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Abstract
Array configuration design is a critical issue for a high quality of the snapshot point spread function (PSF) and restored image in Michelson imaging interferometer. In classic design, the optimized configurations usually address the few specifications and single objective, which is unable to balance the requirements of both non-redundancy and sampling distribution. In this paper, we formalize mathematically the composite metric to trade-off the multiple demands of observation, and propose the hybrid-index-based array layout optimization strategy. The simulation results demonstrate that, in comparison with the typical distribution, the optimized array using the proposed optimization framework enables the acquisition of more comprehensive spectrum information while utilizing an equal number of apertures, providing superior imaging quality in different observation situations. Furthermore, the designed optimized array masks and the compared conventional array masks were fabricated and used for our experimental validation, further verifying the feasibility of this strategy. This array configuration optimization framework may not only find applications to Michelson interferometric imaging, but also provide a positive impact on all u-v sampling-based imaging modes, including radio interferometry, magnetic resonance imaging, and photonic integrated interferometric imaging.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Michelson imaging interferometers using aperture synthesis method can achieve high angular resolution down to milliarcseconds for astrophysical studies by combining light from multiple sub-apertures [1–5]. Through the snapshot or Earth rotation synthesis mode, the spatial frequency of targets in the $u$-$v$ plane can be sampled by cross-correlating sub-aperture pairs, and subsequently be inverted to estimate the original brightness distribution of the observed scene [6,7]. The configuration of array elements determines the $u$-$v$ coverage, which should be subtly designed. For a low sampled value, a non-optimized $u$-$v$ coverage will not only cause decoherence on redundantly sampled baselines due to phase fluctuations [8] but will also lead to more severe sampling holes and miss of baseline [9], which may result in larger inversion errors and limit the imaging ability of the array. Since the quality of the reconstructed image is governed by the observations with different baselines, determining the optimal array configuration that provides the best $u$-$v$ coverage is a prominent factor for image retrieval. Furthermore, the optimal array configuration may also benefit all $u$-$v$ sampling-based image modes, including radio interferometry, magnetic resonance imaging (MRI), and photonic integrated interferometric imaging (PIII).
The design of interferometric arrays should be driven by the available number of telescopes, scientific purposes and the technical requirements [10,11]. Firstly, for a given number of sub-apertures, the array without repeated baselines, called non-redundance, leads to more filling of $u$-$v$ plane which enhances image sharpness and minimizes artifacts [12]. Secondly, for the special scientific observations, a diluted array with long baselines enables the acquisition of higher frequency components of the target, while a densely populated array has a point-spread function with fewer sidelobes, which makes it easier to observe complex scenes [10]. By establishing a appropriate metric that ensures the specified requirements for $u$-$v$ coverage, the array optimization technique converts the array configuration design into a nonlinear optimization problem, which has became the main solution to the optimum array configuration for scientific observation [13,14].
For decades, many studies have been conducted to address the optimization principles of an interferometric array. In one of the first studies by M. Golay in 1970, an iterative method was used to build up the number of apertures in regular periodic grids and simultaneously defined the redundancy and compactness of the array configuration [15]. Based on this research, the special one-dimensional data structures were mapped to a two-dimensional space and subsequently constructed completely non-redundant array, which liberated the requirement of massive calculation in the large array elements optimization [16–18]. Considering the impact of array configuration on image reconstruction, two typical $u$-$v$ distribution cases were introduced, namely, the Gaussian and the uniform distributions. Optimizing the distribution of telescopes in the array by employing numerical methods, the optimization procedure showed a tendency to place the telescopes on regular geometric curves, either on the perimeter of a circle [19], or on a Reuleaux triangle, which is a triangle with rounded sides [20]. Despite achieving the goal of good uniformity, sidelobes in the snapshot PSF are also distributed regularly, which are not suppressed satisfactorily [21]. By contrast, the Gaussian $u$-$v$ distribution shows the imaging advantage of a higher signal-to-noise ratio, due to its centrally condensed sampling coverage with many short baselines [22]. In practice, the array configuration using prescribed geometric curves will confront the technical challenges from the terrain of the device [23–26], high costs associated with multiple optical transmission vacuum pipes and the measurement requirements of small contrast fringes [27].To get rid of the limitation of the fixed distribution, some researches directly optimized the positions of the sub-apertures in the two-dimensional free space by constructing a proper index to characterize the given $u$-$v$ coverage distribution [12]. However, due to the single metric index, the state-of-the-art array configuration optimization framework makes it difficult to dynamically trade-off redundancy and the sampling distribution of $u$-$v$ coverage. In addition, current optimizations are mostly driven by the typical $u$-$v$ coverage. Consequently, the reconstructed image quality might not be satisfactory if the observed targets exhibit the frequency distribution significantly different from the assumption.
