Abstract

Atmospheric turbulence corrupts astronomical images formed by ground-based telescopes. Adaptive optics systems allow the effects of turbulence-induced aberrations to be reduced for a narrow field of view corresponding approximately to the isoplanatic angle θ0. For field angles larger than θ0, the point spread function (PSF) gradually degrades as the field angle increases. We present a technique to estimate the PSF of an adaptive optics telescope as function of the field angle, and use this information in a space-varying image reconstruction technique. Simulated anisoplanatic intensity images of a star field are reconstructed by means of a block-processing method using the predicted local PSF. Two methods for image recovery are used: matrix inversion with Tikhonov regularization, and the Lucy–Richardson algorithm. Image reconstruction results obtained using the space-varying predicted PSF are compared to space invariant deconvolution results obtained using the on-axis PSF. The anisoplanatic reconstruction technique using the predicted PSF provides a significant improvement of the mean squared error between the reconstructed image and the object compared to the deconvolution performed using the on-axis PSF.

© 2007 Optical Society of America

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References

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2002 (2)

J. Christou, E. Steinbring, S. Faber, D. Gavel, J. Patience, and E. Gates, "Anisoplanatism within the isoplanatic patch," Am. Astron. Soc. 34, 1257 (2002).

M. Owner-Petersen and A. Goncharov, "Multiconjugate adaptive optics for large telescopes: analytical control of the mirror shapes," J. Opt. Soc. Am. 19, 537-548 (2002).
[CrossRef]

2001 (1)

1999 (1)

1998 (3)

1997 (1)

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, "Improving the resolution of ground-based telescopes," Rev. Mod. Phys. 69, 437-505 (1997).
[CrossRef]

1994 (3)

1990 (1)

1989 (1)

1988 (1)

1982 (1)

1972 (1)

Am. Astron. Soc. (1)

J. Christou, E. Steinbring, S. Faber, D. Gavel, J. Patience, and E. Gates, "Anisoplanatism within the isoplanatic patch," Am. Astron. Soc. 34, 1257 (2002).

Appl. Opt. (4)

J. Opt. Soc. Am. (3)

M. Owner-Petersen and A. Goncharov, "Multiconjugate adaptive optics for large telescopes: analytical control of the mirror shapes," J. Opt. Soc. Am. 19, 537-548 (2002).
[CrossRef]

W. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am. 62, 55-59 (1972).
[CrossRef]

D. L. Fried, "Anisoplanatism in adaptive optics," J. Opt. Soc. Am. 72, 52-61 (1982).
[CrossRef]

J. Opt. Soc. Am. A (7)

Rev. Mod. Phys. (1)

M. C. Roggemann, B. M. Welsh, and R. Q. Fugate, "Improving the resolution of ground-based telescopes," Rev. Mod. Phys. 69, 437-505 (1997).
[CrossRef]

Other (8)

K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1997).

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).

G. Cochran, "Phase screen generation," Tech. Rep. TR-663 (The Optical Sciences Company, Placentia, California, 1985).

T. J. Brennan and P. H. Roberts, "AOTOOLs: the Adaptive Optics Toolbox" (Optical Sciences Company, Anaheim, California, 1999).

J. W. Goodman, Introduction To Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

A. N. Tikhonov, A. V. Goncharsky, V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, 1995).

M. Aubailly and M. C. Roggemann, "Block-processing method for post-detection correction of anisoplanatic image detects in astronomical adaptive optics images," in 2005 AMOS Technical Conference, P. W. Kervin and J. L. Africano, eds. (Maui Economic Development Board Publications, 2005), pp. 131-137.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2002).

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Figures (11)

Fig. 1
Fig. 1

Model of telescope pupil showing primary and secondary, fully illuminated subapertures, master actuator locations, and the images due to each lenslet for the case of a plane wave falling on the lenslet array.

Fig. 2
Fig. 2

(Color online) Example PSFs as a function of field angle for the adaptive optics telescope modeled.

