{'fraction_non_alphanumeric': 0.03842916070376294, 'fraction_numerical': 0.014563515447428234, 'mean_word_length': 4.431870141746685, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Randomized Aperture Imaging

Speckled images of a binary broad band light source (600-670 nm), generated by randomized reflections or transmissions, were used to reconstruct a binary image by use of multi-frame blind deconvolution algorithms. Craft store glitter was used as reflective elements. Another experiment used perforated foil. Also reported here are numerical models that afforded controlled tip-tilt and piston aberrations. These results suggest the potential importance of a poorly figured, randomly varying segmented imaging system.


I. INTRODUCTION
It is well known that angular resolution may be enhanced by use of a larger aperture. In practice, the size of a monolithic aperture is limited by the cost, weight and construction constraints [1,2]. Even if such constrains can be surmounted, adaptive optics methods must be applied to achieve diffraction limited performance [3]. Alternatively, passive approaches that make use of computer post-processing have been successfully employed [4][5][6]. Examples, including an aperture masking system [7], and a multi-aperture system [8] have shown great promise in astronomy and remote sensing [9][10][11],biology [12], clinical trials [13,14], and new types of computational cameras [15][16][17]. In these imaging systems, the burden of hardware control are replaced or greatly alleviated by digital computations. Mathematical tools, such as Fourier analysis [18], constrained optimization [19,20], and Bayesian inference [21] are essential in this approach. Both aperture masking and multi-aperture systems have demonstrated an improved signal-to-noise ratio of the acquired images, a calibrated point spread function (PSF), rejection of atmospheric noise, and closure phase measurements [22]. However, the sparsity of apertures implies a sparse coverage of spatial frequencies and loss of flux. Both adaptive optics and aperture masking systems attempt to improve the images obtained from a relatively well-figured optical system. In this paper we address the case of an ill-figured segmented optical system that varies randomly in time.
To make matters worse, we assume no knowledge of the randomly varying PSF. We ask the following the proof of concept question: Is it possible to reconstruct a near diffraction-limited image from a series of recordings from such a system?
In some respects this study is related to the random lens imaging system [23], where a collection of random reflectors served as a primary camera lens. Similarly, the sparkle vision system [24], simplifies the random lens imaging system by using a lens to better focus the light. However, in these examples, the PSF was not timevarying, and intensive machine learning algorithms were needed to uncover the PSF. In contrast, we aimed to reconstruct the time-varying PSF in a near real-time manner.
This report is organized as follows. In Section 2 we provide a description of a randomized complex aperture system. Two different experimental scenarios are established in Section 3: "far field"and "near field". For the purpose of comparing experimental and modeled results, we next describe in Section 4 the corresponding numerical models. Both the experimental and numerical data are processed by means of the multiframe blind deconvolution scheme described in Section 5. We then report image construction results in Section 6. Concluding remarks are provided in Section 7.

II. BASICS OF RANDOMIZED APERTURE IMAGING
An illustration of the randomized aperture scheme, shown in Fig. 1, depicts a point source (or sources) an infinite distance from elements that are distributed across a quasi-parabolic surface of baseline D and a detector at the mean focal distance f . For a continuous parabolic reflector, a diffraction-limited image would appear on the detector. However, tip,tilt and piston errors associated with each reflecting element produces aberrated speckles. Further, the distribution may be evolving in time, providing a sequence of N speckled images. Here, we ignore motion blur. Postprocessing techniques such a multi-frame blind deconvolution [25][26][27][28] may be used to recover a near-perfect image. This approach is particularly suitable in cases where it is not practical to make repeated measurement of time varying PSF. What is more, multi-frame blind deconvolution is a self-heuristic algorithm with less computational cost than machine learning, and may be simpler to implement than phase retrieval methods when the PSF is time varying. An advantage of this approach is (see Section 5) that whereas a single-image blind deconvolution scheme prescribes a ratio of unknowns (the PSF and the recovered object) to measurements as 2 : 1, the multi-frame blind deconvolution scheme improves the ratio to N + 1 : N . In general, the system shown in Fig. 1 may be shift-variant, in which case shift-variant multi-frame blind deconvolution algorithms [27] are better suited. For the cases examined below, however, the small shift-variance may be neglected, and we find the shift-invariant multi-frame blind deconvolution algorithm is sufficient to achieve good reconstructed image of a pair of point sources.

