Distributed aperture synthesis

Distributed aperture synthesis

David Rabb*, Douglas Jameson, Andrew Stokes, and Jason Stafford

Air Force Research Laboratory – Sensors Directorate, AFRL/RYJM B622, 3109 Hobson Way, WPAFB, OH, 45433-

7700, USA

*[email protected]

Abstract: Distributed aperture synthesis is an exciting technique for

recovering high-resolution images from an array of small telescopes. Such a

system requires optical field values measured at individual apertures to be

phased together so that a single, high-resolution image can be synthesized.

This paper describes the application of sharpness metrics to the process of

phasing multiple coherent imaging systems into a single high-resolution

system. Furthermore, this paper will discuss hardware and present the

results of simulations and experiments which will illustrate how aperture

synthesis is performed.

OCIS codes: (090.1995) Digital Holography; (100.3010) Image Reconstruction Techniques.

References and links

1. J. C. Marron, and R. L. Kendrick, “Distributed Aperture Active Imaging,” Proc. SPIE 6550, 65500A (2007).

2. R. A. Muller, and A. Buffington, “Real-time correction of atmospherically degraded telescope images through

image sharpening,” J. Opt. Soc. Am. 64(9), 1200–1210 (1974).

3. J. R. Fienup, and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc.

Am. A 20(4), 609–619 (2003).

4. R. L. Kendrick, and J. C. Marron, “Analytic Versus Adaptive Image Formation Using Optical Phased Arrays,”

Proc. SPIE 7468, 75680N (2009).

5. N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 5933–

5943 (2007).

6. Q. Wu, L. Qian and W. Shen, “Image Recovering for Sparse-aperture Systems,” Proc. SPIE 5642, 478–486

(2005).

7. R. G. Paxman, and J. C. Marron, “Aberration Correction of Speckled Imagery with an Image Sharpness

Criterion,” Statistical Optics,” Proc. SPIE 976, 37–47 (1988).

8. S. T. Thurman, and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A 25(4), 983–

994 (2008).

9. J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Höft, “Atmospheric turbulence correction

using digital holographic detection: experimental results,” Opt. Express 17(14), 11638–11651 (2009).

1. Introduction

Coherent aperture synthesis is driven by a need for both high resolution imagery and small

sensor form-factors. These systems seek to replace large, monolithic optical apertures with

arrays of smaller sub-apertures as shown in Fig. 1.

Fig. 1. Aperture synthesis initially allows large monolithic apertures, (a), to be replaced with

dense-packed distributed arrays, (b). Over time, the array patterns will become sparser and

system depth can be minimized by utilizing pupil-plane imaging techniques, (c).

#124416 - $15.00 USD Received 18 Feb 2010; revised 14 Apr 2010; accepted 19 Apr 2010; published 4 May 2010(C) 2010 OSA 10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 10334

Initially, a densely-packed array of sub- apertures can provide significant savings in system

volume by reducing the required focal length of the individual telescopes. Future systems

may further expand this technique and combine sparse arrays with conformal aperture

technology.

The architecture of coherent aperture synthesis, like most imaging systems, can be broken

down into hardware and software elements. However, unlike traditional imaging systems,

aperture synthesis relies heavily upon software algorithms to form a single high-resolution

image from the output of each of the sub-apertures. An illustration of an aperture synthesis

system is shown in Fig. 2. In this system, a laser is used as master oscillator and is the source

for the transmitter and for the local oscillator, LO, for each receiver. The transmitter

subsystem (TX) is used to flood illuminate the target, at right, with coherent laser light. The

receiver subsystems (RX1-3) record the holographic fringes created from the mixing of the

LO with the backscattered return from the target. The intensity values of the fringes are

passed on to the CPU, where the amplitude and phase of the backscattered field are recovered

for multiple coherent apertures using a Fast Fourier Transform (FFT).

Fig. 2. Illustration of coherent aperture synthesis architecture. The hardware consists of a

transmitter Tx, coherent receivers Rx1-Rx3 which capture holographic fringes across a camera

array, and a computer which forms images in software.

The hardware from Fig. 2 may utilize digital holography in the same manner as the

systems described by Marron and Kendrick [1]. Each of the field measurements is used to

populate a digital pupil plane, in which a series of algorithms is used to sharpen the image

produced by each sub-aperture, register their imagery and then simultaneously sharpen across

multiple sub-apertures. The former steps are sometimes referred to as “phasing” the apertures

and often utilize image sharpness metrics [2–4]. If speckle is ignored the imagery which

results from an ideal set of algorithms will have a resolution determined by the array

geometry [5,6].

Sharpening synthesized imagery is a difficult task due to the presence of speckle and poor

image SNR [7]. In particular, the inter-aperture piston sharpening algorithms are highly

sensitive to speckle noise, due to the piston largely affecting higher spatial frequencies where

speckle content may dominate image content. In this paper we will describe how image

sharpness metrics that incorporate speckle averaging are used to synthesize the field captured

at multiple apertures into a single high-resolution image.

