Abstract

Methods for extrapolating gradient data outside a circular aperture from measurements obtained within a circular aperture are presented. The proposed methods are required to be computationally efficient and to avoid the excitation of additional waffle modes in Fried alignment. It is shown that, using an octagon as an intermediate step from the circle to the square in the extrapolation process, the computations or residual reconstruction error can be reduced. The resulting computational cost is as low as O(N1/2), where N is the number of measurement points. The performances of the extrapolation methods are studied in connection with a recently developed O(N) wavefront reconstruction algorithm based on wavelet filter banks [IEEE J. Sel. Top. Signal Process. 2, 781 (2008)] Experiments indicate that, as expected, there is a significant reconstruction error if no extrapolation is used. Further, the proposed extrapolation techniques lead to a reconstruction with data that are marginally different from a pupil masked reconstruction using data from a square aperture.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  23. B. P. Wallace, P. J. Hampton, C. H. Bradley, and R. Conan, “Evaluation of a MEMS deformable mirror for an adaptive optics testbench,” Opt. Express 14, 10132-10138 (2006).
    [CrossRef] [PubMed]
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2008 (3)

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781-792(2008).
[CrossRef]

D. R. Andersen, M. Fischer, R. Conan, M. Fletcher, and J.-P. Veran, “VOLT: the Victoria Open Loop Testbed,” Proc. SPIE 7015, 70150H (2008).
[CrossRef]

P. J. Hampton, R. Conan, O. Keskin, P. Agathoklis, and C. Bradley, “Self-characterization of linear and nonlinear adaptive optics systems,” Appl. Opt. 47, 126-134 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (5)

2005 (1)

2004 (1)

D. Gavel, “Tomography for multiconjugate adaptive optics systems using laser guide stars,” Proc. SPIE 5490, 1356-1373 (2004).
[CrossRef]

2003 (1)

D. T. Gavel, “Suppressing anomalous localized waffle behavior in least-squares wavefront reconstructors,” Proc. SPIE 4839, 972-980 (2003).
[CrossRef]

2002 (3)

2001 (1)

2000 (1)

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

1996 (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

1993 (1)

1988 (2)

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
[CrossRef]

T. Nakajima, “Signal-to-noise ratio of the bispectral analysis of speckle interferometry,” J. Opt. Soc. Am. A 5, 1477-1491(1988).
[CrossRef]

1986 (1)

1980 (1)

1977 (2)

1976 (1)

Agabi, A.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Agathoklis, P.

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781-792(2008).
[CrossRef]

P. J. Hampton, R. Conan, O. Keskin, P. Agathoklis, and C. Bradley, “Self-characterization of linear and nonlinear adaptive optics systems,” Appl. Opt. 47, 126-134 (2008).
[CrossRef] [PubMed]

Andersen, D. R.

D. R. Andersen, M. Fischer, R. Conan, M. Fletcher, and J.-P. Veran, “VOLT: the Victoria Open Loop Testbed,” Proc. SPIE 7015, 70150H (2008).
[CrossRef]

Bierden, P. A.

S. A. Cornelissen, P. A. Bierden, and T. G. Bifano, “Development of a 4096 element deformable mirror for high contrast astronomical imaging,” Proc. SPIE 6306, 630606(2006).
[CrossRef]

Bifano, T. G.

S. A. Cornelissen, P. A. Bierden, and T. G. Bifano, “Development of a 4096 element deformable mirror for high contrast astronomical imaging,” Proc. SPIE 6306, 630606(2006).
[CrossRef]

Borgnino, J.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Bradley, C.

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781-792(2008).
[CrossRef]

P. J. Hampton, R. Conan, O. Keskin, P. Agathoklis, and C. Bradley, “Self-characterization of linear and nonlinear adaptive optics systems,” Appl. Opt. 47, 126-134 (2008).
[CrossRef] [PubMed]

Bradley, C. H.

Brase, J. M.

Chellappa, R.

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
[CrossRef]

Conan, R.