Fig. 3
Fig. 3

Schematic representation of the off-axis PSFs for a beacon located at the origin of the FOV reference system ( θ x , θ y ) . The position of the PSF h θ , α ( u , u ) in the FOV is given in polar coordinates by ( θ , α ) , where θ corresponds to the field angle and α is its orientation. The reference system ( u , u ) is relative to the PSF with its origin at the center of the PSF. Two geometrical properties are assumed for the PSF: symmetry with respect to the axis u and u , and invariance by rotation around the beacon.

Fig. 4
Fig. 4

(Color online) PSF model coefficients { a i } and { b i } for i = 1 , … ,  7 as functions of the field angle θ.

Fig. 5
Fig. 5

(Color online) Cross sections of the predicted PSFs (dotted line) compared to the simulated PSFs (solid curve) at different field angles.

Fig. 6
Fig. 6

(Color online) PSF model coefficients { a i } and { b i }   for i = 1 , … ,  7 as functions of the zenith angle for a fixed field angle θ = 20   μRad .

Fig. 7
Fig. 7

(Color online) PSF model coefficients { a i } and { b i } for i = 1 , … ,  7 as functions of the visual magnitude for a fixed field angle θ = 20   μRad .

Fig. 8
Fig. 8

(Color online) Reconstruction error metric ϵ recons for anisoplanatic image recovery based on matrix inversion and Tikhonov regularization as a function of the Tikhonov parameter α. Results are given for a star field with visual magnitude VM of 1 and observation in the zenith direction ( θ z = 0 ) .

Fig. 9
Fig. 9

(Color online) Reconstruction error metric ϵrecons for anisoplanatic image recovery based on Lucy–Richardson deconvolution shown as a function of the number of iterations. Results are given for a star field with visual magnitude VM of 1 and observation in the zenith direction ( θ z = 0 ) .

Fig. 10
Fig. 10

(Color online) Reconstruction error as a function of the zenith angle for anisoplanatic image recovery using the Tikhonov regularization-based and the LR-based techniques.

Fig. 11
Fig. 11

(Color online) Reconstruction error as a function of the visual magnitude for anisoplanatic image recovery using the Tikhonov regularization-based technique and the LR-based techniques.

Tables (2)

Tables Icon

Table 1 Improvement Factor ξ from Isoplanatic to Anisoplanatic Reconstruction of a Star Field Using the Tikhonov Regularization-Based Recovery Technique

Tables Icon

Table 2 Improvement Factor ξ from Isoplanatic to Anisoplanatic Reconstruction of a Star Field Using the LR-Based Recovery Technique

Equations (14)

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θ 0 = 58.1 × 10 3 λ 6 / 5 [ ( sec θ z ) 8 / 3 0 L d z C n 2 ( z ) z 5 / 3 ] 3 / 5 ,
r 0 = 0.185 [ 4 π 2 sec θ z k 2 z 1 z 2 d z C n 2 ( z ) ] 3 / 5 ,
c , θ mod ( u ) = i = 0 N a i T i ( u )   for   u 0 ,
c θ mod ( u ) = i = 0 N b i T i ( u )   for   u 0 ,
h θ mod ( ρ , β ) = [ 1 γ ( β ) ] c θ mod ( ρ ) + γ ( β ) c , θ mod ( ρ ) ,
ϵ fit ( θ ) = u u h θ mod ( u , u ) h θ ( u , u ) 2 u u h θ ( u , u ) 2 .
K ¯ V M + 1 = K ¯ V M 2.51 .
i = H θ mod o + n ,
H θ mod = [ H 0 H N iso 1 H 1 H 1 H 0 H 2 H N iso 1 H N iso 2 H 0 ] ,
H i = [ h i , 0 h i , N iso 1 h i , N iso 1 h i , 1 h i , 0 h i , N iso 2 h i , N iso 1 h i , N iso 2 h i , N iso 0 ] ,
o ˜ = [ ( H θ mod ) T H θ mod + α I ] 1 ( H θ mod ) T i ,
o ˜ ( k + 1 ) = o ˜ ( k ) × [ h θ mod ( i h θ mod o ˜ ( k ) ) ] ,
ϵ recons = x y o ˜ ( x , y ) o ( x , y ) 2 x y o ( x , y ) 2 ,
ξ = ϵ on ϵ off ϵ on ,

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