Cosine Matching Imaging Method for Rotating Scanning Synthesis Aperture Imaging Radiometer

Millimeter-wave synthesis aperture imaging radiometer (SAIR) can realize high-resolution observations without requiring the real large aperture antenna. Among the SAIRs, the Rotating Scanning SAIR (RS-SAIR) with linear sparse array is a popular systemwith low redundancy and high reliability. However, due to the lack of matched imaging methods, its imaging precision is usually low. For improving its imaging precision, a novel Cosine Matching Imaging (CMI) method is proposed in this paper. In CMI method, according to the characteristic of rotating projection imaging, the angle-orientation image is constituted by the 1Dprojection images measured by RS-SAIR in a series of angles firstly.Then, according to the trajectory of the target in angle-orientation image, the pixel values of the brightness temperature image are extracted by cosine matchingmethod from the angle-orientation image one by one. The simulation results demonstrate that the proposed CMI imaging method has higher reconstruction accuracy for the RS-SAIR.


Introduction
Millimeter-wave synthesis aperture imaging radiometers (SAIR) are powerful sensors for high-resolution observations in various fields [1][2][3].By the aperture synthesis technique, SAIR can utilize some small antennas to constitute a large aperture antenna and realize high-resolution imaging observation.Each complex correlation of pairwise antennas is a sample of visibility function, which can be utilized to reconstruct the brightness temperature image of the target scene.Thus, the antenna array configuration is an extremely important component for the SAIR.It directly defines the distribution of the visibility samples and further affects the imaging performance of SAIR system.In fact, how to utilize fewer antennas to achieve the better imaging performance (i.e., array optimization) is always a difficult problem for the SAIR design.Since the first SAIR (ESTAR) developed in 1980s [4,5], various array configurations have been proposed.Such as 2D rectangular array, sparse "Y" array, "U" array, circular array, and other arrays [6][7][8][9][10][11][12][13][14][15].All of these arrays are looking for achieving the optimal array with low redundancy and larger synthesis aperture constituted by fewer antennas.Unfortunately, there is no effective optimization array which was proposed for 2D SAIR at present.Many antennas are needed for those 2D arrays, which result in high redundancy and system costs.
For utilizing antennas effectively and reducing systematic complexity, the scanning SAIRs with sparse array are proposed and applied in some SAIRs [13][14][15][16][17].Among these SAIRs, the Rotating Scanning SAIR (RS-SAIR) with linear sparse array is the most popular system with low redundancy and high reliability.However, the existing imaging methods may not be accurate enough for some special applications.The sampling points of RS-SAIR is circularly distributed, which is unsuitable for traditional FFT-based methods.In order to deal with these visibility functions with irregular distribution, the Gridding and Nonuniform FFT (NUFFT) methods are proposed for the irregular SAIRs [18][19][20][21].But due to the problem of insertion and resampling, some unnecessary errors will be introduced to the reconstructed images inevitably.Therefore, the effective high-precision imaging method is to find the appropriate inverse method to reconstruct the target images from the irregular visibility function directly.Fortunately, the sampling points of RS-SAIR in radial direction are equispaced and suitable for applying FFT.The 1D radial projection images can be reconstructed by FFT-methods firstly, for example, the back projection (BP) method first to reconstruct the 1D projection images by FFT-methods and then compound the 2D scene images by the back projection operation according to the measuring angle of 1D images [22][23][24].But due to the limitations of back projection, BP method cannot effectively deal with the aliasing problem between targets, especially for nearfield with wider point spread function (PSF) and serious interference between targets [24][25][26].For improving the imaging precision of RS-SAIR further, we study the synthesis aperture projection imaging principle deeply and find that the trajectory of targets in angle-orientation image meets cosine distribution.Based on this fact, a novel Cosine Matching Imaging (CMI) algorithm based on the CLEAN idea is proposed in this paper.In CMI method, the 1D projection images of 2D target scene are measured by the RS-SAIR with linear sparse array in a series of angles firstly.Then, the 1D projection images are arranged to constitute an angleorientation image according the corresponding array angle.Finally, the pixel values of the brightness temperature image are extracted by the cosine matching method one by one from the angle-orientation image.The numerical simulation results demonstrate that the proposed CMI method can well reconstruct the millimeter images from RS-SAIR with fewer antennas.Its imaging accuracies is better than the BP method under the same conditions.

