Abstract

The purpose of this paper is to clarify the relationship between the isolated zero points in modulus distribution and the least-squares phase estimation from the phase difference. The concepts of phase and phase difference are reaffirmed. In addition, a necessary condition that makes the least-squares phase estimation feasible is derived by applying the concept of complete observability in estimation theory to the measurement of the phase difference. The occurrence of isolated zero points causes the conventional least-squares phase estimation to fail because the phase difference defined by this concept does not satisfy the necessary condition when isolated zero points occur. This condition also generates a new type of least-squares phase estimation that is feasible for phase retrieval even if zero points exist. One algorithm for realizing this new type of least-squares phase estimation is proposed, and its effectiveness is verified by using computer simulations. Two types of phase unwrapping are also presented: one is the exponential function type; the other results from the proposed least-squares phase-estimation algorithm.

© 1988 Optical Society of America

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References

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  1. J. R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  2. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One dimensional considerations,” Optik 62, 131–142 (1982);W. R. Fright, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik 62, 219–230 (1982).
  3. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
    [CrossRef]
  4. K. T. Knox, “Image retrieval from astronomical speckle patterns,” J. Opt. Soc. Am. 66, 1236–1239 (1976).
    [CrossRef]
  5. J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [CrossRef]
  6. A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
    [CrossRef]
  7. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  8. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  9. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  10. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  11. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  12. M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
    [CrossRef]
  13. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  14. K. Honda, “Speckle image reconstruction: discussions on Knox–Thompson method and on image improvement by means of least square phase estimation,” M.Sc. Eng. thesis (Kyushu Institute of Technology, Kyushu, Japan, 1985).
  15. T. Takahashi, K. Honda, H. Takajo, S. Kawanaka, M. Tsukamura, “Phase estimation algorithm for speckle imaging,” in 1985 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1985), Paper no. 308, p. 128.
  16. N. E. Nahi, Estimation Theory and Application (Krieger, New York, 1976), Sec. 3.7, pp. 103–105.
  17. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 53.

1986

1985

1982

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One dimensional considerations,” Optik 62, 131–142 (1982);W. R. Fright, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik 62, 219–230 (1982).

J. R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1981

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

1980

1979

1977

1976

1974

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One dimensional considerations,” Optik 62, 131–142 (1982);W. R. Fright, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik 62, 219–230 (1982).

Cubalchini, R.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 53.

Fiddy, M. A.

Fienup, J. R.

Hardy, J. W.

Herrmann, J.

Honda, K.

T. Takahashi, K. Honda, H. Takajo, S. Kawanaka, M. Tsukamura, “Phase estimation algorithm for speckle imaging,” in 1985 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1985), Paper no. 308, p. 128.

K. Honda, “Speckle image reconstruction: discussions on Knox–Thompson method and on image improvement by means of least square phase estimation,” M.Sc. Eng. thesis (Kyushu Institute of Technology, Kyushu, Japan, 1985).

Hudgin, R. H.

Hunt, B. R.

Kawanaka, S.

T. Takahashi, K. Honda, H. Takajo, S. Kawanaka, M. Tsukamura, “Phase estimation algorithm for speckle imaging,” in 1985 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1985), Paper no. 308, p. 128.

Knox, K. T.

K. T. Knox, “Image retrieval from astronomical speckle patterns,” J. Opt. Soc. Am. 66, 1236–1239 (1976).
[CrossRef]

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Koliopoulos, C. L.

Lefebvre, J. E.

Lim, J. S.

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 53.

Nahi, N. E.

N. E. Nahi, Estimation Theory and Application (Krieger, New York, 1976), Sec. 3.7, pp. 103–105.

Oppenheim, A. V.

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Scivier, M. S.

Southwell, W. H.

Takahashi, T.

T. Takahashi, K. Honda, H. Takajo, S. Kawanaka, M. Tsukamura, “Phase estimation algorithm for speckle imaging,” in 1985 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1985), Paper no. 308, p. 128.

Takajo, H.

T. Takahashi, K. Honda, H. Takajo, S. Kawanaka, M. Tsukamura, “Phase estimation algorithm for speckle imaging,” in 1985 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1985), Paper no. 308, p. 128.

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Tsukamura, M.

T. Takahashi, K. Honda, H. Takajo, S. Kawanaka, M. Tsukamura, “Phase estimation algorithm for speckle imaging,” in 1985 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1985), Paper no. 308, p. 128.

Wackerman, C. C.

Appl. Opt.

Astrophys. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Optik

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik 61, 247–262 (1982);K. L. Garden, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One dimensional considerations,” Optik 62, 131–142 (1982);W. R. Fright, R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik 62, 219–230 (1982).

Proc. IEEE

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Other

K. Honda, “Speckle image reconstruction: discussions on Knox–Thompson method and on image improvement by means of least square phase estimation,” M.Sc. Eng. thesis (Kyushu Institute of Technology, Kyushu, Japan, 1985).

T. Takahashi, K. Honda, H. Takajo, S. Kawanaka, M. Tsukamura, “Phase estimation algorithm for speckle imaging,” in 1985 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1985), Paper no. 308, p. 128.

N. E. Nahi, Estimation Theory and Application (Krieger, New York, 1976), Sec. 3.7, pp. 103–105.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 53.

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Figures (9)

Fig. 1
Fig. 1

Original object.

