Abstract

The resolution of imaging by space and earth-based telescopes is often limited by the finite aperture of the optical systems. We propose a novel synthetic aperture-based imaging system with two physical subapertures distributed only along the perimeter of the synthetic aperture. The minimum demonstrated two-subaperture area is only 0.43% of a total full synthetic aperture area. The proposed optical configuration is inspired by a setup in which two synchronized satellites move only along the boundary of the synthetic aperture and capture a few light patterns from the observed scene. The light reflected from the two satellites interferes with an image sensor located in a third satellite. The sum of the entire interfering patterns is processed to yield the image of the scene with a quality comparable to an image obtained from the complete synthetic aperture. The proposed system is based on the incoherent coded aperture holography technique in which the light diffracted from an object is modulated by a pseudorandom coded phase mask. The modulated light is recorded and digitally processed to yield the 3D image of the object. A laboratory model of imaging with two synchronized subapertures distributed only along the border of the aperture is demonstrated. Experimental results validate that sampling along the boundary of the synthetic aperture is enough to yield an image with the resolving power obtained from the complete synthetic aperture. Unlike other schemes of synthetic aperture, there is no need to sample any other part of the aperture beside its border. Hence, a significant saving of time and/or devices are expected in the process of data acquisition.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

In general, imaging with a synthetic aperture is a technique in which a relatively small physical aperture scans a relatively large (much larger than the physical aperture) synthetic aperture over time [1]. The accumulated data in time are processed to yield an image with qualities equivalent to that of an image acquired by a single exposure of the complete synthetic aperture [2]. Since the minimal resolvable detail of an image is inversely proportional to the aperture size, the image resulting from the synthetic aperture has better resolution than the image obtained from the physical aperture by direct imaging. Commonly, the resolution improvement is equal to the ratio between the sizes of the synthetic and the physical apertures.

Usually, in order to avoid information loss, the synthetic aperture is sampled by the physical aperture, with a sampling rate higher than the sampling limits [3,4]. This sampling restriction is valid for synthetic aperture radar (SAR) in the radio frequencies [1], or for Michelson stellar interferometry [5], very large baseline interferometer (VLBI) [6], and synthetic aperture with Fresnel elements (SAFE) [7] in the optical regime. Sparse SAFE (S-SAFE) is an example in which the synthetic aperture is sampled according to the rules of compressed sensing [8,9]. Nevertheless, the entire synthetic aperture is sampled also in the case of S-SAFE. Herein, for the first time to the best of our knowledge, we propose to sample the synthetic aperture only along its perimeter with a much smaller pair of physical apertures with respect to the total synthetic aperture area at a time. Although only the margin is sampled, the resulting image maintains the resolution and other qualities similar to the image obtained by sampling the complete synthetic aperture.

Synthetic aperture techniques such as SAR [1] and VLBI [6] are indirect imaging techniques, where the image is not directly obtained on the sensor, but exhaustive digital signal processing is implemented to retrieve the image in the computer. Another well-known indirect, multistep but simpler, imaging method is digital holography [10]. First, a hologram is recorded, usually by an interference of the light diffracted from an object with a reference wave. Following a digital process of the hologram, the next step comes in, in which the image is reconstructed from the processed hologram. The space between the stages of recording and reconstruction offers ample opportunities to apply synthetic aperture procedures. Techniques of imaging by synthetic aperture using coherent [1113] and incoherent digital holography such as SAFE [7,14,15] and S-SAFE [8] are constrained to sample aperture regions inside the borderline of the synthetic aperture with relatively larger aperture ratio.

In the present study, we introduce an incoherent digital holographic system denoted as a synthetic marginal aperture with revolving telescopes (SMART). SMART is based on two concepts: namely, synthetic aperture imaging [18] and interferenceless coded aperture correlation holography (I-COACH) [16]. However, in SMART, the recorded intensity pattern is obtained as an interference of light coming from two separated subapertures. Therefore, SMART, which does make use of two-wave interference, cannot be considered as an interferenceless system like I-COACH. Nevertheless, some of the principles and insights of I-COACH are still valid for SMART. I-COACH is an incoherent digital holography technique in which the light emitted from an object is recorded by a camera after passing through or reflecting from a pseudorandom coded phase mask (CPM). The recorded intensity pattern contains the complete 3D information of the object, and therefore the complicated interferometers are not necessary. The current study is a continuation of the project of the partial aperture imaging system (PAIS) [17] in which high-resolution imaging has been demonstrated with an annular or a ring-shaped aperture only. However, the whole annular aperture of PAIS is the physical and not the synthetic aperture of the imaging system, since no scanning is involved in PAIS. In SMART, on the other hand, the annular aperture is sampled in space and scanned on time by a physical subaperture in the synthetic aperture mode, and it can be easily realized for practical purposes. In this study, SMART is investigated with the vision for a possible application of two synchronized orbiting satellites used as a space telescope, as is shown in Fig. 1. The couple of satellites modulate the incident stellar light by pseudorandom phase masks. From the satellites, the two scattered beams are projected toward a third satellite, which records the interference pattern of the two beams. The intensity images recorded by the third satellite at different mutual positions of the two modulating satellites are digitally processed in order to reconstruct the image of the observed scene. As a proof of principle, in this study, a table-top miniature laboratory model of SMART configuration is constructed with aperture shapes that mimic the scheme of Fig. 1.

 figure: Fig. 1.

Fig. 1. Schematic of the space-based telescope for the implementation of SMART.

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2. METHODOLOGY

A schematic of the laboratory model of SMART is shown in Fig. 2. Incoherent light from a light-emitting diode (LED) is used to illuminate a point object by a lens L0. The light emitted from the point object is used as a guidestar for the proposed method. It is assumed that the light arrives from a far-field source, and therefore the incident wavefront from each point source can be approximated to a plane wave. This condition is optically simulated by collimating the light diffracted from the point object with a second refractive lens L1. The collimated light is incident on a spatial light modulator (SLM) whose aperture function is engineered with diffractive optical functions to match the scenario of Fig. 1. A synthetic aperture grid consisting of eight points distributed with equal angular separations along a ring is used. In the case of SMART, the aperture is synthesized by distributing two relatively small circular pseudorandom CPMs on the annular aperture grid. This arrangement imitates the two orbiting satellites in all different 28 (N=8 in N(N1)/2) possible permutations. Every CPM pair is synthesized using the modified Gerchberg–Saxton algorithm (GSA) [18] to obtain a uniform magnitude over a predefined region of the spatial spectrum domain, as shown in Fig. 3. The predefined region was selected as the central 600pixels×600pixels out of 1080pixels×1080pixels in the spectrum domain of the CPM. The GSA is used also to synthesize the CPMs for PAIS of four subapertures and eight subaperture cases. This GSA condition plays a crucial role in suppressing the background noise during image reconstruction. The Fourier relation between the two domains of the GSA is satisfied in the optical configuration by multiplying the CPM with a diffractive lens with a focal length of zh, the distance between the SLM and the sensor plane. Through this lens, it is guaranteed that a Fourier transform of the CPM is obtained on the sensor plane. The SLM reflectivity is engineered further to deviate the light, which is not incident on the pseudorandom subapertures away from the sensor using diffractive optical elements (more details in Supplement 1). Only the light from the subapertures is recorded by the image sensor, while the light from all other areas of the SLM is deflected away from the sensor plane.

 figure: Fig. 2.

Fig. 2. Laboratory model of SMART for image acquisition. CPM, coded phase mask; L0 and L1, refractive lens; LED, light-emitting diode; and DOE, diffractive optical element.