In this paper, an array configuration optimization strategy based on hybrid-index is proposed. On the one hand, the strategy aims to obtain the maximum amount of spatial frequency information of the object, which can be realized by non-redundant sampling. On the other hand, the strategy aims to sample the dominant spatial frequency of the observed scene, also with the spatial frequency as high as possible, which can be realized by designs of specialized $u$-$v$ coverage distribution. By dynamically modulating the proportion of these two aspects, the hybrid-index is formalized mathematically to trade-off the multiple demands of the observation. Deploying the hybrid-index in the Particle Swarm Optimization (PSO) algorithm, the array optimization problem is transformed into a nonlinear optimization process. Simulations and experiments demonstrate that the optimized array enables the acquisition of more comprehensive spectrum information, providing superior imaging quality while utilizing an equal number of apertures. This array configuration optimization framework may not only benefit optical imaging, but also provide a positive impact on all $u$-$v$ sampling-based imaging modes [28–31].
This paper is organized as follows. In Section 2, the imaging model of the Michelson imaging interferometer is analyzed, and three separated $u$-$v$ coverage metrics are subsequently described. Through weight coefficient modulation, the hybrid-index is constructed to balance the sampling rate and distribution characteristics of the $u$-$v$ coverage. In Section 3, the 33-element arrays optimized by hybrid-index are designed, and then compared to typical arrays configured by fixed distribution. Further improvements in imaging quality were achieved by incorporating prior information about the object and optimizing the array based on spectral feature distribution. In Section 4, we designed and manufactured multiple 33-element array masks to validate the feasibility and effectiveness of this method.
2. Principle
The reconstructed image of a Michelson imaging interferometer closely depends on the array configuration. However, the relationship between the array configuration and its image quality is not directly apparent. Simply put, the array configuration determines its $u$-$v$ coverage, which, in turn, influences its image quality.
2.1 Imaging model and $u$-$v$ coverage
The imaging principle of the interferometer can be traced back to the van Cittert–Zernike theorem, putting the concept of Fourier optics in the context of stellar interferometry. In the context of interferometric imaging, a number of assumptions are made (including 1. Fresnel approximation 2. Spatially incoherent light sources 3. The quasi-monochromatic approximation), under which the van Cittert–Zernike theorem can be simplified and expressed as
The coordinate vector in the source plane is denoted by $x$, and the plane of observation has the coordinate vector $\xi$. The size of the source and the area of interest in the plane of observation are much smaller than the distance $z_0$ allowing for small angle approximations $\alpha = x/z_0$. According to the van Cittert–Zernike theorem, the Fourier transformed spatial spectrum of an object’s intensity distribution $O_b(\alpha )$ can be measured by the coherence function $\Gamma _{qm}$ of the aperture plane.
An interferometric array consisting of the combination of two sub-apertures is shown in Fig. 1, which can capture the frequency information of the target object at $\mathbf {R}_B=\mathbf {r}/\lambda$ and $-\mathbf {R}_B=-\mathbf {r}/\lambda$. The projections of the baseline vectors onto the sky plane perpendicular to the line of sight are effective baselines $\boldsymbol {\mathbf {r}}$, and the wavelength of light is $\lambda$. The $u$-$v$ coverage represents all the target spatial frequency locations captured.
2.2 Main metrics for array optimization
According to the imaging principle of the Michelson imaging interferometer, the $u$-$v$ sampling is determined by the layout of sub-apertures. Determining the optimal array configuration that provides the best $u$-$v$ coverage is a prominent factor for image reconstruction. In addition to image quality, resolution should also be taken into account. For specific scientific observation scenarios, the limiting resolution of interferometric arrays will accordingly have a certain standard, so this should also be included in our consideration. Existing research shows that the design of a Michelson imaging interferometric array can be driven by three metrics: redundancy, $u$-$v$ coverage distribution, and the full width at half maximum (FWHM) of the PSF.
