MTF improvement for optical synthetic aperture system via mid-frequency compensation

Yong Wu; Mei Hui; Weiqian Li; Ming Liu; Liquan Dong; Lingqin Kong; Yuejin Zhao

doi:10.1364/OE.420512

1. Introduction

Due to the humans’ ambitions to explore the universe, optical synthetic aperture imaging is proposed to enhance the spatial resolution by increasing the effective aperture diameter. Meanwhile, it is also desirable to minimize the total image collection area because of the volume and weight of large monolithic mirror. The optical synthetic aperture system has successfully solved the manufacturing, installation and launching problems caused by the fully filled large aperture system [1–3]. The system consists of several in-phase circular sub-mirrors. The synthetic aperture system combines the radiation from these small sub-mirrors to realize a resolution equivalent to that of a monolithic aperture, whose area is greater than the sum of the individual sub-mirrors [3]. However, due to the discrete distribution of the sub-apertures, the pupil function is no longer a connected domain, which further leads to the attenuation or loss of the mid-frequency modulation transfer function (MTF) [4–5]. It means that the mid-band spatial frequency contains lower energy, therefore the image information at the corresponding spatial frequency will be suppressed or lost. Thus, the mid-frequency MTF compensation is a key focus in designing a good synthetic aperture system.

Fortunately, there are two main ways to compensate the mid-frequency MTF [5]. The first method is to improve mid-frequency by image restoration method. In general, a high-speed and efficient image restoration method employs deconvolution algorithms, which include Wiener-Helstrom filter, linear least-squares filter, or wavelet transformation, and so on [6–8]. The second method is to optimize the aperture configuration. By using optimization algorithm, some aperture configurations with excellent performance are obtained, which maximize the spatial cutoff frequency while simultaneously maintain a minimum MTF level over all frequencies up to the cutoff [9–11]. However, in both two methods, the decrease of the mid-frequency MTF caused by the dispersion and sparsity is still inevitable. That is, the deteriorated or lost part of the mid-frequency information cannot be recovered.

For a monolithic aperture, the MTF decreases approximately linearly before the cutoff frequency, which means that the MTF approximates 0.5 at half the cutoff frequency [12]. But for optical synthetic aperture system, the MTF quickly drops below 0.5 at low frequency and slowly decays at middle frequency until it becomes zero at cutoff frequency. Therefore, Chenghao Zhou and Zhile Wang proposed an image fusion algorithm to combine the images of the optical synthetic aperture system and the monolithic aperture system [5]. However, in their method, the images of the two systems are added directly, which leads to many special features of the system not being utilized. In this paper, based on the above analysis, a complete mid-frequency MTF compensation algorithm based on MTF characteristics of the synthetic aperture system and monolithic aperture system is proposed. The spatial frequency information according to the MTF is extracted and fused. Then the whole spatial frequency information is recovered to full-frequency enhanced image. The proposed method can not only expand the spatial frequency cutoff, but also compensate the attenuation of MTF in the middle frequency and maintain an appropriate MTF level at the full frequency. Both simulation and experiment prove that compensating the mid-frequency MTF by synthesizing two systems is effective.

The rest of paper is organized as follows. Section 2 derives the spatial frequency equivalence point and presents the process of frequency extraction and synthesis. The improved Wiener algorithm is also introduced in the section. Section 3 analyzes the performance of the proposed method. Section 4 details the experiment results. Finally, section 5 summarizes the conclusions.

2. Method

In this section, in order to compensate the mid-frequency MTF of optical synthetic aperture, another monolithic aperture system is introduced. The size of the monolithic aperture is designed and the spatial frequency equivalence point is derived. Then the spatial frequency information is extracted and synthesized. The Wiener-Helstrom filter is improved to adapt to the specific case.

2.1. MTF analysis

As the concept of the baseline of astronomical telescope, the baseline of the optical synthetic aperture system refers to the lines that can be obtained over the image collection area [13]. As shown in Fig. 1, a short baseline A1A2 corresponds to the low-frequency of the acquired information, which is usually the contour of the image. The long baseline C1C2 is responsible for the high frequency information, which is the detail of the image. The longest baseline D_span determines the cutoff frequency, which corresponds to the resolution of the system. The cutoff frequency ρ_c can be expressed as:

(1)$${\rho _c} = \frac{{{D_{span}}}}{{\lambda l}}\textrm{ }where\textrm{ }{\rho _c} = \sqrt{{f_x}^2 + {f_y}^2}$$

where f_x and f_y represent the spatial frequency coordinates, and λ is the wavelength and l denotes the distance from the camera to the object being imaged. Thus the longest baseline of the system is proportional to the resolution of the system.