P. J. Hampton, R. Conan, O. Keskin, P. Agathoklis, and C. Bradley, “Self-characterization of linear and nonlinear adaptive optics systems,” Appl. Opt. 47, 126-134 (2008).
[CrossRef] [PubMed]

D. R. Andersen, M. Fischer, R. Conan, M. Fletcher, and J.-P. Veran, “VOLT: the Victoria Open Loop Testbed,” Proc. SPIE 7015, 70150H (2008).
[CrossRef]

B. P. Wallace, P. J. Hampton, C. H. Bradley, and R. Conan, “Evaluation of a MEMS deformable mirror for an adaptive optics testbench,” Opt. Express 14, 10132-10138 (2006).
[CrossRef] [PubMed]

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Cornelissen, S. A.

S. A. Cornelissen, P. A. Bierden, and T. G. Bifano, “Development of a 4096 element deformable mirror for high contrast astronomical imaging,” Proc. SPIE 6306, 630606(2006).
[CrossRef]

Duncan, B. D.

Ellerbroek, B.

Ellerbroek, B. L.

Fischer, M.

D. R. Andersen, M. Fischer, R. Conan, M. Fletcher, and J.-P. Veran, “VOLT: the Victoria Open Loop Testbed,” Proc. SPIE 7015, 70150H (2008).
[CrossRef]

Fletcher, M.

D. R. Andersen, M. Fischer, R. Conan, M. Fletcher, and J.-P. Veran, “VOLT: the Victoria Open Loop Testbed,” Proc. SPIE 7015, 70150H (2008).
[CrossRef]

Frankot, R. T.

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
[CrossRef]

Freischlad, K.

Fried, D. L.

Gavel, D.

D. Gavel, “Tomography for multiconjugate adaptive optics systems using laser guide stars,” Proc. SPIE 5490, 1356-1373 (2004).
[CrossRef]

Gavel, D. T.

D. T. Gavel, “Suppressing anomalous localized waffle behavior in least-squares wavefront reconstructors,” Proc. SPIE 4839, 972-980 (2003).
[CrossRef]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with the Fourier tranform,” J. Opt. Soc. Am. A 19, 2100-2111 (2002).
[CrossRef]

Gilles, L.

Hampton, P. J.

Harris, S. R.

Herrmann, J.

Hudgin, R.

Keskin, O.

Koliopoulos, C.

Kovesi, P.

P. Kovesi, “Shapelets correlated with surface normals produce surfaces,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (IEEE, 2005), pp. 994-1001.

Macintosh, B. A.

Martin, F.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Nakajima, T.

Noll, R. J.

Poyneer, L. A.

Ragazzoni, R.

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

Ribak, E. N.

Roddier, C.

Roddier, F.

Talmi, A.

Tokovinin, A.

A. Tokovinin and E. Viard, “Limiting precision of tomographic phase reconstruction,” J. Opt. Soc. Am. A 18, 873-882 (2001).
[CrossRef]

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Trinquet, H.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Veran, J.-P.

D. R. Andersen, M. Fischer, R. Conan, M. Fletcher, and J.-P. Veran, “VOLT: the Victoria Open Loop Testbed,” Proc. SPIE 7015, 70150H (2008).
[CrossRef]

Véran, J.-P.

Viard, E.

Vogel, C.

Vogel, C. R.

Wallace, B. P.

Widiker, J. J.

Yang, Q.

Ziad, A.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Appl. Opt. (3)

Astron. Astrophys. Suppl. Ser. (1)

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astron. Astrophys. Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

IEEE J. Sel. Top. Signal Process. (1)

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781-792(2008).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
[CrossRef]

J. Mod. Opt. (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (10)

C. Roddier and F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277-2287 (1993).
[CrossRef]

A. Tokovinin and E. Viard, “Limiting precision of tomographic phase reconstruction,” J. Opt. Soc. Am. A 18, 873-882 (2001).
[CrossRef]

B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803-1816 (2002).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817-1822 (2002).
[CrossRef]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with the Fourier tranform,” J. Opt. Soc. Am. A 19, 2100-2111 (2002).
[CrossRef]

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A 22, 1515-1526(2005).
[CrossRef]

L. A. Poyneer, B. A. Macintosh, and J.-P. Véran, “Fourier-transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A 24, 2645-2660 (2007).
[CrossRef]

K. Freischlad and C. Koliopoulos, “Modal estimation of a wave-front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852-1861 (1986).
[CrossRef]

