Minimize SMART

SMART (Synthetic Marginal Aperture with Revolving Telescopes)

Ben-Gurion University (BGU) researchers of Beer-Shiva, Israel have developed a new satellite imaging system that could revolutionize the economics and imagery available from space-based cameras and even earth-based telescopes. "This is an invention that completely changes the costs of space exploration, astronomy, aerial photography, and more," says Angika Bulbul, a BGU Ph.D. candidate under the supervision of Prof. Joseph Rosen in the BGU Department of Electrical and Computer Engineering. 1)

In a paper published in the December issue of Optica, the researchers demonstrate that nanosatellites the size of milk cartons arranged in a spherical (annular) configuration were able to capture images that match the resolution of the full-frame, lens-based or concave mirror systems used on today's telescopes. 2)

"Several previous assumptions about long-range photography were incorrect," Bulbul says. "We found that you only need a small part of a telescope lens to obtain quality images. Even by using the perimeter aperture of a lens, as low as 0.43 percent, we managed to obtain similar image resolution compared to the full aperture area of mirror/lens-based imaging systems. Consequently, we can slash the huge cost, time and material needed for gigantic traditional optical space telescopes with large curved mirrors."

To demonstrate the SMART system capabilities, the research team built a miniature laboratory model with a circular array of sub-apertures to study the image resolution and compare them with full lens imagery.

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Figure 1: Ben-Gurion University's SMART nanosatellite (image credit: BGU)

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. 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. 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) This sampling restriction is valid for SAR (Synthetic Aperture Radar) in the radio frequencies, or for Michelson stellar interferometry, 4) very large baseline interferometer (VLBI), and SAFE (Synthetic Aperture with Fresnel elements) in the optical regime. 5) Sparse SAFE (S-SAFE) is an example in which the synthetic aperture is sampled according to the rules of compressed sensing.6) 7) 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 and VLBI 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 . 8) 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 and incoherent digital holography such as SAFE 9) 10) and S-SAFE (Ref. 6) 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 SMART (Synthetic Marginal Aperture with Revolving Telescopes). SMART is based on two concepts: namely, SAR and I-COACH (Interferenceless Coded Aperture Correlation Holography) . 11) 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 CPM (Coded Phase Mask). 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 PAIS (Partial Aperture Imaging System) 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. 12)

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 Figure 2. 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 the SMART configuration is constructed with aperture shapes that mimic the scheme of Figure 2.

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Figure 2: Schematic of the space-based telescope for the implementation of SMART (image credit: BGU)

 

Methodology

A schematic of the laboratory model of SMART is shown in Figure 3. Incoherent light from a LED (Light-Emitting Diode) 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 SLM (Spatial Light Modulator) whose aperture function is engineered with diffractive optical functions to match the scenario of Figure 2.

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Figure 3: 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), image credit: BGU

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 (N -1)/2) possible permutations. Every CPM pair is synthesized using the modified GSA (Gerchberg–Saxton Algorithm) to obtain a uniform magnitude over a predefined region of the spatial spectrum domain, as shown in Figure 4.

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Figure 4: 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 (image credit: BGU)

The predefined region was selected as the central 600 pixels x 600 pixels out of 1080 pixels x 1080 pixels 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, 13)). 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.

 

Experiments

The schematic of the experimental setup is shown in Figure 4. The setup consists of two illumination channels with identical LEDs (Thorlabs LED635L, 170 mW, λc = 635 nm, and Δλ=15 nm). 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 NBS (National Bureau of Standards) resolution targets (NBS 1963A Thorlabs), namely, 14 lp/mm and 16 lp/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, 1920 pixels x 1080 pixels, 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, 2048 pixels x 2048 pixels, 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).

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Figure 5: 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), image credit: BGU

 

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 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.

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Figure 6: (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 (image credit: BGU)

The intensity patterns recorded for the point object and for 14 lp/mm of the NBS resolution target, for eight subapertures and for three different CPMs, are shown in Figures 6 (a)–6 (c) and 6 (g)–6 (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 [(Ref. 11), Ref. 12)] as an effective tool to increase the signal-to-noise ratio (SNR). The set of three intensity patterns of Figures 6 (a)–6 (c) and Figures 6 (g)–6 (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 complexvalued object response hologram (hOR). The magnitude and phase of hIR and hOR are shown in Figures 6 (d) and 6 (e), and 6 (j) and 6 (k), respectively. hOR is cross-correlated with a phase-only filtered [14)] version of hIR, and the reconstructed image is shown in Figures 6 (f). The direct imaging result using a lens function over only the eight subapertures is shown in Figure 6 (l). It is evident from Figures 6 (f) and 6 (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.