The Basic Synthesis Aperture Imaging Theory
Before the description of CMI method for RS-SAIR, the basic synthesis aperture imaging theory is demonstrated here.As Figure 1 shows, the antennas are located on OXY, the extend radiation source S is located on oxy, and its brightness temperature image is T(x,y).The radiation source S is dispersed into N small parts.The distances between the -th radiation source   and antennas  and  are    and    , respectively.
The correlation between the received signals   and   of pairwise antennas (c, l) is a sample of the so-called visibility function.According to [27], the visibility samples measured by pairwise antennas (c, l) can be expressed as where < • > denotes time integration operation, (  ,   ) is the coordinate of the point target   , T is the modified brightness temperature, F(•) is the normalized antenna pattern, k = 2/ is angular wavenumber, and  , is the so-called fringe-wash function [28].
The exp[−(   −    )] denotes the phase difference of two antennas, which is the key factor for the synthetic aperture imaging.According to Figure 1, the distance    and    through Taylor expansion can be expressed as Substitute ( 2) and (3) into (1), and define the spatial coordinates u=k(  -  )/R, v=k(  -  )/R.We can get the relation between visibility function and the brightness temperature image as where )/2R is the phase-modified item, which can improve the imaging accuracy for the nearfield SAIR effectively.For the ideal SAIR with identical receivers and antennas, the decorrelation effects  , ≈1 and antenna pattern F(•) can be ignored simply.Then (4) can be rewritten as follows: where T O (x, y) is the approximate brightness temperature image.For the complete visibility function V(u, v) measured by the SAIR with "T" array, the brightness temperature image T O (x,y) can be reconstructed by the FFT-based methods as an approximate solution of T(x,y).Such as the Modified-FFT (MFFT) imaging method, the brightness temperature images T MF are reconstructed by Fourier transform from the visibility function with phase-modified item directly [26].

Antennas
Source

Description of CMI Method for RS-SAIR
For utilizing the antennas more effectively, RS-SAIR only uses a sparse linear array with fewer antennas to detect the 2D target scene.Its basic imaging theory is the aperture synthesis technology also, but the signal processing and imaging steps are different to other SAIRs.Its measuring principle is illustrated in Figure 2. The spares array is placed in the XOY plane and rotated around the center point O.   is the array rotation angle.
For the sparse array with given   , its spatial coordinates u=k(  -  )/R, v=k(  -  )/R=0 under X r OY r coordinates, and the coordinates of targets should be transformed to the projection coordinates (  ,   ), which are the change of rotation angle   .Then the visibility function  .(, V) measured by the linear array with given   can be rewritten as where   is the projection coordinate on the X r axis (paralleling to the sparse array at angle   );   is the projection coordinate on the Y r axis.   (  ) = ∫ (  ,   )  is the 1D projection image, which can be reconstructed by the 1D MFFT method as the following formula: However, the value of 1D projection image   is a synthetic value containing all the pixels with the same   coordinate under angle   .The targets distributed along the Y r axis will be concealed for each other and hardly to be distinguished.Thus, the large reflectors are applied in the 1D SAIR to compress the antenna beam for achieving the high-resolution on the Y r axis [16].However, the SAIRs with the large reflector usually have a larger volume and are not suitable for many applications.The BP method uses the back projection method to perform the cumulative imaging with array rotating and can reconstruct the target image effectively.But for the near-field RS-SAIR with wider PSF and serious stacking interference between targets, the back projection method cannot deal with these problems effectively; its recovery precision is not good.
In this paper, we seek a better solution from another angle.For the given target S i , when the array is rotating from 1 to 180 ∘ (rotating step is 1 ∘ ), its projection coordinate   can be expressed as For the given target   , its R  and   are the constants, which depended on the target coordinates (  ,   ) only.Thus, the projection coordinate   will change with the angle   of antenna array, and its curve meets the cosine distribution.As Figure 3 shows, the images T P are measured by RS-SAIR at a series of angle   firstly (Figures 3(a) and 3(b)).Then, T P are arranged to constitute the completed angle-orientation image (Figure 3(c)).Obviously, there are two cosine curves in the angle-orientation image.
Based on this fact, we can arrange the 1D projection images (with different  a ) to constitute the angle-orientation image according the sequence of angle  a .Then the target will be extracted by the cosine matching method from angleorientation image one by one.Due to the fact the weak targets are usually hidden in the projection of higher targets, the higher targets will be first s extracted partly for exposing the weak targets.Considering the interference noise between targets is higher for near-field RS-SAIR, the nearby targets of the higher target will be extracted at the same time.In addition, each target S i will be extracted repeatedly in CMI method for maintaining the target continuity.Its detailed process can be summarized as follows: (1) At the start of the CMI method, initialize the brightness temperature image   = 0 and draw the cosine (5) Perform the subtraction of all selected targets in the kth step to form the new remaining image  -(+1) (6) Repeat steps (2)-( 5) until all the residual CV S-i are below the noise threshold V noise .The final image   is the brightness temperature image.
The kth flowchart of CMI method is given in Figure 4.The core of this CMI method is the selection of , N, and V noise .As a benefit from the parameter , target Si will be extracted repeatedly for reducing the aliasing impact between targets and maintaining the target continuity.We know that a higher value of  can be further reduced the complexity with fewer steps to reach the noise floor.However, the higher

Record relevant data
Step 2 Step 3 Step 4 Step 5 Choose highest target S k0 in remaining image T ao-k

Subtract target S k0 from image T ao-k in advance
Search the neighbourhood of target S k0 .

