1. Introduction

With the rapid development of science and technology, various deep-space exploration missions have increasingly higher requirements for the observation capability and detailed resolution of optical imaging systems. According to the Rayleigh criterion, the angular resolution of an optical telescope system is proportional to the wavelength of light and ratio of the telescope aperture $\theta = 1.220{{f\lambda } / D}$. Increasing the aperture of the telescope can significantly improve the resolution; however, it also increases the difficulty and cost of manufacturing. To resolve this, distributed optical synthetic aperture (DOSA) technology has emerged [1–3].

In 1971, Golay presented a non-redundant layout result for a small number of sub-apertures [4]. Subsequently, scholars conducted in-depth research on Golay-type arrays with multiple sub-apertures, in which Golay3- and Golay6-type arrays were studied extensively. In 1988 and 1998, Cornwell proposed uniform frequency-domain coverage and Gaussian frequency-domain coverage, respectively, and proposed a logarithmic spiral array for the first time to realize this; however, the information of high-frequency components in the array was very serious, possibly leading to poor image detail and making it impossible to observe the target [5]. In 2007, Villiers used the projection method to obtain the optimized array layout corresponding to different degrees of Gaussian frequency domain coverage [6]. In 2011, Liu et al. proposed a multi-circular MCA array to increase the cutoff frequency and equivalent aperture [7]. In 2017, Aimin Liu proposed a high-cutoff frequency IMCA array based on the study conducted by Liu et al. (2011) [8]. In 2019, Hao et al. proposed a fractal-structure sparse aperture array and analyzed its imaging performance [9]. In 2022, Wang et al. improved the sub-aperture array of SPIDER and proposed an ESPIA array to improve the spatial frequency uniformity of multi-sub diameter arrays [10]. However, our research focuses on improving the imaging performance of multi-sub path arrays. Achieving ultra-high-resolution observations with a 3-4 sub-diameter DOSA system has so far only been discussed in the Terrestrial Planet Finder (TPF) program proposed by the Jet Propulsion Laboratory in 2004 [11–14]. However, TPF was suspended owing to fiscal and other reasons.

The aperture of a single telescope, limited by technology, is generally only 2–4 m. Taking the Golay3 array of 3 m diameter as an example, if the observation angle resolution reaches 0.01 arcsec, the DOSA system needs to achieve the imaging effect of a 200 m aperture telescope, and the filling factor of the array is only 0.0675%. According to Abbe's imaging theorem and Fourier optics, the sub-diameter is equivalent to a low-pass filter [15]. When the fill factor is reduced, the cutoff frequency of the sub-path is also reduced, resulting in a loss of information in the middle- and high-frequency bands. Traditional DOSA arrays increase the frequency domain coverage by rotating the array and increasing the baseline length. However, such configurations including spiral arrays cannot avoid the existence of zero points. Therefore, it is necessary to study how a DOSA system with a very low fill ratio can achieve high coverage in the frequency domain and improve the cutoff frequency of the array using a position-transformed method, implying that only the baseline (location of sub-diameter) of the array is changed, but the configuration of the array remains the same.

To improve the spatial frequency coverage of an array, the problem of sparse and uneven frequency coverage in the variable array of a traditional DOSA system must be solved. Based on the Archimedean spiral, this study proposes a position-transformation method for DOSA systems with high-frequency domain coverage called High-Frequency domain-Covering discrete Archimedean Spiral Arrays (HFCASA). HFCASA integrates array rotation and array baseline length into the Archimedean helix, so that the frequency domain can be uniformly covered and contains no zero point under certain parameter settings.

The remainder of this paper is organized as follows. Section 2 introduces problems with the position-transformed DOSA array and the influence of a very low filling ratio on the position-transformed array. In the third section, the design method of the HFCASA is introduced in detail, and the design results of Golay3 and the uniform four-diameter HFCASA are presented. In Section 4, the imaging performance of the HFCASA is introduced, considering the Golay3 array as an example, and is compared with that of the traditional position-transformed array method. Finally, Section 5 presents the conclusion of the study.

