Abstract

Obtaining robust phase estimates from phase differences is a problem common to several areas of importance to the optics and signal-processing communities. Specific areas of application include speckle imaging and interferometry, adaptive optics, compensated imaging, and coherent imaging such as synthetic-aperture radar. We derive in a concise form the equations describing the phase-estimation problem, relate these equations to the general form of elliptic partial differential equations, and illustrate results of reconstructions on large M by N grids, using existing, published, and readily available FORTRAN subroutines.

© 1989 Optical Society of America

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References

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1988 (4)

1979 (1)

1978 (1)

1977 (3)

1970 (1)

B. L. Busbee, G. H. Golub, C. W. Nielson, SIAM J. Numer. Anal. 7, 627 (1970).
[CrossRef]

Brown, W. D.

Busbee, B.

B. Busbee, Source and Development of Mathematical Software, W. R. Cowell, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Busbee, B. L.

B. L. Busbee, G. H. Golub, C. W. Nielson, SIAM J. Numer. Anal. 7, 627 (1970).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (U. Cambridge Press, Cambridge, 1986).

Fried, D. L.

Ghiglia, D. C.

Golub, G. H.

B. L. Busbee, G. H. Golub, C. W. Nielson, SIAM J. Numer. Anal. 7, 627 (1970).
[CrossRef]

Hardy, J. W.

Hudgin, R. H.

Hunt, B. R.

Koliopoulos, C. L.

Lefebvre, J. E.

Nielson, C. W.

B. L. Busbee, G. H. Golub, C. W. Nielson, SIAM J. Numer. Anal. 7, 627 (1970).
[CrossRef]

Noll, R. J.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (U. Cambridge Press, Cambridge, 1986).

Swartztrauber, P.

P. Swartztrauber, R. Sweet, Rep. NCAR-TN/1A-109 (National Center for Atmospheric Research, Boulder, Colo., 1975).

Sweet, R.

P. Swartztrauber, R. Sweet, Rep. NCAR-TN/1A-109 (National Center for Atmospheric Research, Boulder, Colo., 1975).

Takahashi, T.

Takajo, H.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (U. Cambridge Press, Cambridge, 1986).

Vandevender, W. H.

W. H. Vandevender, Rep. SAND84-0281 (Sandia National Laboratories, Albuquerque, N.M., 1984).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (U. Cambridge Press, Cambridge, 1986).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (4)

SIAM J. Numer. Anal. (1)

B. L. Busbee, G. H. Golub, C. W. Nielson, SIAM J. Numer. Anal. 7, 627 (1970).
[CrossRef]

Other (5)

B. Busbee, Source and Development of Mathematical Software, W. R. Cowell, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1984).

W. H. Vandevender, Rep. SAND84-0281 (Sandia National Laboratories, Albuquerque, N.M., 1984).

SLATEC software is available through the National Energy Software Center, Department of Energy, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (U. Cambridge Press, Cambridge, 1986).

P. Swartztrauber, R. Sweet, Rep. NCAR-TN/1A-109 (National Center for Atmospheric Research, Boulder, Colo., 1975).

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Figures (1)

Fig. 1
Fig. 1

A, Arbitrary phase function scaled for display. B, Phase differences in the x direction with added noise. C, Phase differences in the y direction with added noise. D, Reconstructed phase from noisy phase differences. A few constant phase contours are superimposed.

Equations (18)

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ψ i , j = ϕ i , j + 2 π k , k an integer , 0 ψ i , j < 2 π , i = 1 M , j = 1 N .
α i + 1 / 2 , j = ψ i + 1 , j ψ i , j , i = 1 M 1 , j = 1 N ,
β i , j + 1 / 2 = ψ i , j + 1 ψ i , j , i = 1 M , j = 1 N 1 .
ϕ i + 1 , j ϕ i , j = α i + 1 / 2 , j , i = 1 M 1 , j = 1 N ,
ϕ i , j + 1 ϕ i , j = β i , j + 1 / 2 , i = 1 M , j = 1 N 1 .
i = 1 M 1 j = 1 N ( ϕ i + 1 , j ϕ i , j α i + 1 / 2 , j ) 2 + i = 1 M j = 1 N 1 ( ϕ i , j + 1 ϕ i , j β i , j + 1 / 2 ) 2 .
a i ( ϕ i + 1 , j ϕ i , j ) a i 1 ( ϕ i , j ϕ i 1 , j ) + b j ( ϕ i , j + 1 ϕ i , j ) b j 1 ( ϕ i , j ϕ i , j 1 ) = a i α i + 1 / 2 , j a i 1 α i 1 / 2 , j + b j β i , j + 1 / 2 b j 1 β i , j 1 / 2 , i = 1 M , j = 1 N ; a i = 1 , i = 1 M 1 ; a 0 = a M = 0 ; b j = 1 , j = 1 N 1 ; b 0 = b N = 0 .
Φ = { ϕ i , j } ,
Γ = { α i + 1 / 2 , j , β i , j + 1 / 2 } .
Φ , Φ ˆ 1 = i = 1 M j = 1 N ϕ i , j ϕ ˆ i , j
Γ , Γ ˆ 2 = i = 1 M 1 j = 1 N α i + 1 / 2 , j α ˆ i + 1 / 2 , j + i = 1 M j = 1 N 1 β i , j + 1 / 2 β ˆ i , j + 1 / 2 .
H Φ = Γ ,
α i + 1 / 2 , j = ϕ i + 1 , j ϕ i , j , i = 1 M 1 , j = 1 N , β i , j + 1 / 2 = ϕ i , j + 1 ϕ i , j , i = 1 M , j = 1 N 1 .
H * H Φ = H * Γ 0 ,
Γ , H Φ 2 = i = 1 M 1 j = 1 N ( ϕ i + 1 , j ϕ i , j ) α i + 1 / 2 , j + i = 1 M j = 1 N 1 ( ϕ i , j + 1 ϕ i , j ) β i , j + 1 / 2 .
Γ , H Φ 2 = i = 1 M j = 1 N ( a i 1 α i 1 / 2 , j a i α i + 1 / 2 , j + b j 1 β i , j 1 / 2 b j β i , j + 1 / 2 , ϕ i , j ,
a i = 1 , i = 1 M 1 ; a 0 = a M = 0 ; b j = 1 , j = 1 N 1 ; b 0 = b N = 0 .
H * Γ = Φ , ϕ i , j = a i 1 α i 1 / 2 j a i α i + 1 / 2 , j + b j 1 β i , j 1 / 2 b j β i , j + 1 / 2 , i = 1 M , j = 1 N ; a i = 1 , i = 1 M 1 ; a 0 = a M = 0 ; b j = 1 , j = 1 N 1 ; b 0 = b N = 0 .

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