Abstract

The resolution of a diffraction-limited imaging system is inversely proportional to the aperture size. Instead of using a single large aperture, multiple small apertures are used to synthesize a large aperture. Such a multi-aperture system is modular, typically more reliable and less costly. On the other hand, a multi-aperture system requires phasing sub-apertures to within a fraction of a wavelength. So far in the literature, only the piston, tip, and tilt type of inter-aperture errors have been addressed. In this paper, we present an approach to correct for rotational and translational errors as well.

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References

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2010 (1)

2009 (1)

2008 (1)

2007 (1)

2006 (1)

2003 (1)

1996 (1)

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5(8), 1266–1271 (1996).
[CrossRef] [PubMed]

1988 (1)

R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image sharpness criterion,” Proc. SPIE 976, 37–47 (1988).

1976 (1)

1974 (1)

1962 (1)

Buffington, A.

Chatterji, B. N.

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5(8), 1266–1271 (1996).
[CrossRef] [PubMed]

Dierking, M. P.

Duncan, B. D.

Fienup, J. R.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2004).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2006).

Grow, T. D.

Hoft, T. A.

Jameson, D.

Kendrick, R. L.

Leith, E. N.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 2007).
[CrossRef]

Marron, J. C.

Miller, J. J.

Miller, N. J.

Muller, R. A.

Noll, R. J.

Paxman, R. G.

R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image sharpness criterion,” Proc. SPIE 976, 37–47 (1988).

Rabb, D.

Reddy, B. S.

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5(8), 1266–1271 (1996).
[CrossRef] [PubMed]

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (SPIE, 2010).

Seldomridge, N.

Stafford, J.

Stokes, A.

Thurman, S. T.

Upatnieks, J.

Appl. Opt. (1)

IEEE Trans. Image Process. (1)

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5(8), 1266–1271 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Express (3)

Proc. SPIE (1)

R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image sharpness criterion,” Proc. SPIE 976, 37–47 (1988).

Other (4)

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2006).

D. Malacara, Optical Shop Testing (Wiley, 2007).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2004).

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (SPIE, 2010).

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Figures (7)

Fig. 1
Fig. 1

(a) Single aperture configuration; magnitude of the pupil field ak(x, y) is shown. (b) Three-aperture configuration; magnitude of the pupil field ak(x, y) is shown. (c) One realization b(u,v) from single aperture. (d) Average of 15 realizations from single aperture. (e) Average of 60 realizations from single aperture. (f) Average of 60 realizations from three-aperture configuration.

Fig. 2
Fig. 2

Zernike polynomials up to fifth degree are shown.

Fig. 3
Fig. 3

The phase of a sub-aperture in Cartesian and polar coordinates is shown. A rotation in Cartesian coordinate system corresponds to a circular shift along the θ axis in the polar coordinate system.

Fig. 4
Fig. 4

Illustration of the domains to do different corrections. The indices of the complex fields and their Fourier transforms are not included in this illustration.

Fig. 5
Fig. 5

Focused image, obtained by averaging 60 realizations, in the three sub-aperture design is shown.

Fig. 6
Fig. 6

(a)(b)(c) Average sub-aperture realizations. (d)(e)(f) Intra-aperture correction. (g) Composite without any inter-aperture correction. (h) After piston/tip/tilt correction. (i) Result with piston/tip/tilt and rotation/shift correction.

Fig. 7
Fig. 7

Zoomed regions from the previous figure. (a) Averaged first sub-aperture. (b) Composite before any inter-aperture correction. (c) Result with piston/tip/tilt correction. (d) Result with piston/tip/tilt and rotation/shift correction.

Tables (1)

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Algorithm 1: Multi-aperture phasing algorithm.

Equations (8)

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I k ( x , y ) = | a k ( x , y ) + r ( x , y ) | 2 ,
( I k ( x , y ) ) = ( | a k ( x , y ) + r ( x , y ) | 2 ) = ( | a k ( x , y ) | 2 ) + ( | r ( x , y ) | 2 ) + ( a k ( x , y ) r * ( x , y ) ) + ( a k * ( x , y ) r ( x , y ) ) , = R ( a ^ k ( u , v ) ) + R ( r ^ ( u , v ) ) + a ^ k ( u u 0 , v v 0 ) + a ^ k * ( u + u 0 , v + v 0 ) ,
w ^ k , 1 , w ^ k , P = argmin { S ( | ( a k ( x , y ) e j p = 1 P w k , p Z p ( x , y ) ) | 2 ) } ,
a comp ( 1 , k ) ( x , y ) = a 1 ( x x 1 , y y 1 ) + a k ( x x k , y y k ) e j p = 1 P w k , p Z p ( x , y ) ,
w ^ k , 1 , w ^ k , P = argmin { S | ( a comp ( 1 , k ) ( x , y ) ) | 2 ) } .
w ^ k = argmin { S ( | ( a comp ( 1 , k ) ( x , y ) ) | 2 ) } ,
a comp ( 1 , k ) ( x , y ) = a 1 ( x x 1 , y y 1 ) + 𝒫 1 [ 1 ( ( 𝒫 [ a k ( x x k , y y k ) ] ) e j w k Z 1 1 ( u ρ , v ϕ ) ) ] ,
( I k ( x + δ x k , y + δ y , k ) ) = ( | a k ( x + δ x k , y + δ y k ) + r ( x + δ x k , y + δ y k ) | 2 ) = ( | a k ( x + δ x k , y + δ y k ) | 2 ) + ( | r ( x + δ x k , y + δ y k ) | 2 ) + ( a k ( x + δ x k , y + δ y k ) r * ( x + δ x k , y + δ y k ) ) + ( a k * ( x + δ x k , y + δ y k ) r ( x + δ x k , y + δ y k ) ) , = R ( a ^ k ( u , v ) e j 2 π ( δ x k u + δ y k v ) ) + R ( r ^ ( u , v ) e j 2 π ( δ x k u + δ y k v ) ) + a ^ k ( u u 0 , v v 0 ) e j 2 π ( δ x k ( u u 0 ) + δ y k ( v v 0 ) ) e j 2 π ( δ x k u 0 + δ y k v 0 ) + a ^ k * ( u + u 0 , v + v 0 ) e j 2 π ( δ x k ( u + u 0 ) + δ y k ( v + v 0 ) ) e j 2 π ( δ x k u 0 + δ y k v 0 ) .

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