1. Introduction

In recent years, many new ultra-light and ultra-thin imaging systems have been proposed. For example, the space partitioning deployable optical imaging technology [1,2] divides the primary mirror of the large aperture (such as 3 meters or more) optical system into several small lightweight sub-mirrors, which can be folded and carried, deployed and locked in orbit, and then spliced to form the primary mirror in a certain way. Space-based thin film diffraction imaging technology [3,4], using lightweight foldable films to replace traditional optical system lenses to solve the aperture bottleneck problem of optical imaging satellites in high orbit (such as geostationary orbit); And space-based optical synthetic aperture imaging technology [5], using precisely arranged multiple small-aperture optical elements or optical systems enables light to achieve interference imaging on the sensor, which achieves the imaging accuracy of the large-aperture optical system and avoid the difficulty of its optical system manufacturing. The segmented planar imaging detector for electro-optical reconnaissance (SPIDER) is a typical ultra-thin computational photoelectric imaging system. In SPIDER [6], the large optical devices and mounting structure of the conventional system are replaced by a dense interferometer array composed of lens array and PIC components, which can provide large effective apertures for fine-resolution imaging, while minimizing system volume, mass, and cost.

The development of SPIDER imaging technology involves micro-nano manufacturing technology, PICs, spatial frequency domain undersampling image inversion and so on. A lot of relevant researches have been done in recent years. Duncan et al. have developed three generations of SPIDER system so far, and each generation has made corresponding structural changes [7–11]. The working principle of PIC was demonstated by Thurman et al. [12–13]. Chu et al. verified that the imaging quality could be effectively improved by adjusting the Nyquist sampling density, optimizing the baseline pairing method and increasing the spectral channel of demultiplexer [14]. An optimal scheme of baseline matching method was proposed by Lv Guo-mian et al. [15]. Moreover, Qinghua Yu et al. presented a “checkerboard” imaging system design using a square grid aperture arrangement [16]. Although uniform sampling can be achieved in the horizontal and vertical directions, the spatial sampling along the baseline direction is not uniform. Weiping Gao et researched a segmented planar imager with a novel hierarchical multistage sampling lens array [17–18], which added medium and short radial lens arrays on the basis of the original lens array. Although the sampling at medium and low frequencies was increased, the extra lenses almost doubled the power consumption of the system. The lens array, baseline pairing and spatial frequency sampling have been optimized to some extent, but there are still some defects in the reconstructed image, such as artifacts of ideal image [7,9,10,11], low objective evaluation index of PSNR, less optimization space, serious noise and poor image clarity after sampling. All these will be key issues in the development and optimization of ultra-light, ultra-thin and high-resolution segmented planar imaging system.

Based on the segmented planar imaging system problems mentioned above, a novel dense azimuth sampling lens array is proposed, which can effectively weaken the ideal image artifacts and improve the upper limit of the image PSNR. Under the condition that the resolution of the system remains unchanged, reorganizing the spatial frequency can reduce the effective sampling radius, increase the sampling of the medium and low frequency information, and effectively improve the actual image quality. Meanwhile, a full-chain imaging theoretical model and simulation test were established to explore the impact of the number of radial-spoke PICs and the effective frequency sampling radius on image quality. The research results provide certain theoretical support for the high performance segmented planar imaging system design and spatial frequency sampling image inversion.

2. Structure design of dense azimuth sampling segmented planar imaging system

The structure of the dense azimuth sampling segmented planar imaging system is shown in Fig. 1[11]. The system is mainly composed of multiple 1D interferometer arrays arranged in a radial-spoke pattern. Each 1D interference arm is mainly divided into three layers: a lens linear array at the top, the transmission module PIC chips in the middle and the signal processing system at the bottom. The traditional SPIDER system lens array resembles the shape of a wheel, with the length of the interference arms equal and odd, and the number of lenses in a 1D radial lens linear array is even and the filling factor is 1. The number of radial lens arrays of the dense azimuth sampling system is twice that of the conventional SPIDER system. In order to avoid increasing additional power consumption, the even-numbered interference arms only have lenses at even-numbered positions, and at the same the odd-numbered interference arms have lenses at odd-numbered positions. In addition, the number of lenses in the 1D dense azimuth-sampling lens array is odd and the radial filling factor is 0.5 (Suppose the distance between adjacent lenslets is D and the diameter of the lenslets is d. The radial fill factor of the lens array is defined as:$FF = {d / D}$[17].). In the baseline matching, there is an independent lens A which does not pair with any of the others, the light received by the lens directly detects the current intensity through the photodiode, corresponding to the central zero frequency amplitude of the spectrum. Other lenses are matched to form interference arms of different lengths (analogous to small-scale Michelson stellar interferometers). The light from the scene is coupled into the optical waveguide on the PIC through several separate lens pairs, and then the interference fringes are formed through the grating beam splitter, phase retarder and coupler. The complex visibility (amplitude and phase) is demodulated by the detector, and then, the intensity distribution of the scene can be restored.