Fig. 1. Baselines of optical synthetic aperture system.

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However, due to the reduction of the image collecting area, the sampling of information at different frequencies is not completely compared with the monolithic aperture. In general, the smaller the image collecting area is, the more information will be lost. It can be quantified by the fill factor α which is the ratio of the area of the sub-apertures to the area of a single filled aperture with equivalent image resolution, given by:

(2)$$\alpha \textrm{ = }\frac{{N(\pi {{(d/2)}^2})}}{{\pi {{({D_{circ}}/2)}^2}}} = \frac{{N{d^2}}}{{{D_{circ}}^2}}$$

where N is the number of sub-apertures, and d and D_circ represent the diameter of the sub-aperture and the circumscribed circle, respectively. It proves that a decrease in fill factor will result in a reduction in the MTF at middle frequency.

The MTF is the modulus of the optical transfer function (OTF), which is defined as the normalized Fourier transform of the point spread function (PSF) intensity. The PSF intensity can be obtained by taking the magnitude squared of the amplitude point spread function (ASF). The ASF is the Fourier transform of the system’s pupil function P(x, y):

(3)$$h({u,v} )= \frac{A}{{\lambda f}}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {P({x,y} ){e^{ - j\frac{{2\pi }}{{\lambda f}}({ux + vy} )}}dxdy} }$$

where (u, v) represents the image plane coordinate. A is the pupil area, and f denotes the focal length of the system [12]. The pupil function of the optical synthetic aperture system is:

(4)$$P(x,y) = \sum\limits_{n = 1}^N {{P_{sub}}(x - {x_n},y - {y_n}){e^{j{\phi _n}(x,y)}}}$$

where (x_n, y_n) is the center of the n-th sub-aperture, and ϕ_n denotes the phase of each sub-aperture. The binary pupil function of each sub-aperture P_sub depends on the sub-mirror, which is one inside the image collecting area of the sub-mirror and zero outside the image collecting area. For a circular aperture, its pupil function is a circle function. Thus, the ASF of the optical synthetic aperture system is:

(5)$$h(u,v) = {h_{sub}}(u,v)\sum\limits_{n = 1}^N {{e^{ - j2\pi (\frac{u}{{\lambda f}}{x_n} + \frac{v}{{\lambda f}}{y_n})}}} = \left( {\frac{{\pi {d^2}}}{{4\lambda f}}} \right)\frac{{2{J_1}\left( {\pi d\sqrt{{u^2} + {v^2}} /\lambda f} \right)}}{{\pi d\sqrt{{u^2} + {v^2}} /\lambda f}}\sum\limits_{n = 1}^N {{e^{ - j2\pi (\frac{u}{{\lambda f}}{x_n} + \frac{v}{{\lambda f}}{y_n})}}}$$

where h_sub(u,v) denotes the ASF of the sub-aperture, and J₁(•) is the first order Bessel function, which is the Fourier transform of the circle function.

After the ASF is obtained, the PSF intensity of the optical synthetic aperture system can be calculated by:

(6)$$PSF({u,v} )= {|{h({u,v} )} |^2} = {(\frac{{\pi {d^2}}}{{4\lambda f}})^2}{(\frac{{2{J_1}(\pi d\sqrt{{u^2} + {v^2}} /\lambda f)}}{{\pi d\sqrt{{u^2} + {v^2}} /\lambda f}})^2}{\left|{\sum\limits_{n = 1}^N {{e^{ - j2\pi (\frac{u}{{\lambda f}}{x_n} + \frac{v}{{\lambda f}}{y_n})}}} } \right|^2}$$

The configurations and their corresponding PSF of two typical optical synthetic aperture systems are shown in Fig. 2.

Fig. 2. (a) The configuration of Golay-3 and the corresponding PSF; (b) the case of Golay-6.

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The optical transfer function is defined as the normalized Fourier transform of the PSF intensity:

(7)$$OTF({{f_x},{f_y}} )= \frac{{{\cal{F}}\{{PSF({u,v} )} \}}}{{\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {PSF({u,v} )dudv} } }}$$

where f_x and f_y are the spatial frequency coordinates, and Φ denotes the Fourier transform. The MTF is the modulus of the optical transfer function, which can be calculated as:

(8)$$MTF({{f_x},{f_y}} )= MT{F_{sub}}({{f_x},{f_y}} )\ast \left[ {\delta ({{f_x},{f_y}} )+ \frac{1}{N}\sum\limits_{n = 1}^{{{N({N - 1} )} / 2}} {\delta \left( {{f_x} \pm \frac{{{x_{n + 1}} - {x_n}}}{{\lambda f}},{f_y} \pm \frac{{{y_{n + 1}} - {y_n}}}{{\lambda f}}} \right)} } \right]$$

where $\delta $ denotes the Dirac delta function and * is the convolution operator. MTF_sub(f_x, f_y) is the MTF of a single circle aperture diffraction system, which is given by:

(9)$$MT{F_{sub}}(\rho )= \left\{ \begin{array}{ll} \frac{2}{\pi }\left[ {\arccos (\frac{{\lambda f}}{d}\rho ) - (\frac{{\lambda f}}{d}\rho )\sqrt{1 - {{(\frac{{\lambda f}}{d}\rho )}^2}} } \right] &\rho \le \frac{D}{{\lambda f}}\\ 0 &\rho \ge \frac{D}{{\lambda f}} \end{array} \right.$$

where ρ represents $\sqrt{{f_x}^2 + {f_y}^2} $. Therefore, Eq. (8) shows that the MTF of the optical synthetic aperture system includes two terms: the central part and sidelobes. The typical Golay configurations consist of a series of two-dimensional point array geometries having compact, non-redundant autocorrelations [14]. Since the MTF is normalized, the maximum MTF level of the sidelobes of the Golay array is 1/N, except the unit valued zero frequency point. The derivation of the MTF is verified by the simulation results of ZEMAX. The parameters of the telescopes are detailed in Table 1. As shown in Fig. 3, the MTF of the Golay-3 and Golay-6 contain the sidelobes with maximum values of 1/3 and 1/6 respectively, which is consistent with expectation.

Fig. 3. (a) - (b) 3D model of the Golay-3 and Golay-6 in ZEMAX; (c) - (d) their corresponding MTF.

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Table 1. Parameters of Cassegrain telescope system.

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From Eq. (9), it is proved that the MTF of a fully filled circular aperture has unity modulation at zero spatial frequency, and the MTF decreases almost linearly until it reduces to zero at the cutoff frequency. Therefore, the MTF of a monolithic aperture is simplified to a linear function. When the MTF of the Golay-N system with a spatial frequency cutoff ρ_c1 is equal to that of the fully filled aperture system with a spatial frequency cutoff ρ_c2, the spatial frequency can be calculated by:

(10)$$MT{F_{Golay}}({{\rho_{eq}}} )+ \frac{{{\rho _{eq}}}}{{{\rho _{c2}}}} - 1 = 0$$

For example, the spatial frequency equivalence point of Golay-3 is calculated as:

(11)$${\rho _{eq}} \cong \frac{{4{\rho _{c1}}{\rho _{c2}}}}{{3({{\rho_{c1}} + {\rho_{c2}}} )}} = \frac{{4{D_{Monolithic}}{D_{Golay3}}}}{{3\lambda f({{D_{Monolithic}} + {D_{Golay3}}} )}}$$

where D_Monolithic and D_Golay are the diameter of the monolithic aperture and Goaly3 configuration, respectively. Optical design experiments have proved that the MTF of the mid-frequency level should be at least 0.3. Thus, for this monolithic aperture system, the MTF is reduced to 0.3 in the half of the spatial frequency cutoff ρ_c1. The diameter of the monolithic aperture can be calculated by:

(12)$$\frac{{{D_{Monolithic}}/\lambda f - {D_{Golay3}}/2\lambda f}}{{{D_{Monolithic}}/\lambda f}}\textrm{ = }0.3$$

Thus, the diameter of the monolithic aperture satisfies:

(13)$${D_{Monolithic}}\textrm{ = }\frac{5}{7}{D_{Golay3}}$$

Then, Eq. (10) can be rewritten as:

(14)$${\rho _{eq}} \cong \frac{{4{D_{Monolithic}}{D_{Golay3}}}}{{3\lambda f({{D_{Monolithic}} + {D_{Golay3}}} )}} = \frac{{5{D_{Golay3}}}}{{9\lambda f}} = \frac{5}{9}{\rho _{c1}}$$

Figure 4 illustrates the spatial frequency equivalence point ρ_eq of the Golay-3 and Golay-6.

Fig. 4. The spatial frequency equivalence point of the monolithic aperture system with Golay-3 and Golay-6 respectively.

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After the spatial frequency equivalence point is obtained, the spatial frequency information of this two systems is extracted and fused to compensate the mid-frequency.

