Abstract

A novel method is presented for estimating a phase sequence from an ensemble of measured phase sequences that have noise and ambiguities of integral multiples of 2π rad. The maximum likelihood equation is approximately solved at a small measurement-error limit to obtain the new estimator. Statistical performance of the new estimator and the result of a numerical simulation of high-resolution imaging through the turbulent atmosphere are compared with those of the conventional phase unwrapping method which eliminates stepwise the 2π ambiguities by assuming some continuity constraints.

© 1983 Optical Society of America

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References

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  1. J. R. Fienup, Opt. Eng. 18, 529 (1979).
    [CrossRef]
  2. J. G. Walker, Appl. Opt. 21, 3132 (1982).
    [CrossRef] [PubMed]
  3. B. L. McGlamery, NASA Spec. Publ, SP-256 (U.S. GPO, Washington, D.C., 1971), p. 167.
  4. K. T. Knox, B. J. Thompson, Astrophys. J. 193, L45 (1974).
    [CrossRef]
  5. A. H. Greenaway, J. C. Dainty, Opt. Acta 25, 181 (1978).
    [CrossRef]
  6. C. Roddier, F. Roddier, in Image Formation from Coherence Function in Astronomy, C. van Schooneveld, Ed. (Reidel, Dordrecht, 1979), p. 175.
    [CrossRef]
  7. L. N. Mertz, Appl. Opt. 18, 611 (1979).
    [CrossRef] [PubMed]
  8. K. Itoh, Y. Ohtsuka, Appl. Opt. 20, 4239 (1981).
    [CrossRef] [PubMed]
  9. E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, Opt. Acta 29, 701 (1982).
    [CrossRef]
  10. J. F. Walkup, J. W. Goodman, J. Opt. Soc. Am 63, 399 (1973).
    [CrossRef]
  11. J. C. Wyant, Appl. Opt. 14, 2622 (1975).
    [CrossRef] [PubMed]
  12. K. Itoh, Y. Ohtsuka, J. Opt. Soc. Am. 73, 479 (1983).
    [CrossRef]
  13. H. L. Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York, 1968), Sec. 2.4.
  14. K. Itoh, Appl. Opt. 21, 2470 (1982).
    [CrossRef] [PubMed]
  15. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Wave from Rough Surfaces (Pergamon, New York1963), Sec. 7.1.
  16. See, for example, B. R. Hunt, J. Opt. Soc. Am. 69, 393 (1979).
    [CrossRef]

1983 (1)

1982 (3)

K. Itoh, Appl. Opt. 21, 2470 (1982).
[CrossRef] [PubMed]

J. G. Walker, Appl. Opt. 21, 3132 (1982).
[CrossRef] [PubMed]

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, Opt. Acta 29, 701 (1982).
[CrossRef]

1981 (1)

1979 (3)

1978 (1)

A. H. Greenaway, J. C. Dainty, Opt. Acta 25, 181 (1978).
[CrossRef]

1975 (1)

1974 (1)

K. T. Knox, B. J. Thompson, Astrophys. J. 193, L45 (1974).
[CrossRef]

1973 (1)

J. F. Walkup, J. W. Goodman, J. Opt. Soc. Am 63, 399 (1973).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Wave from Rough Surfaces (Pergamon, New York1963), Sec. 7.1.

Cocke, W. J.

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, Opt. Acta 29, 701 (1982).
[CrossRef]

Dainty, J. C.

A. H. Greenaway, J. C. Dainty, Opt. Acta 25, 181 (1978).
[CrossRef]

Fienup, J. R.

J. R. Fienup, Opt. Eng. 18, 529 (1979).
[CrossRef]

Goodman, J. W.

J. F. Walkup, J. W. Goodman, J. Opt. Soc. Am 63, 399 (1973).
[CrossRef]

Greenaway, A. H.

A. H. Greenaway, J. C. Dainty, Opt. Acta 25, 181 (1978).
[CrossRef]

Hege, E. K.

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, Opt. Acta 29, 701 (1982).
[CrossRef]

Hubbard, E. N.

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, Opt. Acta 29, 701 (1982).
[CrossRef]

Hunt, B. R.

Itoh, K.

Knox, K. T.

K. T. Knox, B. J. Thompson, Astrophys. J. 193, L45 (1974).
[CrossRef]

McGlamery, B. L.

B. L. McGlamery, NASA Spec. Publ, SP-256 (U.S. GPO, Washington, D.C., 1971), p. 167.

Mertz, L. N.

Ohtsuka, Y.

Roddier, C.

C. Roddier, F. Roddier, in Image Formation from Coherence Function in Astronomy, C. van Schooneveld, Ed. (Reidel, Dordrecht, 1979), p. 175.
[CrossRef]

Roddier, F.

C. Roddier, F. Roddier, in Image Formation from Coherence Function in Astronomy, C. van Schooneveld, Ed. (Reidel, Dordrecht, 1979), p. 175.
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Wave from Rough Surfaces (Pergamon, New York1963), Sec. 7.1.

Strittmatter, P. A.

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, Opt. Acta 29, 701 (1982).
[CrossRef]

Thompson, B. J.

K. T. Knox, B. J. Thompson, Astrophys. J. 193, L45 (1974).
[CrossRef]

Trees, H. L.

H. L. Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York, 1968), Sec. 2.4.

Walker, J. G.

Walkup, J. F.

J. F. Walkup, J. W. Goodman, J. Opt. Soc. Am 63, 399 (1973).
[CrossRef]

Wyant, J. C.