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 figure: Fig. 3.

Fig. 3. Modified GSA for designing the CPM pairs with all possible permutations of eight equally separated circles along the perimeter of the aperture such that every pair is constrained to produce a uniform magnitude on the sensor plane.

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A. Mathematical Formulation

The propagation of light through the optical components and the image reconstruction are described in this section while referring to Fig. 2. Generally, the image of the observed scene is obtained as a cross-correlation between system responses of the object and of the guidestar [16,17]. Hence, in the first step, a point object (a guidestar) is located at the focal plane of the lens L1, and the impulse responses IIR are recorded for the set of all aperture combinations. Later, an object is placed at the same axial location of the point object, and the object responses IOR are recorded using the same set of CPMs.

The analysis begins from the point object with the assumption that the point object is a self-luminous body in space located at (r¯s,zs)=(xs,ys,zs). The SLM consists of the CPM subapertures and the beam deviator. The majority of the optical power is deviated by the beam deviator phase mask. Therefore, the only light arriving at the sensor is diffracted from the CPM given by

MP(r¯)=p=1Pexp[iϕp(r¯)]Circ(|r¯|R)*δ(r¯r¯p),
where “ * ”stands for convolution, δ is the Dirac delta function, Circ(|r¯|/R)=1 for |r¯|R and 0 otherwise, R is the radius of each subaperture centered around r¯p, ϕp(r¯) is the pseudorandom phase of the subapertures calculated by the GSA, and P is the number of subapertures, whereas, in the case of SMART, P is always 2 at any given time. Assuming the sensor plane is the image plane of an object point at (r¯s,zs), the intensity distribution is a magnitude square of a scaled Fourier transform of the CPM [17] as follows:
IIR(r¯0;r¯s,zs)=C0|ν[1λzh]F{MP(r¯)}|2*δ(r¯0zhzsr¯s),
where F is a two-dimensional Fourier transform and ν is the scaling operator such that ν[α]f(x)=f(αx). To avoid cross-correlation between two real-positive functions, resulting in an unacceptable level of background noise, three independent intensity responses to the guidestar, as well as to the object, are recorded for three different CPMs. From three sets of impulse responses, a complex impulse response hIR is calculated by projecting each set onto the complex space with the constant phases θ1,2,3=0, 2π/3 and 4π/3, as the following superposition:
hIR(r¯0;0,zs)=j=13IIR(r¯0;0,zs)exp(iθj).
Next, we show the equivalence between two intensity responses: the SMART of N sub-apertures, in which one couple of subapertures out of N(N1)/2 permutations is active at any given time, and the response of PAIS of N subapertures; all are simultaneously active. Let us consider first the impulse response of PAIS of N subapertures. By substituting Eqs. (1) and (2) into Eq. (3), the obtained complex impulse response is
hIR,PAIS(r¯0;0,zs)=C0j=13exp(iθj)|ν[1λzh]F{MP=N(r¯)}|2=C0j=13exp(iθj)|ν[1λzh]F{p=1Nexp[iϕj,p(r¯)]Circ(|r¯|R)*δ(r¯r¯p)}|2=C0j=13exp(iθj)|p=1NF{exp[iϕj,p(λzhr¯)]}*Jinc(R|ro¯|λzh)exp[i2πλzh(r¯p·r¯o)]|2,
where C0 and C0 are constants, Jinc(r)=J1(2πr)/r, and J1(2πr) is a first-order Bessel function of the first kind. The magnitude square of N addends contains N terms of bias; all are approximately canceled by the superposition of three jth responses. Hence the complex impulse response of Eq. (4) can be rewritten approximately as the sum of 3N(N1) interference terms, without the bias terms as follows:
hIR,PAIS(r¯0;0,zs)C0j=13exp(iθj)k=1N1l=k+1N[(F{exp[iϕj,k(λzhr¯)]}*Jinc(R|r¯o|λzh))×(F{exp[iϕj,l(λzhr¯)]}*Jinc(R|r¯o|λzh))exp[i2πλzh[(r¯kr¯l)·r¯o]]+C.C.],
where C0 is a constant and C.C. stands for the complex conjugate.

Due to the mechanism of eliminating the bias terms of the intensity pattern, we show next that in an ideal system SMART of N subapertures, without noise, has a similar impulse response as of PAIS with N subapertures, given by Eq. (5). In SMART for each camera shot, the CPM is given in Eq. (1) for P=2, and there is a total of 3(N1)N/2 camera shots, whereas N is an even number of the total subapertures of PAIS. By substituting Eqs. (1) and (2) into Eq. (3), for every pair of subapertures and repeating the recording 3(N1)N/2 times, the obtained complex impulse response is

hIR,SMART(r¯o;0,zs)=C0k=1N1l=k+1Nj=13exp(iθj)|ν[1λzh]F{M2(r¯)}|2=C0k=1N1l=k+1Nj=13exp(iθj)×|ν[1λzh]F{exp[iϕk,j(r¯)]Circ(|r¯r¯k|R)+exp[iϕl,j(r¯)]Circ(|r¯r¯l|R)}|2=C0k=1N1l=k+1Nj=13exp(iθj)|F{exp[iϕk,j(λzhr¯)]}*Jinc(R|r¯o|λzh)exp(i2πr¯k·r¯oλzh)+F{exp[iϕl,j(λzhr¯)]}*Jinc(R|r¯o|λzh)exp(i2πr¯l·r¯oλzh)|2.
Recall that for each permutation of pair subapertures, the bias term is removed by the superposition of three jth intensity patterns; the result becomes similar to Eq. (5), as follows:
hIR,SMART(r¯o;0,zs)C0j=13exp(iθj)×k=1N1l=k+1N[(F{exp[iϕk,j(λzhr¯)]}*Jinc(R|r¯o|λzh))×(F{exp[iϕl,j(λzhr¯)]}*Jinc(R|r¯o|λzh))exp(i2π(r¯kr¯l)·r¯oλzh)+C.C.].
Note that, although per permutation SMART has only six bias terms, the total number of bias terms in SMART reflected from Eq. (6) is 3(N1)N, larger than the number of bias terms in PAIS of 3N, as reflected from Eq. (4). However, the total number of cross terms is 3(N1)N in both methods. Hence, since the bias terms are removed in both methods, the complex impulse response of SMART and PAIS are equivalent. Equivalent, but not identical, because first, each permutation of SMART is performed with different pseudorandom phase functions ϕj(r¯) than the experiment of PAIS, and second, because of the camera noise, the SNR is higher for SMART with lower bias term per camera shot than in PAIS. Hence, from now on the expression of Eq. (5) is considered as the impulse response of both the PAIS and SMART systems. Moreover, all the advantages of PAIS over direct imaging described in the Discussion section are valid for SMART as well. Explicitly, the spatial bandwidth of SMART is the same as for PAIS and as for direct imaging, but the shape of its modulation transfer function (MTF) is similar to that of PAIS, and they are both more transparent in the high-frequency regime than the direct imaging, which behaves as a narrow low-pass filter.

