Abstract

We discuss how we can obtain noniteratively an exact solution for the modified normal equation in least-squares phase estimation from the measured phase difference. To achieve this, we introduce the idea of equivalent transformation of the measured phase difference into a periodic one. We perform this tranformation by using a geometric procedure based on mirror reflection. Computer simulation results indicate that the proposed noniterative algorithm works well, as was expected, and can overcome the difficulties caused by such singularities as isolated zero points.

© 1988 Optical Society of America

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  1. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
    [CrossRef]
  2. K. T. Knox, “Image retrieval from astronomical speckle patterns,” J. Opt. Soc. Am. 66, 1236–1239 (1976).
    [CrossRef]
  3. J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [CrossRef]
  4. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  5. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  6. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  7. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  8. R. L. Frost, C. K. Rushforth, B. S. Baxter, “Fast FFT-based algorithm for phase estimation in speckle imaging,” Appl. Opt. 18, 2056–2061 (1979).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. K. R. Freischlad, C. L. 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]
  13. 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]
  14. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  15. S. Shigehiro, “Speckle image reconstruction: phase retrieval using least-squares estimation,” graduation thesis (Kyushu Institute of Technology, Kyushu, Japan, 1987).
  16. M. Tsukamura, T. Takahashi, H. Takajo, “Phase reconstruction algorithm based on least-squares estimation,” in 1986 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1986), paper 642, p. 338.

1988 (1)

1986 (1)

1985 (1)

1980 (2)

1979 (3)

1978 (1)

1977 (3)

1976 (1)

1974 (1)

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

Baxter, B. S.

Cubalchini, R.

Fiddy, M. A.

Freischlad, K. R.

Fried, D. L.

Frost, R. L.

Hardy, J. W.

Herrmann, J.

Hudgin, R. H.

Hunt, B. R.

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.

Noll, R. J.

Rushforth, C. K.

Scivier, M. S.

Shigehiro, S.

S. Shigehiro, “Speckle image reconstruction: phase retrieval using least-squares estimation,” graduation thesis (Kyushu Institute of Technology, Kyushu, Japan, 1987).

Southwell, W. H.

Takahashi, T.

H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
[CrossRef]

M. Tsukamura, T. Takahashi, H. Takajo, “Phase reconstruction algorithm based on least-squares estimation,” in 1986 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1986), paper 642, p. 338.

Takajo, H.

H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
[CrossRef]

M. Tsukamura, T. Takahashi, H. Takajo, “Phase reconstruction algorithm based on least-squares estimation,” in 1986 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1986), paper 642, p. 338.

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.

M. Tsukamura, T. Takahashi, H. Takajo, “Phase reconstruction algorithm based on least-squares estimation,” in 1986 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1986), paper 642, p. 338.

Appl. Opt. (1)

Astrophys. J. (1)

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. (9)

J. Opt. Soc. Am. A (3)

Other (2)

S. Shigehiro, “Speckle image reconstruction: phase retrieval using least-squares estimation,” graduation thesis (Kyushu Institute of Technology, Kyushu, Japan, 1987).

M. Tsukamura, T. Takahashi, H. Takajo, “Phase reconstruction algorithm based on least-squares estimation,” in 1986 Kyushu Branch Convention Record of IEE (Institute of Electrical Engineers, Kyushu, Japan, 1986), paper 642, p. 338.

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Figures (10)

Fig. 1
Fig. 1

Phase-difference sampling geometry and phase mesh points for BD (N = 4). Each line segment indicates a phase difference, and each dot is a phase point. The symbol ↺ indicates the minimum enclosures.

Fig. 2
Fig. 2

Phase difference sampling geometry and phase mesh points that are periodic in each of m and n indices with period N = 4. This figure indicates phase and phase-difference distributions over one period only.

Fig. 3
Fig. 3

Closed loops peculiar to PD.

Fig. 4
Fig. 4

Equivalent transformation of BD (N = 4) into PD.

Fig. 5
Fig. 5

Original object with isolated zero points.

Fig. 6
Fig. 6

Comparisons of least-squares phase-estimation methods in the noise-free case: (a) algorithm A and (b) algorithm A.

Fig. 7
Fig. 7

Reconstructed object for the case in which σn = 0.3 rad: (a) algorithm A, (b) one-path integration, and (c) two-path averaging.

