Active sparse aperture imaging using independent transmitter modulation with improved incoherent Fourier ptychographic algorithm

Zongliang Xie; Haotong Ma; Bo Qi; Ge Ren; Xiaojun He; Li Dong; Yufeng Tan

doi:10.1364/OE.25.020541

1. Introduction

Modern ground-based and space-based observatories are facing enormous challenges of further improving the resolution due to the limitation of the size and weight of monolithic primary mirrors. Allowing the same resolution as a large monolithic aperture to be achieved by synthesizing multiple small sub-apertures, optical sparse aperture imaging has grown out of the quest for higher angular resolution in astronomy, which is promising for the next generation of telescopes with greatly reduced size, weight and cost [1]. For this attractive technique, it is critical that the sub-apertures must be phased within a fraction of the wavelength to achieve optimal performance. With some form of actuation usually designed in sparse aperture systems to allow for adjustments of optical path distances, feedback compensation technique is a common strategy to address this problem, in which the phasing errors involving piston and tip-tilt usually need to be measured as commands to control the actuators. Many researchers have contributed to sensing the phasing errors. Recently, Deprez et al have presented a method of sensing the segmented wavefront using a holes mask and diffraction component [2]. Mourard et al have proposed a piston sensor composed of 3 spectral channel based on dispersed fringes principle [3]. As an image-based wavefront sensing and image restoration technique, phase diversity is also widely used in sparse aperture systems with different modulations on the receiving configurations, including traditional focus diversity [4], sub-aperture piston diversity [5], and spatial diversity [6]. Alternatively, even without any wavefront sensor, phasing errors can be corrected by controlling the actuators with an optimization algorithm [7]. Regardless of the implementation form, such feedback compensation technique is only performed in the receiving path of the multi-aperture system.

Recently, a method of transmitter modulation, termed multi-transmitter aperture synthesis (MAS), has been proposed for special synthetic aperture systems with holographic settings by Rabb [8]. Specifically, it is a coherent multi-aperture imaging scheme using a moving transmitter based on digital holographic detection, which could improve both the resolution and contrast [8], digitally phase sub-apertures [9], and correct aberrations [10,11]. In spite of these inspiring demonstrations, it seems less possible for MAS to be applied to general sparse aperture systems. On the one hand, with the demand of special hardware in the detection path, particularly a spatial heterodyne combining reference and scattered beams to generate holographic fringes, MAS can’t be handled only by independent transmitter modulation, which makes it difficult to be compatible with receiving configurations of other general sparse aperture systems. On the other hand, relying on holography, MAS requires high coherence, while general sparse aperture systems are usually used for incoherent observations.

Motivated by their work, we propose and demonstrate an improved approach of independent transmitter modulation for general sparse aperture systems in this paper, which can digitally correct distortions and further improve resolution. The transmitter is no longer a laser but a speckle pattern projector. A series of raw images are captured by multi-aperture systems suffering phasing errors and aberrations with the transmitter scanned to project the random pattern onto the dark objects. Then the incoherent Fourier ptychographic (IFP) algorithm, which has been introduced and developed for fluorescence microscopy [12] and macroscopic photography [13], iteratively switches between the spatial and Fourier domains based on the acquired data set. The algorithm used here is improved by embedding a model of optical transfer function (OTF) estimation in Fourier domain to address the effect of phase distortions. Similar to deconvolution concept, with adaptive estimation of OTF in the iteration updating procedure, a good-quality sparse aperture image computationally removing phasing errors and aberrations is finally obtained. Furthermore, by mixing the frequency between the object and the illuminated pattern, some higher-frequency information can be shifted within the cutoff frequency of the sparse aperture system, which can be recovered by the improved IFP algorithm. As a result, besides overcoming phasing errors and aberrations, our proposed approach can also improve resolution.

Except for using the same detector, the transmitter modulation technology reported here is absolutely independent of the receiving configurations of sparse aperture systems. Without special design in detection path and influences on actuators and wavefront sensors, it is compatible with existing multi-aperture systems, and can be applied as a good supplement or a backup in case of some unexpected situations where the feedback compensation technique might lose validity. To the best of our knowledge, such active sparse aperture imaging using independent transmitter modulation to eliminate distortions and improve resolution proposed in this paper has never been reported.

This paper is structured as follows. In section 2, the optical model of general sparse aperture imaging and how the transmitter modulation method operates with such systems are introduced. Section 3 presents detailed simulation demonstrations and analysis. Section 4 gives the experimental verification. Finally, this paper is concluded in section 5.