2.2.1 $u$-$v$ coverage and redundancy of array
Given a set $\mathbf {A}=\left \{(x_i,y_i)\right \}, i\in [1,\dots,n ]$ representing the position of an n-element array’s sub-aperture, the array configuration can be represented as:
According to Section 2, a pair of sub-apertures in the Michelson imaging interferometer can capture a set of spatial frequency information from the target object. Its $u$-$v$ coverage can be represented as:
From this, we can obtain the set $\mathbf {B}=\left \{(u_i,v_i)\right \}, i\in [1,\dots,N ]$, representing all baseline vectors of the array configuration. In an ideal scenario, the set $\mathbf {B}$ does not contain any identical elements, and such a $u$-$v$ coverage is considered non-redundant. However, in most cases, there are duplicate elements in the set $\mathbf {B}$, which means that the baselines are repeated, indicating that the array’s $u$-$v$ coverage is redundant. By removing the duplicate elements from the set $\mathbf {B}$, we obtain a new set $\mathbf {B'}=\left \{(u'_i,v'_i)\right \}, i\in [1,\dots,N' ]$ while $N'<n(n-1)$. We evaluate the array redundancy using Eq. (4). The larger the value of $R$, the higher the redundancy of the array.
The achievement of non-redundant $u$-$v$ coverage or the reduction of $u$-$v$ coverage redundancy has always been a focal point in array configuration optimization [15,17,18,22]. Due to the extremely sparse spectrum sampling of the Michelson imaging interferometric array, lower $u$-$v$ coverage redundancy leads to a greater number of target frequency samplings. Simply following Parseval’s theorem, an array with non-redundant $u$-$v$ coverage will have the lowest sidelobe levels in its PSF. Reducing the redundancy of the array’s $u$-$v$ coverage will lower the sidelobe level of the PSF.
2.2.2 Features of $u$-$v$ distribution
In addition to the impact of redundancy on imaging, the $u$-$v$ distribution characteristics of the array’s $u$-$v$ coverage also affect imaging quality. In the absence of prior information about the image or the goals of an observation, $u$-$v$ coverage that satisfies uniform or Gaussian distribution is considered to yield better imaging quality [20,22,32]. To assess the similarity between continuous standard distribution functions and the actual array’s $u$-$v$ coverage. We employed the projection method proposed by Villiers in 2007 [33] to characterize the characteristics of the $u$-$v$ coverage by projecting the $u$-$v$ coverage distribution in various directions. Firstly, it is necessary to calculate the standard coordinate arrays’ $u$-$v$ coverage in various projection directions based on the standard distribution function $T(u,v)$. By projecting the function in the $\theta$ direction, a two-dimensional curve can be obtained:
Integrating it yields the cumulative distribution function (CDF) of the $u$-$v$ coverage projected in the $\theta$ direction.
We consider the function $f_\theta (\rho )$ obtained from Eq. (5) as the probability density function of the $u$-$v$ coverage projected in the $\theta$ direction. Therefore, the CDF $F_\theta \left (\rho \right )$ obtained from Eq. (6) can be viewed as the probability distribution of the $u$-$v$ coverage in the $\theta$ direction. $F_\theta \left (\rho \right )$ indicates that there should be $F_{\theta }\left (\rho \right )$ baselines in the region $[-\infty,\rho ]$. Taking the value of $\rho _0$ when $F_\theta \left (\rho _0\right )=m$ as the standard coordinate position of the $m$th projected baseline, we can obtain an $N$-dimensional array $S_\theta$, where $N=n(n-1)$. In this way, we obtain the standard coordinate array of the $u$-$v$ coverage in the $\theta$ direction.
Figure 2 illustrates the process of obtaining the standard coordinate arrays for a uniform distribution of $u$-$v$ coverage with $N=6$ (array sub-apertures $n=3$) on the $u$-$v$ plane. Figure 2(a1) displays the standard distribution function for a uniform distribution as follows:
with R=60m. Figure 2(a2) represents the projection function $f_0\left (\rho \right )$ calculated according to Eq. (5) in the $\theta =0^{\circ }$ direction. The projection of the standard distribution functions for both uniform and Gaussian distributions is consistent in all directions. Figure 2(a3) shows the CDF $F_0\left (\rho \right )$ for the uniform distribution in the $\theta =0^{\circ }$ direction, along with the standard coordinate array of the projections of the $u$-$v$ coverage.
Fig. 2. Procedures for obtaining standard coordinate arrays of (a) Uniform distribution and (b) Gaussian distribution. Standard distribution function of the (a1)uniform distribution and (b1) Gaussian distribution. (a2) and (b2) the projection of the standard distribution function in the $\theta =0^{\circ }$ direction. (a3) and (b3) CDF and standard coordinate arrays.