Fig. 2
Fig. 2

Phase distribution of the original object.

Fig. 3
Fig. 3

(a) Relation between boundary curves (AA′, BB′, …) from white to black and (b) isolated zero-point pairs (A–A′, B–B′, …).

Fig. 4
Fig. 4

Comparisons of least-squares phase-estimation methods in the noise-free case: (a) the conventional method and (b) our proposed method.

Fig. 5
Fig. 5

Relation of rms error to the number of iterations in our proposed algorithms.

Fig. 6
Fig. 6

Reconstructed object for the case in which σn = 0.2 rad: (a) one-path integration, (b) two-path averaging, and (c) least-squares estimation.

Fig. 7
Fig. 7

Reconstructed object for the case in which σn = 0.4 rad: (a) one-path integration, (b) two-path averaging, and (c) least-squares estimation.

Fig. 8
Fig. 8

Comparison of convergence speed dependent on the choice of initial values in case of σn = 0.2 rad.

Fig. 9
Fig. 9

Reconstructed object for the case in which σn = 0.45 rad: (a) two-path averaging and (b) least-squares estimation.

Tables (1)

Tables Icon

Table 1 Rms Error Characteristics

Equations (49)

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s c + n c = ϕ ( = î x f x + î y f y ) ,
s c = ϕ ,
c s c · d f = 0
× s c = 0 ,
ϕ ( f ) = f 0 f s c · d f + C ,
s c = s c L + s c T ,
× s c L = 0 ,
· s c T = 0 .
1 S s [ ϕ ̌ ( f ) ( f 0 f s c L · d f + C ) ] 2 d S = 0
ϕ ̌ = f 0 f s c L · d f + C = f 0 f ( s c s c T ) · d f + C ( almost everywhere ) ,
ϕ ̌ ( f ) = f 0 f s c · d f + C .
ϕ i j ( i , j = 1 , , N )
S i j 1 = ϕ j i ϕ i + 1 , j ( i = 1 , , N 1 , j = 1 , , N ) , S i j 2 = ϕ i j ϕ i , j + 1 ( i = 1 , , N , j = 1 , , N 1 ) .
s + n = [ P ] ϕ ,
s = [ P ] ϕ ,
[ P ] = [ [ D 1 ] [ D 1 ] [ D 1 ] [ D 1 ] I I I I I I I I ] ,
[ D 1 ] = [ 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 · · · · · · · · · · 0 0 1 1 ]
c S i j l = 0 ,
R i j ( { S p q l } ) = S i j 1 + S i + 1 , j 2 S i , j + 1 1 S i j 2 .
R i j ( { S p q l } ) = 0 ( i , j = 1 , 2 , , N 1 ) .
ϕ ̌ = { [ P ] T [ P ] } 1 [ P ] T [ s + n ] .
ϕ ̌ = ¼ ( ϕ ̂ i 1 , j + ϕ ̌ i , j 1 + ϕ ̌ i + 1 , j + ϕ ̌ i , j + 1 + S i j 2 S i , j 1 2 + S i j 1 S i 1 , j 1 )
ϕ n = ϕ p + 2 π n ( n = 0 , ± 1 , ± 2 , )
S m = ϕ 1 ϕ 2 p + 2 π m ( m = 0 , ± 1 , ± 2 , )
c 1 S i j l = 2 π
c 2 S i j l = 2 π
c 3 S i j l = 0 .
c s c · d f = i 2 π m i
× s c = i 2 π n ̂ i δ ( f f i ) ,
s c = ϕ + × A .
c ( × A ) · d f = c s c · d f = i 2 π m i
s c T ( f ) = × A = i n ̂ 1 × ( f f i ) ( f f i ) 2 .
ϕ ̌ ( f ) = f 0 f [ s c i n ̂ 1 × ( f f i ) ( f f i ) 2 ] · d f + C .
R i j ( { S p q l } ) = 2 π M i j ,
S i j l = S i j l + n i j l + 2 π L i j l ,
S i j l = S i j l + 2 π K i j l = S ¯ i j l + n i j l
S ¯ i j l = S i j l + 2 π L i j l + 2 π K i j l .
R i j ( { S ¯ p q l } ) = 0 .
R i j ( { K p q l } ) = D i j ,
2 π D i j = R i j ( { S p q l } + { 2 π L p q l } ) .
K i j 1 = 0 for 1 i N 1 , 1 j N , K 1 j 2 = 0 for 1 j N 1.
K i + 1 , j 2 = l = 1 i D l j , 1 i N 1 , 1 j N 1.
| R i j ( { n p q l } ) | < π
2 π D i j = R i j ( { S p q l } ) arg { exp [ i R i j ( { S p q l } ) ] }
σ ϕ = { N 2 i , j [ ( ϕ ̌ i j N 2 l , k ϕ ̌ l k ) ( ϕ ¯ i j N 2 l , k ϕ ¯ l k ) ] 2 } 1 / 2 .
2 π D i j = R i j ( { S p q l } ) R i j ( { n p q l } )
arg { exp [ i R i j ( { S p q l } ) ] } = arg ( exp { i [ 2 π D i j + R i j ( { n p q l } ) ] } )
2 π D i j = 2 π M i j + R i j ( { 2 π L p q l } ) ;
arg { exp [ i R i j ( { S p q l } ) ] } = arg { exp [ i R i j ( { n p q l } ) ] } = R i j ( { n p q l } ) .

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