2. Operation principle

2.1 Optical model of general sparse aperture imaging

General sparse aperture imaging provides a possibility of obtaining the same resolution as a large monolithic system by optically combining beams from smaller sub-apertures onto a common focal plane. The synthetic image captured on the focal detector can be modeled as [14]:

d (x, y) = o (x, y) \otimes h (x, y) + n (x, y)

where

d (x, y)

is the recorded sparse aperture image,

o (x, y)

is the ideal image intensity,

h (x, y)

represents the point spread function (PSF) of the multi-aperture system,

n (x, y)

is the corresponding additive noise,

\otimes

denotes a convolution, and

(x, y)

are the coordinates of image plane. In the incoherent case, the PSF is magnitude squared of coherent impulse response, which can be expressed as

h (x, y) = {| F (P (u, v)) |}^{2}

where

P (u, v)

is the generalized pupil function,

(u, v)

are the pupil plane coordinates and

F (\cdot)

presents Fourier transform.

In general, there are two implementations of a sparse aperture system shown as Fig. 1, which are the segmented mirror and sub-aperture array, respectively. No matter which form it takes, ideal $P (u, v)$ without aberrations can be represented by a summation over all the sub-aperture functions

P (u, v) = \sum_{n = 1}^{N} p (u - u_{n}, v - v_{n})

where N is the total number of sub-apertures,

p (u, v)

is the binary pupil function of an elementary,

(u_{n}, v_{n})

is the center position of the nth sub-aperture. If the sub-apertures are not aligned well, the synthetic system won’t achieve the optimal performance, suffering from artificial aberrations including piston and tilt in x and y directions, also called phasing errors. Considering such errors, we can rewrite the generalized pupil function as

P (u, v) = P (u - u_{1}, v - v_{1}) + \sum_{n = 2}^{N} P (u - u_{n}, v - v_{n}) \exp [\frac{2 π i}{λ} (p_{n} + α_{n} (u - u_{n}) + β_{n} (v - v_{n}))]

where the first sub-aperture is take as a reference, λ is the wavelength,

p_{n}

is the relative piston error between the nth sub-aperture and the reference one,

α_{n}

and

β_{n}

are the relative subaperture gradients in the u and v directions, respectively. Like other common imaging systems, sparse aperture systems are also influenced by system aberrations or atmospheric turbulence. When these distortion effects are taken into account, the generalized pupil function can be modeled as

\begin{array}{l} P (u, v) = P (u - u_{1}, v - v_{1}) \exp (i φ_{1} (u, v)) + \sum_{n = 2}^{N} P (u - u_{n}, v - v_{n}) \\ \exp [\frac{2 π i}{λ} (p_{n} + α_{n} (u - u_{n}) + β_{n} (v - v_{n}))] \exp (i φ_{n} (u, v)) \end{array}

where

φ_{n} (u, v)

is the phase distribution of the nth sub-aperture aberrations.

Fig. 1 Two implementations of a sparse aperture imaging system: (a) segmented mirror and (b) sub-aperture array.

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The effects of the phasing errors and sub-aperture aberrations in Fourier domain can be described by OTF, which reflect the capability of an incoherent imaging system of transferring object spatial frequency to an image [14], given by

O T F (f_{u}, f_{_{v}}) = \frac{F (h (x, y))}{\iint h (x, y) d x d y}

where

(f_{u}, f_{_{v}})

are spatial frequencies.

2.2 Reconstruction with improved IFP algorithm

As for such a general sparse aperture system described above, our proposed transmitter modulation technology can be used to overcome the phasing errors, correct aberrations and further improve its resolution. As diagrammed in Fig. 2, the transmitter is first moved in two dimensions, projecting an unknown random pattern onto the object being observed by a sparse aperture system, and then corresponding images modulated by the scanning pattern are captured without any effect on the receiving configuration, which are synthesized by all the sub-images. Here the relationship between the object o and the transmitting pattern $T_{n}$ is modeled as a multiplication, which results in a target image $I_{t n} = o \cdot T_{n}$ in spatial domain. Ignoring the noise, Eq. (1) becomes $d_{n} = (o \cdot T_{n}) \otimes h$ , and correspondingly its Fourier transform is $F (d_{n}) = F (o \cdot T_{n}) \cdot O T F$ . Based on the acquired raw images, iteratively inverting these two multiplication relationships in both the spatial and Fourier domains with IFP algorithm can extract the hidden information beyond the cutoff frequency brought by the pattern modulation and realize super-resolution imaging, which has been demonstrated in fluorescence microscopy [12] and macroscopic photography [13]. Here OTF estimation is embedded into the algorithm. Then the whole iteration processing seems a procedure combining the super-resolution reconstruction and the establishment of degeneration model. Therefore, the final result synthesizing the two advantages is obtaining a good-quality image with distortions corrected and resolution improved.