Next, we analyze the process of recording the object response and reconstructing the image. A 2D object located at the same axial location as the point object can be considered as a collection of M uncorrelated point objects given by

o(r¯s)=mMamδ(r¯sr¯m),
where am is the intensity of the mth object point at r¯m. Assuming the system is linear and space-invariant, the overall object intensity response on the sensor plane is a sum of the shifted and scaled impulse responses, given by
IOR(r¯0;zs)=mamIIR(r¯0zhzsr¯m;0,zs).
Equation (9) is valid since the object is a spatially incoherent self-luminous body, such that the response of each individual point does not interfere with the response of any other object point. Three object intensity responses are superposed into a complex object hologram following the same formula of Eq. (3) as follows:
hOR,PAIS(r¯0;zs)=j=13IOR(r¯0;zs)exp(iθj)=j=13mamIIR(r¯0zhzsr¯m;0,zs)exp(iθj)=mamhIR,PAIS(r¯0zhzsr¯m;zs).
In the case of SMART, the object intensity responses for different permutations of subapertures are superposed into a complex object hologram as follows:
hOR,SMART(r¯0;zs)=k=1N1l=k+1Nj=13IOR(r¯0;zs)exp(iθj)=k=1N1l=k+1Nj=13mamIIR(r¯0zhzsr¯m;0,zs)exp(iθj)mamhIR,SMART(r¯0zhzsr¯m;zs).
In order to reduce the background noise, the images are reconstructed by correlating hOR(r¯0;zs) with phase-only filtered [17,19] hIR(r¯0;zs) given by h˜IR(r¯)=F1{exp[i·arg(F{hIR(r¯)})]}. The image reconstruction, for both PAIS and SMART, can be expressed as follows:
IIM(r¯R)=hOR(r¯0;zs)h˜IR*(r¯0r¯R;zs)dr¯0=mamhIR(r¯0zhzsr¯m;zs)h˜IR*(r¯0r¯R;zs)dr¯0=mamΛ(r¯Rzhzsr¯m)o(r¯sMT),
where Λ is a δ-like function used as the PSF of the entire imaging system, approximately equal to 1 at (0,0) and to small negligible values elsewhere. The reconstructed image has a transverse magnification of MT=zh/zs. In Eq. (12), hOR and hIR stand for the responses of either PAIS or SMART, depending on the method with which we image the object. The cross-correlation of Eq. (12) reconstructs the object image at the axial plane of the point object. However, in order to reconstruct objects at different depths, hIR is recorded at those depths and cross-correlated as above to retrieve the object at various depths. As the object image is reconstructed using cross-correlation, the lateral and axial resolutions are given by the lateral and axial correlation lengths, respectively [19].

B. Experiments

The schematic of the experimental setup is shown in Fig. 4. The setup consists of two illumination channels with identical LEDs (Thorlabs LED635L, 170 mW, λc=635nm, and Δλ=15nm). The two optical channels never overlap laterally, and therefore the beams never interfere with each other. A pinhole with a diameter of 25 μm is used for recording the impulse responses. Two negative National Bureau of Standards (NBS) resolution targets (NBS 1963A Thorlabs), namely, 14 l p/mm and 16 l p/mm, are illuminated in the two optical channels via two identical lenses, L0A and L0B, located at a distance of 3 cm away from the LEDs. Light beams diffracted from the objects in the two channels were combined by a beam splitter BS1 and collimated using a biconvex lens L1 with a diameter of 2.5 cm and a focal length of 20 cm. A polarizer P is located beyond L1 to pass only the light with an orientation parallel to the active axis of the SLM (Holoeye PLUTO, 1920pixels×1080pixels, 8 μm pixel pitch, phase-only modulation). The SLM was mounted at a distance of 10 cm from the lens L1. Three independent sets of CPMs were synthesized by the GSA for each permutation of the subaperture pairs. Subapertures were synthesized with different diameters in order to investigate the influence of the CPM size. The radii of subapertures were r=0.8, 0.4, 0.28, and 0.2 mm, with aperture area ratios (defined as the ratio between the area of pair subapertures to the total synthetic aperture area) 6.8%, 1.7%, 0.84%, and 0.43%, respectively, where the maximum radius of the full aperture on the SLM is 4.32 mm. The numerical aperture (NA) of the system is 0.0216, considering the full aperture of the SLM as the entrance pupil. The light modulated by the aperture is recorded by the image sensor (Hamamatsu ORCA-Flash4.0 V2 Digital CMOS, 2048pixels×2048pixels, 6.5 μm pixel pitch, monochrome), placed at a distance of 25 cm from the SLM. (Another experiment with two distinct SLMs performed to verify the feasibility of SMART is described in Supplement 1).

 figure: Fig. 4.

Fig. 4. Experimental setup for demonstration of SMART. BS1 and BS2, beam splitters; SLM, spatial light modulator; NBS, National Bureau of Standards; L0A, L0B and B1, refractive lenses; LED1 and LED2, identical light-emitting diodes; and P, polarizer.

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3. RESULTS

Different investigations were carried out to understand the SMART system in comparison to other similar imaging systems. Before the experiments of the synthetic aperture, we tested the concept of sampling the annular aperture of PAIS in a mode of physical, nonsynthetic aperture. First, the reconstruction results for full clear aperture are compared with that of PAIS with the aperture of eight subapertures distributed along a ring, each of which with a pseudorandom CPM and with a radius of 0.4 mm. The eight subaperture CPMs are synthesized using the GSA [18] to produce a uniform magnitude in the spectrum domain as much as possible. The use of GSA has reduced the background noise during the reconstruction of the object image. The aperture transparency outside the eight subaperture CPMs is engineered with diffractive optical elements to deviate the light away from the image. Therefore, only the light beams modulated by eight subapertures are incident on the image sensor.

The intensity patterns recorded for the point object and for 14 l p/mm of the NBS resolution target, for eight subapertures and for three different CPMs, are shown in Figs. 5(a)5(c) and 5(g)5(i), respectively. The method of acquiring three intensity patterns by three camera shots, for the object as well as for the point, has been proved [16,17] as an effective tool to increase the signal-to-noise ratio (SNR). The set of three intensity patterns of Figs. 5(a)5(c) and Figs. 5(g)5(i) are superposed with three phase constants (θ1,2,3=0, 2π/3, and 4π/3) to generate a complex-valued impulse response hologram (hIR) and a complex-valued object response hologram (hOR). The magnitude and phase of hIR and hOR are shown in Figs. 5(d) and 5(e), and 5(j) and 5(k), respectively. hOR is cross-correlated with a phase-only filtered [19] version of hIR, and the reconstructed image is shown in Fig. 5(f). The direct imaging result using a lens function over only the eight subapertures is shown in Fig. 5(l). It is evident from Figs. 5(f) and 5(l) that direct imaging through an aperture of eight subapertures cannot resolve the fine details of the object as PAIS with the same aperture does. This experiment reveals that PAIS with eight subapertures has a higher imaging resolution compared to an equivalent direct imaging system.

 figure: Fig. 5.

Fig. 5. (a)–(c), (g)–(i) Intensity patterns recorded for a point object and a resolution target for eight subapertures, respectively; (d) magnitude and (e) phase of hIR; (j) magnitude and (k) phase of hOR; (f) reconstructed image of PAIS; (l) direct imaging result using eight subapertures with diffractive lens; (m)–(o), (s)–(u) intensity patterns recorded for a point object and a resolution target for full aperture, respectively; (p) magnitude and (q) phase of hIR; (v) magnitude and (w) phase of hOR; (r) reconstructed image of full aperture imaging system; (x) direct imaging result using a full aperture with a diffractive lens.