III. LABORATORY DEMONSTRATION
F ar F ield. In the case shown in Fig 2, a "far field" arrangement was constructed whereby a transmissive mask was positioned at the surface of a convex lens of focal length f = 200mm. A set of N = 50 thin foil masks was produced, each having M = 50 randomly placed pinholes(radius r ≈ 0.1mm ) distributed across a D = 3mm diameter. The foil was covered by a layer of wrinkled cellophane to randomize the phase at each pinhole. The close proximity of the cellophane and foil allowed for a nearly shift-invariant system. An Energetiq laser-driven white light source was spatially and spectrally filtered to produce a collimated polychromatic beam with a wavelength range λ = 600 − 670nm. Light transmitted through the mask was recorded at the back focal plane of the lens on the detector array of a Canon 5D Mark III camera having a pixel pitch of 6.25µm and detector size of 24×36mm. We call this arrangement "far field" because the detected light is governed by Fraunhofer diffraction from the pinholes. Imaging information is encoded in the interference of the diffracting beamlets. An effective second mutually incoherent light source was introduced by transmitting the beam through the system at angle 2θ 0 . The two images were added together to produce a single image of a binary source. The ground truth image can be obtained using the same setup, but without the phase and aperture masks.
N ear F ield. A schematic of the second experimental setup is shown in Fig 3. for a "near field" arrangement. As above, spatially and spectrally filtered light was formed into a collimated beam. In this case, however, reflecting elements were used to divert beamlets toward the detector. We used the front surface reflection from a glass wedge to produce a binary light source. The reflecting elements were comprised of square "fine size" craft store glitter, with dimensions of roughly 0.3×0.3mm. The random aperture condition was achieved by randomly sprinkling glitter across a blackened concave surface having a radius of curvature 65.4mm. After each image was recorded, the glitter was washed off, and a new random surface was prepared. The number of sub-apertures across the full D = 10mm beam diameter ranged from M = 200 to 400. The pellicle beam splitter was used to collect the reflected light and direct it toward the detector array of the same camera. Owing to space constraints, the detector could not be placed directly in the focal plane of the concave surface (f = 32.7mm from the surface), and thus, a 150mm relay lens was used. This configuration is called "near field" because the beamlets from each reflecting element undergo Fresnel diffraction upon reaching the detector. The diffraction length of a single reflecting element is roughly 450mm, which is much greater than f . We assert that the system is nearly shift invariant since the elements roughly conform to a small patch of diameter D on the concave surface. As seen in the inset of Fig. 3, the elements display significant tip-tilt errors.
The reconstruction results of experimental data are discussed in Section 6. Parameters of both schemes are listed in Table 1.