Fig. 2 Schematic of operation principle of our proposed technique. A sparse aperture system, with a transmitter of pattern projection, creates a set of pattern-modulated raw images by scanning the transmitter to different positions. Then, the timely collected images are input to the improved IFP iteration algorithm and ultimately a high quality image is obtained with distortions computationally corrected, and resolution further improved.

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The iteration procedure starts with inputs of initial estimation of $o_{0}$ , $T_{0}$ , and $O T F_{0}$ . Then in each iteration, the captured raw images are used as support constraints to renew these initial guesses. The processing stops until the solution converges. Each iteration cycle for the improved algorithm can be described as follows.

First, in the nth iteration, a target image is generated in spatial domain as follows:

I_{t n} = o_{_{n - 1}} \cdot T_{n - 1} (x - x_{n})

where

x_{n}

(n = 1, 2, 3…) denotes the scanning position of the speckle pattern, which is a prior knowledge.

Secondly, in the Fourier domain, the captured image $d_{n}$ is used to update the target image as follows:

F (I_{t n}^{u p d a t e}) = F (I_{t n}) + O T F_{n - 1} \cdot [F (d_{n}) - O T F_{n - 1} \cdot F (I_{t n})]

where

I_{t n}^{u p d a t e}

is the renewed target image. This step can be also regarded as a degeneration model, where OTF can be adaptively estimated in the following procedure. As an initial guess,

O T F_{0}

is considered here as the OTF of an ideal sparse aperture system without phasing errors and aberrations, which can be calculated using the optical model described above.

Then in the spatial domain the object image is updated according to

o_{n}^{u p d a t e} = o_{n - 1} + \frac{T_{n - 1}}{{(\max (T_{n - 1}))}^{2}} \cdot (I_{t n}^{u p d a t e} - o_{n - 1} \cdot T_{n - 1})

which inverts the first multiplication relationship between object and illuminated pattern.

Next, the unknown pattern is renewed taking the similar form

T_{n} = T_{n - 1} + \frac{o_{n}^{u p d a t e}}{{(\max (o_{n}^{u p d a t e}))}^{2}} \cdot (I_{t n}^{u p d a t e} - I_{t n})

Finally, the important estimation of OTF is performed using the following equation:

O T F_{n} = O T F_{n - 1} + α \frac{conj (F (I_{t n}))}{{(\max (F (I_{t n})))}^{2}} \cdot (F (d_{n}) - O T F_{n - 1} \cdot F (I_{t n}))

where

conj (\cdot)

presents the conjugate operator and α is weight of the updated components, which is selected to provide a subjectively pleasing image. In this paper,

α =1

is used in the simulations and experiments. Based on the multiplication relation

F (d_{n}) = F (o \cdot T_{n}) \cdot O T F

in Fourier domain, the model of the OTF update is motivated by the iteration projection algorithm, taking the similar form of the hybrid input-output algorithm using difference map [15]. Such a form is very common in phase retrieval algorithms like FP [16], and the conventional ptychography [17]. It works as follows: at one step of the iteration, the current spectra estimation of the speckle modulated image is obtained by multiplying the present OTF and spectra of the target image. We then subtract the estimation from the real spectra of the speckle modulated images, resulting a difference function. Divided by the current spectra of the target images, the difference function is then fed back to correct our running estimate of the OTF.

3. Simulation analysis

To demonstrate the effectiveness and evaluate the performance of our proposed active sparse aperture imaging framework, we conduct a series of simulations in this section. Simulations are performed based upon a Golay-3 sparse aperture system consisting of 3 sub-apertures with diameters of 10 mm, shown as Fig. 3(a), which can equivalently denote not only the segmented mirror, but also the telescope array. The neighboring subaperture is close to each other. Note that although there are gaps between each sub-aperture in pupil function, the OTF of the system is continuous within the cut-off frequency. In the IFP algorithm, it is the continuous OTF that serves as the support constrains in Fourier domain rather than the pupil function, so the gaps here have nearly no influences on our reconstruction. The equivalent focus length is 300 mm and the pixel size of the detector is 3.45 μm. The parameters of the simulated multi-aperture system closely match our experimental setup. When all the sub-apertures are phased well, a synthetic image is attained as shown in Fig. 3(b1) and its spectrum is shown as Fig. 3(b2). Then a realization of phasing errors are yielded and loaded on the system as shown in Fig. 3(c), where the upper sub-aperture is taken as the reference, the relative pistons of the other two are 0.3 λ and 0.2 λ, respectively, and their relative peak-valley (PV) values of tilt in x direction are 0.42 λ and 0.34 λ, and those in y direction are 0.42 λ and 0.23 λ, respectively. The root-mean-square (RMS) of the generated phasing errors is 0.23 λ. Correspondingly, the Golay-3 image is blurred as shown in Fig. 3(d1), suffering from these errors, and Fig. 3(d2) presents its spectrum.