2. Theory

2.1 Hardware

The hardware presented here utilizes digital holography to measure the backscattered target

field, as shown in Fig. 3. An afocal telescope is used to increase the relative size of the

aperture in order to capture more of the target’s angular spectrum as well as signal photons. A

tilted LO is introduced via a beam splitter and serves to separate reconstructed images from


the on-axis correlation terms. The LO and backscattered field then mix and are measured at

the camera.

Fig. 3. Digital holography using spatial heterodyne technique. The pupil field, Ut(x,y), is

imaged onto the CMOS array using the afocal telescope formed by lenses L1 and L2. A non-

polarizing beam splitter is used to insert a tilted LO reference, ULO(x,y), which is interfered

with pupil field. The resulting fringes are then captured across the CMOS array.

The intensity I(x,y) incident upon the camera is given by

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 * *

, , , , , , , ,LO t LO t LO tI x y U x y U x y U x y U x y U x y U x y= + + + (1)

where ULO(x,y) is the local oscillator field at the camera, Ut is the backscattered target field at

the camera and * denotes the complex conjugate. Assuming a distant target, the field at the

target plane can be recovered by simply taking the Fourier transform of Eq. (1) such that

( ){ } ( ){ } ( ){ }

( ) ( ){ } ( ) ( ){ }

2 2

* *

, , ,

, , , , .

LO t

LO t LO t

I x y U x y U x y

U x y U x y U x y U x y

= +

+ +

F F F

F F

(2)

The first two terms of Eq. (2) can be recognized as the autocorrelation of the incident LO

field and backscattered field, respectively. It can be assumed that the LO is a plane wave with

uniform amplitude given by ALO and that the tilt angle between the wavefront and the optic

axis are given by θx and θy. For small tilt angles, substitution of these values yields

( ){ } ( ) ( ){ }( )( ) ( ){ }( )( ) ( ){ }

22, , ,

*exp ,

exp , ,

LO x y t

LO x y t

LO x y t

I x y A f f U x y

A jk x y U x y

A jk x y U x y

δ

θ θ

θ θ

= +

+ +

+ − +

F F

F

F

(3)

where k is the wavevector number and δ is the Dirac-delta function. Further simplification

yields

( ){ } ( ) ( ){ }( ){ } ( )( ){ } ( )

22

*

0 0

0 0

, , ,

, ,

, , ,

LO x y t

LO t x x y y

LO t x x y y

I x y A f f U x y

A U x y f f f f

A U x y f f f f

δ

δ

δ

= +

+ ∗ − −

+ ∗ + +

F F

F

F

(4)

where fx0 and fy0 are the spatial heterodyne offset frequencies determined by the tilt angles θx

and θy and given by fx0,y0 = θx,y/λ. This process is valuable in that it provides direct access to

the backscattered field at the target plane, ( ){ },TU x yF . Note that there is gain from the local


oscillator present in the recovered image which is proportional to ALO, such that the

reconstructed image is stronger than the non-localized backscattered field autocorrelation

term.

2.2 Processing

The theory which drives the image synthesis algorithms relies on the fifth image sharpness

metric S described by Muller and Buffington as

( ), ,S I x y dxdyγ= ∫ (5)

where I(x,y) is the image intensity and the power term γ is defined so that γ > 2 [2]. The work

is later expanded into coherent imaging while attributing small γ values to enhancing the

darker pixels and larger values of γ to enhancing the brighter pixels [3,8]. Marron et al. apply

sharpness metrics to correct atmospheric aberrations through a single telescope utilizing pupil

plane architecture [9]. The current research applies these same techniques to coherent image

synthesis where multiple, corrected pupil fields have been recovered, but the relative phase

between the captured fields is unknown.

Unfortunately, speckle is present throughout the imaging process due to the surface

roughness of realistic targets. Speckle interferes with the ability of the sharpness metric to

accurately describe the synthesized images, especially when iterating on piston errors

between the apertures. The algorithm used to apply the sharpness metric to the captured

imagery must be improved to allow for speckle averaging so that the sharpness metric can be

applied to the speckle-averaged image. An algorithm containing two major steps is proposed.

The first step will sharpen the imagery taken at the individual apertures while the second step

will sharpen the synthesized imagery.

The first step of the improved algorithm operates on the imagery captured at the

individual sub-aperture. This step is largely an implementation of the work described by

Thurman and Fienup [8]. It is assumed here that field values are captured simultaneously at

the individual apertures and that speckle realizations are independent between sequential data

collects. The output from each of the sub-apertures is averaged across the total number of

speckle realizations to create an initial version of the speckle-averaged, single-aperture image.