IV. NUMERICAL SIMULATION OF OPTICAL SYSTEM
F ar F ield. A numerical model of the far field experiment may be represented by a distribution of complex circular sub-aperture functions (see Fig 4): is an arbitrary vector in the plane of the aperture, c i = (c x,i , c y,i ) indicates the center of the i-th sub-aperture, r s is the radius of the sub-apertures, | · | 2 is l 2 -norm, and β = 100 is the power of Super-Gaussian function that defines the sub-apertures. Phase aberrations, φ i = φ p,i + φ t,i , across each subaperture are represented, respectively, by piston and tiptilt terms: where k = 2π/λ is wavenumber, a i is a pairwise independent uniformly distributed random variable with in- terval equals to ∆z, b i is a pairwise independent Gaussian random variable with mean 0 and variance given by < a 2 i >= σ 2 0 , and < − > denotes an ensemble average. The complete system aperture function may be expressed as where M is the number of sub-apertures, c 0 = (c x,0 , c y,0 ) indicates the center of the baseline, and R the radius of the full effective aperture, and γ = 100 is the power of Super-Gaussian function that defines the baseline. We further impose a non-overlapping condition to the sub-apertures: An example of the complex pupil function is shown in Fig 4.(a) for M = 100 sub-apertures. For two monochromatic point sources at infinity having an apparent angular separation of 2θ 0 along the xdirection, the electric field in the back focal plane of the system may be expressed as Fourier transforms of the field from each source, incident upon the imaging system(assuming it is shift-invariant): where for paraxial rays, k x ≈ 2πθ 0 /λ. Assuming the two light sources are mutually incoherent, we write the intensity in the plane of the detector as: N ear F ield In the near field regime, the numerical model ignores diffraction of the reflected light as it propagates from the aperture to the focal plane detector. That is, we assume the characteristic diffraction distance is much longer than the effective focal length: πr 2 s /λ >> f . The geometric model of near field scheme is illustrated in Fig 5. For a single source of collimated light, we assume a beam of parallel ray aligned along the unit vector d 0 . Rays incident upon the i th reflector centered at the point c i are reflected along the unit vector d r,i : where the Cartesian components of the unit normal vector of the i th reflector may be expressed in terms of direction cosines as: n i = (cos θ i ), where θ i = (θ x,i , θ y,i , θ z,i ). If the reflectors conformed to a paraboloid where f is the focal length. Each of the i th rays directed along d r,i would coincide at the focal point o = (0, 0, 0). In that special case the direction angles of the normal vector of each reflector may be expressed as θ i,0 = (θ x,i,0 , θ y,i,0 , θ z,i,0 ), where and p = ((x/2f ) 2 +(y/2f ) 2 +1) 1/2 . We note theses angles are related by cos 2 (θ x,i,0 ) + cos 2 (θ y,i,0 ) + cos 2 (θ z,i,0 ) = 1.
In general, the normal vector of each reflector will suffer from random tip-tilt errors ∆θ i = (θ x,i ∆a x,i a y,i ∆θ y,i , a z,i ∆θ z,i ), and thus the directional angles becomes where a x,i , a y,i , a z,i are pairwise independent Gaussian random variables variable with mean 0 and variance equal to σ 0 . Piston error, w i ∆z i may be introduced by displacing the center of the i th mirror from the paraboloid surface defined by c i = (c i,x , c i,y , f + (c 2 i,x + c 2 i,y )/4f ), where w i is pairwise independent uniform random variable with interval of ∆z i Let us consider a ray incident upon an arbitrary point x = (x, y, z) on the i th flat reflecting element. The plane of the element is represented by the equation n i (x − c i ) = 0. For a circular reflector of radius r s we impose |x − c i | 2 ≤ r s 2 . The intersection points of the incident beam with the plane of the i th circular reflector can be represented parametrically as: where t ∈ [0, 2π], α ∈ [0, r s ], e 1 and e 2 are unit vectors that satisfy e 1 = (0, 1, cosθ x /cosθ y )/(1 + (cosθ x /cosθ y ) 2 ) 1/2 and e 2 = e 1 × n.
Next we wish to determine where the reflected beam intersects the detector plane. In general, the detector plane can be defined by a center point q (usually the focal point of the paraboloid) and a normal vector n d . The projection of the points m along the reflected beam onto the detector plane may then be expressed as: The electric field within each of the i th projected beams in the detector plane may be represented as a tilted plane wave scaled by an obliquity factor from both the mirror d 0 · n i , and the detector d r · n d .
With two point sources at infinity, we repeat the above procedure by including two bundles of incoming rays reflected from an element along unit vectors d + 0 and d − 0 respectively. The two fields in the detection plane are labelled E + i and E − i : where A + i and A − i are zero valued outside the projection area of the i th mirror. L + i and L − i are full path length of rays that travel from the binary source to the i th mirror, and then reflected onto the detector. k = 2π/λ is the wavenumber. For small angular deviations in the detector plane and for equally luminous point sources, we make the approximation of setting the A + i and A − i within the interior regions of each bundle of rays equal for all u i . Here we consider two mutually incoherent sources subtending an angle 2θ 0 and bisecting the z-axis: d + ·d − = cos(2θ 0 ). Defining the postion vector on detector plane as x d = (x d , y d ), the measured irradiance may be expressed: The ground truth image is captured by imaging the objects using a monolithic mirror of the baseline size. The value numerical parameters are given in Table 2.