Fig. 8
Fig. 8

Original object with π discontinuity.

Fig. 9
Fig. 9

Comparisons of least-squares phase-estimation methods in noise-free case: (a) algorithm A and (b) algorithm A.

Fig. 10
Fig. 10

Comparison of noise propagations between algorithms A and B.

Tables (2)

Tables Icon

Table 1 Noise Dependency of each Quantity in Phase-Difference Measurement Equation

Tables Icon

Table 2 Root-Mean-Square Error Characteristics for σn = 0.3 rad

Equations (59)

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ϕ m , n , m , n = 1 , , N .
S m n 1 = ϕ m n ϕ m + 1 , n p m = 1 ,…, N 1 , n = 1 ,…, N , S m n 2 = ϕ m n ϕ m + 1 , n p m = 1 ,…, N , n = 1 ,…, N 1 ,
S ˜ m n l = S m n l + n m n l + 2 π L m n l = S ˜ m n l + n m n l , l = 1 , 2 ,
S ¯ m n l S ˜ m n l n m n l = S m n l + 2 π L m n l ;
2 π D m n R m n ( { S ¯ i j l } ) = R m n ( { S i j l } ) + R m n ( { 2 π L i j l } ) = 0 for m , n = 1 , , N 1 ,
R m n ( { S ¯ i j l } ) = S ¯ m n l + S ¯ m + 1 , n 2 S ¯ m , n + 1 1 S ¯ m n 2 .
R m n ( { S i j l } ) = 2 π M m n ,
| S m n l + n m n l | > π .
S ˜ ˜ m n l = S ˜ m n l + 2 π K m n l = S ¯ ¯ m n l + n m n l ,
S ¯ ¯ m n l S ˜ ˜ m n l n m n l = S m n l + 2 π L m n l + 2 π K m n l .
K m n l = 0 , 1 m N 1 , 1 n N , K 1 n 2 = 0 , 1 n N 1 , K m + 1 , n 2 = l = 1 m D ln , 1 m N 1 , 1 n N 1 ,
R m n ( { S ¯ ¯ i j l } ) = 0 for m , n = 1 ,…, N 1 ,
4 ϕ ˇ m n ϕ ˇ m 1 , n ϕ ˇ m , n 1 ϕ ˇ m + 1 , n ϕ ˇ m , n + 1 = S ˜ ˜ m n 2 S ˜ ˜ m , n 1 2 + S ˜ ˜ m n 1 S ˜ ˜ m 1 , n 1 , 2 m N 1 , 2 n N 1 ,
3 ϕ ˇ 1 , n ϕ ˇ 1 , n 1 ϕ ˇ z , n ϕ ˇ 1 , n + 1 = S ˜ ˜ 1 , n 2 S ˜ ˜ 1 , n 1 2 + S ˜ ˜ 1 , n 1 m = 1 , 2 n N 1 ,
3 ϕ ˇ N , n ϕ ˇ N 1 , n ϕ ˇ N , n 1 ϕ ˇ N , n + 1 = S ˜ ˜ N , n 2 S ˜ ˜ N , n 1 2 + S ˜ ˜ N 1 , n 1 , m = N , 2 n N 1 ,
3 ϕ ˇ m , 1 ϕ ˇ m 1 , 1 ϕ ˇ m + 1 , 1 ϕ ˇ m , 2 = S ˜ ˜ m , 1 2 + S ˜ ˜ m , 1 1 S ˜ ˜ m 1 , 1 1 , 2 m N 1 , n = 1 ,
3 ϕ ˇ m , N ϕ ˇ m 1 , N ϕ ˇ m , N 1 ϕ ˇ m + 1 , N = S ˜ ˜ m , N 1 2 + S ˜ ˜ m , N 1 S ˜ ˜ m 1 , N 1 , 2 m N 1 , n = N ,