Fig. 3 The simulation structure and the generated images. (a) The Golay-3 configuration with each sub-pupil close to each other, (b1) the ideal Golay-3 image without aberrations, and (b2) its spectrum. (c) The loaded phasing errors, (d1) the corresponding blurred Golay-3 image, and (d2) its spectrum.

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Our proposed transmitter modulation technology can be utilized to overcome the phasing errors and further improve resolution. Specifically, the transmitter modulation is realized by scanning a speckle pattern projector in two dimensions to illuminate the object and at each position, the modulated object is incoherently imaged by the Golay-3 system. In the simulation, we use a fully randomized distribution as the unknown illumination pattern, shown as Fig. 4(a), and the transmitter is shifted to 64 positions, which are equally spaced in 2 dimensional square grid. Figures 4(b1)-4(b4) show 4 out of the 64 raw images modulated by the pattern. Based on the acquired images, the improved IFP algorithm works by iteratively stitching in spatial and Fourier domains and estimating OTF adaptively as described in section 2. Figure 5(a) shows the recovered high-quality image, which presents greatly enhanced sharpness and contrast with respect to the blurred one shown in Fig. 3(d1). From the comparison of the reconstruction and the phased image presented as Fig. 3(b1), it can be found that the resolution has been improved with smaller textures resolvable. To give a more explicit resolution improvement in Fourier domain, Fig. 5(b) gives the spectrum of the recovery, in which more spatial frequencies beyond the cut-off frequency shown as Figs. 3(b2) and (d2) are attained. The modulus of the OTF (MTF) is used as a metric to test the accuracy of the OTF estimation. Figures 5(c) and 5(d) show the loaded and estimated MTFs, respectively. That they present a satisfied agreement demonstrates the validity of the OTF estimation model. To evaluate the reconstruction quality quantitatively, the correlation coefficient (Co) is used here as a metric, which is also used in [18] and defined below:

Co (A, B) = cov (A, B) {(σ_{A} \cdot σ_{B})}^{- 1}

where

cov (A, B)

is the cross-covariance between the input image A and the recovery B, and

σ

denotes the standard deviation. Ranging in values of zero and one, the bigger the Co value, the better the reconstruction. The corresponding Co value of the recovered image is 98.76%, while those of Golay-3 image with and without phasing errors are 78.92% and 92.65%, respectively. In this set of simulations, we basically demonstrate the validity of our transmitter modulation method for phasing error correction and resolution improvement in sparse aperture systems.

Fig. 4 Simulated raw images modulated by the scanning pattern. (a) The used unknown illumination pattern, which is translated to 64 different locations. (b1)-(b4) Four frames of the captured raw images.

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Fig. 5 Reconstruction results. (a) The reconstructed image and (b) its spectrum. (c) The loaded and (d) estimated MTFs

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We also explore the influence of the speckle size on the reconstructions. Previous work has demonstrated that the speckle size determines the resolution [12], while here its influence on the convergence speed is studied. Different illumination patterns with different speckle sizes are generated by imaging the one shown as Fig. 4(a) with different systems of f-numbers of 60, 50, 40, 30, 20, and 10, which are shown as Figs. 6(a1)-6(a6), respectively. Based on the same setup above, 6 simulations are performed. Their Co convergence curves are shown as Fig. 6(b), from which it can be indicated that the smaller speckle size might slow the convergence speed but can generate higher-resolution images. Therefore, the choice of optimal speckle size should keep a balance between the demands of the resolution and imaging speed. The speckle patterns presented here are only used in this set of simulations, while the following simulations still use the fully randomized one shown as Fig. 4(a).

Fig. 6 Results of simulations using different speckle sizes. (a1)-(a6) The illumination patterns with different speckle sizes corresponding to f-numbers of 60, 50, 40, 30, 20 and 10, respectively. (b) The Co value convergence curves with different speckle sizes.

