Abstract

Phase differences in the far field of a coherently illuminated object are used to estimate the two-dimensional phase in the measurement plane of an imaging system. A previously derived phase-correlation function is used in a minimum-variance phase-estimation algorithm to map phase-difference measurements optimally to estimates of the phase on a grid of points in the measurement plane. Theoretical and computer-simulation comparisons between the minimum-variance phase estimator and conventional least-squares estimators are made. The minimum-variance phase estimator produces a lower aperture-averaged mean-square phase error for all values of a sampling parameter β.

© 1998 Optical Society of America

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  16. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  17. D. K. C. Mac Donald, “Some statistical properties of random noise,” Proc. Cambridge Phil. Soc. 45, 368–372 (1949).
  18. J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).
  19. M. C. Roggemann, “Limited degree-of-freedom adaptive optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
  20. D. G. Voelz, J. D. Gonglewski, P. S. Idell, “SCIP computer simulation and laboratory verification,” in Digital Image Recovery and Synthesis II, P. Idell, ed., Proc. SPIE2029, 169–176 (1993).
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1996 (1)

1995 (1)

1994 (2)

1993 (1)

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging,” Proc. IEEE 2029, 161–168 (1993).

1992 (2)

D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Elect. Eng. 18, 451–466 (1992).

1991 (1)

1989 (1)

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am A 6, 1913–1923 (1989).

1988 (1)

1985 (1)

1983 (1)

1979 (1)

1977 (2)

1970 (1)

1949 (1)

D. K. C. Mac Donald, “Some statistical properties of random noise,” Proc. Cambridge Phil. Soc. 45, 368–372 (1949).

Arsenault, H.

Brown, S. M.

S. M. Brown, D. C. Ehn, J. W. Hardy, R. Hutchin, P. J. Maihot, A. S. Menikoff, M. B. Michalik, S. Paley, “Reconstructor development study,” Tech. Rep. F30602-86-C-0010 (Itek Optical Systems, Lexington, Mass., 1987).

Craig, A. T.

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics (Macmillan, New York, 1978).

Ehn, D. C.

S. M. Brown, D. C. Ehn, J. W. Hardy, R. Hutchin, P. J. Maihot, A. S. Menikoff, M. B. Michalik, S. Paley, “Reconstructor development study,” Tech. Rep. F30602-86-C-0010 (Itek Optical Systems, Lexington, Mass., 1987).

Ellerbroek, B. L.

B. L. Ellerbroek, “Comparison of least squares and minimal variance wavefront reconstruction for turbulence compensation in the presence of noise,” Tech. Rep. TR-721R (Optical Science Company, Placentia, Calif., 1986).

Fiddy, M. A.

Fornaro, G.

Franceschetti, G.

Freund, I.

Fried, D. L.

Gardner, C. S.

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am A 6, 1913–1923 (1989).

Ghiglia, D. C.

Gonglewski, J. D.

D. G. Voelz, J. D. Gonglewski, P. S. Idell, “SCIP computer simulation and laboratory verification,” in Digital Image Recovery and Synthesis II, P. Idell, ed., Proc. SPIE2029, 169–176 (1993).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1975).

Hardy, J. W.

S. M. Brown, D. C. Ehn, J. W. Hardy, R. Hutchin, P. J. Maihot, A. S. Menikoff, M. B. Michalik, S. Paley, “Reconstructor development study,” Tech. Rep. F30602-86-C-0010 (Itek Optical Systems, Lexington, Mass., 1987).

Hogg, R. V.

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics (Macmillan, New York, 1978).

Hudgin, R. A.

Hunt, B. R.

Hutchin, R.

S. M. Brown, D. C. Ehn, J. W. Hardy, R. Hutchin, P. J. Maihot, A. S. Menikoff, M. B. Michalik, S. Paley, “Reconstructor development study,” Tech. Rep. F30602-86-C-0010 (Itek Optical Systems, Lexington, Mass., 1987).

Hutchin, R. A.

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging,” Proc. IEEE 2029, 161–168 (1993).

Idell, P. S.

D. G. Voelz, J. D. Gonglewski, P. S. Idell, “SCIP computer simulation and laboratory verification,” in Digital Image Recovery and Synthesis II, P. Idell, ed., Proc. SPIE2029, 169–176 (1993).

Lanari, R.

Lowenthal, S.

Mac Donald, D. K. C.

D. K. C. Mac Donald, “Some statistical properties of random noise,” Proc. Cambridge Phil. Soc. 45, 368–372 (1949).

Maihot, P. J.