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For comparison purposes, a similar experiment with a full clear aperture of 4.32 mm radius, is depicted in Figs. 5(m)5(w). The intensity patterns recorded for the point object, and for the NBS target, for full aperture are shown in Figs. 5(m)5(o) and 5(s)5(u), respectively. The magnitude and phase of the complex hIR for full aperture are shown in Figs. 5(p) and 5(q), respectively. The magnitude and phase of hOR for full aperture are shown in Figs. 5(v) and 5(w), respectively. The hOR is cross-correlated with phase-only filtered hIR, and the reconstructed image is shown in Fig. 5(r). The direct imaging result using a lens function with full aperture is shown in Fig. 5(x). By comparing Figs. 5(f), 5(l), 5(r), and 5(x), one can conclude that although the SNR of PAIS [Fig. 5(f)] is lower than those of the full aperture and direct imaging, the imaging resolution is the same, whereas in direct imaging through the partial aperture, the resolution is poorer. These comparisons indicate that the resolution limit of PAIS is the same as of direct imaging with clear full aperture, although the area of the PAIS aperture (the total area of the eight circles) in this experiment is less than 7% of the full aperture. However, as explained later, in order to maintain the resolution performance of the full aperture, the partial aperture should be distributed along the perimeter of the full aperture.

In the next experiment, we investigate the influence of the subaperture size on the quality of the reconstructed images. This comparative investigation is done with four arrangements of the subapertures. In Figs. 6(a) and 6(b), there are two subapertures on the perimeter, in Figs. 6(c)6(e) and 6(f)6(h), there are four and eight subapertures, respectively, distributed with equal gaps on the perimeter, and in Fig. 6(i), there is a single subaperture in the center. In all these aperture configurations, four sizes of the subapertures with the radii of r=0.2, 0.28, 0.4, and 0.8 mm are tested. Three imaging methods are compared in Fig. 6: direct imaging [Figs. 6(b), 6(d), 6(g), and 6(i)], PAIS [Figs. 6(a), 6(c), and 6(f)], and SMART [Figs. 6(e) and 6(h)]. Unlike PAIS, in which all the subapertures are involved in the imaging at any given time, in SMART only a single pair of subapertures reflects the light onto the sensor at any given time. Based on Fig. 6, it is clear that SMART is capable of reconstructing the object with more visual details in comparison to direct imaging, as well as PAIS, in both four and eight symmetric points. However, as expected, a decrease in the visibility and some increase in noise is noticed when the subaperture radius is decreased from 0.8 to 0.2 mm. With a radius of 0.2 mm, even though the reconstruction result is better than direct imaging and PAIS, the visibility is lower, and the noise level is higher. Furthermore, it is clear and expected that SMART with eight subapertures yields a better-quality image than that of four subapertures. (See Supplement 1 for a similar experiment with a large field of view to demonstrate the technique with larger objects).

 figure: Fig. 6.

Fig. 6. Reconstruction results for r=0.2, 0.28, 0.4, and 0.8 mm of (a) PAIS with a pair of subapertures; (b) direct imaging results through a pair of subapertures with a diffractive lens; (c) reconstruction results of PAIS with four subapertures; (d) direct imaging results through four subapertures with a diffractive lens; (e) reconstruction results of SMART with all possible permutations of subaperture pair over the four locations; (f) reconstruction results of PAIS with eight subapertures; (g) direct imaging results through eight subapertures with a diffractive lens; and (h) reconstruction results of SMART with all possible permutations of subaperture pair over the eight locations; (i) direct imaging with a single aperture at the center.

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The next experiment was performed with two different NBS resolution charts, 14 lp/mm and 16 lp/mm, mounted in two different optical channels. The 3D imaging capabilities of SMART were studied by shifting the location of the 16 lp/mm target away from the 14 lp/mm by 1 cm and capturing the object hologram of the multiplane scene. The two objects located in the different axial planes were reconstructed by cross-correlating the appropriate complex impulse responses with the complex hologram of the targets. The reconstruction results for PAIS, SMART, and direct imaging are shown in Fig. 7 for four radii of the subaperture, r=0.2, 0.28, 0.4, and 0.8 mm. The reconstruction results of PAIS with eight subapertures for the two planes are shown in Figs. 7(a) and 7(b), respectively. The images of direct imaging with eight subapertures for the two planes are shown in Figs. 7(c) and 7(d), respectively, whereas the reconstruction results of SMART for the same two planes are shown in Figs. 7(e) and 7(f), respectively. It is evident that the quality of 3D imaging by SMART is higher than that of PAIS, and they are both much better in a sense of resolution than the direct imaging. Once again, larger subapertures yield better results than smaller subapertures in all the imaging methods.

 figure: Fig. 7.

Fig. 7. Reconstruction results for r=0.2, 0.28, 0.4, and 0.8 mm of PAIS with eight subapertures at (a) z=0cm; (b) z=1cm, direct imaging results through eight subapertures with a diffractive lens at (c) z=0cm; (d) z=1cm; reconstruction results of SMART at (e) z=0cm; (f) z=1cm.

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4. DISCUSSION

From Fig. 5, it can be seen that when the full aperture is used, the grating lines of the object 14 l p/mm are resolvable by both direct imaging as well as I-COACH. However, direct imaging is unable to resolve 14 l p/mm at all in the case of eight subapertures, each with r=0.4mm, whereas in PAIS with the same aperture area, the grating lines could be perceived. The above behavior was demonstrated earlier with an annular aperture [17], but in the present case, the area has been reduced by using only eight subapertures with a total area of only 6.8% of the full aperture and 20% of the annular aperture with a width of 0.8 mm.

A more rigorous explanation of the resolution superiority of PAIS (and SMART) over direct imaging is obtained by comparing the imaging processes of the two methods. The image of direct imaging is given by the relation f*|h|2, where f is the object and h is the point spread function (PSF) of the imaging system under coherent illumination, obtained by a scaled inverse Fourier transform of the aperture function [20]. In the spatial spectral domain, the spectrum of the image is F(HH), where F and H are Fourier transforms of f and h (hence, H is the scaled aperture function), respectively, and stands for correlation. Therefore, when the aperture function H is a ring of diameter D and a thickness a, or a set of isolated subapertures distributed along this ring, the bandwidth is 2D (the diameter of the autocorrelation of H), but the ratio between the central peak and the averaged level of sidelobes increases linearly with the ratio D/a. In other words, as much as the ratio D/a is increased, the decay of the high frequencies of the object spectrum is increased. The MTF equal to |HH| gets the shape of a peak with an effective width of 2a. The effect on the object spectrum is like a low-pass filter with a width of 2a, and consequently, the imaging resolution is reduced. Note that the bandwidth of an annular aperture system is defined as the diameter of the autocorrelation area of the annular aperture, whereas this area includes the entire autocorrelation values that are different from zero. Under this definition, the bandwidth is not the dominant parameter, which dictates the image resolution. In case the ratio D/a is high, the relative transparency in the high spatial frequencies is negligible, and hence, the resolution is not affected by the bandwidth defined above. Next, we show that this loss of resolution does not happen with PAIS because of its different mechanism of imaging.