The image sharpness metric of the incoherent speckle average SA can be written as

( )1

1, ,

N

A n

n

S dxdy I x yN

γ

=

= ∑∫ (6)

where N is the number of speckle realizations, n denotes an individual speckle realization and

In(x,y) is the target image intensity for realization n. At this point the algorithm focuses on

correcting the averaged sub-aperture pupil plane data by correcting the phase of the individual

pupil realizations using Zernike polynomials. The algorithm corrects an individual telescope

realization so that the final, averaged image reaches maximum sharpness and then iterates to

the next realization. This process is illustrated in Fig. 4. The algorithm will result in N

corrected pupil field values at each sub-aperture.


Fig. 4. The first stage of the image sharpening algorithm relies on sharpening the data taken

across a single sub-aperture. This algorithm corrects the phase across a single pupil realization

such that the speckle-averaged image is sharpened.

The second step of the algorithm synthesizes a larger effective aperture by combining the

previously corrected pupils into a single digital pupil plane according to the realization

number n. Phase corrections are then made to an individual pupil within the now synthesized

array using Zernike polynomials, which are centered on the pupil being corrected. This is

done for each pupil in the array, and for each realization of the three pupils. The sharpness

metric SA is maximized for the synthesized, averaged image as shown in Fig. 5.

Fig. 5. The second step of the image sharpening algorithm combines the pupils into a single

digital pupil plane based on speckle realization. The individual pupil functions are then

corrected so that the synthesized and speckle-averaged imagery is appropriately sharpened.

Simulations are performed to predict a value for the power γ which will yield the best

image synthesis results. As previously stated, the value of γ is highly dependent on the target

type. In this paper the target of interest is an ISO 12233 target, Fig. 6, which features high

contrast features. The simulation models the theory described above and includes a randomly

generated, but known, piston phase between each aperture. Three horizontally arrayed

apertures are modeled with 32 independent speckle realizations captured across the array. The

synthesis algorithm is then applied for multiple values of γ which results in a final, sharpened

image for each value of γ. Additionally, the piston error, which is estimated by the algorithm,


is captured and compared to the original known piston error for each value of γ. This process

is repeated for 64 random piston realizations and the RMS error between the known piston

error and the synthesis algorithm’s predicted error is shown as a function of γ in Fig. 7.

Fig. 6. A portion of the ISO 12233 Target.

It can be seen in Fig. 7 that the RMS error in estimated piston reaches a minimum when

0.4 ≤ γ ≤ 0.5. This would tend to hold for generally bright images with high-contrast as is the

case with the ISO 12233 target shown in Fig. 6. Recall that Thurman and Fienup showed that

sharpness metrics less than 1 tend to make dark points darker and the ISO target is composed

of dark lines on a white background. For a different type of target the optimal value of γ used

for phasing may be different. Note that the dots in Fig. 7 are from the numerical simulation

and the solid line represents a fourth degree polynomial fit.

Fig. 7. The average RMS piston error for values of 0.025 ≤ γ ≤ 2 from simulated results are

shown as dots and the solid line represents a fourth degree polynomial fit.

3. Experiment

An experiment was designed to validate the predicted performance for the sharpness

algorithm described above. The experiment will show that the synthesis algorithm can

achieve improved resolutions based on the synthesized array geometry. The aperture

synthesis experiment consists of three afocal telescope systems, each with the same

configuration as seen in Fig. 3, which are arrayed horizontally. The horizontal aperture

pattern is chosen to maximize the synthesized resolution in the horizontal dimension while

maintaining sub-aperture limited resolution in the vertical dimension.


3.1 Hardware

The laser source used in this experiment is a HeNe laser operating at the traditional 632.8 nm

line. The HeNe is fiber-coupled to a large area single mode fiber which is connected to a 95/5

fiber splitter which splits the power into transmit and LO paths, respectively. The transmitter

path illuminates the target using a bistatic architecture while the LO path is split into three

channels which are used to provide the tilted LO at each of the sub-apertures.

The aperture synthesis experiment consists of three horizontally arrayed receivers of the

type shown in Fig. 3. The cameras are Lumenera LU120M models, which are based on a

CMOS sensor and have 6.7 µm square pixels in a 1280x1024 format. This small pixel pitch,

while useful in a standard imaging situation, yields a relatively large effective FOV when

imaging the pupil plane. In other words, the pixel pitch allows the system to resolve high

frequency fringes that result from the interference of the LO and fields which are captured

from the edge of target space. An adjustable aperture stop located at the common focus of the

afocal telescope can be used to limit the FOV to an area of interest. The afocal telescope is

designed with a two-inch front lens and system magnification of 8X to ensure that the

entrance pupil of the telescope is imaged onto the CMOS array without vignetting. The sub-

aperture entrance pupil clear aperture is approximately 4.83 cm.

