and scene layout are shown in Figure 5. The red “x” represents transmitting antenna and blue ”o” is receiving antenna, as shown in Figure 5a. The simulation experiment used MIMO plane scanning system. The images reconstructed by different imaging algorithms are shown in Figure 6.

Since a relatively ideal scene was used, all methods can reconstruct the simulated target well as shown in Figure 6. Especially the original BP and the last three methods show the contour of the target well with fewer artifacts, since the chunked nonlinear normalized weights can be treated separately for each of the three divided regions. It is observed that although the artifacts are significantly reduced, the energy of target region is also partially attenuated. When the SNR-based multichannel fusion technique is applied, different channels carry different information in the same region. The data from one of the channels can be used to compensate for the signals that are mistakenly attenuated in the other channel. From the results, the energy of the reconstructed image is restored, and the noise is well-suppressed. The metrics of the simulation experiment are shown in

Table 2. Indexes for reconstruction of simulation experiment.

	Original BP	Original RMA	AHI-LFM	RSA	Chunked Nonlinear Normalization	SNR-Based Multichannel Fusion
IE	1.6748	2.2370	2.1113	1.8723	1.5391	1.5335
IC	8.7311	7.1208	7.5966	8.3009	10.5111	9.9105
CORR	0.9414	0.8015	0.8635	0.9244	0.9861	0.9724
PSNR	35.25	27.51	29.16	32.46	34.99	35.46
Time	25 s	1.47 s	4.61 s	7.52 s	2.08 s	8.12 s

, where the IE value decreases gradually, and the IC value increases gradually. We choose the “lemon” graphic imported when generating the point target as the reference image. By comparing the image correlation and PSNR, the proposed algorithms have the relatively highest value indicating the best reconstruction quality. These indexes show that the proposed algorithms have achieved better performances. Except for BP, the time taken by all other algorithms is not much different. The proposed algorithms do not need much more computational burden.

4.3. System Configuration

The radar system is a self-developed platform with a commercial radar IWR1443 BOOST and data acquisition card DCA1000EVM, which they are all produced by Texas Instruments located in Dallas, TX, USA, as shown in Figure 7. The wave-absorbing sponge is added around the target for shielding and protection to minimize the impact of ambient noise. All the practical experiments are conducted on the workstation equipped with Inter(R) Xeon(R) W-2145 CPU.

The raw data are extracted and rearranged to form the imaging data. The data format is m × n × p, where m and n are the number of sampling positions of the synthetic aperture in the horizontal and vertical directions, and p is the sampling point. More accurately, for the commercial radar based on FMCW signals adopted in this project, p is the product of the sampling point per cycle, the number of cycles, and the number of chirps in a cycle. IWR1443BOOST can be up to three transmitters and four receivers, and a channel can be established with any pair of transmitter antenna and receiver antenna.

4.4. Rectangular Reference Target

A rectangular reference target without any object occlusion was first imaged. As shown in Figure 8, the size of the reference target is 150 mm × 100 mm. The width of the largest rectangular hole is 10 mm with a decreasing step of 1 mm until 4 mm as the smallest rectangular hole. The three groups of rectangular holes are oriented in different directions to test the resolution of the algorithms in each direction. The distance between the target and the antennas is 0.24 m. The imaging results obtained by various algorithms are shown in Figure 9.

According to Figure 9, the original BP images well, with uniformly distributed energy without much artifact. However, the contours of the rectangular holes are poorly imaged. The shapes of the smaller holes are distorted and cannot be distinguished to be rectangular. The original RMA and the AHI-LFM can only roughly image the peripheral contour of the holes because there exist a lot of artifacts. As the size of the holes decreases, the imaging effect of these two algorithms on the holes becomes worse. The RSA can image the peripheral contour of the rectangles, and the edges of the rectangular holes are also well-imaged. There are still obvious artifacts around the rectangular holes. Since the chunked nonlinear normalized weights can be treated separately for each of the three divided regions. Artifacts in the background region and the transition region are effectively suppressed, and the suppression intensity in the background region is higher than that in the transition region. It is observed that although the artifacts are significantly reduced, the energy of target region is also partially attenuated. The artifacts inside the holes are well-suppressed to the level of background. When the SNR-based multichannel fusion technique is applied, different channels carry different information in the same region. The data from one of the channels can be used to compensate for the signals that are mistakenly attenuated in the other channel to maintain the structure of the target. With the addition of SNR-based multichannel fusion, the energy of the reconstructed image is substantially enhanced while keeping the surrounding artifacts nearly unchanged. The comparison of the IE and IC indexes obtained by these algorithms is shown in

Table 3. Indexes for reconstruction of rectangular reference target.