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Then we analyze the effect of additive noises using the same simulation setting. Since noise meets the Gaussian model in high-level photon or CCD readout circumstance, Gaussian noise is considered in this set of simulations. Different amounts of Gaussian noises (variance $σ^{2}$ = 0.1%, 0.5% and 1% of the variance of signal values) are added. The phased and non-phased Golay-3 images under influence of these different amount noises are presented in Figs. 7(a1)-7(c1) and Figs. 7(a2)-7(c2), respectively. For the different noise levels, our reported scheme also outputs high-quality images, as shown in Figs. 7(a3)-(c3), with corresponding Co values of 98.73%, 98.49%, and 98.17%, respectively. From this set of simulations, we can see that the proposed method is robust against noises.

Fig. 7 Results of simulations against different levels of noises. (a1)-(c1) The phased Goaly-3 images with 0.1%, 0.5%, and 1% additive Gaussian noises, respectively. (a2)-(c2) The non-phased images blurred by the loaded phasing errors with the corresponding noises. (a3)-(c3) The corresponding reconstructed images using our proposed approach.

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We also test the performance of our proposed technology with different realizations of phasing errors. To improve the realism of the simulations, 1% Gaussian noise describing the CCD readout noise is considered. The different groups of piston and tip-tilt errors are specifically listed in Table 1, generating different wavefronts with RMSs of 0.26 λ, 0.29 λ, 0.34 λ, and 0.39 λ, respectively. The corresponding blurred Golay-3 images are shown as Figs. 8(a1)-8(d1) with Co values of 76.01%, 74.21%, 70.92% and 68.84%, respectively. Figures 8(a2)-8(d2) show the reconstructions. It is easy to recognize that the sharpness and contrast of the recoveries are obviously enhanced and more bar elements become resolvable. Also, their corresponding Co values are improved to 98.54%, 98.30%, 97.35% and 97.28%, respectively. This set of simulations further demonstrate the validity of our reported method to digital correction of phasing errors.

Table 1. Different realizations of phasing errors

View Table

Fig. 8 Results of simulations with different realizations of phasing errors. (a1)-(d1) the non-phased images blurred by loaded different phasing errors with RMSs of 0.26 λ, 0.29 λ, 0.34 λ, and 0.39 λ, respectively. (a2)-(d2) The corresponding reconstructed images.

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Finally, another set of simulations taking into account the sub-aperture aberrations and phasing errors simultaneously are performed. A global aberration distribution of the imaging beam with RMS of 0.19 λ is generated by summing up Zernike polynomials with individual weights from the fourth to eleventh order (the first three Zernike polynomials representing piston, x and y tilt are excluded), which are defined and shown in Table 1 of [19]. With the imaging beam propagating thorough the Golay-3 system, the corresponding part of the global aberrations segmented by each pupil are used to denote the sub-aperture aberrations, as shown in Fig. 9(a). A set of phasing errors shown as Fig. 9(b) with RMS of 0.23 λ is also loaded on the Glay-3 system. Then the overall phase distortions are obtained by combining the two aberrations above, which are presented in Fig. 9(c), and the corresponding RMS is 0.27 λ. We also add 1% Gaussian noise to improve the realism. To decrease simulation fortuity, here a star resolution target instead of USAF 1951 test chart is used. Figures 10(a1) and 10(b1) show the aberrated and aberration-free images with Co values of 75.83% and 93.46%, respectively. The reconstructed result is shown in Fig. 10(c1) and the corresponding Co value is increase to 96.12%. It is also easy to recognize that the sharpness and contrast of the recovery are obviously improved. Figures 10(a2)-10(c2) show their spectrums, from which it can been directly seen that the resolution is improved. The loaded and estimated MTFs, shown respectively as Figs. 10(d1) and (d2), present almost the same shape and profile, indicating that we get an accurate OTF. By this set of simulations, our reported technology’s capabilities of computationally overcoming phasing errors, correcting aberrations and improving resolution are further verified.

Fig. 9 Loaded distortions combing sub-aperture aberrations and phasing errors. (a) The global wavefront generated by the first 11 Zernike polynomials (piston, x and y tilt excluded). (b) The loaded phasing errors. (c) The final overall distortions combining both (a) and (b).

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Fig. 10 Results of simulations considering both the sub-aperture aberrations and phasing errors with a star resolution target. (a1) The blurred image aberrated by the loaded distortions, (b1) the ideal Golay-3 image without phase distortions, and (c1) the reconstructed image using our proposed technology. (a2)-(c2) The spectrums of (a1)-(c1), respectively. (d1) The loaded and (d2) estimated MTFs.