S. M. Brown, D. C. Ehn, J. W. Hardy, R. Hutchin, P. J. Maihot, A. S. Menikoff, M. B. Michalik, S. Paley, “Reconstructor development study,” Tech. Rep. F30602-86-C-0010 (Itek Optical Systems, Lexington, Mass., 1987).

Marroquin, J. L.

Menikoff, A. S.

S. M. Brown, D. C. Ehn, J. W. Hardy, R. Hutchin, P. J. Maihot, A. S. Menikoff, M. B. Michalik, S. Paley, “Reconstructor development study,” Tech. Rep. F30602-86-C-0010 (Itek Optical Systems, Lexington, Mass., 1987).

Michalik, M. B.

S. M. Brown, D. C. Ehn, J. W. Hardy, R. Hutchin, P. J. Maihot, A. S. Menikoff, M. B. Michalik, S. Paley, “Reconstructor development study,” Tech. Rep. F30602-86-C-0010 (Itek Optical Systems, Lexington, Mass., 1987).

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Paley, S.

S. M. Brown, D. C. Ehn, J. W. Hardy, R. Hutchin, P. J. Maihot, A. S. Menikoff, M. B. Michalik, S. Paley, “Reconstructor development study,” Tech. Rep. F30602-86-C-0010 (Itek Optical Systems, Lexington, Mass., 1987).

Rodriguez-Vera, R.

Roggemann, M. C.

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Elect. Eng. 18, 451–466 (1992).

M. C. Roggemann, “Limited degree-of-freedom adaptive optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).

Romero, L. A.

Sansosti, E.

Scivier, M. S.

Servin, M.

Takahashi, T.

Takajo, H.

Tapia, M.

Vaughn, J. L.

Voelz, D. G.

D. G. Voelz, J. D. Gonglewski, P. S. Idell, “SCIP computer simulation and laboratory verification,” in Digital Image Recovery and Synthesis II, P. Idell, ed., Proc. SPIE2029, 169–176 (1993).

Wallner, E. P.

E. P. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).

E. P. Wallner, “Comparison of wavefront sensor configurations using optimal reconstruction and correction,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds., Proc. SPIE351, 42–53 (1982).

Welsh, B. M.

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am A 6, 1913–1923 (1989).

Appl. Opt. (2)

Comput. Elect. Eng. (1)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Elect. Eng. 18, 451–466 (1992).

J. Opt. Soc. Am A (1)

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am A 6, 1913–1923 (1989).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (6)

Proc. Cambridge Phil. Soc. (1)

D. K. C. Mac Donald, “Some statistical properties of random noise,” Proc. Cambridge Phil. Soc. 45, 368–372 (1949).

Proc. IEEE (1)

R. A. Hutchin, “Sheared coherent interferometric photography: a technique for lensless imaging,” Proc. IEEE 2029, 161–168 (1993).

Other (9)

S. M. Brown, D. C. Ehn, J. W. Hardy, R. Hutchin, P. J. Maihot, A. S. Menikoff, M. B. Michalik, S. Paley, “Reconstructor development study,” Tech. Rep. F30602-86-C-0010 (Itek Optical Systems, Lexington, Mass., 1987).

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

D. G. Voelz, J. D. Gonglewski, P. S. Idell, “SCIP computer simulation and laboratory verification,” in Digital Image Recovery and Synthesis II, P. Idell, ed., Proc. SPIE2029, 169–176 (1993).

B. L. Ellerbroek, “Comparison of least squares and minimal variance wavefront reconstruction for turbulence compensation in the presence of noise,” Tech. Rep. TR-721R (Optical Science Company, Placentia, Calif., 1986).

E. P. Wallner, “Comparison of wavefront sensor configurations using optimal reconstruction and correction,” in Wavefront Sensing, N. Bareket, C. L. Koliopoulos, eds., Proc. SPIE351, 42–53 (1982).

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics (Macmillan, New York, 1978).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1975).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988).

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Figures (5)

Fig. 1
Fig. 1

Top: Asymmetric brightness function for an object subtending 1.733 μdeg. The measurement-plane-to-object distance is 106 m, and the illuminating center wavelength is 1.3 μm. Bottom: The reconstructed image, assuming that the object’s diffracted field is perfectly known on a 33 × 33 point grid in the measurement plane of the optical system. As such, the bottom figure represents a 3 m × 3 m diffraction-limited image of the object illustrated in the top figure.

Fig. 2
Fig. 2

(a) Noise-free minimum-variance reconstruction of the object illustrated at the top of Fig. 1. The measurement-plane sampling grid was 8 × 8 points evenly spaced across the 3 m × 3 m collecting aperture. The number of independent image realizations used to construct the indicated time-averaged image was 100. (b) Minimum-variance reconstruction with additive Gaussian noise with a standard deviation of π/3 rad. (c) Minimum-variance reconstruction with a noise standard deviation of π/2 rad. (d) Reconstruction with a noise standard deviation of 2π/3.