Figure 2(b) illustrates the process of obtaining standard coordinates for the Gaussian distribution in the $\theta =0^{\circ }$ direction. In Fig. 2(b1), the standard distribution function for the Gaussian distribution is shown as follows:
with $\sigma =25$. Figure 2(b3) provides the standard coordinate array of the $u$-$v$ coverage projected in the $\theta =0^{\circ }$ direction under the Gaussian distribution.Next, calculate the projection coordinate array of the array’s $u$-$v$ coverage. Using Eq. (3) to calculate the $u$-$v$ coverage of the array. Then, calculate the projection coordinates of the $u$-$v$ coverage and arrange them in descending order to obtain the $u$-$v$ coverage projection coordinate array $S'_{0}=\left \{ \rho '_{0i} \right \}_{1\le i\le N}=\left \{\rho '_{01},\dots,\rho '_{0N} \right \}$, as determined by Eq. (9).
Finally, the $u$-$v$ distribution characteristic evaluation function is obtained by subtracting the standard coordinate arrays from the $u$-$v$ coverage projection coordinate arrays in $M$ projection directions and performing a weighted summation.
with $C_{ij}$ is the weight coefficient. When optimizing the $u$-$v$ coverage for a uniform distribution, considering that low-frequency components better highlight the source features in objects, the value of $C_{ij}$ follows a Gaussian function, as shown in Eq. (11). However, $C_{ij}$ is set to 1 when optimizing for a Gaussian distribution.2.2.3 Spatial resolution
Interferometers were originally designed to achieve the resolution of large aperture systems using small aperture optics. The first successful measurement of a stellar diameter with separate apertures was performed 30 years later by A.A. Michelson and F.G. Pease on Mt. Wilson, determining the diameter of $\alpha$ Orionis to $0.047$ arcsec [34] when the smallest diameter that could be measured with a full aperture was about 1 arcsec at this time.
The resolution of the interferometer can be estimated from the full width at half maximum (FWHM) of its PSF without considering the effect of sidelobes. According to Eq. (12), the PSF of the interferometer can be obtained directly by performing Fourier transforming on its u-v coverage.
with $b_0(x,y)$ the PSF of the interferometer. The inhomogeneity of the optimized arrays in all directions makes it difficult to directly evaluate the spatial resolution of the arrays using the FWHM of the cross section, so in order to take into account the resolution of the arrays in all directions the area of the region where the intensity of the main peak of the array is higher than half maximum is used to indicate the spatial resolution.2.2.4 Hybrid-index optimisation strategy
The redundancy and distribution characteristics of $u$-$v$ coverage significantly impact the restored image quality and FWHM can describe the spatial resolution of the array. This paper proposed a Michelson imaging interferometric array configuration optimization strategy based on the hybrid-index. This strategy allows the $u$-$v$ coverage of the optimized array to meet the specified $u$-$v$ distribution characteristics while considering redundancy and spatial resolution. Combining Eq. (4) and Eq. (10), we can obtain a hybrid-index evaluation function for $u$-$v$ coverage redundancy and distribution characteristics as well as the spatial resolution of interferometer.
with $c_R$, $c_x$ and $c_F$ the weighting coefficients for $u$-$v$ coverage redundancy, $u$-$v$ distribution characteristics, and spatial resolution, used to flexibly adjust the optimization focus of the array.In addition to main distribution characteristics of the u-v coverage, there might be some special limits , according to our specific needs. Based on this, we can further rewrite Eq. (13) as:
In the actual optimization process, it seems a difficult task to balance all the three indexes for improvement of the imaging performance of the array. In the following research, the hybrid-index optimization scheme is performed with two of the indexes. The choice of the index depends on the requirements of applications.
Considering that $u$-$v$ coverage is determined by the overall array distribution, adjusting individual specific baseline can be quite challenging. Changes in the position of a single sub-aperture of the array can lead to variations in the positions of $N-1$ baselines. Therefore, introducing additional constraints or making vague adjustments to the array may not necessarily be effective. Hence, we ultimately employ the PSO.
Figure 3 presents the flowchart of the optimization algorithm. In the Standard Coordinate Generation Module, standard coordinates are generated based on Eq. (5) and Eq. (6), and this process is only executed once during the entire optimization. In the Fitness Calculation Module, fitness values for each particle are calculated based on Eq. (13). In the Iterated Module, the array configuration undergoes iterative optimization.
3. Simulation
We used the hybrid-index optimization strategy mentioned in Section 2 to optimize 33-element arrays. The spatial domain is discretized into $101*101$ rectangular grids, each representing an area of $1*1m^2$ in size. The sub-aperture of the array can only be located in the center of each grid, ensuring that the total size of the interferometric array was controlled to be within an area of $101*101m^2$ and that each sub-aperture of the interferometric array was at least one meter apart.