For comparison purposes, a similar experiment with a full clear aperture of 4.32 mm radius, is depicted in Figures 6 (m)–6 (w). The intensity patterns recorded for the point object, and for the NBS target, for full aperture are shown in Figures 6 (m)–6 (o) and 6 (s)–6 (u), respectively. The magnitude and phase of the complex hIR for full aperture are shown in Figures 6 (p) and 6 (q), respectively. The magnitude and phase of hOR for full aperture are shown in Figures 6 (v) and 6 (w), respectively. The hOR is cross-correlated with phase-only filtered hIR, and the reconstructed image is shown in Figure 6 (r). The direct imaging result using a lens function with full aperture is shown in Figure 6 (x). By comparing Figures 6 (f),6 (l), 6 (r), and 6 (x), one can conclude that although the SNR of PAIS [Figure 6 (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 Figures 7 (a) and 7 (b), there are two subapertures on the perimeter, in Figures 7 (c)–7 (e) and 7 (f )–7 (h), there are four and eight subapertures, respectively, distributed with equal gaps on the perimeter, and in Figures 7 (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 Figure 7: direct imaging [Figures 7 (b), 7 (d),7 (g), and 7 (i)], PAIS [Figures 7 (a), 7 (c), and 7 (f )], and SMART [Figures 7 (e) and 7 (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 Figure 7, 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, Ref. 13)).

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 Figure 8 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 Figures 8 (a) and 8 (b), respectively. The images of direct imaging with eight subapertures for the two planes are shown in Figures 8 (c) and 8 (d), respectively, whereas the reconstruction results of SMART for the same two planes are shown in Figures 8 (e) and 8 (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.

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Figure 7: 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 (image credit: BGU)

 

Discussion

From Figure 6, it can be seen that when the full aperture is used, the grating lines of the object 14 lp/mm are resolvable by both direct imaging as well as I-COACH. However, direct imaging is unable to resolve 14 lp/mm at all in the case of eight subapertures, each with r = 0.4 mm, whereas in PAIS (Partial Aperture Imaging System) with the same aperture area, the grating lines could be perceived. The above behavior was demonstrated earlier with an annular aperture (Ref. 12), 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 x ∣h∣2, where f is the object and h is the PSF (Point Spread Function) of the imaging system under coherent illumination, obtained by a scaled inverse Fourier transform of the aperture function. 15)

In the spatial spectral domain, the spectrum of the image is F(H ⊗ H), 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 (Modulation Transfer Function) equal to ∣H ⊗ H∣ 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 x t ⊗ r, and the spectrum of the image is F x T x 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 x e(i x phase {T }) the spectrum of the image is actually F ·x ∣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 (Coded Phase Mask).

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 H'k , and each H'k 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 ∣H ⊗ H∣, 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 [Ref. 11), Ref. 12)]. The MTF profiles for the ring-shaped aperture and eight subaperture cases were computed based on simulated data. The MTF plots are shown in Figure 9.

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Figure 8: Reconstruction results for r = 0.2, 0.28, 0.4, and 0.8 mm of PAIS with eight subapertures at (a) z = 0 cm; (b) z = 1 cm, direct imaging results through eight subapertures with a diffractive lens at (c) z = 0 cm; (d) z = 1 cm; reconstruction results of SMART at (e) z = 0 cm; (f) z = 1 cm (image credit:BGU)

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 (Spatial Light Modulator), 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.

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Figure 9: 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 (image credit: BGU)

From Figure 9 (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 9 (b) shows how the response of direct imaging to higher spatial frequencies abruptly drops with decreasing the ring width of the annular aperture. In Figure 9 (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 Figures 7 (a) and 7 (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 Figures7 (a4) and 7(a3) with Figures 7 (b4) and 7 (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 Figures 7 (a2) and 7 (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 Figures 7 (c), PAIS was able to reconstruct the image for r = 0.8, 0.4, and 0.28 mm, as seen in Figures 7 (c4) to 3(c2), respectively, but visibility is lost in the case of r = 0.2 mm. However, the SMART technique was able to reconstruct the object for all cases, as seen in Figures 7 (e1)–7 (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 Figures 7 (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 = 0.2 mm [Figure 7 (f1)]. The results of SMART seen in Figure 7(h) show visible improvement in the reconstruction for all values of the radius of the subapertures. Based on Figures 7 (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 Figure 8 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 Figures 7 and 8, 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 I IR 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 10 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 (σ ≈ 0.4) 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 σ ≈ 0.4, 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 Figure 10 gives quantified measures on the effectiveness of the GSA in reducing the background noise.

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Figure 10: Plot of the normalized SNR and visibility versus scattering rank determined by the effective pixel size of the SLM (Spatial Light Modulator). Inset figures are the reconstruction results using object–element 1 group 3 of USAF 1951 1X negative resolution chart (image credit: BGU)

In summary, the SMART project team has 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.

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 spaceborne 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 of this research was provided by ISF (Israel Science Foundation) of the Ministry of Science and Technology, Israel.

 


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The information compiled and edited in this article was provided by Herbert J. Kramer from his documentation of: "Observation of the Earth and Its Environment: Survey of Missions and Sensors" (Springer Verlag) as well as many other sources after the publication of the 4th edition in 2002. - Comments and corrections to this article are always welcome for further updates (herb.kramer@gmx.net).