 

Fig. 1. Schematic diagram of dense azimuth sampling segmented planar imaging system.

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3. Full-chain theoretical model of segmented planar imaging system

In order to analyze the imaging performance of the segmented planar imaging system, consider the propagation of light in the air, the propagation in the lens array, the propagation in the photonic integrated circuit, the processing of the interference fringes and the image restoration process. According to the theory of diffraction optics, the full-chain theoretical model is established, deriving the relationship between light field distribution at PIC output terminal and on object surface, verifying the feasibility of the imaging scheme and image inversion. Figure 2 [19] is a schematic diagram of light field propagation from object surface to PIC output.

 

Fig. 2. diagram of light field propagation from object surface to PIC terminal.

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3.1 Propagation of light from the light source to detection plane

The propagation of optical signal from object plane to lens array plane is a diffraction process. In the space-based detection platform, the distance from object plane to lens array plane satisfies the Fraunhoffer diffraction condition, so the relation between light field distribution in object plane $S(\zeta ,\eta )$ and lens array plane $P({x,y} )$ is:

For a single lenslet, its transmittance function can be expressed by:

The distance between the back plane of lens array and image plane is short, so the relation between light field distribution in image plane $R({u,v} )$ and the back plane of lens array ${P^{\prime}}({x,y} )= P({x,y} ){t_{lj}}(x,y)$ can be expressed by Fresnel diffraction formula:

After the light beam at the image plane is coupled into the optical waveguide array, it is divided into multiple narrow spectral channels by arrayed waveguide gratings (AWGs), and delayed by the phase modulator. Assuming that the light transmission in the waveguide only generates phase delay $\Delta \varphi $, and does not change the complex amplitude distribution of the beam and other variables, when light is transmitted from image plane to PIC output terminal $Q({{u^{\prime}},{v^{\prime}}} )$ through the optical waveguide, there is a relationship $u = {u^{\prime}},v = {v^{\prime}}$[19]. Using the Gaussian formula, ignoring the constant phase, the light passing through a set of lens pairs $\alpha , \beta $ (where $\alpha , \beta = 1,2,3 \cdots \cdots n,\alpha \ne \beta $) and the phase retarder, the relationship between the mutual intensity of two coherent optical signals and the intensity distribution on the object surface is shown in formula (4):

The optical signal after interference is converted into photocurrent by photoelectric conversion, and the current intensity is proportional to the light intensity, the co-directional component I and the quadrature component Q of the photocurrent can be expressed as [20]:

From the real and imaginary parts of the mutual intensity of the two coherent lights, the intensity and phase of the corresponding object plane spatial frequency $({{f_u},{f_v}} )$ can be calculated:

By increasing the coverage of the two-dimensional spatial frequency sampling, more object surface light information can be obtained, and then the target image can be reconstructed by the inverse Fourier transform, $I({\zeta ,\eta } )= {{\cal F}^{ - 1}}[{J({u_\alpha^{\prime},v_\alpha^{\prime};u_\beta^{\prime},v_\beta^{\prime}} )} ]$.