Appl. Opt. (5)

Astrophys. J. (1)

K. T. Knox, B. J. Thompson, Astrophys. J. 193, L45 (1974).
[CrossRef]

J. Opt. Soc. Am (1)

J. F. Walkup, J. W. Goodman, J. Opt. Soc. Am 63, 399 (1973).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Acta (2)

A. H. Greenaway, J. C. Dainty, Opt. Acta 25, 181 (1978).
[CrossRef]

E. K. Hege, E. N. Hubbard, P. A. Strittmatter, W. J. Cocke, Opt. Acta 29, 701 (1982).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, Opt. Eng. 18, 529 (1979).
[CrossRef]

Other (4)

H. L. Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York, 1968), Sec. 2.4.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Wave from Rough Surfaces (Pergamon, New York1963), Sec. 7.1.

B. L. McGlamery, NASA Spec. Publ, SP-256 (U.S. GPO, Washington, D.C., 1971), p. 167.

C. Roddier, F. Roddier, in Image Formation from Coherence Function in Astronomy, C. van Schooneveld, Ed. (Reidel, Dordrecht, 1979), p. 175.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Mean values of the conventional phase estimator plotted as functions of σΔ for various values of ΔΘ: ΔΘ = 0.1, 0.3, 0.5, 0.7, 0.9 π rad.

Fig. 2
Fig. 2

Standard deviations of the conventional phase estimator (symbols) and the new estimator (solid line) plotted as functions of σΔ for various values of ΔΘ. The same symbols are used as in Fig. 1.

Fig. 3
Fig. 3

Results of numerical simulation of interferometric imaging: (a) the original object, (b) an image reconstructed with a phase component estimated by the conventional method, (c) an image reconstructed with a phase component estimated by the new method.

Equations (33)

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Z k ( n ) = | Z k ( n ) | exp { i [ Θ ( n ) + Ψ k ( n ) + 2 π X k ( n ) ] } ,
W Φ k ( n ) = Θ ( n ) + Ψ k ( n ) + 2 π λ k ( n ) ,
π < W Φ k ( n ) π .
Δ Φ k ( n ) = Φ k ( n ) Φ k ( n 1 ) , ( n > 1 ) .
W ( 2 ) Δ W ( 1 ) Φ k ( n ) = Δ Θ ( n ) + Δ Ψ k ( n ) + 2 π { Δ λ k ( 1 ) ( n ) + λ k ( 2 ) ( n ) } ,
f Λ Φ k Δ Θ ( Λ Φ k Δ Θ ) = 1 2 π σ Δ m = × exp [ ( Λ Φ k Δ Θ 2 m π ) 2 2 σ Δ 2 ] .
k = 1 K f Λ Φ k Δ Θ ( Λ Φ k Δ Θ ) .
Δ Θ [ ln k = 1 K f Λ Φ k Δ Θ ( Λ Φ k Δ Θ ) ] = 0 .
| Δ Θ + Δ Ψ k | < π ,
Δ λ k ( 1 ) + λ k ( 2 ) = 0 .
f Λ Φ k Δ Θ ( Λ Φ k Δ Θ ) = 1 2 π σ Δ exp [ ( Λ Φ k Δ Θ ) 2 2 σ Δ 2 ] .
Δ Θ ˆ 1 = K 1 k = 1 K Λ Φ k .
σ 1 2 = ( Δ Θ ˆ 1 Δ Θ ) 2 = σ Δ 2 / K ,
| Δ Θ | + σ Δ π .
σ Δ π .
m = exp [ ( Λ Φ k Δ Θ 2 m π ) 2 2 σ Δ 2 ] exp [ 1 cos ( Λ Φ k Δ Θ ) σ Δ 2 ] .
f Λ Φ k Δ Θ ( Λ Φ k Δ Θ ) 1 2 π σ δ exp [ 1 cos ( Λ Φ k Δ Θ ) σ Δ 2 ] .
Δ Θ ˆ 2 = arg k = 1 K exp ( i Λ Φ k ) .
Δ Θ ˆ 1 = K 1 k = 1 K σ k 2 Λ Φ k ,
Δ Θ ˆ 2 = arg k = 1 K σ k 2 exp ( i Λ Φ i ) .
F Λ Φ k ( ω ) = m = exp ( i m Δ Θ m 2 σ Δ 2 2 ) sin π ( ω m ) π ( ω m ) .
M 1 = 2 m = 1 ( 1 ) m 1 sin ( m Δ Θ ) exp ( m 2 σ Δ 2 / 2 ) m ,
M 2 = π 2 3 4 m = 1 ( 1 ) m 1 cos ( m Δ Θ ) exp ( m 2 σ Δ 2 / 2 ) m 2 .
η 1 = M 1 ,
σ 1 2 = ( M 2 M 1 2 ) / K .
X ̅ + i Y ̅ = k = 1 K exp [ i ( Λ Φ k Δ Θ ) ] .
η x ̅ = K exp ( σ Δ 2 / 2 ) ,
η y ̅ = 0 ,
σ x ̅ 2 = K [ 1 exp ( σ Δ 2 ) ] 2 / 2 ,
σ y ̅ 2 = K [ 1 exp ( 2 σ Δ 2 ) ] / 2 .
f 2 ( Δ Θ ˆ 2 Δ Θ ) = 1 2 π σ x ̅ σ y ̅ 0 r exp { [ r cos ( Δ Θ ˆ 2 Δ Θ ) η x ̅ ] 2 2 σ x ̅ 2 r 2 sin 2 ( Δ Θ ˆ 2 Δ Θ ) 2 σ y ̅ 2 } d r .
σ x ̅ 2 / σ y ̅ 2 = σ Δ 2 / 2 .
σ 2 2 ( σ y ̅ / η x ̅ ) 2 = sinh ( σ Δ 2 ) / K .

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