	Original BP	Original RMA	AHI-LFM	RSA	Chunked Nonlinear Normalization	SNR-Based Multichannel Fusion
IE	3.0099	3.7596	3.8734	3.7345	2.7722	2.7896
IC	8.6510	6.1609	6.6409	6.1850	7.7705	7.1355
Time	227 s	3.79 s	12.64 s	16.84 s	4.6 s	18.11 s

. The proposed algorithms have the relatively lowest IE values. The application of chunked nonlinear normalization weights can significantly enhance the contrast of the reconstructed image. The IC value is still maintained at a high level after applying SNR-based multichannel fusion. These indicate that the techniques can significantly enhance the resolution of the reconstructed image while keeping the artifacts small. The processing times spent on the chunked nonlinear normalization weights technique is close to the original RMA. There is no obvious difference among the AHI-LFM, the RSA, and the proposed algorithms. This shows that the proposed algorithms have not sacrificed the computation efficiency for better imaging quality.

4.5. Multiple-Targets Imaging

In this section, we try to perform imaging experiments on multiple targets. As shown in Figure 10, the imaging objects are a metal rectangle and a plier. The metal rectangle has a very small circular hole in the center with a caliber diameter of 0.35 cm and four large circles around it with a caliber diameter of 2.8 cm. The distance between the target and the antennas is 0.4 m.

The imaging results of different algorithms are shown in Figure 11. The imaging results for the rectangular metal part were similar to the previous experimental. The first four algorithms have more or less artifacts, and the original BP and original RMA are distorted for imaging the large circular hole parts. All four algorithms do not image the smallest circular part well, and original BP and original RMA even lose it. In contrast, the proposed algorithms can perfectly and efficiently image all parts of a metal object including the smallest circular hole. The chunked nonlinear normalized weights technique removes artifacts perfectly, and artifacts are also still effectively suppressed after multichannel fusion. For the plier section, the first three algorithms were out of focus. The original BP and the original RMA have poor imaging of the plier, especially for the handle part. The RSA imaged the plier better, but some artifacts were still present. In contrast, the imaging results of the proposed algorithms revealed no artifacts around the plier and more concentrated energy. The above shows that the proposed algorithms have a large advantage in artifact removal and maintaining target structure. The metrics comparison for this experiment is shown in

Table 4. Indexes for reconstruction of multiple targets.

	Original BP	Original RMA	AHI-LFM	RSA	Chunked Nonlinear Normalization	SNR-Based Multichannel Fusion
IE	2.8485	3.1032	2.6787	2.6742	1.9265	1.9381
IC	7.9059	7.4774	7.9812	7.7341	8.8298	7.2924
Time	349 s	6.56 s	15.99 s	21.52 s	6.63 s	24.69 s

. IE and IC values showed the same trend as the previous experiment. The proposed algorithms maintain the relatively lowest IE values and highest IC values. The computation time is improved because this experiment uses a larger acquisition range. However, the proposed algorithms still maintain a low computation time.

4.6. Imaging Hidden Plier

In the third practice experiment, we use SAR to image a pincer hidden under eight layers of folded fabric to simulate a more realistic scene, as shown in Figure 12. The experiment‘s purpose is verifying the ability to image targets under high occlusion imaging conditions. The distance between the antennas and the plier is 0.4 m. The imaging results of different algorithms are shown in Figure 13.