For the dense azimuth sampling segmented planar imaging system, the field of view of a single waveguide depends on the waveguide coupling efficiency, which can be expressed as [21]:

The coupling efficiency is 81.33% for the point on the axis (instant $|\alpha |\textrm{ = }0$), and the coupling efficiency drops to 8.15% for the edge field of view $|\alpha |\textrm{ = }\lambda /d$. Therefore, the field of view of a single waveguide behind a single microlens is limited to ${\pm} \lambda /d$, and the imaging system including M optical waveguides has the field of view:

3.2 Spectrum sampling of interference fringes

According to Eq. (4), it can be known when optical signal propagates from object surface to the PIC output terminal, the complex coherence degree of optical signal is proportional to the Fourier transform of the light intensity of object surface. Each spatial coherence measurement is equivalent to a unique sample of the object’s 2D Fourier transform. 2D Fourier fill of the scene is achieved by collecting data from multiple 1D interferometer arrays with different orientations. By applying the inverse Fourier transform of the obtained spectrum, the brightness distribution of the object surface can be obtained. The performance of the image is mainly determined by the Fourier sampling density relative to the Nyquist sampling rate. The dense Fourier sampling frequency at the Nyquist sampling rate ensures robust performance with minimal image artifacts [7]. The conventional SPIDER system design focuses on dense radial sampling and coarse azimuth sampling. The radial sampling rate is mainly related to the selection of the lens-pairs to form the interferometer baselines. Dense radial sampling can be increased by measuring multiple optical wavelengths and spectral bands for each of the baselines in every 1D interferometer arrays [19]. The azimuth sampling is mainly determined by the number of radial-spoke PICs used in the system. In practice, it is usually desirable to use fewer PICs and rely on compressed sensing or iterative algorithms to achieve image reconstruction [7,11,19,22]. Using fewer PICs will increase the appearance of image artifacts. It can be mitigated with further modification to reconstruction algorithm, but if the ideal imaging (It is the image that the two-dimensional Fourier spectrum of the scene is completely detected by the system and then restored by the inverse Fourier transform) has artifacts, and the objective evaluation index of PSNR is not high, after sampling the actual image (It is the image that the two-dimensional Fourier spectrum of the scene is actually detected by the system and then restored by the inverse Fourier transform) has less optimization space and the image quality is poor.

To solve the above problems, a dense azimuth sampling lens array is proposed. The number of radial-spoke PICs of the dense azimuth sampling segmented planar system is twice that of the traditional SPIDER system, but the number of lenses in the individual 1D interferometer array is half of the traditional one. The total number of lenses and baselines in the system remain the same, and no additional power consumption is added. Increasing azimuth sampling can effectively improve the ideal image quality and the upper limit of PSNR, and weaken image artifacts. However, the reduction of radial sampling in a single 1D interferometer array deteriorates the actual image after sampling and aggravates image noise and artifacts. In this regard, a discrete spectral matrix reconstruction method is designed for the dense azimuthal sampling 2D lens array structure. Under the condition that the longest baseline remains unchanged, the spatial frequency sampling is concentrated to the medium and low frequencies to improve the actual image quality.

In the dense azimuth sampling 2D lens array structure, the center distance between two adjacent lenses is twice of the lens diameter d. Assuming that the number of radial-spoke PICs of the system is P, and the number of lenses on a single PIC is N ($N$ is an odd number), it can be obtained that the angle between adjacent 1D radial interferometer arrays is $\alpha \textrm{ = }2\pi /P$, and the radius of the circle enclosed by the inner layer of the lens array can be expressed as:

The distance from the $j$th lens on the $i$th column PIC with an even number to the center is ${L_j} = R + j\ast 2\ast d$, then the horizontal and vertical coordinates of the lens can be calculated as:

If the lens on a single interference arm are paired first-to-tail, the pairing method is $(1,N),(2,N - 1),(3,N - 2),(4,N - 3) \cdots $, an independent lens and $(N - 1)/2$ optical element pairs are obtained. The independent lens dose not pair with any of others, the light received is detected directly by the photodiode, corresponding to the amplitude of central zero frequency of the spectrum. As for the other lenses, the baseline lengths are ${B_j} = [4,8,12,16, \cdots \cdots ,2(N - 1)]d$, it can be seen that the distance between the radial spectral point and the center of sample is a discrete integer, so the final spectrum sampled is a series of discrete concentric rings. The coordinates of discrete spatial frequency points $({u_j},{v_j})$ can be expressed as:

For the reconstruction of the discrete spectrum matrix, the shortest baseline on a single interference arm is ${B_{\min }} = 4d$, and the minimum spatial frequency sampled (i.e. the fundamental frequency) during the imaging process in the case of single wavelength is:

Using the Pythagorean theorem, we can calculate the horizontal and vertical coordinates of the spatial frequency points relative to the fundamental frequency in the reconstructed spectrum matrix:

From Eq. (13), it can be known that the distance between the spatial frequency points and the center zero frequency is:

In dense azimuth sampling segmented planar system, the increase of the number of radial-spoke PICs can rise the upper limit of the ideal image PSNR and weaken image artifacts. At the same time, combined with the new 2D lens structure, reconstructing the discrete spectrum matrix, reducing the spatial sampling point spacing and the effective frequency sampling radius, under the condition that the longest baseline remains unchanged, that is, the resolution of the system remains unchanged, increasing the low frequency sampling rate and zero frequency, the actual image quality can be effectively improved after sampling.