In PAIS, the image is obtained in two stages: first, the camera records the convolution of the object with the impulse response of the recording system, and then the image is reconstructed by a digital correlation with a reconstructing function. Formally, the image is given by f*tr, and the spectrum of the image is F·T·R*, where t is the impulse response of the recording system and r is the reconstructing function. T and R are the Fourier transforms of t and r, respectively. Because of using a phase-only filter (i.e., R=exp(i·phase{T}) the spectrum of the image is actually F·|T|. In a three-shot PAIS, T is a superposition of three functions, each of which is the Fourier transform of the camera intensity obtained for a different independent CPM. In a formal notation, T=F{k=13bk|hk|2}=k=13bkF{|hk|2}=k=13bkHkHk, where hk is the coherent PSF of the recording system for the kth CPM, Hk=F{hk} and bk is the kth constant of the superposition. Recall that b1,2,3=1, exp(i2π/3), exp(i4π/3), the value of the spectrum of |T| in the zero frequency is not higher than the average magnitude of |T|, assuming the average distribution of each intensity response |hk|2 is approximately the same for all k. The above-mentioned effect, shown for the direct imaging, of the high ratio between the zero-frequency peak and the averaged level of spectral sidelobes does not exist in PAIS. The bandwidth of 2D is the same as in the direct imaging because |T| is the sum of autocorrelations of Hk, and each Hk has the same dimension as H. However, the effect of a low-pass filter, typical of direct imaging with annular aperture, does not exist in PAIS because the zero-frequency value is not higher than the average value of the sidelobes. In other words, the spatial bandwidth of the two imaging systems is the same, but the MTF of PAIS does not have the peak shape of the direct imaging with an annular aperture. And since the MTF of PAIS, |T|, spreads over the spectrum more uniformly than the MTF of the direct imaging |HH|, PAIS is superior with regard to the aspect of resolution over the direct imaging. Note that in order to guarantee maximum spatial bandwidth and hence optimal resolution, the shape of the PAIS aperture should be in the form of a ring with maximal diameter, or at least a sampled version of this ring. Obviously, in PAIS, as much as the ratio D/a is increased, the quality of the reconstructed image deteriorates, because the uniformity of |T| over the entire spectral region is decreased. However, the quality reduction can be compensated by adding more camera shots with independent CPMs [16,17]. The MTF profiles for the ring-shaped aperture and eight subaperture cases were computed based on simulated data. The MTF plots are shown in Fig. 8.

 figure: Fig. 8.

Fig. 8. MTF profile for PAIS and direct imaging for ring thickness of (a1) 0.2 mm and (a2) 0.4 mm; (b) MTF plots for direct imaging with various annular widths; (c1) 2D and (c2) mesh profile of MTF of direct imaging; and (d1) and (d2) PAIS with eight subapertures.

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Note that the x axis in the entire plots are in unit cycles per millimeter limited by ±D/(λzh), where D is the diameter of the lens displayed on the SLM, i.e., 8 mm, λ (= 635 nm) is the wavelength of the incoherent source used in the experiment, and zh is the distance between the SLM and the sensor plane, which is equal to 20 cm. From Fig. 8(a1 and a2), it is seen that in the case of ring-shaped PAISs, the higher spatial frequencies are attenuated much less than the case of the direct imaging system. Thus, direct imaging with partial or ring-shaped apertures acts as a low-pass filter, and higher frequencies are attenuated more in direct imaging than in the case of PAIS. As a result, PAISs have better lateral resolution than direct imaging does. Figure 8(b) shows how the response of direct imaging to higher spatial frequencies abruptly drops with decreasing the ring width of the annular aperture. In Fig. 8(c1-c2) for PAIS, simulated 2D and 3D MTF profiles are shown for eight equidistant marginal circular subapertures with radii of 0.4 mm. The magnitude and area covered by the higher spatial frequencies are higher in PAIS than in the direct imaging system, and these differences explain the superiority of PAIS and SMART over direct imaging in the sense of lateral resolution.

In Figs. 6(a) and 6(b), the reconstruction results of PAIS and direct imaging are shown for a pair of subapertures for r=0.2, 0.28, 0.4, and 0.8 mm, respectively. Comparing Figs. 6(a4) and 6(a3) with Figs. 6(b4) and 6(b3) shows that PAIS has a higher resolution compared to direct imaging. This difference between PAIS and direct imaging for the same area of the aperture has been explained in the previous paragraphs. However, when the radius of the subapertures is decreased further to r=0.28 and 0.2 mm, PAIS also fails to image the target, as seen in Figs. 6(a2) and 6(a1), respectively. This behavior is also understood in view of the explanation of the previous paragraph. The MTF of PAIS is obtained from a superposition of the autocorrelations of the CPMs, and hence narrowing the area of the CPMs reduces the MTF spectral cover of PAIS. For the same reasons, when the number of subapertures is increased from two to four, as seen in Figs. 6(c), PAIS was able to reconstruct the image for r=0.8, 0.4, and 0.28 mm, as seen in Figs. 6(c4) to 3(c2), respectively, but visibility is lost in the case of r=0.2mm. However, the SMART technique was able to reconstruct the object for all cases, as seen in Figs. 6(e1)–6(e4).

The reasons for the superiority of SMART over PAIS is explained next. In SMART, at any given time there is only interference between two subapertures. Therefore, the intensity pattern on the camera includes only two bias terms, which are eliminated by the superposition and two useful interference terms. On the other hand, in PAIS of N subapertures, the interference yields N bias terms, N/2 times more than in SMART. Consequently, in SMART, the dynamic range of the camera for the useful interference terms and the power ratio of the interference to bias terms is higher than in PAIS. Accordingly, for the same level of noise in the camera, the SNR of the interference signals detected by SMART is higher than in PAIS. In Figs. 6(f) and 6(g), the results show that when the number of subapertures is increased further from four to eight, parts of the image of the object are reconstructed using PAIS even for r=0.2mm [Fig. 6(f1)]. The results of SMART seen in Fig. 6(h) show visible improvement in the reconstruction for all values of the radius of the subapertures. Based on Figs. 6(a), 6(c), and 6(f), it is evident that in order to maintain an acceptable SNR in PAIS, in which the visual information of the object can be recognized, the percentage of the subaperture area should be above some value. When the subaperture area is increased either by an additional number of subapertures or by an increased radius of every subaperture, the SNR is improved.

Recalling the fundamental design principle of SMART with two orbiting reflectors, the above-mentioned results show that it is possible to obtain superresolution with lesser resources. Explicitly, two reflectors with a lower time resolution yield a better image than the case of PAIS with more resources, such as four or eight reflectors, but with a higher time resolution. The results also confirm that with SMART, it is possible to completely retrieve the image by sampling only along the perimeter of the synthetic aperture, and the reconstructed image maintains the same resolution of the complete synthetic aperture. The 3D reconstruction results of SMART shown in Fig. 7 shows that SMART is able to reconstruct objects at different depths with more noticeable details than can be achieved by direct imaging and even by PAIS.

There are three parameters, namely, the total diameter of each annular aperture D, the radius of each subaperture r, and the pixel size of the SLM, which influence the optical characteristics of imaging by PAIS and SMART. As explained in this section, in a similar way as direct imaging with a clear aperture, the diameter D dictates the lateral resolution limit of SMART and PAIS such that the minimal resolved size is about λzs/D. The parameter r determines the cover area of the MTF in the spatial frequency domain. As reflected from the comparative results of Figs. 6 and 7, as r is increased, the MTF cover becomes larger, and the image quality is improved. The pixel size of the SLM controls on the maximum scattering angle of the CPMs. Hence, the area size of the impulse response IIR is determined by the pixel size. In the opposite direction, reducing the area size of IIR means reducing the scattering rank of the CPM and hence increasing the effective pixel size of the SLM. The area size of IIR is determined by one of the constraints of the GSA: the constraint that dictates the area of the magnitude of the CPM Fourier transform. In other words, the area size of IIR, SIR, defines the CPM scattering rank σ given by σ=SIR/max{SIR}, and SIR is inversely proportional to the pixel size of the SLM. Figure 9 shows the normalized SNR and visibility plots of reconstructed images versus the scattering rank. As expected, increasing the scattering rank up to a certain point (σ0.4) improves the SNR and also the visibility, because as the area size of IIR (and consequently of hIR) is increased, the cross-correlation of the form of Eq. (12) becomes sharper. However, above the scattering rank of σ0.4, the same optical power is scattered over a larger area; hence, the SNR of the recorded camera pattern is reduced, and therefore the quality of the reconstruction is gradually deteriorated. The plot of Fig. 9 gives quantified measures on the effectiveness of the GSA in reducing the background noise.

 figure: Fig. 9.