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Besides the piston and tip-tilt errors which could be characterized by OTF, there is another type of wavefront error for sparse aperture systems, which is known as pupil mapping errors caused by that the exit pupil isn’t an exact scaled replica of the entrance pupil [20]. Leading to the piston varying with the field, such errors limit the valid field of view (FOV) of the sparse aperture system. Luckily, in our proposed scheme, this problem seems to be able to be circumvented by field sampling. Similar to the approach commonly used in blind-deconvolution, this strategy separates a large FOV into several small ones, in which the piston differences of every field should be nearly within λ/10. Only in this case, could the segmented field be regarded as an isoplanatic region with the same piston errors. Then we can perform our improved IFP reconstructions for every separated field. As our proposed algorithm could estimate the OTF adaptively, that the segmented fields have different pistons won’t influence our reconstructions. With all independent segmented field reconstructions finished, we then can get a large FOV image by stitching separated recovered images. In this way, the inevitable pupil mapping errors in sparse aperture systems might be digitally bypassed at an expense of large computation with our proposed technology.

4. Experimental results

In this section, we experimentally validate our reported active multi-aperture imaging technology using transmitter modulation by correcting distortions and improving resolution. An experimental platform closely matching the simulation settings is built, as shown schematically in Fig. 11. A 630 nm LED source and a semitransparent diffuser made by spraying white paint on a slice of ground glass are mounted on a motorized XY translation stage, which is used for realization of scanning modulation. A projection lens is utilized to image the diffuser illuminated by the LED onto the target. The components described above compose the transmitter. Then the scattered beams from the lighted object (a flat printed pattern) are received by a Golay-3 system using a pupil-mask to simulate a 3 sub-aperture telescope and a phase screen that is a glass plate of 28 mm diameter with varying thickness to characterize both the phasing errors and sub-aperture aberrations, and the speckle-modulated target is finally incoherently imaged onto the CCD camera (Point Grey GS3-U3-50S5M). The sub-pupils are arranged close to each other with the diameter of the sub-pupil of 10 mm, and the focal length of the imaging lens is 300 mm. The object is about 3.9 m away from the illumination and Golay-3 systems. The pixel size of 3.45 um guarantees the raw images to be oversampled. From this setup, it is noted that the transmitter modulation is absolutely independent of the receiving path except sharing the same detector, so the proposed approach is compatible with existing multi-aperture systems without additional impact and changes in receiving configurations.

Fig. 11 The configuration of the concept-demonstration experiment. A transmitter consists of a 2D translation stage, a 630 nm LED, a diffuser and a projection lens. A Golay-3 telescope is experimentally simulated by using a pupil-mask against an imaging lens, whose phasing errors and sub-aperture aberrations are characterized with a phase screen.

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In the experiment, the XY translation stage is scanned to 64 positions, and 64 frames of raw images are captured corresponding to different locations, 4 of which are shown in Figs. 12(a1)-12(a4). Based on the acquired data set, the reconstruction procedure employing the improved IFP algorithm starts. Figure 12(b) shows the Golay-3 image distorted by the phase screen under uniform illumination (without the diffuser in Fig. 11) for comparison, which obviously exhibits blurring caused by the introduced distortions. The object image seems a flower pattern with 4 petals but it is difficult for us to distinguish the texture of each petal. Luckily, with the help of our proposed transmitter modulation technology, we obtain a good-quality Golay-3 image with sharpness and contrast greatly improved, as shown in Fig. 12(c). The flower pattern presents a clearer profile and more details, especially the feature of horizontal and vertical bars in petals. From the comparison between the reconstruction and the blurred image, we demonstrate that our reported method is effective against phase distortions. Also, to highlight the improvement of our employed IFP algorithm with OTF estimation model embedded, a contrastive image obtained by using the previous IFP with the same 64 frames of raw images is presented in Fig. 12(d). From the comparison, it is easy to recognize that the improved algorithm generates superior results in terms of both sharpness and resolution.

Fig. 12 Experimental results of distortion correction. (a1)-(a4) Four frames of 64 raw images. (b) The low-quality Golay-3 image blurred by the phase screen. (c) The reconstructed image using our reported technique with 64 raw images. (d) The reconstructed image using previous IFP with 64 raw images.