For a better assessment of the array’s performance, imaging and PSF are used as the main metrics. Imaging can be simulated by specifying a source with a known brightness distribution based on the following Eq. (15). Figure 4 represents the source images used in the simulation, which include three celestial sources and three remote sensing sources.
with $f^*(x,y)$ represents the "dirty image", $f(x,y)$ is the original source image, and $\mathrm {Sample}(u,v)$ represents the $u$-$v$ coverage of the interferometer.Fig. 4. Pictures of sources used in the simulation. (a) The Helix Nebula (NGC 7293), (b) Curious spiral spotted by ALMA around red giant star R Sculptoris, (c) Dark matter. (d) Resolution chart, (e) storage tank, (f) ship.
PSF can be obtained by the Fourier transform of the $u$-$v$ coverage. With the $u$-$v$ plane similarly discretized into grids, the resolution of the array can be determined by the baseline length and grid size given by:
with $B$ the length of the baseline, and $\triangle u$ the size of the grid which is transformed according to the wavelength of the source. Therefore, in subsequent experiments, the size of the pixel point area occupied by FWHM is used to evaluate the spatial resolution of the arrays. The PSFs were fitted using Gaussian beam to show their sideslobe levels and non-Gaussianity.3.1 Redundancy+uniform distribution for $u$-$v$ coverage
Firstly, we optimized the 33-element array using the optimization index of ’redundancy + uniform distribution’ with 50 particles and 100 rounds of iteration. This results in the optimized array, which will be referred to as the optimized uniform array, and then compared to the Reuleaux triangle-shaped array. In the case of typical arrays, the $u$-$v$ coverage of the Reuleaux triangle-shaped array is considered to have the best uniformity but inevitably encounters the issue of missing baselines. Figure 5(b1) shows the normalized optimization iteration curve of the global best fitness value using the rewriting fitness function defined in Eq. (13). The optimal design appears after the 97th iteration.
Fig. 5. (a1) Array configuration and (a2)$u$-$v$ coverage of the Reuleaux triangle-shaped array (n=33). (a3)-(a4) PSF and its cross section compared to the fitted Gaussian beam. (a5)-(a10) Imaging results. (b1) Optimization iteration curve, (b2) array configuration and (b3) $u$-$v$ coverage of optimized uniform array (n=33). (a4)-(a5) PSF and its cross section compared to the fitted Gaussian beam. (a6)-(a11) Imaging results.
The 33-element Reuleaux triangle-shaped array configuration, $u$-$v$ coverage, and simulated imaging results for the sources in Fig. 4 are shown in Fig. 5(a). The $u$-$v$ coverage of the Reuleaux triangle-shaped array is illustrated in Fig. 5(a2), where the number of longer baselines is greater than the number of other baselines. This results in the reversal of grey values in some areas of the simulated imaging result compared to the original source images, as seen in the central area of Fig. 5(a7). The presence of artifacts and aliasing makes the images clearly not very impressive, as dirty images are used directly in the comparison to highlight the influence of the array itself in the imaging process. Most modern interferometric imaging needs reconstruction algorithms such as CLEAN or MEM [35], which can improve image quality by supplementing the information in undersampled images with information derived from imaging systems such as PSF or from the images themselves.
For the 33-element optimized uniform array, its optimization iteration curve, array configuration, $u$-$v$ coverage, and simulated imaging results for sources in Fig. 4 are presented in Fig. 5(b). The $u$-$v$ distribution characteristics of the optimized uniform array are evaluated based on Eq. (10), with the introduction of the weighting factor. During the optimization process, the array prioritizes satisfying distribution positions of shorter baselines. This results in fewer artifacts and superior simulated imaging quality compared to the Reuleaux triangle-shaped array.
As shown in Fig. 5(a4) and Fig. 5(b5), the smaller FWHM of the Reuleaux triangle-shaped array allows for greater spatial resolution, while the higher level of sidelobe results in inferior image quality compared to optimized uniform array.
Table 1 provides the numerical metrics for the simulated imaging of Fig. 4 images using the Reuleaux triangle-shaped array and the optimized uniform array. These metrics include peak signal-to-noise ratio (PSNR), structure similarity (SSIM), fidelity, FWHM, and peak sidelobe. The expressions for PSNR, SSIM and fidelity are shown in Eqs. (17), (18) and (19), where $I(x,y)$ is the measured image and $J(x,y)$ is the model image. The optimized uniform array contains fewer missed baselines. More baselines and an increase in the number of short baselines compared to the triangular array result in an improvement in the numerical metrics of image quality at the expense of resolution.