4. Simulation results and analysis

In order to verify the full-chain theoretical model, a space-based imaging simulation experiment is designed. Aiming at the typical application requirements of $500km$ orbit height, working wavelength $800nm$, angular resolution ${1.2^{^{\prime\prime}}}$ (under-satellite point resolution better than $3m$), field-of-view ${1.5^ \circ }$, according to the dense azimuth sampling segmented planar system structure design: It can be seen from Eq. (6) that the system angular resolution ${R_{\min }}$ is determined by the highest spatial sampling frequency ${\mu _{\max }}$ of the system, corresponding to the longest baseline ${B_{\max }}$ and the shortest wavelength ${\lambda _{\min }}$:

In order to meet the resolution ${R_{\min }} = {1.2^{^{\prime\prime}}}$, the longest baseline of the system should be ${B_{\max }} \ge 137.58mm$, and ${B_{\max }}\textrm{ = 138}mm$ is taken. If the lens diameter $d = 1mm$, the field-of-view of a single waveguide can be known from Eq. (8), $FO{V_{\sin gle}} = 1.6mrad$, for an imaging system containing M optical waveguides, to obtain a 1.5° field-of-view, each lens corresponding to a $17{\ast }17$ optical waveguide array. The Nyquist sampling spacing is related to the image $FO{V_{\sin gle}}$ by $\Delta u = {1 / {FO{V_{\sin gle}}}}$, then $\Delta u = 0.625cycles/mrad$.

The traditional SPIDER imaging system consists of 37 interference arms, 138 lenses on a single interference arm, and the center distance between adjacent lenses is d. The lens on the same interferometer arm adopts first-to-tail paired uniform sampling method (i.e. $(1,138),(2,137),(3,136) \cdots \cdots (67,72),(68,71),(69,70)$), and 69 groups of baselines with different lengths were respectively $137,135,133, \cdots \cdots 5,3,1$ (unit: mm). With the same number of lenses and other conditions being equal, the dense azimuth sampling system consists of 74 interference arms and 69 lenses on a single interference arm. The center distance of adjacent lenses is $2d$. Using the same sampling method (i.e. $(1,69),(2,68),(3,67) \cdots \cdots (32,38),(33,37),(34,36)$), 34 groups of baselines with different lengths were respectively $136,132,128, \cdots \cdots 12,8,4$ (unit: mm). The imaging parameter design of the two systems is shown in Table 1.

Table 1. System parameters used for the simulations

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By comparing with the conventional SPIDER system, considering the ideal image, the spatial frequency distribution, the actual image and the objective evaluation index PSNR and MSE values, the optimization performance of the dense azimuth sampling segmented planar imaging system is further verified. Select the resolution board shown in Fig. 3 as the target image and the field-of-view is limited to $512{\ast }512$ pixel. Perform MATLAB simulation on the target image according to the parameters in Table 1. The lens array collects the light information from the target, couples the light signal into the optical waveguide array. After passing through the phase retarder, the optical signal is converted into the current component through the information processing module, and the restored image is obtained after the inverse Fourier transform.

 

Fig. 3. Target image

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In Fig. 4 and Fig. 5, (a) and (b) are the lens array of the traditional SPIDER and the dense azimuth sampling system, as well as the local magnification. The ideal frequency spectrum can be obtained by formula (4) as shown in part (c). Part (d) is the ideal image recovered from part (c). Compared with the target image, the ideal image of the conventional SPIDER system has obvious artifacts, while the ideal image of the dense azimuth sampling system has almost no artifacts. In order to compare the imaging quality, the common peak signal-to-noise ratio (PSNR) and mean square error (MSE) based on image pixel statistics are used as evaluation criteria [23]. The PSNR and MSE values of the traditional SPIDER system and the dense azimuth sampling system are: 23.6927dB, 44.9359dB and 0.0043, 0.00003, respectively. The larger the value of PSNR, the smaller the image distortion, and the smaller the value of MSE, the better the image quality. It is obvious that the dense azimuth sampling system can effectively mitigate the artifacts of ideal image and increase the upper limit of the PSNR.