2.2. Frequency synthesis method

The combination of different object spatial frequencies information is the imaging process, which can be simultaneous or not. According to Fourier optics, for one dimensional continuous image, the imaging process of a diffraction-limited incoherent imaging system can be modeled by:

(15)$$g(x )= \int\limits_0^{{\rho _c}} {A(\rho )MTF(\rho )} \cos ({\rho x + {\theta_\rho }} )d\rho$$

where A(ρ) denotes the amplitude of the original image and θ_ρ denotes the phase, respectively [5]. The amplitude and phase can be obtained by Fourier transform of images taken by the system. In the same way, two dimensional images are similar to one dimensional images, which are the Fourier transform of the captured two dimensional images. This integral can be divided into two terms, given by:

(16)$$g(x )= \int\limits_0^{{\rho _{eq}}} {A(\rho )MTF(\rho )} \cos ({\rho x + {\theta_\rho }} )d\rho + \int\limits_{{\rho _{eq}}}^{{\rho _c}} {A(\rho )MTF(\rho )} \cos ({\rho x + {\theta_\rho }} )d\rho$$

Therefore, the image whose whole spatial frequencies are high can be obtained by fusing the images from a monolithic aperture and optical synthetic aperture system. The fusion process can be expressed as:

(17)$$g(x )= \int\limits_0^{{\rho _{eq}}} {A(\rho )MT{F_{Monolithic}}(\rho )} \cos ({\rho x + {\theta_\rho }} )d\rho + \int\limits_{{\rho _{eq}}}^{{\rho _c}} {A(\rho )MT{F_{Golay}}(\rho )} \cos ({\rho x + {\theta_\rho }} )d\rho$$

Equation (17) can be realized by a low pass filter and a band pass filter. Then, the fusion process can be rewritten as:

(18)$$g = {\cal{F}^{ - 1}}({{\cal{F}}(i )OT{F_{Monolithic}}{H_{low}} + {\cal{F}}(i )OT{F_{Golay}}{H_{band}}} )$$

where ${\cal{F}}$ is the Fourier transform and ${{\cal{F}}^{-1}}$ is the inverse Fourier transform. g and i are the fused image and the original image, respectively. H_low and H_band are the low pass filter and band pass filter respectively, which are given by:

(19)$${H_{low}}({u,v} )= \left\{ \begin{array}{ll} 1, &\sqrt{{u^2} + {v^2}} \le {\rho_{eq}}\\ 0, &\sqrt{{u^2} + {v^2}} \ge {\rho_{eq}} \end{array} \right.$$

and,

(20)$${H_{band}}({u,v} )= \left\{ \begin{array}{ll} 1, &{\rho_{eq}} \le \sqrt{{u^2} + {v^2}} \le {\rho_{c1}}\\ 0, &\sqrt{{u^2} + {v^2}} < {\rho_{eq}}\\ 0, &\sqrt{{u^2} + {v^2}} > {\rho_{c1}} \end{array} \right.$$

According to the imaging model of the optical system, Eq. (18) can be summarized as: the images collected by the monolithic aperture system and the optical synthetic aperture system are first processed by Fourier transform. Then the information of separate frequencies is extracted by using filters in the frequency space, and finally the fused image information on the frequency space is obtained. The detailed process of the algorithm is shown in Fig. 5. The separate frequency information with higher MTF level is extracted and synthesized, and the result is shown in Fig. 6.

Fig. 5. Illustration of the flow chart of the algorithm.

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Fig. 6. Frequency extraction and synthesis. (a) The frequency of the original objective image. (b) The frequency of the image obtained by the monolithic aperture system. (c) The partial frequency extracted from the monolithic aperture system. (d) The frequency of the image obtained by the Golay-3. (e) The partial frequency extracted from the Golay-3 system. (f) The frequency of the final synthesized image.

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2.3. Image restoration

Post-processing is necessary to restore the synthesized images, which can make the reconstructional images sharp and improve the contrast. In theory, the post-processing method is to take the deconvolution of the image collected with the known PSF of the imaging system. In frequency space, the reciprocal of the OTF is therefore a rudimentary example of a restoration filter. However, the finite OTF size and the noise existence make the rudimentary reciprocal OTF restoration filter problematic. The improved Wiener-Helstrom filter is a typical post-processing method for optical synthetic aperture system because of its excellent performance and good anti-noise ability [15]. For the Wiener deconvolution, the transfer function of the ﬁlter can be expressed as:

(21)$$W({{f_x},{f_y}} )= \frac{{OT{F^\ast }({{f_x},{f_y}} )}}{{{{|{OTF({{f_x},{f_y}} )} |}^2} + K}}$$

where (f_x, f_y) represents the coordinate of the frequency space, and OTF*(f_x, f_y) is the complex conjugate of OTF(f_x, f_y), and the K value is given by:

(22)$$K = \frac{{{S_n}({{f_x},{f_y}} )}}{{{S_f}({{f_x},{f_y}} )}}$$

where S_n(f_x, f_y) and S_f(f_x, f_y) are the power spectrum of the noise and the original image respectively. Therefore, K value is reciprocal of the signal-to-noise ratio (SNR). Generally, the priori information about S_n(f_x, f_y) and S_f(f_x, f_y) is unknown, therefore K is generally estimated based on experience in practice.