Fig. 9. Plot of the normalized SNR and visibility versus scattering rank determined by the effective pixel size of the SLM. Inset figures are the reconstruction results using object–element 1 group 3 of USFA 1951 1X negative resolution chart.

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Extensive comparison results of direct imaging, PAIS, and SMART using mean square error (MSE) is given here [structural similarity index (SSIM) and visibility comparison are mentioned in Supplement 1]. MSE, SSIM, and visibility comparison further validate the superiority of SMART over direct imaging and PAIS. First, MSE versus the number of permutations of SMART is calculated. The MSE is defined as

MSE=1MNm=1Mn=1N|OD(m,n)γOM(m,n)|2,
where OD and OM are M×N matrices of the desired and measured reconstructions of the object, respectively. A reconstruction close to direct imaging with full aperture is desirable; hence, OD is selected as the image of the object recorded by direct imaging with full aperture and
γ=m=1Mn=1NOD(m,n)OM(m,n)m=1Mn=1N|OM(m,n)|2.
The plots of MSE versus the number of SMART permutations for the subaperture radius of r=0.2, 0.28, 0.4, and 0.8 mm are shown in Fig. 10.

 figure: Fig. 10.

Fig. 10. Plots of MSE of SMART versus the number of permutations of subaperture pairs for subaperture radii r=0.2, 0.28, 0.4, and 0.8 mm.

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It is clear from Fig. 10 that increasing the number of permutations or increasing the radius of the subaperture reduces the MSE. Figure 11 shows MSE variation when four different radii are used for different cases of SMART with 28 permutations, and PAIS and direct imaging with subapertures at eight equally spaced marginal points. From Fig. 11 it is evident that, in most of the cases, SMART results are better than PAIS results and far better than the conventional direct imaging.

 figure: Fig. 11.

Fig. 11. MSE of PAIS and direct imaging for eight subapertures and SMART with 28 permutations of subaperture pair.

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5. CONCLUSION

We have proposed a novel incoherent synthetic aperture technique with a pair of subapertures having an area as low as 0.43% of total synthetic aperture area. The subaperture pair moves only along the perimeter of the complete synthetic aperture. As a preliminary test, we investigated the combinations of subaperture pairs located on a grid of two to eight equally separated points along the annular perimeter of the complete aperture. Although the synthetic aperture is sampled only along its margin, at least in cases where each subaperture is wide enough, the resolution and the SNR are comparable to the image obtained by the complete aperture. By the proposed method, the image is obtained as a cross-correlation between the system response to the object and the system impulse response. Three impulse intensity responses are recorded for each of the three independent CPMs and for each of the entire subaperture permutations distributed on a grid of N equally separated points. The three impulse intensity responses are superposed with different phase constants into complex-valued holograms. A similar recording process is repeated for the observed object. Finally, images of the different planes of the object are reconstructed by cross-correlating the object hologram with the corresponding complex-valued impulse responses. Although the pair of subapertures has 0.43% of the total synthetic aperture area, the aperture ratio can be decreased further by increasing the number of camera shots acquired with different CPMs and averaging over the obtained images. Probably in the future, the aperture ratio will be further reduced by improving the algorithms of synthesizing the CPMs. Analysis of the system limitations is also a task that will be investigated in the future.

The results of SMART are compared to those of direct imaging and of PAIS, both with the same system aperture. The results of SMART are found to be always better than those of PAIS reconstruction and much better than those of conventional direct imaging with the same apertures. In direct imaging, when the subapertures are distributed along the perimeter of the complete aperture, the PSF has relatively high sidelobes that blur the image. In the spatial spectrum, the effect is of using narrow low-pass filtering. SMART and PAIS do not suffer from this problem because both these techniques are indirect imaging methods, and consequently, the image is not obtained as a result of direct convolution between the object and a PSF dictated by the system aperture. From the spectral perspective, the input images of PAIS and SMART are not filtered by the low-pass filter of the direct imaging. Instead, their MTF is chaotic pseudorandomly distributed, and their transmission is distributed over the entire spectral domain more uniformly than the direct imaging with the same aperture, resulting in superresolution for the case of SMART and PAIS.

In addition to the superiority of SMART over other techniques of the synthetic aperture in terms of the aperture coverage, SMART is also an inherent 3D imaging technique. The imaging quality can be further simply improved by using more than eight points in the position grid and by averaging over many independent imaging results. In summary, we have shown in [17] that a full clear aperture of a conventional imaging system can be replaced by an annular aperture without losing image resolution of the original full aperture system. But for practical purposes, in the present study of SMART and PAIS, we obtained two new findings: first, the annular aperture can be sampled in space or can be replaced by several isolated subapertures. Second, the annular aperture can be sampled in time by a pair of subapertures. In both options, the image resolution of the full clear aperture can be maintained utilizing minimal subaperture area. We believe that the demonstrated idea of SMART with minimal marginal subaperture ratios can be adapted for implementation in space-based and ground-based telescopes over conventional telescopes. The preliminary results shown here using a laboratory model are highly promising and might be a significant contribution to the field of imaging in general and astronomical telescopes in particular. However, further challenges such as atmospheric turbulence, scattering by aerosol, lower light intensity, and finding a stable satellite orbit are anticipated upon scaling up the system for satellite telescope applications, and are topics of future research.

Funding

Israel Science Foundation (ISF) (1669/16); Ministry of Science and Technology, Israel.

Acknowledgment

Part of this study was done during a Research Stay of JR at the Alfried Krupp Wissenschaftskolleg Greifswald.

 

See Supplement 1 for supporting content.

REFERENCES

1. K. Tomiyasu, “Tutorial review of synthetic-aperture radar (SAR) with applications to imaging of the ocean surface,” Proc. IEEE 66, 563–583 (1978). [CrossRef]  

2. M. Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms (Wiley, 1999).

3. A. E. Tippie and J. R. Fienup, “Gigapixel synthetic-aperture digital holography: sampling and resolution considerations,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2011), paper CWB1.

4. A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011). [CrossRef]  

5. P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011). [CrossRef]  

6. P. R. Lawson, Principles of Long Baseline Stellar Interferometry (NASA Jet Propulsion Laboratory, 2000).

7. B. Katz and J. Rosen, “Could SAFE concept be applied for designing a new synthetic aperture telescope?” Opt. Express 19, 4924–4936 (2011). [CrossRef]  

8. Y. Kashter, Y. Rivenson, A. Stern, and J. Rosen, “Sparse synthetic aperture with Fresnel elements (S-SAFE) using digital incoherent holograms,” Opt. Express 23, 20941–20960 (2015). [CrossRef]  

9. R. Zhu, J. T. Richard, D. J. Brady, D. L. Marks, and H. O. Everitt, “Compressive sensing and adaptive sampling applied to millimeter wave inverse synthetic aperture imaging,” Opt. Express 25, 2270–2284 (2017). [CrossRef]  

10. J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009). [CrossRef]  

11. V. Mico, Z. Zalevsky, P. García-Martínez, and J. García, “Synthetic aperture superresolution with multiple off-axis holograms,” J. Opt. Soc. Am. A 23, 3162–3170 (2006). [CrossRef]  