Table 1. Numerical Metrics for Array with Uniform Distribution for $u$-$v$ Coverage
3.2 Redundancy+Gaussian distribution for $u$-$v$ coverage
Similarly, we used the 'redundancy + Gaussian distribution’ index to optimize the array with 50 particles and 100 rounds of iteration. This results in the optimized array, which will be referred to as the optimized Gaussian array, and compared to the power-law Y-shaped array. In typical arrays, the $u$-$v$ coverage of the power-law Y-shaped array is closest to the Gaussian distribution, with shorter baselines being much more numerous than longer baselines. This leads to relatively good image quality, but the array has a very serious miss of baseline.
Because of the severe miss of baseline, the number of baselines in the 33-element power-law Y-shaped array is similar to that of the 27-element optimized gaussian array and much less than that of the 33-element optimized Gaussian array. A comparison was made in the simulation. The 33-element power-law y-shaped array configuration, $u$-$v$ coverage, and simulated imaging results for the sources in Fig. 4 are shown in Fig. 6(a). It can be observed that there is significant distortion in the details of the images, particularly evident in Fig. 6(a6).
Fig. 6. (a1) Array configuration and (a2)$u$-$v$ coverage of the power-law Y-shaped array (n=33). (a3)-(a4) PSF and its cross section compared to the fitted Gaussian beam. (a5)-(a10) Imaging results. (b1) Optimization iteration curve, (b2) array configuration and (b3) $u$-$v$ coverage of optimized Gaussian array (n=27). (a4)-(a5) PSF and its cross section compared to the fitted Gaussian beam. (a6)-(a11) Imaging results.(c1) Optimization iteration curve, (c2) array configuration and (c3) $u$-$v$ coverage of optimized Gaussian array (n=33). (c4)-(c5) PSF and its cross section compared to the fitted Gaussian beam. (c6)-(c11) Imaging results.
The optimized Gaussian array’s iteration curve, array configuration, $u$-$v$ coverage, and simulated imaging results for the sources as well as the PSFs are presented in Fig. 6(b). The optimal design appears after the 75th and 41st iterations, respectively. In terms of image quality, the 27-element optimized Gaussian array is comparable to the 33-element power-law Y-shaped array, with further improvements in details for the 33-element optimized Gaussian array. In the power-law Y-shaped array, the details in the images are very blurry. In comparison, the imaging results of the optimized Gaussian array show better performance in the details section.
The FWHM does not reflect a large disparity overall, but the PSF of the power-law Y-shaped array has a high level of sidelobes, especially in the Y direction, which causes significant distortion of the image in the simulation results.
Table 2 provides the numerical metrics for the power-law Y-shaped array and the optimized Gaussian array. The 33-element optimized Gaussian array has 902 different baselines, which is an increase of 218 baselines to the 684 different baselines in the 33-element power-law Y-shaped array. In other metrics, the 27-element optimized Gaussian array has already outperformed the 33-element power-law Y-shaped array, while the 33-element optimized Gaussian array shows significant improvements in the numerical metrics. Power-law Y-shaped array has a little higher resolution, while optimized Gaussian arrays have lower peak sidelobes.
Table 2. Numerical Metrics for Array with Gaussian-like Distribution for $u$-$v$ Coverage
3.3 Redundancy+spectral feature distribution for $u$-$v$ coverage
Uniform and Gaussian distributions are common optimization probability distribution shapes, but they do not take into account the spectral distribution characteristics of the detected objects themselves. When prior information about the image or the goal of observation is available, it is possible to replace the standard distribution in Eq. (5) with a spectral feature distribution that incorporates prior information about the target. This allows for the design of an array configuration tailored to a specific target, resulting in improved image quality for that specific target.
To verify the feasibility of this idea, we used the spectral information of the storage tank in Fig. 4(e) and the spectral information of the resolution chart in Fig. 4(d) as spectral feature distributions to replace the standard distribution function $T(u,v)$ in Eq. (5). We then obtained the standard coordinate arrays based on Eq. (6), as illustrated in Fig. 7. As shown in Fig. 7(a3) and Fig. 7(b3), the CDF based on the spectral feature distribution of natural objects is similar to that obtained from a Gaussian distribution but has steeper slopes in the end regions of the curve, indicating a higher contribution of high-frequency information.
Fig. 7. The same as the Fig. 2. Replace the standard distribution function with the spectral feature function.