2. Problem of position-transformed DOSA array with a very low fill ratio

2.1 Theory of position-transformed DOSA array

In this section, the Golay3 array is considered as an example to explain the theory of the position-transformed array in the DOSA system, where the sub-diameter of the Golay3 array is 3 m and the target aperture is 200 m, and the filling factor of the array is $F = {{n \cdot {d^2}} / {{D^2}}} = 0.0675\%$. With the center of the outer circle set as the origin of the Cartesian coordinate system, the initial phase angle of the Golay3 is set to $\theta = 0$, the baseline length is $BL = \sqrt 3 \rho $, and $\rho $ is the distance from the center of the sub-diameter to the origin (center distance), as shown in Fig. 1. The pupil function of Golay3 is calculated as follows:

 

Fig. 1. Schematic of Golay3 array in (a) the time domain and (b) frequency domain.

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Here $({{\xi_k},{\eta_k}} )$ represents the center coordinates of the six outside circles in the Golay3 array frequency domain and $\sqrt {\xi _k^2 + \eta _k^2} = BL$. $OT{F_o}({\xi ,\eta } )$ and $MT{F_o}({\xi ,\eta } )$ represent the OTF and MTF of a 3 m single diameter, respectively.

Figure 1(b) shows that when the Golay3 array baseline is long, there are many zero points (light green area) between the six outside circles and the center circle in the frequency domain. This leads to an incomplete collection of spatial frequency information of the array and reduces its imaging performance. To reduce the number of zero points, the position of the sub-diameter must be changed. When the position of the sub-diameter of Golay3 changes, the center of the six outside circles in the frequency domain also changes correspondingly. The MTF after T-position transformation can be expressed as follows:

There are three traditional methods to change the position of the arrays: multi-arm, spiral, and wave configurations. Figure 2 shows the integrated array location diagram and MTF color distribution diagram for the three methods. The multi-arm configuration has no zero point in the direction of the maximum MTF, and the high-frequency distribution is sparsely limited by the configuration. The MTF of the spiral and wave configurations are evenly distributed in all frequency bands; however, there are still a large number of zero points in the radial direction. If the interval between two position-transformed points is reduced, the repeated acquisitions in the frequency domain increase significantly, and the DOSA system becomes more complex. Therefore, there is an urgent need to develop a position-transformed array method that can simultaneously have the advantages of the three configurations. Based on the advantages of these three configurations, the HFCASA was designed in this study.

 

Fig. 2. (a) Three configurations and (b) MTF color distribution map.

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2.2 Drawback of a very low filling ratio array in a position-transformed array

The principle of the DOSA imaging system follows the Abbe imaging theorem. The frequency component of the object cannot completely pass through because the sub-aperture size is limited. The component outside the maximum cutoff frequency is truncated by the aperture. The OTF in incoherent systems is the normalized autocorrelation of the point-diffusion function. Therefore, the aperture information acquisition disconnection caused by a decrease in the filling ratio leads to a decrease and loss of the high-frequency components in the OTF of the distributed optical synthesis; that is, there are a large number of zero points in the MTF of the DOSA system. The cutoff frequency at the first zero point of the MTF is called the minimum cutoff frequency, which can be used to conservatively estimate the imaging performance of the array. Figure 3 shows the MTF diagram of the Golay3 array with different filling ratios and the same baseline length.

 

Fig. 3. Filling factor versus MTF.

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The minimum cutoff frequency of the Golay3 array decreased rapidly with a decrease in the filling factor. The MTF of the Golay3 distributed optical synthetic aperture imaging system can be described by the MTF and fill factor F of the single-aperture imaging system.

Equation (5) describes the Abbe imaging theory, and ${f_R}$ represents the sub-path cutoff frequency. When the frequency exceeds the cutoff frequency, the sub-diameter does not receive high-frequency information from the light source, and the imaging quality of the DOSA imaging system is poor. Figure 4 shows the imaging results of the resolution plate with Golay3 array with a target aperture of 200 m and baseline of 50 m with fill factors of 30%, 3%, and 0.3%, respectively. It is evident that when the filling factor is reduced, the quality of the resolution degradation graph is significantly reduced.