12. L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16, 161–169 (2008). [CrossRef]  

13. L. Granero, V. Micó, Z. Zalevsky, and J. García, “Synthetic aperture superresolved microscopy in digital lensless Fourier holography by time and angular multiplexing of the object information,” Appl. Opt. 49, 845–857 (2010). [CrossRef]  

14. B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18, 962–972 (2010). [CrossRef]  

15. Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22, 20551–20565 (2014). [CrossRef]  

16. A. Vijayakumar and J. Rosen, “Interferenceless coded aperture correlation holography-a new technique for recording incoherent digital holograms without two-wave interference,” Opt. Express 25, 13883–13896 (2017). [CrossRef]  

17. A. Bulbul, A. Vijayakumar, and J. Rosen, “Partial aperture imaging by systems with annular phase coded masks,” Opt. Express 25, 33315–33329 (2017). [CrossRef]  

18. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

19. A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography system with improved performance [Invited],” Appl. Opt. 56, F67–F77 (2017). [CrossRef]  

20. J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman, 2017).

References

  • View by:

  1. K. Tomiyasu, “Tutorial review of synthetic-aperture radar (SAR) with applications to imaging of the ocean surface,” Proc. IEEE 66, 563–583 (1978).
    [Crossref]
  2. M. Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms (Wiley, 1999).
  3. A. E. Tippie and J. R. Fienup, “Gigapixel synthetic-aperture digital holography: sampling and resolution considerations,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2011), paper CWB1.
  4. A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011).
    [Crossref]
  5. P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
    [Crossref]
  6. P. R. Lawson, Principles of Long Baseline Stellar Interferometry (NASA Jet Propulsion Laboratory, 2000).
  7. B. Katz and J. Rosen, “Could SAFE concept be applied for designing a new synthetic aperture telescope?” Opt. Express 19, 4924–4936 (2011).
    [Crossref]
  8. Y. Kashter, Y. Rivenson, A. Stern, and J. Rosen, “Sparse synthetic aperture with Fresnel elements (S-SAFE) using digital incoherent holograms,” Opt. Express 23, 20941–20960 (2015).
    [Crossref]
  9. R. Zhu, J. T. Richard, D. J. Brady, D. L. Marks, and H. O. Everitt, “Compressive sensing and adaptive sampling applied to millimeter wave inverse synthetic aperture imaging,” Opt. Express 25, 2270–2284 (2017).
    [Crossref]
  10. J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
    [Crossref]
  11. V. Mico, Z. Zalevsky, P. García-Martínez, and J. García, “Synthetic aperture superresolution with multiple off-axis holograms,” J. Opt. Soc. Am. A 23, 3162–3170 (2006).
    [Crossref]
  12. L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16, 161–169 (2008).
    [Crossref]
  13. L. Granero, V. Micó, Z. Zalevsky, and J. García, “Synthetic aperture superresolved microscopy in digital lensless Fourier holography by time and angular multiplexing of the object information,” Appl. Opt. 49, 845–857 (2010).
    [Crossref]
  14. B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18, 962–972 (2010).
    [Crossref]
  15. Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22, 20551–20565 (2014).
    [Crossref]
  16. A. Vijayakumar and J. Rosen, “Interferenceless coded aperture correlation holography-a new technique for recording incoherent digital holograms without two-wave interference,” Opt. Express 25, 13883–13896 (2017).
    [Crossref]
  17. A. Bulbul, A. Vijayakumar, and J. Rosen, “Partial aperture imaging by systems with annular phase coded masks,” Opt. Express 25, 33315–33329 (2017).
    [Crossref]
  18. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  19. A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography system with improved performance [Invited],” Appl. Opt. 56, F67–F77 (2017).
    [Crossref]
  20. J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman, 2017).

2017 (4)

2015 (1)

2014 (1)

2011 (3)

A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011).
[Crossref]

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

B. Katz and J. Rosen, “Could SAFE concept be applied for designing a new synthetic aperture telescope?” Opt. Express 19, 4924–4936 (2011).
[Crossref]

2010 (2)

2009 (1)

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
[Crossref]

2008 (1)

2006 (1)

1978 (1)

K. Tomiyasu, “Tutorial review of synthetic-aperture radar (SAR) with applications to imaging of the ocean surface,” Proc. IEEE 66, 563–583 (1978).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Andelson, P.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Brady, D. J.

Brooker, G.

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
[Crossref]

Bulbul, A.

Covey, K. R.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Czeszumska, A.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Edelstein, J.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Erskine, D. J.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Everitt, H. O.

Fienup, J. R.

A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011).
[Crossref]

A. E. Tippie and J. R. Fienup, “Gigapixel synthetic-aperture digital holography: sampling and resolution considerations,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2011), paper CWB1.

García, J.

García-Martínez, P.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman, 2017).

Granero, L.

Halverson, S. P.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Hamren, K.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Indebetouw, G.

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
[Crossref]

Javidi, B.

Kashter, Y.

Katz, B.

Kelner, R.

Kimber, D.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Kumar, A.

Lawson, P. R.

P. R. Lawson, Principles of Long Baseline Stellar Interferometry (NASA Jet Propulsion Laboratory, 2000).

Lloyd, J. P.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Marks, D. L.

Martínez-León, L.

Mercer, T.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Mico, V.

Micó, V.

Mondo, D.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Muirhead, P. S.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Muterspaugh, M. W.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Richard, J. T.

Rivenson, Y.

Rosen, J.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Shaked, N. T.

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
[Crossref]

Soumekh, M.

M. Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms (Wiley, 1999).

Stern, A.

Tippie, A. E.

A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011).
[Crossref]

A. E. Tippie and J. R. Fienup, “Gigapixel synthetic-aperture digital holography: sampling and resolution considerations,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2011), paper CWB1.

Tomiyasu, K.

K. Tomiyasu, “Tutorial review of synthetic-aperture radar (SAR) with applications to imaging of the ocean surface,” Proc. IEEE 66, 563–583 (1978).
[Crossref]

Vanderburg, A.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Vijayakumar, A.

Wishnow, E. H.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Wright, J. T.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Zalevsky, Z.

Zhu, R.

Appl. Opt. (2)

J. Hologr. Speckle (1)

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (9)

L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16, 161–169 (2008).
[Crossref]

B. Katz and J. Rosen, “Could SAFE concept be applied for designing a new synthetic aperture telescope?” Opt. Express 19, 4924–4936 (2011).
[Crossref]

Y. Kashter, Y. Rivenson, A. Stern, and J. Rosen, “Sparse synthetic aperture with Fresnel elements (S-SAFE) using digital incoherent holograms,” Opt. Express 23, 20941–20960 (2015).
[Crossref]

R. Zhu, J. T. Richard, D. J. Brady, D. L. Marks, and H. O. Everitt, “Compressive sensing and adaptive sampling applied to millimeter wave inverse synthetic aperture imaging,” Opt. Express 25, 2270–2284 (2017).
[Crossref]

A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011).
[Crossref]

B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18, 962–972 (2010).
[Crossref]

Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22, 20551–20565 (2014).
[Crossref]

A. Vijayakumar and J. Rosen, “Interferenceless coded aperture correlation holography-a new technique for recording incoherent digital holograms without two-wave interference,” Opt. Express 25, 13883–13896 (2017).
[Crossref]

A. Bulbul, A. Vijayakumar, and J. Rosen, “Partial aperture imaging by systems with annular phase coded masks,” Opt. Express 25, 33315–33329 (2017).
[Crossref]

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. IEEE (1)

K. Tomiyasu, “Tutorial review of synthetic-aperture radar (SAR) with applications to imaging of the ocean surface,” Proc. IEEE 66, 563–583 (1978).
[Crossref]

Publ. Astron. Soc. Pac. (1)

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Other (4)

P. R. Lawson, Principles of Long Baseline Stellar Interferometry (NASA Jet Propulsion Laboratory, 2000).

M. Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms (Wiley, 1999).