According to Eq. (18), the OTF(f_x, f_y) of fused image of the monolithic aperture system and the optical synthetic aperture system can be calculated as:

(23)$$OT{F_{Fused}} = OT{F_{Monolithic}}{H_{low}} + OT{F_{Golay}}{H_{band}}$$

After the OTF_Fused of the system is obtained, the SNR of the system needs to be estimated, so that the Wiener filter can have a good performance. The relationship between the SNR, MTF, exposure time, and fill factor for an optical synthetic aperture system has been analyzed, which can be expressed as [4]:

(24)$$SNR({{f_x},{f_y}} )= \frac{{\left( {{\phi_o}\alpha \left( {\frac{{\pi {D_{circ}}^2}}{4}} \right){\eta_t}\left( {\frac{T}{M}} \right)} \right) \cdot \mu ({{f_x},{f_y}} )\cdot MTF({{f_x},{f_y}} )}}{{{{\left[ {({{\phi_o} + {\phi_b}} )\alpha \left( {\frac{{\pi {D_{circ}}^2}}{4}} \right){\eta_t}\left( {\frac{T}{{{M^2}}}} \right) + {\gamma_{dc}}T + {\sigma_r}^2} \right]}^{1/2}}}}$$

Where ϕ_o and ϕ_b are the number of image photons and bias photos per unit area per unit time respectively. α is the fill factor. η_t is the total efficiency of the system, which includes system throughput and the detector quantum efficiency. T is the exposure time. M is the size of the image collected. μ(f_x,f_y) is the normalized scene Fourier transform. γ_dc and σ_r are the dark current and the RMS readout noise respectively.

This expression shows that SNR is proportional to MTF. Since the maximum MTF level of the sidelobes of the Golay-N array is 1/N, the SNR of the sidelobes is 1/N of the center part. Thus the K value can be rescaled by filters. The transfer function of the Wiener ﬁlter can be rewritten as:

(25)$$W({{f_x},{f_y}} )= \frac{{{{({OT{F_{Monolithic}}{H_{low}} + OT{F_{Golay}}{H_{band}}} )}^\ast }}}{{{{|{OT{F_{Monolithic}}{H_{low}} + OT{F_{Golay}}{H_{band}}} |}^2} + \left( {\frac{1}{N}K{H_{low}} + K{H_{band}}} \right)}}$$

Then K can finally get the optimal value according to peak signal-to-noise ratio [15].

3. Simulation results

In this section, the effect of the proposed method is demonstrated. The effect of the image restoration algorithm is analyzed under different SNR. And the performance of the mid-frequency MTF compensation is evaluated quantitatively.

To demonstrate the performance in real situations, where the optical synthetic aperture systems are used to capture images of celestial objects through ground-based or space telescopes, the images used in the simulation are based on Hubble space telescope [16]. A total of 60 images are used for tested. Part of the test set is shown in Fig. 7.

Fig. 7. Illustration of the dataset of images from the Hubble Space Telescope.

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In simulation, the PSF of the monolithic aperture system and optical synthetic aperture system are both acquired in ZEMAX. In practice, the phase differences between the sub-mirrors can be reduced to a small fraction of the wavelength by using interferometry. Thus, the sub-mirrors are co-phased in the simulation. The Gaussian white noise is added because it is proved that the total noise of all noise mentioned in Eq. (24) can be modeled as Gaussian white noise [17,18]. But real systems will have many multiplicative noise which depends on the state of the systems. They generally do not obey the Gaussian distribution and may affect image quality. In order to evaluate the noise resistance of the proposed method, the signal to noise ratio is set to 30 dB and 40 dB respectively. The peak signal to noise ratio (PSNR) and the structural similarity index (SSIM) are selected to evaluate quantitatively, which are widely applied to the analysis of image restoration of optical synthetic aperture system [19]. The PSNR can be defined as:

(26)$$PSNR = 10{\log _{10}}\left( {\frac{{{{255}^2} \times M \times N}}{{\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {{{|{E^{\prime}(i,j) - {E_0}(i,j)} |}^2}} } }}} \right)$$

where M and N are the size of the image. E'(i, j) denotes the gray values of pixel in i-th row and j-th column of restored image, and E₀(i, j) is the gray values of the corresponding pixel of original clear image.