 

Fig. 4. Imaging results of resolution plates by Golay3 array with different filling factors.

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Equations (4) and 5 show that when the filling factor of the Golay3 array is reduced, the minimum cutoff frequency of the sub-diameter decreases rapidly, and the minimum cutoff frequency is equal to the minimum cutoff frequency of a single sub-diameter of Golay3. Although the minimum cutoff frequency can be increased by changing the array position, a low fill factor implies that it must be changed more times. Therefore, when studying the frequency-domain coverage of a very low-filling-ratio DOSA system, the frequency of the variable matrix is also the focus of the design of the variable matrix method.

3. Design of discrete Archimedean spiral array with high-frequency domain coverage

3.1 HFCASA design principle

The HFCASA proposed in this study has the characteristics of a multi-arm configuration with no zero point in the direction of the maximum MTF and a spiral wave configuration with a uniform distribution in all frequency bands. Therefore, the HFCASA has a higher imaging capability and a lower array frequency when the fill ratio is extremely low. The design principle of the HFCASA is described in detail below.

The previously mentioned helical configuration is based on Conway's logarithmic spiral array. According to section 2.1, it has a uniform coverage in the low-frequency band; however, due to the limitation of the logarithmic spiral itself, the distribution in the high-frequency band is sparse, and the high-frequency response is poor. If the number of transformations increases in the high-frequency band based on the logarithmic spiral, it also affects the uniformity of the overall spatial frequency distribution. To ensure the uniformity of the full frequency band, the array position transformation requires the baseline to increase slowly while rotating slowly. Assuming that the baseline growth rate is $w$ and the angular velocity is $v$, the uniformity condition is satisfied when ${w / v}$ is constant. An equal-velocity ratio (equidistant) spiral is one such spiral with ${v / w} = $ constant, which can be viewed as a superposition of linear and circular motions and moves the same distance along the line in each rotation period. In this study, a spiral with a constant velocity ratio, namely, the Archimedean spiral, which moves straight through the center of the circle, is introduced into the DOSA position-transformation method with a very low filling ratio, and the Archimedean spiral is improved to satisfy the high coverage characteristics in the frequency domain.

The Archimedean spiral can be expressed as follows:

Here, $\rho $ represents the polar radius, $\theta $ represents the polar angle, $r$ represents the base circle radius, $w$ represents the linear motion speed, $v$ represents the circular motion speed, and ${w / v}$ is a constant. According to Eq. (6), for any two adjacent points, $({\rho ,\theta } )$ and $({{\rho_2},\theta + {\theta_c}} )$, on the Archimedean spiral with a polar angle difference of ${\theta _c}$, the difference in the polar radius between the two points is a constant ${\rho _2} - {\rho _1} = {\rho _c}$. To ensure that the Archimedean spiral also has the advantage of a multi-arm configuration, it can satisfy the frequency domain without zero points in the direction of the maximum MTF. Therefore, in this study, numerous discrete points on an Archimedean spiral with equal polar angle differences of ${\theta _c}$ were used to form a discrete Archimedean spiral. The polar coordinate representation of the discrete Archimedean spiral is expressed as follows:

The expression $({{\rho_k},{\theta_k}} )$ represents the k-th discrete point, with ${\theta _c}$ being a constant. For a circle, any section taken along the circumference will require a full rotation of the circle to achieve complete coverage. Similarly, for a DOSA array with an increased baseline, multiple rotations are required to fully cover the frequency range on the current baseline. However, for the discrete Archimedes helix represented using Eq. (7), it is difficult to satisfy the requirement of full coverage owing to the large discrete distance after several rotation periods. When the rotation period is large, adjacent discrete points can continue to be sampled at equal intervals. Therefore, Eq. (7) can be modified to obtain an improved discrete Archimedean spiral. The Cartesian coordinates of the HFCASA are as follows:

Here, $({{x_n},{y_n}} )$ represents the sub-radius coordinates after the n-th Golay3 array transformation, and $({{x_{n + 1}},{y_{n + 1}}} )$ represents the sub-radius coordinates after the (n + 1)-th Golay3 array transformation. ${r_n}$ represents the radial step size of the sub-radius center after the n-th array transformation. According to the HFCASA design principle, the spacing of adjacent discrete points is the same, so ${r_n}$ is constant. $k{\theta _n}$ represents the circumferential step size of the sub-radius center after the n-th transformation, and ${\theta _n}$ varies periodically, with $k{\theta _n}$ being constant between adjacent discrete points. Figure 5 shows schematics of the Archimedean spiral and Golay3 HFCASA. In Fig. 5, A$({{\rho_k},{\theta_k}} )$ and B $({{\rho_{k + 1}},{\theta_{k + 1}}} )$ represent discrete Archimedes spiral sampling points, and C$({{x_n},{y_n}} )$ and D $({{x_{n + 1}},{y_{n + 1}}} )$ represent uniform sampling points, $|{AC} |= |{CD} |= |{DB} |= {r_n}$.

 

Fig. 5. (a) Archimedean spiral. (b) Golay3 HFCASA.

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The HFCASA can determine a reasonable design according to the actual target equivalent aperture requirements and the given initial array sub-diameter. The design steps of the HFCASA for different initial arrays are as follows: 1) determine the initial array; 2) use the frequency domain characteristics of the initial array to determine the uniform sampling parameter ${\theta _c}$ from the Archimedes Spiral; 3) determine the parameters ${r_n}$ and ${\theta _n}$ to satisfy the equivalent aperture expansion under the condition of high coverage in the frequency domain; and 4) adjust the parameters ${r_n}$ and ${\theta _n}$ based on the accuracy requirements of the image processing algorithm for frequency domain coverage and uniformity. The design flowchart is shown in Fig. 6. The second step is the most important in the design process. This directly determines the complexity of the entire system and the subsequent parametric analytical solution.

 

Fig. 6. HFCASA design flowchart.

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3.2 Design of Golay3 HFCASA

Based on Eqs. (7) and 8, the detailed design parameters of the Golay3 HFCASA are discussed by using the Golay3 array as an example. It is necessary to ensure that there are no zero points between the kth and nth, n + 1th frequency domain circles, as well as between the nth and n + 1th, and k-1th frequency domain circles to maintain the no-zero point property in the frequency domain. Figure 7 shows the time- and frequency-domain diagram of the Golay3 HFCASA after 10 variable arrays in ${\theta _c} = {\pi / 3}$.

 

Fig. 7. Schematic of Golay-3 HFCASA after 10 position-transformed operations in (a) time domain and (b) frequency domain.

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For the kth, nth, and n + 1-th cases, based on the uniform sampling rule and the frequency-domain-to-time-domain correspondence of the Golay3 array, the spacing between the three frequency-domain circles is ${\theta _c} = {\pi / 3}$. To ensure that no zero points exist between the three frequency-domain circles, it is necessary to satisfy the following condition:

As the outer six circles of the Golay6 array in the frequency domain form a strictly regular hexagonal structure, it is possible to discuss the case of ${({k - 1} )^\prime }$ in the outer six circles at the k-1-th discrete point. For the k-1-th case, the spacings between the frequency-domain circle and the kth and nth circles are ${r_k} - {r_{k - 1}}$ and $\sqrt {x_n^2 + y_n^2} - {r_{k - 1}}$, respectively. Because ${x_n} = {x_k} + {r_n}\cos ({{\theta_n}} )$ and ${y_n} = {y_k} + {r_n}\sin ({{\theta_n}} )$, the two spacings are ${r_k} - {r_{k - 1}}$ and $\sqrt {r_k^2 + 2{r_k}{r_n}\cos (2\pi k{\theta _c} - {\theta _n}) + r_n^2} $, respectively. To ensure that no zero points exist between the three frequency-domain circles, it is necessary to satisfy the following condition:

Because ${r_k}$, ${r_n}$, and ${\theta _n}$ all vary with changes in k and n, it is difficult to solve Eq. (1)0. To simultaneously satisfy Eqs. (9) and 10, and with reference to the characteristic of the initial Golay3 array in which the frequency domain has no zero points and the outer six circles tightly enclose the central circle, the formula for the discrete Archimedean spiral points in Eq. (8) is modified. The new formula for discrete Archimedean spiral points, which ensures the equidistant spiral condition, is as follows:

Here, ${\theta _c}$, ${\theta _{o1}}$, and ${\theta _{o2}}$ are constants, and ${r_n}$ represents the distance between two discrete points, $({{x_k},{y_k}} )$ and $({{x_{k - 1}},{y_{k - 1}}} )$, which are uniformly sampled. By substituting Eqs. (9) and 10 into Eq. (11) and satisfying the equality condition, with the initial value of ${\theta _n}$ as ${\pi / 3}$, ${\theta _{o1}} = {\pi / 3},{\theta _{o2}} = {\pi / 6}$ can be solved. In the actual design, several parameters can be modified according to frequency domain coverage requirements. Figure 8 shows the Golay3 HFCASA and the corresponding MTF color distribution map under this solution condition.

 

Fig. 8. Golay3 HFCASA and MTF color distribution map.

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Note that the general equation for the HFCASA is given by Eqs. (7) and (8). Equation (1) is obtained according to the frequency-domain distribution characteristics of the Golay3 array and is only applicable to the analysis of the Golay3 array. Special equations for other arrays can be derived from Eqs. (7) and (8) and the frequency domain distribution of the array. The value of each parameter in Eq. (11) can be set based on prior knowledge. For example, ${\theta _c} = {\theta _n} = {\pi / 3}$ can be set according to the Golay3 array frequency–domain distribution in a regular hexagon. Therefore, the values of other parameters can be parsed. When the initial value set changed, the value of the parsed parameter also changed. Many experiments have proven that when the parameters conform to the distribution characteristics of the array frequency domain, the parameters are optimal.

3.3 Design of uniform four sub-aperture HFCASA

To illustrate the universality of the HFCASA, this study presents an equation for a uniform four sub-aperture HFCASA designed according to the flowchart shown in Fig. 6.

Because the frequency domain distribution of the uniform four sub-aperture has two parts, the four central sub-diameters and the four outer sub-diameters in the frequency domain are regular quadrilaterals. When transforming the array position, designing with the four center sub-paths resulted in increased repeat coverage. Therefore, two discrete Archimedean spirals were used to reduce repetitive coverage in the four uniform sub-apertures. Figure 9 shows the frequency domain distribution of the uniform four sub-aperture and uniform four sub-aperture HFCASA under the condition of zero in the frequency domain in ${r_n} = d,{r_m} = \sqrt 2 d$. Figure 9(b) shows the position transformation of one of the four uniform HFCASA sub-diameters. This is because if all four sub-diameters are displayed on the graph, it would be so dense that it would be impossible to determine which sub-diameter took which path.

 

Fig. 9. (a) Uniform four sub-aperture and (b) their HFCASA.

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4. Imaging performance of HFCASA

This section considers the Golay3 array as an example to discuss the number of transforming array positions and the imaging capability of the Golay3 HFCASA.