Target S k1
Target S k2 Target S kN Subtract all selected targets to form new remaining image T ao-(k+1) If CV S-k1 > V noise ? is only suitable for simple point targets.For the real-life extended target, the higher  will lead to the result that some smaller targets will be canceled before be extracted.For our simulations,  is selected by the following formula adaptively: where mean(•) represents to calculate the mean of normalized image   .Clearly, the value of  is smaller for real-life target with continuous distribution, and is larger for discrete isolated target.The parameters  and V noise are the important factor for the search step (4).The larger  and smaller V noise can reduce the time and complexity of CMI method, but the accuracy will be reduced too.Based on a large number of simulation experiments, we have come to the conclusion that N is usually selected as 2 times of the PSF's main-lobe width (N = 10 in this paper).And the noise threshold V noise is usually set from 0.1 to 0.2. Figure 3(d) gives the images reconstructed by CMI method for the above two-point scene.It can be seen that two-point targets are reconstructed clearly by CMI method.

Simulation Experiments and Results
To verify the validity of the proposed CMI method, many numerical simulation experiments with various scenes have been conducted, and only a few representative ones are demonstrated in this section.The main simulation parameters are listed in Table 1.As Table 1 show, the RS-SAIR only needs 16 antennas to constitute the sparse linear array with the larger synthesis aperture.The antennas locations are set based on [29]; its arrangement is shown in Figure 5.The minimum baseline spacing is about 1.1cm, and the synthesis aperture D SA is about 1 m.Then the array rotated around the center to achieve two-dimensional scene detection.That is to say RS-SAIR can achieve the equivalent imaging performance with fewer antennas (about 15% of the "T" array) in theory, and the ratio will be lower for the larger synthesis aperture [29,30].In order to simulate RS-SAIR imaging process accurately, the received signals of antennas are obtained by integral operation of the radiation waves generated by all discrete sources, and the visibility samples are calculated by cross-correlated calculation between antennas.The spacing (ΔL) between sources is set to half of the spatial resolution.
In the reconstruction process, the 1D projection images are recovered by the MFFT method firstly.Then the proposed CMI method and BP method are applied to reconstruct the scene images from the 1D projection images.In addition, the NUFFT method [19] is also be applied to reconstruct the images from the irregular visibility function for comparison.All the experiments are performed in MATLAB software.The packaged presentation files used to support the findings of this paper are available from the corresponding author upon request.

Simulation Experiments of the Point Targets.
Here, a series of simulation experiments with point targets are carried out to validate the imaging performance of the proposed CMI method for RS-SAIR.It is well known that the point spread function (PSF) is an important parameter for evaluating the resolution of imaging systems.We first make a simulation experiment with a point target.The corresponding results are demonstrated in Figure 6.
It can be seen from the PSF images that the shape of three PSFs is different.The NUFFT's PSF is a circular distribution (Figure 6(a)) with a fewer noise near the point.The BP's PSF is similar to the NUFFT's PSF distribution, but the noise near the point is higher.This is because that the 1D projection images reconstructed by the FFT method usually have higher side-lobe near the targets, then the 1D images are directly used to generate the final image in the BP method by the back projection method.This makes the side-lobe will also focus on the point target just as shown in Figure 6(b).In the proposed CMI method, these side-lobes will be eliminated when the higher target's cosine images  - are subtracted from image   with its nearby targets.As shown in Figure 6(c), the CMI's PSF is similar to a point distribution compared with the ones of NUFFT and BP, and the noise is eliminated well.For comparing specifically, the 1D sectional view of three PSF images is demonstrated in Figure 6(d).Obviously, CMI's PSF is closer to a point target.Since the interpolation operation when resampling the visibility function, there is a fewer Gaussian noise near the point target in NUFFT's PSF.The BP's PSF is the worst one with the higher cumulative side-lobe noise.
For further testing the resolution of the proposed CMI method, a simulation experiment with two-point targets is performed as follows.The space of two-point targets is just the system resolution (i.e., 2ΔL), and the simulation results are shown in Figure 7.
As Figure 7 shows, the two-point targets have two cosine curves in the angle-orientation image clearly (Figure 7(a)).
However, the images reconstructed by NUFFT and BP method are not very clear; there is a bigger side-lobe noise between two-point targets.It is difficult to distinguish the two points from the Figures 7(b) and 7(c).Actually, as the number of targets increases, this side-lobe noise will increase further.Through the multiple subtraction of higher targets, this side-lobe noise can be eliminated well by the CMI method, the two-point targets can be distinguished from the CMI reconstructed image (Figure 7(d)).This does mean that the imaging accuracy of RS-SAIR can be enhanced by the proposed CMI method effectively.