 

Fig. 4. (a) is the lens array of the traditional SPIDER system; (b) is the enlarged view of the center of (a); (c) is the ideal frequency spectrum; (d) is the ideal image recovered from (c); (e) is the space spectrum distribution; (f) is the actual spectrum sampled from the ideal spectrum by (e); (g) is the actual image recovered by (f) through the inverse Fourier transform; (h) is the actual image with photon noise.

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Fig. 5. (a) is the lens array of the dense azimuth sampling system; (b) is the enlarged view of the center of (a); (c) is the ideal frequency spectrum; (d) is the ideal image recovered from (c); (e) is the space spectrum distribution; (f) is the actual spectrum sampled from the ideal spectrum by (e); (g) is the actual image recovered by (f) through the inverse Fourier transform; (h) is the actual image with photon noise.

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Based on the baseline pairing method of first-to-tail pairing and the traditional interference fringe sampling mode as shown in formula (11), the spatial frequency distributions of the conventional SPIDER system and the dense azimuth sampling system are shown in Fig. 4(e) and Fig. 5(e). The effective spatial sampling radius of the two systems are $R\textrm{ = }171.3$ and $R\textrm{ = }170$ respectively. Part (f) is the actual spectrum sampled from the ideal spectrum by part (e) and part (g) is the actual image recovered by part (f) through the inverse Fourier transform. As shown in Table 2, the PSNR and MSE of the actual image objective evaluation values of the traditional SPIDER system and the dense azimuth sampling system are 12.8048dB, 8.5129dB and 0.0524, 0.1405 respectively. Since the radial sampling interval of the two systems is discrete, and the spatial frequency distributions are a series of discrete concentric rings, the uneven sampling leads to side lobes after inverse Fourier transform. The actual image recovered after sampling contains a lot of noise, and the definition is low, which is far from the ideal image. Although the dense azimuth sampling system can weaken ideal image artifacts and increase the upper limit of PSNR, the reduction of the number of lenses on a single interference arm makes the radial sampling interval more discrete, and the actual image is blurred, which is a “dirty picture” almost completely submerged by noise. Further consider the impact of noise in the detector on the image, Fig. 4(h) and Fig. 5(h) are the actual images containing photon noise. Comparing of part (g) and part (h), it can be seen that photon noise causes subtle changes in the brightness and clarity of the image.

Table 2. Comparison of imaging effects of conventional SPIDER system and dense azimuth sampling system

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According to the theoretical analysis in Sect. 3.2, the distance from the reorganized spatial frequency point to the sampling center calculated from Eq. (13) is ${L_j} = [0, 1,2,3,4 \cdots \cdots ,(N - 1)/2],(N = 69)$ (where $L\textrm{ = }0$ represents the center zero frequency interval collected by an independent lens A) and the longest sampling distance $L\textrm{ = }34$. It can be seen that the sampling distance is a continuous integer, so the final spectrum distribution should be a series of continuous concentric rings, as shown in Fig. 6(a). The effective sampling radius $R\textrm{ = }34$ is consistent with the theory. The reorganized spatial frequency distribution radius is less than ${1 / 4}$ of the traditional interference sampling mode radius. While keeping the longest baseline unchanged, the resolution of the system is the same, and the low frequency sampling rate is nearly doubled, which effectively improves the image quality. Figure 6(b) is the actual spectrum sampled from the ideal spectrum by Fig. 6(a). The actual image recovered from the spectrum in Fig. 6(b) is shown in Fig. 6(c), there is almost no artifacts, the noise is effectively reduced, and the overall contour structure is clearer. The PSNR and MSE values are 22.9055dB and 0.0118, respectively. The actual image objective evaluation value of the dense azimuth sampling system is close to the ideal image value of the traditional SPIDER system, and the imaging effect is good. Figure 6(d) is the actual image containing photon noise. Compared with Fig. 6(c), the brightness of the image has changed slightly, and the sharpness has not changed significantly. The simulation results prove that the dense azimuth sampling system can weaken the ideal image artifacts; under the condition that the system resolution remains unchanged, reorganizing the spatial frequency sampling matrix to reduce the effective sampling radius can optimize the actual image quality after sampling.