2 ϕ ˇ 1 , 1 ϕ ˇ 2 , 1 ϕ ˇ 1 , 2 = S ˜ ˜ 1 , 1 2 + S ˜ ˜ 1 , 1 1 , m = n = 1 ,
2 ϕ ˇ N , 1 ϕ ˇ N 1 , 1 ϕ ˇ N , 2 = S ˜ ˜ N , 1 2 + S ˜ ˜ N 1 , 1 1 , m = N , n = 1 ,
2 ϕ ˇ 1 , N ϕ ˇ 1 , N 1 ϕ ˇ 2 , N = S ˜ ˜ 1 , N 1 2 + S ˜ ˜ 1 , N 1 , m = 1 , n = N ,
2 ϕ ˇ N , N ϕ ˇ N 1 , N ϕ ˇ N , N 1 = S ˜ ˜ N , N 1 2 S ˜ ˜ N 1 , N 1 , m = n = N .
| R m n ( { n i j l } ) | < π
2 π D m n = R m n ( { S ˜ i j l } ) arg { exp [ i R m n ( { S ˜ i j l } ) ] } .
S ˜ ˜ m n l p = S ¯ ¯ m n l p + n m n l ,
R m n ( { S ¯ ¯ i j l p } ) = 0 for m , n = 1 ,…, N
m = 1 N S ¯ ¯ m n 1 p = 0 for some integer value of n , n = 1 N S ¯ ¯ m n 2 p = 0 for some integer value of m .
4 ϕ ˇ mn p ϕ ˇ m 1 , n p ϕ ˇ m , n 1 p ϕ ˇ m + 1 , n p ϕ ˇ m , n + 1 p = S ˜ ˜ m n 2 p S ˜ ˜ m , n 1 2 p + S ˜ ˜ m n 1 p S ˜ ˜ m 1 , n 1 p ,
S ˜ ˜ m n l p = S ˜ ˜ m + γ N , n + δ N l p ,
[ 4 2 cos ( 2 π N u ) 2 cos ( 2 π N υ ) ] Φ ˇ uv p = [ 1 exp ( 2 π i N u ) ] S ˜ ˜ u υ 1 p + [ 1 exp ( 2 π i N υ ) ] S ˜ ˜ u υ 2 p ,
Φ ˇ uv p = 1 N m = 1 N n = 1 N ϕ ˇ mn p exp [ 2 π i N ( m u + n υ ) ] ,
S ˜ ˜ u υ l p = 1 N m = 1 N n = 1 N S ˜ ˜ m n l p exp [ 2 π i N ( m u + n υ ) ] .
Φ ¯ uv p = [ 1 exp ( 2 π i N u ) ] S ˜ ˜ u υ 1 p + [ 1 exp ( 2 π i N υ ) ] S ˜ ˜ u υ 2 p 4 2 cos ( 2 π N u ) 2 cos ( 2 π N υ ) .
Φ ˇ 00 p = 0.
ϕ ˇ m n = ϕ ˇ mn p | { m , n ; m , n = 1 ,…, N } ,
S ˜ ˜ m n 1 p = S ˜ ˜ m n 1 , 1 m N 1 , 1 n N , S ˜ ˜ m n 2 p = S ˜ ˜ m n 2 , 1 m N , 1 n N 1.
S ˜ ˜ N , n 1 p = S ˜ ˜ 2 N , n 1 p = 0 , 1 n 2 N , S ˜ ˜ m , N 2 p = S ˜ ˜ m , 2 N 2 p = 0 , 1 m 2 N .
S ˜ ˜ 2 N m , n 1 p = S ˜ ˜ m , n 1 p , 1 m N 1 , 1 n N , S ˜ ˜ ( 2 N + 1 ) m , n 2 p = S ˜ ˜ m , n 2 p , 1 m N , 1 n N 1.
S ˜ ˜ m , ( 2 N + 1 ) n 1 p = S ˜ ˜ m n 1 p , 1 m 2 N 1 , 1 n N , S ˜ ˜ m , 2 N n 2 p = S ˜ ˜ m n 2 p , 1 m 2 N , 1 n N 1.
4 ϕ ˇ mn p ϕ ˇ m 1 , n p ϕ ˇ m , n 1 p ϕ ˇ m + 1 , n p ϕ ˇ m , n + 1 p = S ˜ ˜ m n 2 p S ˜ ˜ m , n 1 2 p + S ˜ ˜ m n 1 p S ˜ ˜ m 1 , n 1 p ,
S ˜ ˜ m n l p = S ˜ ˜ m + 2 γ N , n + 2 δ N l p .