V. SYSTEM ESTIMATION AND IMAGE RECOVERY
Multi-frame blind deconvolution algorithms are used to recover a target scene from a set of blurry, noisy and distorted observations. They are generally categorized into two types: (1) non-blind deconvolution, where the target scene is reconstructed based on complete or partial knowledge of the point spread function(PSF) of the imaging system; and (2) blind deconvolution, where the unknown target scene and system point spread function are recovered simultaneously without a priori knowledge. Since the work of Ayers and Dainty [29], multiframe blind deconvolution has become an important tool for image recovery, resulting in numerous research efforts and applications. Common approaches involve: (1) Batch mode multi-frame blind deconvolution [25,30], where all the distorted observations are processed at the same time; and (2) Serial mode multi-frame blind deconvolution [26,31,32], where degraded inputs are sequentially processed. Compared with the batch mode, the serial approach is more memory efficient and can in principle be done at the same time as the image acquisition. In this paper image restoration was accomplished by use of a serial multi-frame blind deconvolution [31] scheme.
Here we give a brief review of the online multi-frame blind deconvolution algorithm. We assume that at each time point n = 1, 2, · · · , N , the random aperture mirror system records a blurred image {g n }. Assuming the imaging system is shift-invariant(or approximate shiftinvariant), the image formation process can be modeled as convolution of the target image and the system PSF, where the recorded n th image g n (x, y) may be expressed: where ⊗ denotes the two-dimensional convolution operator, and f (x, y) and q n (x, y) respectively represent the ideal image and its random noise, and h n (x, y) represents the shift-invariant PSF of the n th time frame. We assume the PSF changes from frame to frame owing to time varying tip, tilt, and piston errors, as well as the location of each mirror. The goal for multi-frame blind deconvolution is to recover the ideal image f and the temporally varying PSF h n from a set of degraded images {g n }. A simple but effective choice for the blind deconvolution can be achieved by solving the following non-negatively constrained problem [26] {f, h n } = min (18) whereG i ,F ,H i are the Fourier transforms of the observed images g i , ideal image f , and PSFs h i respectively, and u is the position vector in frequency space.
Solutions to Eq.18 are commonly solved using either: (1) batch mode optimization using the constrained conjugate gradient descent of cost function with respect to f and {h n } in an alternating manner [25,28,30]; and (2) serial mode optimization using the multiplicative updates [26,31], as is shown in the method we employed, outlined below in Algorithm 1,estimated image and PSF are updated respectively using multiplicative method in each iteration: and where denote the component-wise multiplication.

VI. RESULTS AND DISCUSSION
Analyses of our reconstructed images resulting from multi-frame blind deconvolution are presented below. First we describe the experimentally measured data for far field and near field schemes using a polychromatic (10% bandwidth) light source. Numerically generated data is then used to demonstrate the reconstruction of a monochromatic binary light source under different tiptilt and piston errors. Quantitative comparisons between ground truth and reconstructed images are evaluated based on two metrics: spatial separation error E D = |D − D |/D and the peak intensity ratio error E p = |I p,1 /I p,2 − I p,1 /I p,2 |/(I p,1 /I p,2 ), where D and D are the respective distances between the peaks of the ground truth and restored images respectively, and where I p and I p are the respective peak intensities of the ground truth and restored results.

A. Experimental Results
Reconstructed images were obtained by use of N = 50 images for both the far field and near field schemes. Details of the experimental setup and parameters are discussed in Section 4 and Table 1. The restored images of a binary light source and examples of speckle images are shown in Fig 6. Immediately we see in both cases that the reconstructed images are superior to the speckle images, i.e, qualitative agreement with the ground truth is acheived. Quantitatively, we find the distance between the intensity peaks are in good agreement, with E D ≈ 5%. Furthermore the intensity peaks are equal to within E p ≈ 15%. This is remarkable considering the 10% bandwidth of the light source, and the estimated 15λ/D tip-tilt error and a likely piston error of at least several waves. The good agreement between the ground truth and reconstructed images may be attributed to the high degree of shift invariance of the imaging systems. That is, the speckle data in both Fig 6.(b) and (d) contains multiple overlapping pairs of binary images. We note that in the far field (diffracted) case the pairs dis-placed from the central region of the speckle image are diffused owing to the broad bandwidth of the light source. In contrast, for the near field case, the beamlets from each reflecting element experience little diffraction, and thus no chromatic spreading of the beamlets. We believe the multiframe BD scheme is successful at recovering the binary light source in both cases because the underlying imaging systems are well described by a shift invariant convolution process.