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To verify the resolution improvement, the sharp Golay-3 image formed by the experimental setup without the diffuser and phase screen distortion is given in Fig. 13 (a1) as a reference. Figure 13(b2) is the recovery using 64 raw images, which shows higher resolution with more details presented compared with Fig. 13(a1). Pixel trace plots of the same two-line textures of Figs. 13(a1) and 13(b1) marked in red and green dotted lines are shown respectively as Figs. 13(a2) and 13(b2) to make the resolution enhancement clear. The spatial frequency of the selected two-line texture is about 72 lp/mm. The trace line in Fig. 13(a2) exhibits flat variation, indicating the two-line feature is unresolvable in Fig. 13(a1), while that in Fig. 13(b2) shows a fluctuation, indicating the corresponding texture is resolved. A higher-resolution image yielded by a full-filled aperture of 25 mm diameter is also presented in Fig. 13(c1) and the trace line of the same feature is given as Fig. 13(c2), which shows a similar profile to that of our recovery, demonstrating that the resolvable two-line texture is not a product artificially brought by the algorithm but the result of resolution improvement.

Fig. 13 Experimental results of resolution improvement. (a1) The Golay-3 image under uniform illumination without blurring (removing the phase screen). (b1) The recovered image. (c1) The higher-resolution image formed by a full-filled aperture of 25 mm diameter. (a2)-(c2) The trace lines of the same two-line features of (a1)-(c1) marked with dotted lines, respectively.

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Furthermore, we also evaluate the performance of the reconstructions with different numbers of transmitter scanning locations. Recoveries using different numbers of raw images created by moving the transmitter to 16, 25, 36, and 49 positions are shown in Figs. 14(a)-14(d), respectively. From this set of images, it can be recognized that the reconstruction presents better quality with more raw images used to constrain the iterations, and seems to converge with 36 raw images. Additionally, this series of recoveries seem to show a fluctuating background, which might be an artifact of algorithm due to the experimental degenerations such as pupil and speckle location errors. In our future work, we will improve the robustness of the algorithm against practical errors by introducing some advanced models, such as Poisson maximum-likelihood objective function and Truncated Wirtinger gradient, which have been successfully used in conventional FP [21].

Fig. 14 Recoveries using our proposed technology with (a) 16, (b) 25, (c) 36 and (d) 49 frames of raw images.

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5. Conclusion

The active sparse aperture imaging using transmitter modulation to computationally overcome phasing errors, correct aberrations and improve resolution is proposed and demonstrated in this paper. To start with, a series of simulations are performed to validate the effectiveness. In the simulations, we first basically demonstrate the capability of the technology of digitally overcoming phasing errors, then we evaluate its performance with several levels of noises and different realizations of phasing errors, and finally more complicated phase distortions combining both sub-aperture aberrations and phasing errors are considered. Then we conduct an experiment, where a pupil-mask is used to simulate a 3 aperture telescope and a phase screen is applied to characterize the sub-aberrations and phasing errors. The experimental result that the reconstructed Goaly-3 imagery shows enhanced sharpness and higher resolution further proves the validity. We also study the effect of different numbers of raw images and find that the reconstruction converges to 36 raw images.

Unlike MAS only used for coherent multi-aperture systems based on holographic detection, the technique reported here using modified IFP algorithm to recover images can be applied for general sparse aperture systems. The proposed transmitter modulation is independent of the configurations of beam combining, so it is compatible with existing sparse-aperture systems and can be used as a good supplement or a backup. By providing a new insight to phase sub-apertures and restore images in transmitting path of sparse aperture systems, it may find wide applications in the next generation of segmented or telescope array imaging systems.

Funding

National Natural Science Foundation of China (61205144); The Youth Innovation Promotion Association, CAS.

References and links

1. J. S. Fender, “Synthetic apertures: an overview,” Proc. SPIE 440, 2–7 (1984). [CrossRef]

2. M. Deprez, C. Bellanger, L. Lombard, B. Wattellier, and J. Primot, “Piston and tilt interferometry for segmented wavefront sensing,” Opt. Lett. 41(6), 1078–1081 (2016). [CrossRef] [PubMed]

3. D. Mourard, W. Dali Ali, A. Meilland, N. Tarmoul, F. Patru, J. M. Clausse, P. Girard, F. Henault, A. Marcotto, and N. Mauclert, “Group and phase delay sensing for cophasing large optical arrays,” Mon. Not. R. Astron. Soc. 445(2), 2082–2092 (2014). [CrossRef]

4. D. Yue, S. Xu, and H. Nie, “Co-phasing of the segmented mirror and image retrieval based on phase diversity using a modified algorithm,” Appl. Opt. 54(26), 7917–7924 (2015). [CrossRef] [PubMed]