A. E. Tippie and J. R. Fienup, “Gigapixel synthetic-aperture digital holography: sampling and resolution considerations,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2011), paper CWB1.

J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman, 2017).

Supplementary Material (1)

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Figures (11)

Fig. 1.
Fig. 1. Schematic of the space-based telescope for the implementation of SMART.
Fig. 2.
Fig. 2. Laboratory model of SMART for image acquisition. CPM, coded phase mask; L0 and L1, refractive lens; LED, light-emitting diode; and DOE, diffractive optical element.
Fig. 3.
Fig. 3. Modified GSA for designing the CPM pairs with all possible permutations of eight equally separated circles along the perimeter of the aperture such that every pair is constrained to produce a uniform magnitude on the sensor plane.
Fig. 4.
Fig. 4. Experimental setup for demonstration of SMART. BS1 and BS2, beam splitters; SLM, spatial light modulator; NBS, National Bureau of Standards; L0A, L0B and B1, refractive lenses; LED1 and LED2, identical light-emitting diodes; and P, polarizer.
Fig. 5.
Fig. 5. (a)–(c), (g)–(i) Intensity patterns recorded for a point object and a resolution target for eight subapertures, respectively; (d) magnitude and (e) phase of hIR; (j) magnitude and (k) phase of hOR; (f) reconstructed image of PAIS; (l) direct imaging result using eight subapertures with diffractive lens; (m)–(o), (s)–(u) intensity patterns recorded for a point object and a resolution target for full aperture, respectively; (p) magnitude and (q) phase of hIR; (v) magnitude and (w) phase of hOR; (r) reconstructed image of full aperture imaging system; (x) direct imaging result using a full aperture with a diffractive lens.
Fig. 6.
Fig. 6. Reconstruction results for r=0.2, 0.28, 0.4, and 0.8 mm of (a) PAIS with a pair of subapertures; (b) direct imaging results through a pair of subapertures with a diffractive lens; (c) reconstruction results of PAIS with four subapertures; (d) direct imaging results through four subapertures with a diffractive lens; (e) reconstruction results of SMART with all possible permutations of subaperture pair over the four locations; (f) reconstruction results of PAIS with eight subapertures; (g) direct imaging results through eight subapertures with a diffractive lens; and (h) reconstruction results of SMART with all possible permutations of subaperture pair over the eight locations; (i) direct imaging with a single aperture at the center.
Fig. 7.
Fig. 7. Reconstruction results for r=0.2, 0.28, 0.4, and 0.8 mm of PAIS with eight subapertures at (a) z=0cm; (b) z=1cm, direct imaging results through eight subapertures with a diffractive lens at (c) z=0cm; (d) z=1cm; reconstruction results of SMART at (e) z=0cm; (f) z=1cm.
Fig. 8.
Fig. 8. MTF profile for PAIS and direct imaging for ring thickness of (a1) 0.2 mm and (a2) 0.4 mm; (b) MTF plots for direct imaging with various annular widths; (c1) 2D and (c2) mesh profile of MTF of direct imaging; and (d1) and (d2) PAIS with eight subapertures.
Fig. 9.
Fig. 9. Plot of the normalized SNR and visibility versus scattering rank determined by the effective pixel size of the SLM. Inset figures are the reconstruction results using object–element 1 group 3 of USFA 1951 1X negative resolution chart.
Fig. 10.
Fig. 10. Plots of MSE of SMART versus the number of permutations of subaperture pairs for subaperture radii r=0.2, 0.28, 0.4, and 0.8 mm.
Fig. 11.
Fig. 11. MSE of PAIS and direct imaging for eight subapertures and SMART with 28 permutations of subaperture pair.

Equations (14)

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MP(r¯)=p=1Pexp[iϕp(r¯)]Circ(|r¯|R)*δ(r¯r¯p),
IIR(r¯0;r¯s,zs)=C0|ν[1λzh]F{MP(r¯)}|2*δ(r¯0zhzsr¯s),
hIR(r¯0;0,zs)=j=13IIR(r¯0;0,zs)exp(iθj).
hIR,PAIS(r¯0;0,zs)=C0j=13exp(iθj)|ν[1λzh]F{MP=N(r¯)}|2=C0j=13exp(iθj)|ν[1λzh]F{p=1Nexp[iϕj,p(r¯)]Circ(|r¯|R)*δ(r¯r¯p)}|2=C0j=13exp(iθj)|p=1NF{exp[iϕj,p(λzhr¯)]}*Jinc(R|ro¯|λzh)exp[i2πλzh(r¯p·r¯o)]|2,
hIR,PAIS(r¯0;0,zs)C0j=13exp(iθj)k=1N1l=k+1N[(F{exp[iϕj,k(λzhr¯)]}*Jinc(R|r¯o|λzh))×(F{exp[iϕj,l(λzhr¯)]}*Jinc(R|r¯o|λzh))exp[i2πλzh[(r¯kr¯l)·r¯o]]+C.C.],
hIR,SMART(r¯o;0,zs)=C0k=1N1l=k+1Nj=13exp(iθj)|ν[1λzh]F{M2(r¯)}|2=C0k=1N1l=k+1Nj=13exp(iθj)×|ν[1λzh]F{exp[iϕk,j(r¯)]Circ(|r¯r¯k|R)+exp[iϕl,j(r¯)]Circ(|r¯r¯l|R)}|2=C0k=1N1l=k+1Nj=13exp(iθj)|F{exp[iϕk,j(λzhr¯)]}*Jinc(R|r¯o|λzh)exp(i2πr¯k·r¯oλzh)+F{exp[iϕl,j(λzhr¯)]}*Jinc(R|r¯o|λzh)exp(i2πr¯l·r¯oλzh)|2.
hIR,SMART(r¯o;0,zs)C0j=13exp(iθj)×k=1N1l=k+1N[(F{exp[iϕk,j(λzhr¯)]}*Jinc(R|r¯o|λzh))×(F{exp[iϕl,j(λzhr¯)]}*Jinc(R|r¯o|λzh))exp(i2π(r¯kr¯l)·r¯oλzh)+C.C.].
o(r¯s)=mMamδ(r¯sr¯m),
IOR(r¯0;zs)=mamIIR(r¯0zhzsr¯m;0,zs).
hOR,PAIS(r¯0;zs)=j=13IOR(r¯0;zs)exp(iθj)=j=13mamIIR(r¯0zhzsr¯m;0,zs)exp(iθj)=mamhIR,PAIS(r¯0zhzsr¯m;zs).
hOR,SMART(r¯0;zs)=k=1N1l=k+1Nj=13IOR(r¯0;zs)exp(iθj)=k=1N1l=k+1Nj=13mamIIR(r¯0zhzsr¯m;0,zs)exp(iθj)mamhIR,SMART(r¯0zhzsr¯m;zs).
IIM(r¯R)=hOR(r¯0;zs)h˜IR*(r¯0r¯R;zs)dr¯0=mamhIR(r¯0zhzsr¯m;zs)h˜IR*(r¯0r¯R;zs)dr¯0=mamΛ(r¯Rzhzsr¯m)o(r¯sMT),
MSE=1MNm=1Mn=1N|OD(m,n)γOM(m,n)|2,
γ=m=1Mn=1NOD(m,n)OM(m,n)m=1Mn=1N|OM(m,n)|2.

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