B. Numerical Results
To explore how well multiframe blind deconvolution restores a binary image from random aperture mirror images, numerically simulated far field and near field data were generated for various degrees of tip-tilt and piston error. Details of the system were described in Sec 4 and Table 2.
First we examine numerical cases that closely resemble the experimental system. For both the near and far field schemes we numerically generated speckled images with 50 non-overlapping sub-apertures. Gaussian random tiptilt error with σ 0 = 10λ/D and uniform random piston error with ∆z = 1λ were assumed for each of the N = 60 images of the generated sequence. Shown in Fig 7 are the ground truth and example speckle images for the far field and near field scenarios. The insets show a zoomed image of the ground truth in Fig 7.(a), and the corresponding reconstructed images in Fig 7.(b,c). As in the experimental case, the qualitative agreement between the ground truth and reconstructed images is good. Quantitatively the spatial separation error is E D ≈ 10% for both cases, and the magnitude ratio error is E p is less than 4%. These errors are smaller than the experimental values because the numerical cases are noise free and monochromatic. We found that the near field values of E D and E p are somewhat larger than the far field values. We attributed this difference to the lower degree of shift invariance in the near field case owing to different projections of the beamlets on the detector plane.
To examine the fidelity of reconstructed image as a function of tip-tilt and piston error, we modeled the far field system with σ 0 = 10, 20, 40[λ/D] with no piston error (see Fig 8), and then ∆z = 0.5, 1.0, 1.5λ with no tip tilt error (see Fig 9). In these cases we set the number of sub-apertures to M = 100, and the number of images to N = 50. As expected, the quality of the reconstructed images deteriorates with increasing phase error. The reconstruction errors increase from E D ≈ 8.9, 9.1 to 9.4%, and E P ≈ 2, 24 to 34% as the tip-tilt error increase from σ 0 = 10 to 20 to 40[λ/D]. On the other hand, as the increment of piston error increases from ∆z = 0.5 to 1.0 to 1.5λ, the errors for the restoration increase from E D ≈ 13, 16 to 22%, and E P increase from ≈ 6, 27 to 58%. For both tip-tilt and piston error we find the magnitude ratio error suffers more significantly than the spatial separation error. We attribute this to the increased intensity of speckles and interference patterns that are resulted from sufficiently large phase errors.

VII. CONCLUSION
We numerically and experimentally explored the concept of random aperture mirror telescope for both monochromatic binary light sources and polychromatic binary light sources, in both far field and near field schemes. For an approximate shift-invariant system, binary light sources can be restored using Multi-frame blind deconvolution techniques from both experimental and numerical data with reconstruction error kept in a tight tolerance. The numerical results further demonstrate that for an approximate shift-invariant system, a near diffraction limit resolution (1.5λF #) can be achieved in the presence of tip-tilt of 40λ/D and piston phase up to 1.0λ for monochromatic sources.
Several interesting aspects remain yet to be analyzed.
We would like to quantifying the phase errors by the use of spatial light modulator in the experiments. Also, further investigation of a few system parameters, i.e. fill factor of sub-apertures, varying F #, the number of light sources, the magnitude ratio among sources, as well as noise will be conducted for both experiments and numerical simulation to better evaluate the system performance.

VIII. ACKNOWLEDGMENT
This work was funded by the US National Science Foundation (ECCS-1309517) and by the NASA Innovative Advanced Concepts (NIAC) program. We are grateful to Alexandra B. Artusio-Glimpse (RIT) for valuable suggestions about the experiment. We are also very grateful to Dr. Marco B. Quadrelli (NIAC Principal Investigator) and Dr. Scott A. Basinger (Jet Propulsion Laboratory) for discussions about future space telescopes.