Figure 8(a) shows the spectral feature array optimized for the storage tank image, referred to as the 'optimized spectral feature array1’, and its imaging performance on the storage tank image in Fig. 4(e). The optimization iteration curve of the optimized spectral feature array1 is shown in Fig. 8(a1). Compared to the optimization iteration curves of the optimized uniform array and optimized Gaussian array, the optimized spectral feature array1’s convergence rate is noticeably slower, and it spends more time trapped in local minima, eventually achieving the optimal design after the 93rd iteration. The $u$-$v$ coverage of the array changes significantly, with more short baselines in the horizontal direction than in the vertical direction, giving its $u$-$v$ coverage a rectangular shape in the center, and more long baselines in the vertical direction. This brings the $u$-$v$ coverage of the optimized array closer to the spectral feature of the storage tank picture.
Fig. 8. (a1) Optimization iteration curve, (a2) array configuration and (a3) $u$-$v$ coverage of optimized spectral feature array1. (a4) Results of imaging the Storage tank in Fig. 4(e). (a5)-(a6) PSF and its cross section compared to the fitted Gaussian beam. (b1) Optimization iteration curve, (b2) array configuration and (b3) $u$-$v$ coverage of optimized spectral feature array2. (b4) Results of imaging the resolution chart in Fig. 4(d). (b5)-(b6) PSF and its cross section compared to the fitted Gaussian beam.
Figure 8(b) presents the 'optimized spectral feature array2’ for the resolution chart image and its imaging performance. The optimization iteration curve for the optimized spectral feature array2, as shown in Fig. 8(b1), indicates that, similarly, the array converges slower than the optimized uniform array and optimized Gaussian array, eventually reaching optimal design after 52nd rounds of iteration. As shown in Fig. 8(b3), the baselines of optimized spectral feature array2 are mainly concentrated in the vertical and horizontal directions, with a smaller number of baselines in the tilted direction, which is consistent with the spectral feature of the resolution chart.
Table 3 provides the numerical metrics of the simulated imaging for the optimized spectral feature arrays, optimized uniform array, and optimized Gaussian array. Whether it’s the image of the storage tank or the resolution chart, the optimized spectral feature arrays show an improvement of nearly 10% in terms of PSNR, SSIM and fidelity compared to the other two arrays. Compared to both the optimized Gaussian array and optimized uniform array, the optimized spectral feature arrays have a large increase in limiting resolution while the level of sideslobs have also improved significantly.
Table 3. Numerical Metrics for Optimized Spectral Feature Array
3.4 Spatial resolution+Gaussian distribution for $u$-$v$ coverage
This paper presents some results in terms of image quality optimization in Section 3.1,3.2 and 3.3. However, the improvement in image quality of the Gaussian and uniform arrays in the previous section comes at the expense of spatial resolution, which is unacceptable in some of the observation scenarios. According to the studies in Section 3.2 for imaging the source of Fig. 4, the $u$-$v$ coverages with the Gaussian-like distribution achieve better image quality compared to those with the uniform distribution. To balance the image quality and spatial resolution, we use the 'FWHM+Gaussian distribution' index to optimize the array with 50 particles and 100 rounds of iteration. This results in the optimized array, which will be referred to as the 'optimized FWHM+Gaussian array'.
The optimized FWHM+Gaussian array’s iteration curve, array configuration, $u$-$v$ coverage, and simulated imaging results for the sources in Fig. 4 as well as the PSF are presented in Fig. 9(a). The optimal design appears after the 59th iterations. As shown in Fig. 9(a4) and (a5) there is a significant increase in the level of sidelobe in its PSF compared to other optimized arrays. An increase in spatial resolution is obtained, with a better imaging performance because the spatial resolution has been taken into account in the optimization indexes.
Fig. 9. (a1) Optimization iteration curve, (a2) array configuration and (a3)$u$-$v$ coverage of optimized FWHM+Gaussian array. (a4)-(a5) PSF and its cross section compared to the fitted Gaussian beam. (a6)-(a11) Imaging results.
Table 4 provides the numerical metrics for the optimized FWHM+Gaussian array. The introduction of the spatial resolution metrics enables further improvements in imaging performance for sources such as resolution chart and R Sculptoris, which have more details, and the optimized FWHM+Gaussian array has advantages in spatial resolution, peak sidelobe level, and imaging quality compared to the power-law Y-shaped array, which also has a Gaussian-like distribution.
Table 4. Numerical Metrics for Optimized FWHM+Gaussian Array
4. Experimental result
To validate the simulations, we set up two experimental setups. Based on the array configurations in Figs. 5(a1),(b2) and Figs. 6(a1),(c2), we created masks as depicted in Fig. 10(a1)-(a4).