T. Nakajima, “Signal-to-noise ratio of the bispectral analysis of speckle interferometry,” J. Opt. Soc. Am. A 5, 1477-1491(1988).
[CrossRef]

A. Talmi and E. N. Ribak, “Wavefront reconstruction from its gradients,” J. Opt. Soc. Am. A 23, 288-297 (2006).
[CrossRef]

Opt. Express (1)

Proc. SPIE (4)

D. T. Gavel, “Suppressing anomalous localized waffle behavior in least-squares wavefront reconstructors,” Proc. SPIE 4839, 972-980 (2003).
[CrossRef]

S. A. Cornelissen, P. A. Bierden, and T. G. Bifano, “Development of a 4096 element deformable mirror for high contrast astronomical imaging,” Proc. SPIE 6306, 630606(2006).
[CrossRef]

D. R. Andersen, M. Fischer, R. Conan, M. Fletcher, and J.-P. Veran, “VOLT: the Victoria Open Loop Testbed,” Proc. SPIE 7015, 70150H (2008).
[CrossRef]

D. Gavel, “Tomography for multiconjugate adaptive optics systems using laser guide stars,” Proc. SPIE 5490, 1356-1373 (2004).
[CrossRef]

Other (1)

P. Kovesi, “Shapelets correlated with surface normals produce surfaces,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (IEEE, 2005), pp. 994-1001.

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Figures (20)

Fig. 1
Fig. 1

Fried alignment of gradient measurements. Phase pixels are represented by squares. Measurement is modeled as the discrete gradient of the phase pixels in each of the two Cartesian directions. The measurement point is at the crossing of the two arrows.

Fig. 2
Fig. 2

Extrapolation process of converting a circular gradient data set to any larger sized square. White solid lines indicate simulated mirror locations. Black solid lines indicate the new data set edge. The figures display the logarithm of the magnitude of the gradients and tend to look like pencil sketches for image gradient data. Reflections are to be processed on each of the two data sets individually.

Fig. 3
Fig. 3

Reconstruction of data represented by Fig. 2e. The result resembles a kaleidoscope.

Fig. 4
Fig. 4

Pupil masked representation of the image shown in Fig. 2.

Fig. 5
Fig. 5

Smallest closed path of measured gradients in Fried alignment. Example of process in quadrant 2. The variable name convention holds for all quadrants.

Fig. 6
Fig. 6

Fundamental relationships of curved mirror reflection.

Fig. 7
Fig. 7

Example of desired measurement point between four measurement grid positions. Indices for measurement points increase toward the right for p and downward for q in Eq. (19).

Fig. 8
Fig. 8

Relationship between extrapolated grid points (x) and the corresponding required measurement positions (o). Measurement data (·) do not coincide with (o), so the process requires interpolation of the actual measurement points by matrix A.

Fig. 9
Fig. 9

Example of relationship between coordinate systems.

Fig. 10
Fig. 10

Lower right quadrant of the octagon. The octagon is symmetric about the x and y axes.

Fig. 11
Fig. 11

Circular and octagonal apertures on a square 15 × 15 grid. For this resolution there are only 8 total points (2 per quadrant) that are fully within the octagon but outside the circular aperture. Points on the pupil edge are considered outside the pupil.

Fig. 12
Fig. 12

Decision flow chart for presented extrapolation options.

Fig. 13
Fig. 13

Atmospheric turbulence phase screen downsampled to 512 × 512 . Amplitude normalized to 1 unit rms. r 0 = 0.25 m , L 0 = 22 m , 32 m width, 500 nm wavelength, and 25 m / s wind speed.

Fig. 14
Fig. 14

Residual error from using the proposed extrapolation methods and the Hampton–Agathoklis reconstruction method [16]. For each pair of similar lines, the black line is superior at high gradient SNR, and the gray line is inferior. Figure 15 shows performance at low gradient SNR.

Fig. 15
Fig. 15

Zoomed representation of Fig. 14 showing residual error at low gradient SNR. The curved mirror extrapolation was the worst method at high gradient SNR and is shown here to provide superior reconstruction for low gradient SNR.

Fig. 16
Fig. 16

Residual error from using the proposed extrapolation methods and the Hampton–Agathoklis reconstruction method with all waffle modes suppressed. For each pair of similar lines, the black line is superior at high gradient SNR, and the gray line is inferior. Figure 17 shows performance at low gradient SNR.