 

Fig. 6. (a) is the reorganized spatial frequency distribution of the dense azimuth sampling system; (b) is the actual frequency spectrum sampled from the ideal frequency spectrum by (a); (c) is the actual image recovered by (b) through the inverse Fourier transform; (d) is the actual image with photon noise.

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5. Performance optimization of dense azimuth sampling system

From the full-chain model of Sect. 3 and the simulation results of Sect. 4, it can be seen that the imaging quality of the dense azimuth sampling system is related to the number of radial-spoke PICs and the effective space sampling radius. In order to further understand the influence of the two performances on image quality, the imaging system was optimized by considering ideal imaging, actual spatial frequency distribution, recovered images after sampling, objective evaluation indexes PSNR and MSE.

5.1 Number of radial-spoke PIC

Theoretically, a decrease in the number of radial-spoke PICs will increase the occurrence of image artifacts. In the experimental process, only the number of radial-spoke PICs was changed, and the effective sampling radius remained unchanged after the recombination of the discrete spectral matrix. The imaging quality of the dense azimuth sampling system is numerically simulated, considering the influence of the number of radial-spoke PICs within a certain range on the ideal image and the actual image. Figure 7(a) and (b) respectively show the curve relationship between the number of radial-spoke PICs and the imaging PSNR and MSE, where the actual image A and the actual image B respectively represent the actual image of the traditional sampling mode and the actual image of the spectral matrix reconstruction.

 

Fig. 7. (a) and (b) respectively show the curve relationship between the number of radial-spoke PICs and the imaging PSNR and MSE

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The results show that with the increase of the number of radial-spoke PICs, the PSNR of the ideal image overall rises and the MSE tends to stabilize after the decline, the image quality was effectively improved. The PSNR of the actual image restored by the traditional sampling mode is small and almost unchanged, and the MSE begins to decrease slowly, and then tends to be stable overall. After spectral matrix reconstruction, the PSNR and MSE of the actual image and the ideal image changed in the same trend, which means that increase the number of radial-spoke PICs is a better optimization method.

The PSNR of ideal image is approximately in the range of 5-50dB, and the interval is divided into 5 parts. Five points with PSNR of 11.05dB, 22.58 dB, 25.37 dB, 35.11 dB and 48.02 dB were randomly selected, corresponding the number of radial-spoke PICs is: 17, 26, 38, 50, 78, to simulate the imaging process of the dense azimuth sampling system, and further observe the image artifacts. The PSNR of the actual image restored by the traditional sampling mode is low, and the noise is serious, so we don't analyze it here. We mainly observe the ideal image and the actual image recovered from the reconstructed spectral matrix. The results are shown in Fig. 8, (a)-(e) are ideal images, and (f)-(j) are actual images. Due to the small number of radial-spoke PICs, Fig. 8(a) and Fig. 8(f) have serious ringing effects and the brightness distribution is uneven. As the number of radial-spoke PICs increases, the image artifacts in Fig. 8(b) and Fig. 8(c) are gradually weakened, and there are almost no artifacts in Fig. 8(d) and Fig. 8(e); the corresponding actual image Fig. 8(g) has obvious noise and artifacts, the artifact in Fig. 8(h) is similar to the ideal image Fig. 8(c), there are some noises in Fig. 8(i) and Fig. 8(j) and almost no artifacts.

 

Fig. 8. (a) - 8(e) and Fig. 8(f) – 8(j) are the ideal images and the actual image recovered by reorganizing the spatial sampling spectrum of the dense azimuthal sampling system with the number of radial-spoke PICs of 17, 26, 38, 50 and 78.

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In the dense azimuth sampling system, with the increase of the number of radial-spoke PICs, the ideal image artifacts can be effectively weakened, and the upper limit of PSNR value can be raised. Due to insufficient radial lens filling on a single PIC, the spatial spectrum distribution is discrete and the low-frequency information is not sampled enough. The actual image recovered by the traditional sampling mode has serious noise, which is unsuitable for the dense azimuth sampling system. According to the system structure design, the actual image recovered by reorganizing the spatial sampling spectrum, the increase in the number of radial-spoke PICs, the PSNR value change trend is the same as the ideal image, which can effectively optimize the imaging quality of the system.