The SSIM can be calculated as:

(27)$$SSIM(f,g) = \frac{{({2{\mu_f}{\mu_g} + {c_1}} )({2{\sigma_{fg}} + {c_2}} )}}{{({{\mu_f}^2 + {\mu_g}^2 + {c_1}} )({{\sigma_f}^2 + {\sigma_g}^2 + {c_2}} )}}$$

where f, g denote the two images for comparison. μ is the mean of the image, and σ² is the variance of the image. σ_fg is the covariance of the two images. c₁, c₂ are constants. ${c_1} = {({{k_1}L} )^2}\; ,{c_2} = {({{k_2}L} )^2}$, where L is dynamic range of the image, which is 255. k₁, k₂ are constants in the SSIM index formula, which defaults to 0.01 and 0.03, respectively.

The performance is compared with the image recovery results based on single system, which is illustrated in Fig. 8 and Fig. 9. Figure 8(a) and Fig. 9(a) are the original clear images, which are taken as the ideal benchmarks. And the PSNR and SSIM of other images are calculated from these benchmarks. The degraded images of column 2 and column 3 in Fig. 8 and Fig. 9 are the results of the original images being processed by different systems. And the degraded images contain different levels of noise. In order to make the comparison fair, the optimal value of Wiener parameter K of other method is designed based on the PSNR according to the method in Ref. [15]. Although, for example, the narrow galaxies in Fig. 8(d) and Fig. 8(e) are still fuzzy. Meanwhile, the Fig. 8(f) obtained by synthesizing two systems is visually close to the original image. Besides, the maximum PSNR and SSIM values are both achieved by Fig. 8(f).

Fig. 8. Illustration of the results of the final restored images when the SNR is 40 dB. (a) The original clear images. (b) The degraded images and their corresponding Golay-N array configurations. (c) The degraded images of the monolithic aperture system. (d) The restoration results of the degraded images of the Golay-N system via Wiener algorithm. (e) The restoration results of the degraded images of the monolithic aperture system via Wiener algorithm. (f) The restoration results of the proposed method.

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Fig. 9. Illustration of the results of the final restored images when the SNR is 30 dB.

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Apart from the PSNR and SSIM indices, several other matrices are adopted, which make the evaluation more comprehensive and objective. These matrices include information fidelity criterion (IFC), and visual information fidelity (VIF), weighted peak signal-to noise ratio (WPSNR), and multi-scale structure similarity index (MSSSIM) [20]. They are all commonly used in the field of image quality assessment. And the closer the restored image is to the original image, the larger the values of all these evaluation indexes are. The final results are shown in Table 2. All these qualitative evaluations get the highest value in the proposed method, which demonstrates that restoring by using two systems can successfully improve the image restoration effect of an individual system.

Table 2. Comprehensive assessment of the recovery results.

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In order to directly analyze the effect of the mid-frequency MTF compensation of optical synthetic aperture system, an edge method proposed in Ref. [21] is applied to assert the MTF. The edge spread function of the system can be obtained from an angled edge image. The line spread function is determined by the edge spread function. Thus the MTF is estimated because it is the zero frequency normalized modulus of the Fourier Transform of the line spread function. The results are shown in Fig. 10. It can be seen that the MTF curves of the proposed method are similar to that of monolithic aperture system in the middle and low frequencies, which improves the performance of a separate synthetic aperture at the middle and low frequencies. It means that the proposed approach can better maintain the contour and texture of the image. And the cutoff frequency of the MTF curve is still the same as that of synthetic aperture. Thus the results of two systems preserve the high frequency information of the optical synthetic aperture system. Because the MTF curve of the Golay-6 decreases more rapidly than that of the Golay-3, the mid-frequency MTF compensation of the Golay-6 aperture system is more obvious.

Fig. 10. The estimation of the MTF via the edge method.

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The compensation effect of the mid-frequency MTF is quantitatively evaluated. The middle spatial frequency can be represented by a single value MTF_{mid freq}, which is the mean MTF value over spatial frequencies from ρ_d to ρ_c, where ρ_d denotes the spatial cutoff frequency corresponding to individual sub-aperture [7,22]. Here, ρ_d is approximately 0.2. The MTF_{mid freq} can be expressed as the following equation:

(28)$$MT{F_{mid\textrm{ }freq}} = \frac{{\int_0^{2\pi } {\int_{{\rho _d}}^{{\rho _c}} {MTF({\rho ,\phi } )\rho d\rho d\phi } } }}{{\int_0^{2\pi } {\int_{{\rho _d}}^{{\rho _c}} {\rho d\rho d\phi } } }}$$

According to the Fig. 10, in the range from ρ_d to ρ_c, the ratio of the area enclosed by the MTF curve and the coordinate axis to the difference between the two frequencies is the value of MTF_{mid freq}. It can be calculated that the MTF_{mid freq} of Golay-3 increases from 0.12 to 0.16. And the MTF_{mid freq} of Golay-6 increases from 0.06 to 0.18. Thus, the proposed method not only realizes the cutoff frequency determined by the longest baseline of the optical synthetic aperture system, but also successfully compensates its mid-frequency MTF.