5. M. R. Bolcar and J. R. Fienup, “Sub-aperture piston phase diversity for segmented and multi-aperture systems,” Appl. Opt. 48(1), A5–A12 (2009). [CrossRef] [PubMed]

6. H. Ma, Z. Xie, X. Long, B. Qi, G. Ren, J. Shi, Z. Cui, Y. Jiang, and X. Xu, “Synthetic aperture imaging by using spatial modulation diversity technology with stochastic parallel gradient descent algorithm,” Opt. Express 23(11), 14836–14849 (2015). [CrossRef] [PubMed]

7. I. Paykin, L. Yacobi, J. Adler, and E. N. Ribak, “Phasing a segmented telescope,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(2), 023302 (2015). [CrossRef] [PubMed]

8. D. J. Rabb, D. F. Jameson, J. W. Stafford, and A. J. Stokes, “Multi-transmitter aperture synthesis,” Opt. Express 18(24), 24937–24945 (2010). [CrossRef] [PubMed]

9. D. Rabb, D. Jameson, A. Stokes, and J. Stafford, “Distributed aperture synthesis,” Opt. Express 18(10), 10334–10342 (2010). [CrossRef] [PubMed]

10. D. J. Rabb, J. W. Stafford, and D. F. Jameson, “Non-iterative aberration correction of a multiple transmitter system,” Opt. Express 19(25), 25048–25056 (2011). [CrossRef] [PubMed]

11. B. K. Gunturk, D. J. Rabb, and D. F. Jameson, “Multi-transmitter aperture synthesis with Zernike based aberration correction,” Opt. Express 20(24), 26448–26457 (2012). [CrossRef] [PubMed]

12. S. Dong, P. Nanda, R. Shiradkar, K. Guo, and G. Zheng, “High-resolution fluorescence imaging via pattern-illuminated Fourier ptychography,” Opt. Express 22(17), 20856–20870 (2014). [CrossRef] [PubMed]

13. S. Dong, P. Nanda, K. Guo, J. Liao, and G. Zheng, “Incoherent Fourier ptychographic photography using structured light,” Photon. Res. 3(1), 19–23 (2015). [CrossRef]

14. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2004).

15. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009). [CrossRef] [PubMed]

16. X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22(5), 4960–4972 (2014). [CrossRef] [PubMed]

17. F. Hüe, J. M. Rodenburg, A. M. Maiden, and P. A. Midgley, “Extended ptychography in the transmission electron microscope: possibilities and limitations,” Ultramicroscopy 111(8), 1117–1123 (2011). [CrossRef] [PubMed]

18. Y. Shi, T. Li, Y. Wang, Q. Gao, S. Zhang, and H. Li, “Optical image encryption via ptychography,” Opt. Lett. 38(9), 1425–1427 (2013). [CrossRef] [PubMed]

19. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]

20. S. J. Chung, D. W. Miller, and O. L. Weck, “ARGOS testbed: study of multidisciplinary challenges of future spaceborne interferometric arrays,” Opt. Eng. 43(9), 2156 (2004). [CrossRef]

21. L. Bian, J. Suo, J. Chung, X. Ou, C. Yang, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient,” Sci. Rep. 6(1), 27384 (2016). [CrossRef] [PubMed]

Left sub-aperture			Right sub-aperture			RMS	Co values
Piston	X-tilt (PV)	Y-tilt (PV)	Piston	X-tilt (PV)	Y-tilt (PV)	RMS	Blurred image	Recovery
0.25 λ	0.45 λ	0.49 λ	0.3 λ	0.47 λ	0.51 λ	0.26 λ	76.01%	98.54%
0.35 λ	0.44 λ	0.45 λ	0.3 λ	0.48 λ	0.43 λ	0.29 λ	74.21%	98.30%
0.35 λ	0.53 λ	0.51 λ	0.4 λ	0.49 λ	0.53 λ	0.34 λ	70.92%	97.35%
0.4 λ	0.55 λ	0.49 λ	0.5 λ	0.51 λ	0.53 λ	0.39 λ	68.84%	97.28%

Active sparse aperture imaging using independent transmitter modulation with improved incoherent Fourier ptychographic algorithm

Abstract

1. Introduction

2. Operation principle

2.1 Optical model of general sparse aperture imaging

2.2 Reconstruction with improved IFP algorithm

3. Simulation analysis

4. Experimental results

5. Conclusion

Funding

References and links

Cited By

Figures (14)

Tables (1)

Equations (12)

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