Fig. 17
Fig. 17

Zoomed representation of Fig. 16 showing residual error at low gradient SNR when waffle modes are suppressed.

Fig. 18
Fig. 18

Comparison between the full reconstruction and waffle suppression. Both are combined with the diagonal-octagonal extrapolation method.

Fig. 19
Fig. 19

The residual reconstruction error of various sizes of data sets. The extrapolation method is diagonally flat. The reconstruction method is the Hampton–Agathoklis method [16]. The unit of diameter, D, is pixels. The physical diameter of the pupil is a constant 32 m .

Fig. 20
Fig. 20

Amplification of mean square noise versus the number of sensor points. Error bars indicate maximum and minimum amplification in 30 noisy data sets at each resolution.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

Φ = Φ x u ^ x + Φ y u ^ y = Φ ˙ x u ^ x + Φ ˙ y u ^ y ,
ϕ ˙ x , p , q = 0.5 ( ϕ i , j + ϕ i , j + 1 ϕ i + 1 , j + ϕ i + 1 , j + 1 ) ,
ϕ ˙ y , p , q = 0.5 ( ϕ i , j ϕ i , j + 1 + ϕ i + 1 , j + ϕ i + 1 , j + 1 ) ,
i = p - 0.5 ,
j = q - 0.5 ,
ϕ ˙ x , p , q ϕ ˙ y , p , q + ϕ ˙ x , p , q + 1 + ϕ ˙ y , p , q + 1 = ϕ ˙ x , p + 1 , q + 1 ϕ ˙ y , p + 1 , q + 1 + ϕ ˙ x , p + 1 , q + ϕ ˙ y , p + 1 , q .
ϕ ˙ y , p , q = ϕ ˙ x , p , q
ϕ ˙ y , p , q + 1 = ϕ ˙ x , p , q + 1 ,
ϕ ˙ y , p + 1 , q = ϕ ˙ x , p + 1 , q ,
ϕ ˙ y , p + 1 , q + 1 = ϕ ˙ x , p + 1 , q + 1 ,
Φ ( r out , θ ) = R 2 r out 2 ( Φ ( r in , θ ) r in u ^ r + Φ ( r in , θ ) r in θ u ^ θ ) ,
r in = R 2 r out .
Φ in ( r in , θ ) = Φ out ( r out , θ ) ,
Φ ( r out , θ ) = Φ ( r out , θ ) r out u ^ r + Φ ( r out , θ ) r out θ u ^ θ .
Φ ( r out , θ ) = Φ ( r in , θ ) r out u ^ r + Φ ( r in , θ ) r out θ u ^ θ ,
Φ ( r out , θ ) = ( r in r out ) Φ ( r in , θ ) r in u ^ r + ( r in r o u t ) Φ ( r in , θ ) r in θ u ^ θ .
[ Φ out x Φ out y ] = B 1 G B A d ,
d = [ ϕ ˙ x , p , q ϕ ˙ x , p , q + 1 ϕ ˙ x , p + 1 , q ϕ ˙ x , p + 1 , q + 1 ϕ ˙ y , p , q ϕ ˙ y , p , q + 1 ϕ ˙ y , p + 1 , q ϕ ˙ y , p + 1 , q + 1 ] T .
A = [ a null null a ] ,
a = [ ( 1 Δ x ) ( 1 Δ y ) Δ x ( 1 Δ y ) ( 1 Δ x ) Δ y Δ x Δ y ] .
B = [ cos θ sin θ sin θ cos θ ] ,
G = R 2 r out 2 [ 1 0 0 1 ] .
d in = A d ,
[ Φ ( r , θ ) r Φ ( r , θ ) r θ ] = B [ Φ ( r , θ ) x Φ ( r , θ ) y ] .
[ Φ ( x out , y out ) x Φ ( x out , y out ) y ] = [ cos 2 θ sin 2 θ sin 2 θ cos 2 θ ] [ Φ ( x in , y in ) x Φ ( x in , y in ) y ] ,
x in = x out 2 Δ d cos θ ,
y in = y out 2 Δ d sin θ ,
σ e 2 = α σ n 2 ,
α ( N ) = 0.0261 ln 2 N + 0.2405 ln N + 0.6833 ,

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