4. Experimental results

In this section, a reflective synthetic aperture system is established to validate the proposed approach. The experimental platform is shown in Fig. 11. Different images are collected from the detector by adjusting the sources. Specifically, the helium-neon laser and the pinhole are used to obtain the PSF of the system, where the size of the pinhole is 40 µm. And the LED and the resolution board are used to get the degraded images, where the resolution board (GCG-020101), which consists of different groups of line pairs, has a measuring capacity of 200 cycles per 1 mm. Then the pellicle beam splitter is applied to control the input of different sources. The beams of different sources both pass through the collimator, which is used to produce the parallel beams. Then the parallel beams are reflected by a spherical reflector with a pupil mask. The pupil mask is designed in the form of Golay-3 and a single circle to simulate the Golay-3 synthetic aperture system and the monolithic aperture system respectively. For the pupil mask of Golay-3, the diameter of sub-aperture is 18 mm and that of the ex-circle is 35 mm. For the monolithic aperture, the diameter is 25 mm. Finally, the beams pass through another pellicle beam splitter and are captured by a CCD camera (Point Grey GS2-GE-20S4M), where the pixel size is 4.4 µm × 4.4 µm. Because the focal length of the reflector is 20 mm. Thus the sum of the distance from the reflector to the beam splitter and the beam splitter to the detector is 20 mm.

Fig. 11. Optical setup of the experiment.

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The actual degraded images and the PSF are obtained by the experiment, as shown in Fig. 12. By adjusting the pupil mask, the degraded image and the PSF of the Golay-3 synthetic aperture system and the monolithic aperture system are captured respectively. In order to better display the image changes, the collected images are partially enlarged. As shown in Fig. 12(a1) and Fig. 12(a2), the collected images are visually fuzzy and low contrast. And because the Golay-3 system's MTF decays faster at low and middle frequencies, the collected image of this system has lower contrast than that of the monolithic aperture system. The result is consistent with the simulation above. The restored result in Fig. 12(a3) is obviously sharper and has a higher contrast than these collected images. A comparison of two groups of line pairs of the resolution board is analyzed, as shown in Fig. 12(c1) and Fig. 12(c2). Here, the oblique line pairs are group1, and the vertical line pairs are group2. For the line traces of the restored result of the proposed method, the gaps between the peaks and valleys are widest, which means it has the highest contrast. Therefore, image restoration by synthesizing two systems can effectively improve the image quality.

Fig. 12. Illustration of the captured image of the Golay-3 system (a1) and the monolithic aperture system (a2); (a3) the recovered image by the proposed method; (b1) - (b2) the captured PSF of the two system; (c1) - (c2) the line traces of the line pairs of group1 and group2, respectively.

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5. Conclusion

For a single optical synthetic aperture system, the attenuation of the mid-frequency MTF caused by the dispersion and sparsity is inevitable. In this paper, another monolithic aperture system is combined to compensate the mid-frequency MTF. The dimension of the monolithic aperture is designed, which is much smaller than that of the optical synthetic aperture system as we expected. The MTF of the Golay system is analyzed and the spatial frequency equivalence point of the two systems is determined. Then the separated spatial frequency information is extracted and synthesized by using a low pass filter and a band pass filter. Finally an improved Wiener-Helstrom filter is proposed, which adopts specific parameters based on different sub-aperture arrangements. Both simulation and practical experiment are established to verify the proposed method. The final MTF approximates to that of monolithic aperture system in the middle and low frequencies, which better maintains the contour and texture of the image. Thus the proposed method improves the performance of a separate optical synthetic aperture system at the middle and low frequencies. At the same time, the MTF still preserves the cutoff frequency of the synthetic aperture. And the mid-frequency MTF of Golay-3 increases from 0.12 to 0.16 and that of Golay-6 increases from 0.06 to 0.18. Therefore, synthesizing both two systems not only realizes the spatial resolution determined by the longest baseline of the optical synthetic aperture system, but also successfully compensates its mid-frequency MTF. In addition, in order to be applied to the more complex reality, the proposed method can be further improved. The combination of MTF characteristics of non-aligned optical synthetic aperture system and that of monolithic aperture system is worth further study. We believe that the proposed mid-frequency MTF compensation method will be useful to other researchers in real-world situations.

Funding

National Natural Science Foundation of China (61475018).

Disclosures

The authors declare no conflicts of interest.