Fig. 3
Fig. 3

Comparison of the aperture-averaged mean-squared phase error for the minimum-variance reconstructor (with and without branch cuts). Solid curve, the theoretical aperture-averaged mean-squared phase error without the presence of branch cuts. The knee in the curve near β = 1.5 results from the object size’s being too small (the phase-difference correlation matrix becomes ill conditioned). Dotted–dashed curve, the simulated aperture-averaged mean-squared phase error with branch cuts present. For smaller objects sizes, fewer branch cuts are expected, and a convergence between the aperture-averaged mean-squared phase error without branch cuts and that with branch cuts is expected, as is evident here. Dashed curve, the theoretically expected aperture-averaged mean-squared phase error with branch cuts present. Dotted curve, the uncorrected aperture-averaged mean-squared phase error.

Fig. 4
Fig. 4

Comparison of the simulated aperture-averaged mean-squared phase error for the minimum variance (dotted–dashed curve) and the LS (dashed curve) results with branch cuts present. The theoretical aperture-averaged uncorrected phase variance is also shown for reference purposes (dotted curve). The phase differences were determined from the measurement-plane reference phases. The horizontal axis is the sampling parameter β. The measurement-plane sampling density was 8 × 8 evenly spaced samples in a 3 m × 3 m square collecting aperture. The number of independent measurement-plane frames used to determine 〈∊2〉 were 100. The theoretical average reflectivity profile of the objects used to estimate M jn was Gaussian with infinite extent.

Fig. 5
Fig. 5

Histograms of the simulated aperture-averaged mean-squared phase errors for the LS and the minimum-variance phase- reconstruction methods. The object is 3.47 m in diameter. The total number of independent realizations of the instantaneous aperture-averaged squared phase error for both the LS and the minimum-variance histograms was 100.

Equations (29)

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- W A x d x = 1 ,
Φ n q = Δ ϕ q x n + v x n ,
Δ ϕ q x n = ϕ x n - ϕ x n + s q ,
ϕ ˆ x = j   c j r j x ,
c j = n   M jn Φ n q .
x = ϕ x - ϕ ˆ x = ϕ x - j n   M jn Φ n q r j x .
2 x = ϕ 2 x - 2   j n   M jn r j x ϕ x Φ n q + j n l m   M lm r l x M jn r j x Φ n q Φ m r .
2 = - d x W A x 2 x .
2 = j l n m   M jn M lm Φ n q Φ m r × - d x W A x r j x r l x - 2   j n   M jn - d x W A x r j x Φ n q ϕ x + - d x W A x ϕ 2 x .
R jl = - d x W A x r j x r l x , A jn = - d x W A x r j x Φ n q ϕ x , Φ nm qr = Φ n q Φ m r , 0 2 = - d x W A x ϕ 2 x .
2 = j n l m   M jn M lm R jl Φ nm qr - 2   j n   M jn A jn + 0 2 .
φ = Δ ϕ x x 1 Δ ϕ y x 1 Δ ϕ x x 2 Δ ϕ y x 2 Δ ϕ x x N Δ ϕ y x N ,
ξ nm Φ nm qr = φ n φ m + v x n v x m .
2 = j n l m   M jn M lm R jl ξ nm - 2   j n   M jn A jn + 0 2 .
M jn = l m   R jl - 1 ξ nm - 1   A lm .
2 min = 0 2 - j n   M jn A jn .
2 min = 0 2 - Tr R jl - 1 A lm ξ mn - 1 A nj .
ϕ x ϕ x = π 2 1 + 2 γ - 4 γ 2 + 1 / 12 Ω k 0 - π 2 .
γ 1 2 π sin - 1   k 0 ,
Ω k 0 6 π 2 n = 1 k 0 2 n n 2 .
v x n v x m = k qr σ n 2 δ x n - x m ,
h nj q = s n q c j LS = -   w n x r j x - r j x + s q d x .
Δ = s - H c LS ,
c opt LS = H T H - 1 H T s .
M jn LS = H T H - 1 H T .
r j x = exp - x - x j 2 R 2 - y - y j 2 R 2 .
k 0 = exp - π   R ξ 2 x n - x m 2 λ Z 2 exp - π   R η 2 y n - y m 2 λ Z 2 .
A c = - -   k 0 Δ x ,   Δ y | 2 d Δ x d Δ y = 1 2 λ ¯ Z 2 R ξ R η ,
R x = π D x 2 , R y = π D y 2 .

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