Fig. 10. (a) The masks used in the experiments. (b) Experimental setup for point source experiments. (c) Experimental setup for extended target experiments.
Since it is difficult for us to obtain satisfactory results for complex images due to the limitations of the experimental system, an array optimized with the ’redundancy + spectrum of the handwritten letter E’ metric is used to validate the content of the spectral feature sampling, whose configuration and masks are shown in Fig. 11(e) and Fig. 10(a5) respectively.
Fig. 11. Results of point source experiments, including PSF obtained using different masks and their calculated $u$-$v$ coverage.
We set up two experimental setups: one is the point source experiment, as shown in Fig. 10(b), where a laser point source couples with a collimator through an optical fiber. After refraction through mirrors, the light passes through the array mask and is focused onto a CCD camera using a lens. The PSF is recorded on the CCD, and by Fourier transform, we can obtain the array’s $u$-$v$ coverage.
The extended target experiment, depicted in Fig. 10(c), involves a 2D array source. The light source passes through a digital micromirror device (DMD) and enters the collimator, following a similar configuration as the point source experiment. The source images used for the extended target experiment are very small, and the field of view is constrained to the size of an Airy disk of an individual sub-aperture, so that direct imaging can be used to simulate the imaging procedure of Michelson imaging interferometers.
The experimental setup used a collimator with a diameter of 300mm, a focal length of 3m, and an f-number of 10. For the point source experiment, a monochromatic laser with a wavelength of 639nm was employed. The source used for the extended target experiment was the SOLIS-620D High-Power LED for Microscopy from Thorlabs.
4.1 Point source experiment
The point source experimental results are shown in Figs. 11(a1),(b1),(c1),(d1),(e3). The PSFs of arrays are displayed, which exhibit significant speckle patterns due to the extremely sparse frequency sampling. The Fourier transform of the PSFs, shown in Figs. 11(a2),(b2),(c3),(d3),(e4), yields the $u$-$v$ coverage of the array. The obtained $u$-$v$ coverage matches the structure shown in Section 3 and complies with our simulation.
4.2 Extended target experiment
The experiment using an extended target employed original images as shown in Fig. 12(a). The image obtained by the Reuleaux triangle-shaped array has the lowest contrast, followed by the three optimized arrays, and the image obtained by the power-law Y-shaped array has the highest contrast, which is partly due to the fact that its shortest baseline length is lower than that of all the other arrays and therefore it has more low frequency samples, but this also leads to a blurring of the edges of the image and a certain degree of distortion of the image as a whole.
Fig. 12. Results of extended target experiment. (a) The original image (b)-(f) The imaging results using different masks.
Table 5 gives numerical indicators of the imaging performance of the arrays, with the optimized spectral feature array producing a better image quality of the handwritten letter E than the other arrays, in line with our expectations.
Table 5. Numerical Metrics for the Experiment
5. Conclusion
In this paper, we formalize mathematically the composite metric to trade-off the multiple demands of observation, and propose the hybrid-index-based array configuration optimization strategy. The dynamic regulation of weight coefficients allows for flexible adjustment of the optimization strategy. We design the optimized uniform array and the optimized Gaussian array, and compare their imaging performance with the Reuleaux triangle-shaped array and the power-law Y-shaped array. Simulations demonstrate that the optimized array enables the acquisition of more comprehensive spectrum information, providing superior imaging quality while utilizing an equal number of apertures. Furthermore, incorporating prior information, we can replace the standard distribution with spectral feature distribution, and arrays optimized in this way will have a further improvement in image quality for specific source images than arrays whose $u$-$v$ coverage satisfies the Gaussian or uniform distribution. Combining the spatial resolution and $u$-$v$ coverage metrics we obtain optimized FWHM+Gaussian array which strike a balance between resolution and image quality. Experimental results verifies that our designs are in high agreement with the simulation results, and the actual fabricated arrays have the expected $u$-$v$ coverage and imaging effects. The most important aspect of the hybrid-index optimization strategy is that it provides a way to incorporate external factors into our optimization [36]. This array configuration optimization framework may not only benefit optical imaging, but also provide a positive impact on all $u$-$v$ sampling-based imaging modes, including radio interferometry, magnetic resonance imaging (MRI), and photonic integrated interferometric imaging (PIII).
Funding
National Key Research and Development Program of China (2022YFB3901900); National Natural Science Foundation of China (62175243); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2020372); Outstanding Scientist Project of Tianfu Qingcheng Program.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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