Simulation Experiments of the Extended Targets.
For further demonstrating the imaging performance of the proposed CMI method, the 2D simulation experiments with extended targets are performed here.The target scene radiation intensity (brightness temperature) images are shown in Figures 8(a It can be seen from the simulation results that the images reconstructed by the NUFFT and BP methods have serious noise pollution near the target caused by the serious side-lobe and interference between targets (Figures 8(b), 8(c), 9(b), and 9(c)).Since the BP method reconstructs the images from 1D projection images with higher noise directly, its reconstructed images are fuzzy.Some weaker targets are even covered by higher noises.Compared to BP's images, more target information are reconstructed from the resampling visibility function as shown in Figures 8(b) and 9(b).However, due to the influence of resampling error and lack of corresponding noise filtering means, the serious noise pollution still exists in the NUFFT's images.As a benefit from the multiple extraction operations of block target, these interference noises can be eliminated by the proposed CMI method effectively.As  10. Obviously, the noise pollution around target of CMI method is lower than the NUFFT and BP method, and its target information is closer to the real target.
In order to compare the simulation results objectively, the structural similarity (SSIM), peak signal-to-noise ratio (PSNR) of images (Figures 8 and 9) and the root-mean square errors (RMSE) of the sectional drawing (Figure 10) are calculated by the following formula:

𝑃𝑆𝑁𝑅 (𝑇
where T' is the reconstructed image and  is the original one. ' and   are the mean of image T' and  separately. ' and   are the standard deviation of image T' and , respectively. T'T is the covariance of image T' and T.  1 and  2 are the smallest positive constants.max() denotes to find the max value of the original image T.
The corresponding calculation results are presented in Table 2. Clearly, the images of CMI have the highest SSIM and PSNR, and the RMSE of CMI method is also the lowest one.This indicates that the proposed CMI method can realize a higher imaging accuracy for RS-SAIR and reconstruct the target information accurately while reducing the noise.

Conclusion
RS-SAIR with linear sparse array can utilize some small aperture antennas to realize high-resolution 2D imaging observation.However, due to lack of matched imaging method, its imaging precision is usually low.For improving the imaging accuracy of RS-SAIR further, a novel imaging method (named Cosine Matching Imaging (CMI)) is proposed in this paper.In CMI method, the 1D projection images are reconstructed by the MFFT method at every angle firstly.Then, the 1D projection images are arranged to constitute an angle-orientation image according the measured angle.The brightness temperature image will be extracted by the CMI method from the angle-orientation image in the end.For reducing interference noise of RS-SAIR and maintaining the target continuity, each target is extracted with its nearby targets repeatedly in CMI method.The effectiveness of the proposed CMI is validated by the simulation experiments.
The simulation results demonstrate that the CMI method can realize a higher imaging accuracy for RS-SAIR; the images reconstructed by CMI method are clearer than that by NUFFT and BP methods.The imaging error caused by the interference between targets is eliminated well by the proposed CMI method.In addition, although the proposed CMI method is analyzed and verified in near-field condition in this paper, it is also suitable for the far-field RS-SAIR.

Figure 2 :
Figure 2: Geometry diagram of synthetic aperture projection imaging.

Figure 4 :
Figure 4: The kth flowchart of CMI method.

Figure 5 :
Figure 5: The arrangement of linear sparse array with 16 antennas (abscissa unit is Δu).
) and 9(a), with their gray value as the amplitude of the discrete point targets, and the distance between point targets is set as ΔL.The corresponding reconstructed images are shown in Figures8(b)-8(d) and 9(b)-9(d).

Figure 7 :Figure 8 :Figure 9 :
Figure 7: Simulation results of two-point targets.(a) The angle-orientation image.(b) The result of NUFFT method.(c) The result of BP method.(d) The result of CMI method.

Table 1 :
Simulation parameters of RS-SAIR with a sparse array.

Table 2 :
Comparison of objective data between NUFFT, BP, and CMI methods.