The phase change 2π of the Golay3 array was recorded as the rotation period in which the baseline growth length was ${{2\pi \ast {r_n}} / {{\theta _c}}}$. During ${\theta _c} = {\pi / 3}$, the baseline of the Golay3 array increases by $6{r_n}$. According to the solution of the limit condition, an integral hexagon in the frequency domain of the transformed array is divided into two cases: 1. the target single-particle frequency circle was inscribed on the hexagon; 2. the frequency circle of the target was cut off from the hexagon. When the target single-radius frequency circle is inscribed on the hexagon, the Golay3 array passes through $floor({{{{\rho_R}} / {9{r_n}}}} )$ rotation cycles, and the total number of arrays changes $floor\{{{{[{({{{2{\rho_R}} / {3{r_n}}}} )({{{2{\rho_R}} / {3{r_n}}} + 1} )} ]} / 2}} \}- 1$. Here, $floor({\cdot} )$ implies rounding down. When the target single-particle frequency circle is circumscribed on the hexagon, the Golay3 array passes through $floor\left[ {{{{\rho_R}} / {\left( {6\sqrt 3 {r_n}} \right)}}} \right]$ rotation cycles, and the total number of array changes is $floor\left\{ {{{\left[ {\left( {{{{\rho_R}} / {\sqrt 3 {r_n}}}} \right)\ast \left( {\left( {{{{\rho_R}} / {\sqrt 3 {r_n}}}} \right) + 1} \right)} \right]} / 2}} \right\} - 1$. The total number of position transformations is reduced by 1 because the initial array is calculated as a transformation when calculating the total number of transformation to facilitate the calculation. Table 1 lists the number of array changes in the Golay3 improved discrete Archimedean spiral synthesis array with a sub-aperture of 3 m under different target single-diameter constraints.

Table 1. The number of position-transformation of Golay3 HFCASA under different target single-particle diameter constraints

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It can be observed from Table 1 that with a larger target aperture, the number of array changes is larger, and so is the difference between the number of array changes in cases 1 and 2, which is due to the limitation of the method proposed in this study. For a regular hexagon, there is significant redundancy between the inscribed circle and the regular hexagon; however, these redundancies can be removed. Similarly, the missing frequency domain can be filled in case 2. Without further discussion of the number of variable arrays, the actual number of variable arrays should be close to the average of cases 1 and 2. Figure 10 shows a comparison of the restoration effect of the HFCASA and the three traditional array methods on the resolution plate and galaxy map. The detailed parameters are listed in Table 2. The image-restoration algorithm uses multi-frame Wiener filtering. As shown in Fig. 10 and Table 2, because HFCASA covers most of the frequency space, the mid-low frequency zero points are far fewer than those of the other three methods, and its imaging ability is higher than that of the traditional three methods. In Table 2, CC is the Pearson correlation coefficient that describes the quality of the restored image. The value of the CC was [−1, 1]. The closer the value is to 1, the better the restored image quality.

 

Fig. 10. Equivalent aperture 200 m, sub-aperture 3 m, Golay3 array in different array mode of resolution plate and galaxy plate restoration results.

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Table 2. Equivalent aperture 200 m, sub-aperture 3 m, Golay3 array results under different array modes and the same number of array changes.

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The resolution and galaxy plates test the resolution ability of the DOSA system for different spatial frequency targets and high-contrast targets, respectively. As shown in Fig. 10, the resolution plate background of HFCASA is clearer and more complete than that of the other three methods. In a galaxy plate with excessively strong background light, HFCASA can still clearly distinguish the target, whereas the other three methods are fuzzy. Six groups of remote-sensing images were randomly selected from the NWPU-RESISC45 remote-sensing image dataset. Figure 11 shows the results of these six groups of images. The results verify the generalization of the text method. The HFCASA has a better image performance than the three traditional methods with the same transform-position number.

 

Fig. 11. NWPU-RESISC45 six groups of image original and reconstructed figure results.

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5. Conclusion

In this study, a position-transformed array method HFCASA is proposed with a 3-4 sub-diameter, based on the Archimedean spiral. This study uses the Golay3 array as an example to study the detailed design parameters of the HFCASA. The calculations showed that the number of position-transformation iterations was optimal when the distance of each array position-transformation was equal to the sub-aperture. Compared with traditional methods, this method significantly improved the stability of the mid-to-high-frequency components and the imaging quality of the array. However, the current design of the HFCASA is based on the full coverage of the frequency domain. For practical applications, it needs to be based on the characteristics of the target frequency domain and a high-precision image reconstruction algorithm, which can greatly reduce the number of position transforms, project implementation difficulty, and production cost.