The target in this experiment is composed of a transmissive ISO 12233 chart with chrome

on glass. A diffuser was placed immediately behind the target such that when illuminated it

created a fully realized speckle pattern. The target was placed 10 meters from the imaging

optics and the diffuser was slightly rotated between frames to create multiple speckle

realizations.

3.2 Processing and results

The field is captured at each aperture through the use of the digital holography hardware

described in section 2.1. The field values captured at the individual apertures are corrected

utilizing the methods as described in the first step of our synthesis algorithm. This yields a

sets of three pupil fields for a particular instance of speckle. Multiple field values, with

independent instances of speckle, are collected to assist the algorithm by reducing speckle

noise. Results comparing an aberration-corrected, single-aperture image with 360 speckle

realizations to the same data using three coherently combined sub-apertures are shown in Fig.

8. The first step yields corrected single-aperture imagery which is averaged across 360

speckle realizations and shown in Fig. 8(b). The second step of the algorithm yields corrected

imagery from the coherently synthesized array of sub-apertures described as shown in Fig.

8(d). The theoretical image results of a single aperture with speckle noise equivalent to 360

speckle realizations and three coherently-combined apertures with speckle noise equivalent to

120 speckle realizations are shown in Fig. 8(a) and Fig. 8(c), respectively. Note that the

coherent synthesis is performed across a horizontal array of three apertures which yields a

resolution gain in only the horizontal direction as evident in Fig. 8(c) and Fig. 8(d).


Fig. 8. Imagery from (a) a simulated single aperture with 360 speckle realizations and (b) lab

data from a single, corrected aperture and 360 speckle realizations. Also shown are results for

(c) 120 speckle realizations of a simulated aperture coherently synthesized from three

horizontally arrayed sub-apertures and (d) lab results for an equivalent synthesized aperture.

Given a target distance of 10 m, a wavelength of 632.8 nm, an aperture size of 4.83 cm,

and an array spacing of 5.84 cm (aperture center-to-center) it is possible to find the resolvable

spot size using the Rayleigh criteria. The target lines are labeled in units normalized to

hundreds of line pairs per target height. A target height of 11.5 mm and an array baseline

width of 16.5 cm yield the resolvable number of lines given in Table 1.

Table 1. Predicted resolution for a single aperture and a sparse array.

Aperture Diffraction

Limited Spot Hundreds of Lines per

Target Height Single Aperture 159.8 µm 1.4

Horizontal Array 46.8 µm 4.9

Figure 6 can be used as a key to read the resolution in the images presented in Fig. 8. The

resolution of the image in Fig. 8(b) agrees with the predicted value of 140 lines per image

height, while the resolution of the image in Fig. 8(d) seems to be nearly three times that

achieved in Fig. 8(b), although it appears lower than the predicted value of 490 lines per

image height. This is partly explained by the results of Miller et al. where the image forming

diameter of a sparse aperture array tends to be less than the circumscribed diameter [5]. In


this case, long spatial frequency baselines across the imaging pupil are sampled less

frequently and thus have a greater susceptibility to noise.

4. Conclusions

Coherent aperture synthesis is a technique which promises to enable the use of smaller sub-

aperture systems to collect high-resolution images, significantly reducing the overall volume

of traditional monolithic aperture systems. Such systems can combine the use of traditional

hardware components and newly-developed software algorithms which utilize image

sharpness metrics to synthesize a large aperture via spatial heterodyne techniques.

An overview of the hardware was presented, including the utilization of an off-axis local

oscillator beam which allows for direct access to the return signal when using a digital

holography setup. Image sharpening algorithms were modified to include speckle averaging

because the presence of speckle greatly affects the convergence of the sharpening algorithms,

in particular when considering piston phase error between sub-apertures. Then, a numerical

simulation of the image sharpness metric power coefficient was performed as a function of

RMS piston phase error, which yielded a minimum error when 0.4 ≤ γ ≤ 0.5.

An experiment comprised of three horizontally arrayed afocal telescopes, each with its

own camera and local oscillator, was performed. Images were taken of an ISO 12233 target

with a diffuse scatter immediately behind it. Examples of the speckle-averaged, single-

aperture image, the three-aperture, coherently-synthesized image and a corresponding

theoretical image were presented and their resolution was evaluated. The aberration-

corrected, incoherently combined image matched its predicted 140 lines per target height of

resolution. The coherently combined image yielded an increase of resolution nearly 3X that of

the incoherent image, but had slightly lower resolution than the predicted 490 lines per target

height of resolution. This was attributed, in part, to theoretical system resolution being

defined by the noise cutoff in the MTF rather than the circumscribed array diameter.

Acknowledgments

The authors would like to thank Matt Dierking, Larry Barnes, and John Schmoll for technical

assistance.


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Distributed aperture synthesis

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