ϕ ˇ mn p = ϕ ˇ ( 2 ξ N + 1 ) m , n p , ϕ ˇ mn p = ϕ ˇ m , ( 2 η N + 1 ) n p ,
S ˜ ˜ m n 1 p = S ˜ ˜ m n 1 , S ˜ ˜ m n 2 p = S ˜ ˜ m n 2
4 ϕ ˇ mn p ϕ ˇ m 1 , n p ϕ ˇ m , n 1 p ϕ ˇ m + 1 , n p ϕ ˇ m , n + 1 p = S ˜ ˜ m n 2 S ˜ ˜ m , n 1 2 + S ˜ ˜ m n 1 S ˜ ˜ m 1 , n 1 .
4 ϕ ˇ N , n p ϕ ˇ N 1 , n p ϕ ˇ N , n 1 p ϕ ˇ N + 1 , n p ϕ ˇ N , n + 1 p = S ˜ ˜ N , n 2 p S ˜ ˜ N , n 1 2 p + S ˜ ˜ N , n 1 p S ˜ ˜ N 1 , n 1 p .
S ˜ ˜ N , n 1 p = 0 ,
ϕ ˇ N , n p = ϕ ˇ N + 1 , n p
3 ϕ ˇ N , n p ϕ ˇ N 1 , n p ϕ ˇ N , n 1 p ϕ ˇ N , n + 1 p = S ˜ ˜ N , n 2 p S ˜ ˜ N , n 1 2 p S ˜ ˜ N 1 , n 1 p .
3 ϕ ˇ N , n p ϕ ˇ N 1 , n p ϕ ˇ N , n 1 p ϕ ˇ N , n + 1 p = S ˜ ˜ N , n 2 S ˜ ˜ N , n 1 2 S ˜ ˜ N 1 , n 1 .
4 ϕ ˇ N , N p ϕ ˇ N 1 , N p ϕ ˇ N , N 1 p ϕ ˇ N + 1 , N p ϕ ˇ N , N + 1 p = S ˜ ˜ N , N 2 p S ˜ ˜ N , N 1 2 p + S ˜ ˜ N , N 1 p S ˜ ˜ N 1 , N 1 p .
S ˜ ˜ N , N 1 p = S ˜ ˜ N , N 2 p = 0.
ϕ ˇ N , N p = ϕ ˇ N + 1 , N p , ϕ ˇ N , N p = ϕ ˇ N , N + 1 p .
2 ϕ ˇ N , N p ϕ ˇ N 1 , N p ϕ ˇ N , N 1 p = S ˜ ˜ N , N 1 2 p S ˜ ˜ N 1 , N 1 p = S ˜ ˜ N , N 1 2 S ˜ ˜ N 1 , N 1 .
σ ϕ = { 1 N 2 m , n [ ( ϕ ˇ m n 1 N 2 m , n ϕ ˇ m n ) ( ϕ ¯ m n 1 N 2 m , n ϕ ¯ m n ) ] 2 } 1 / 2
S ˜ ˜ N , n 1 p = m = 1 N 1 S ˜ ˜ m n 1 p , 1 n N , S ˜ ˜ m , N 2 p = n = 1 N 1 S ˜ ˜ m n 2 p 1 m N .
ϕ ˇ m n ( 2 q ) α m n as q , ϕ ˇ m n ( 2 q + 1 ) β m n as q ,
ϕ ˇ ( 100 ) = [ 2.745358 2.429404 2.557477 2.067088 2.055655 3.714018 1.725927 2.669645 1.759197 2.434964 0.868967 1.540443 1.339599 0.092026 0.339499 0.198751 ] ,
ϕ ˇ ( 101 ) = [ 3.088791 2.772838 2.214045 1.723655 1.712222 3.370585 2.069361 3.013079 2.102629 2.778397 0.525535 1.197010 0.996168 0.435458 0.682931 0.144681 ] ,
ϕ ˇ ave = [ 2.917075 2.601121 2.385761 1.895371 1.883938 3.542301 1.897644 2.841362 1.930912 2.606680 0.697251 1.368727 1.167883 0.263742 0.511215 0.027035 ] .
ϕ ˇ = [ 2.917068 2.601117 2.385756 1.895368 1.883935 3.542295 1.897642 2.841356 1.930911 2.606675 0.697252 1.368727 1.167884 0.263741 0.511214 0.027034 ] .

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