5.2 Effect of effective space sampling radius

From the theoretical analysis in Sect. 3.2 and the simulation verification in Sect. 4, it can be seen that the dense azimuth sampling system due to insufficient radial filling of that lenses on a single interference arm, the center distance between two adjacent lenses is $2d$, and the distance between two radially adjacent spatial sampling points is $\frac{{4d}}{{\lambda z}}$. By solving the distance from the spatial frequency point relative to the fundamental frequency to the center zero frequency point, when the longest baseline of the system and the resolution remain unchanged, the spacing of the spatial sampling points is reduced, thereby shortening the effective spatial sampling radius and increasing the low-frequency information sampling, which can improve the quality of restored images. In the case of constant system resolution, analyzing the influence of the effective space sampling radius on the actual image, ${\mu _1}\textrm{ = }\frac{d}{{\lambda z}}, {\mu _2}\textrm{ = }\frac{{2d}}{{\lambda z}}, {\mu _3}\textrm{ = }\frac{{4d}}{{\lambda z}}$ are used as the intermediate conversion frequencies, and the distance from the space frequency point relative to the intermediate conversion frequencies to the center zero frequency point is: ${L_1} = [4,8,12,16 \cdots \cdots ,2(N - 1)]$, ${L_2} = [2,4,6,8 \cdots \cdots ,(N - 1)]$, ${L_3} = [1,2,3,4 \cdots \cdots ,(N - 1)/2],(N = 69)$, the longest sampling distance ${L_{1\max }}\textrm{ = }136,$ ${L_{2\max }}\textrm{ = }68$, ${L_{3\max }}\textrm{ = }34$.

The imaging system parameters are shown in Table 1 in Sect. 4. Figure 9(a), (b), (c) show the spatial frequency distribution after recombination, and the effective radius is ${R_1}\textrm{ = }136, {R_2}\textrm{ = }68, {R_3}\textrm{ = }34$, which are consistent with the longest sampling distance. With the decrease of the space between spatial sampling points, the effective sampling radius of the system lessens, and the sampling of medium and low frequency information improves effectively. When the distance between two radially adjacent spatial sampling points is used as the intermediate conversion frequency (that is, the fundamental frequency), continuous integer multiples of all fundamental frequencies including the zero frequency can be uniformly sampled within the maximum spectrum range. The actual image recovered after spectrum reconstruction is shown in Fig. 9(d), (e) and (f), and the corresponding PSNR and MSE values are 8.4892dB, 12.3120dB, 22.9055dB and 0.1416, 0.0587, 0.0051 respectively. It can be seen that with the decrease of effective spatial sampling radius, the image noise is weakened, the contrast is enhanced and the overall structure contour is clearer.

 

Fig. 9. (a), (b) and (c) are the spatial frequency distribution of the reconstituted discrete spectrum matrix with ${\mu _1}$, ${\mu _2}$ and ${\mu _3}$ as the intermediate frequencies; (d), (e) and (f) are the actual images recovered from the spectrum in (a), (b) and (c).

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6. Summary

This paper introduces the structure design of the dense azimuth sampling system, establishes the full-chain theoretical model of the segmented planar imaging system, and proposes a discrete spectrum matrix reconstruction method for the 2D lens array structure. Based on the full-chain theoretical model, using the resolution board as the detection target, the imaging process of the dense azimuth sampling segmented planar system is numerically simulated, and the influence of the number of radial-spoke PICs and the effective spatial sampling radius on the imaging quality are studied. The following conclusions can be obtained by analyzing the simulation results:

The dense azimuth sampling lens array structure proposed in this paper has a significant effect on improving the imaging quality of the segmented planar system, and does not increase the number of lenses and additional power consumption compared with the conventional SPIDER system. In addition, the discrete spectrum matrix reconstruction method proposed for the 2D lens array can achieve continuous sampling of spatial frequencies, which is of great significance for the collection of low and medium frequency information. The discrete spectrum matrix reconstruction method proposed in this paper is only suitable for single wavelength, and the follow-up work will focus on the optimization of multi-wavelength spatial frequency distribution.