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Parameters	Value
Aperture diameter(mm)	120
Image F number	15.6
Radius of primary mirror	−374.1
Radius of secondary mirror	−82.7
Effective focal length	1890.187

Eval. Mat	Array type	SNR(dB)	Degraded Golay	Degraded Monolithic	Wiener Golay	Wiener Monolithic	Proposed
PSNR(dB)	Golay-3	40	25.17	25.38	26.93	27.08	30.91
	Golay-3	30	23.61	24.07	25.51	25.49	28.76
	Golay-6	40	24.33	24.79	26.60	26.48	29.48
	Golay-6	30	23.12	24.28	25.28	25.63	29.03
SSIM	Golay-3	40	0.6955	0.6991	0.7236	0.7314	0.8934
	Golay-3	30	0.6214	0.6326	0.6772	0.6837	0.8717
	Golay-6	40	0.6328	0.6534	0.6844	0.7166	0.8876
	Golay-6	30	0.6178	0.6411	0.6608	0.7015	0.8602
IFC	Golay-3	40	3.67	3.81	4.68	4.61	5.01
	Golay-3	30	2.92	3.03	3.60	3.84	4.63
	Golay-6	40	3.39	3.62	4.51	4.59	4.90
	Golay-6	30	2.81	3.19	3.73	3.91	4.58
VIF	Golay-3	40	0.61	0.63	0.88	0.87	0.94
	Golay-3	30	0.58	0.60	0.84	0.86	0.87
	Golay-6	40	0.54	0.60	0.84	0.84	0.93
	Golay-6	30	0.51	0.55	0.81	0.80	0.84
WPSNR(dB)	Golay-3	40	35.8	36.2	41.0	42.3	45.2
	Golay-3	30	32.3	34.7	37.9	38.6	43.7
	Golay-6	40	32.9	35.1	41.2	40.9	44.6
	Golay-6	30	31.7	33.8	38.9	39.7	43.0
MSSSIM	Golay-3	40	0.9311	0.9404	0.9533	0.9586	0.9808
	Golay-3	30	0.8967	0.9037	0.9464	0.9507	0.9711
	Golay-6	40	0.9024	0.9216	0.9477	0.9456	0.9763
	Golay-6	30	0.8695	0.8851	0.9461	0.9491	0.9705

Parameters	Value
Aperture diameter(mm)	120
Image F number	15.6
Radius of primary mirror	−374.1
Radius of secondary mirror	−82.7
Effective focal length	1890.187

Eval. Mat	Array type	SNR(dB)	Degraded Golay	Degraded Monolithic	Wiener Golay	Wiener Monolithic	Proposed
PSNR(dB)	Golay-3	40	25.17	25.38	26.93	27.08	30.91
	Golay-3	30	23.61	24.07	25.51	25.49	28.76
	Golay-6	40	24.33	24.79	26.60	26.48	29.48
	Golay-6	30	23.12	24.28	25.28	25.63	29.03
SSIM	Golay-3	40	0.6955	0.6991	0.7236	0.7314	0.8934
	Golay-3	30	0.6214	0.6326	0.6772	0.6837	0.8717
	Golay-6	40	0.6328	0.6534	0.6844	0.7166	0.8876
	Golay-6	30	0.6178	0.6411	0.6608	0.7015	0.8602
IFC	Golay-3	40	3.67	3.81	4.68	4.61	5.01
	Golay-3	30	2.92	3.03	3.60	3.84	4.63
	Golay-6	40	3.39	3.62	4.51	4.59	4.90
	Golay-6	30	2.81	3.19	3.73	3.91	4.58
VIF	Golay-3	40	0.61	0.63	0.88	0.87	0.94
	Golay-3	30	0.58	0.60	0.84	0.86	0.87
	Golay-6	40	0.54	0.60	0.84	0.84	0.93
	Golay-6	30	0.51	0.55	0.81	0.80	0.84
WPSNR(dB)	Golay-3	40	35.8	36.2	41.0	42.3	45.2
	Golay-3	30	32.3	34.7	37.9	38.6	43.7
	Golay-6	40	32.9	35.1	41.2	40.9	44.6
	Golay-6	30	31.7	33.8	38.9	39.7	43.0
MSSSIM	Golay-3	40	0.9311	0.9404	0.9533	0.9586	0.9808
	Golay-3	30	0.8967	0.9037	0.9464	0.9507	0.9711
	Golay-6	40	0.9024	0.9216	0.9477	0.9456	0.9763
	Golay-6	30	0.8695	0.8851	0.9461	0.9491	0.9705

MTF improvement for optical synthetic aperture system via mid-frequency compensation

Abstract

1. Introduction

2. Method

2.1. MTF analysis

2.2. Frequency synthesis method

2.3. Image restoration

3. Simulation results

4. Experimental results

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (12)

Tables (2)

Equations (28)

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