As seen from Figure 13, the original BP and RMA cannot image the plier well with structural deficiencies. Only the metal part with high reflectivity can be imaged with lower imaging resolution with some artifacts. In contrast, AHI-LFM and RSA can image the approximate contour of the plier, with a significant reduction in artifacts. The metal part of the plier can be imaged at a high quality, but the rubber handle part still has some structural deficiencies and cannot be fully reconstructed. After applying the chunked nonlinear normalized technique, it can be seen that the entire contour of the plier can be well-imaged. And the artifacts in the image can be completely removed, but it will also cause the handle part to be mistakenly attenuated. After applying the SNR-based multichannel fusion technique, the energy of the imaged target is significantly enhanced, and the plier contour is completed reconstructed. The handle part is also efficiently imaged due to the complementary effect of the multichannel data. The performance comparison metrics obtained by various algorithms are shown in

Table 5. Indexes for reconstruction of the hidden plier.

	Original BP	Original RMA	AHI-LFM	RSA	Chunked Nonlinear Normalization	SNR-Based Multichannel Fusion
IE	2.7414	1.9983	1.0632	0.9641	0.5777	0.6577
IC	7.7878	8.2565	9.6028	10.3034	11.108	11.3052
Time	264 s	3.58 s	11.61 s	16.27 s	4.34 s	18.25 s

. The proposed algorithms obtain the lowest IE and highest IC, indicating that the proposed algorithms have significant advantages in removing artifacts as well as structure preservation. The time taken by the SNR-based multichannel fusion technique does not increase much compared to the comparison algorithms, indicating that the proposed algorithm can maintain a relatively fast computational speed.

5. Discussion

In this paper, we propose chunked nonlinear normalized weights to suppress the effects of artifacts and noise. Additionally, we propose SNR-based multichannel fusion technique to restore the weakened target signal and maintain target structure. The experiments in this paper include both simulation and practical measurements. Both qualitative images and quantitative metrics validate the high efficiency of the proposed algorithms. The chunked nonlinear normalized weights technique divides imaging data into target region, transition region, and background region based on local statistical values. The improvement in experimental performance metrics stems precisely from this targeted processing mechanism. This technology selectively enhances high-energy target regions while effectively suppressing low-energy artifacts and noise. As shown in Figure 9e, Figure 11e and Figure 13e, the corresponding images reveal a purer background and more pronounced edges in the imaging results. However, the suppression of signals by chunked weights can sometimes weaken the target signal, leading to the loss of target structure. The SNR-based multichannel fusion technique is specifically designed to address this shortcoming. This technique does not simply take an average or select any channels but selects data from multiple channels based on SNR. This fusion strategy successfully compensates for the target structure, as shown in Figure 11f and Figure 13f, and avoids the reappearance of suppressed artifacts. The lower IE value and higher IC value confirm the effectiveness of this technique. The synergy between the two techniques creates high-resolution results and enhances the overall robustness of the imaging system. Compared to DL-based methods, these techniques do not require extensive data training, making it more suitable for scenarios where data acquisition is costly or real-time imaging is required. Compared to RSA and AHI-LFM, these techniques do not consume excessive time to achieve superior results, striking a relative balance between image resolution and computational efficiency.

In addition, the proposed techniques still have limitations. The current algorithms operate exclusively in full-data scenarios, and the performance in large-scale sparse data scenarios remains unexplored. Sparse data acquisition holds significant importance for reducing acquisition time and alleviating storage pressure. Future research will explore integrating these techniques with matrix complementation algorithms to generate high-quality images from under-sampled data. Additionally, we will investigate applying the techniques to larger scenes to assess the effectiveness.

6. Conclusions

In this paper, we propose imaging algorithms that can be applied to limited data or high noise. The chunked nonlinear normalization weights can effectively suppress the artifacts and retain only the target region of interest. The SNR-based multichannel fusion can effectively preserve the integrity of the target structures. By comparing IE and IC of each algorithm, we can see the advantages of the proposed algorithms in suppressing artifacts.

By applying the algorithms proposed in this paper, high-quality images can be obtained while utilizing MIMO arrays for limited data acquisition to reduce acquisition time. The robustness of the algorithms is also improved due to the more efficient use of multichannel data. In addition, it also shows good results in imaging complex scenes.

We have not yet applied the proposed algorithms to sparse data to verify the imaging results under sparse conditions. We believe and will try in our next work, if the proposed algorithms are combined with matrix complementation algorithm or matrix recovery algorithm, that we can still obtain well imaging results under sparse conditions.