Abstract

It is shown that a wave front, or in general any scalar two-dimensional function, can be reconstructed from its discrete differences by a simple multiplicative filtering operation in the spatial-frequency domain by using complex exponentials as basis functions in a modal expansion. Various difference-sampling geometries are analyzed. The difference data are assumed to be corrupted by random, additive noise of zero mean. The derived algorithms yield unbiased reconstructions for finite data arrays. The error propagation from the noise on the difference data to the reconstructed wave fronts is minimal in a least-squares sense. The spatial distribution of the reconstruction error over the array and the dependence of the mean reconstruction error on the array size are determined. The algorithms are computationally efficient, noniterative, and suitable for large arrays since the required number of mathematical operations for a reconstruction is approximately proportional to the number of data points if fast-Fourier-transform algorithms are employed.

© 1986 Optical Society of America

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References

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  1. J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [CrossRef]
  2. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  3. G. Makosch, B. Drollinger, “Surface profile measurement with a scanning differential ac interferometer,” Appl. Opt. 23, 4544–4553 (1984).
    [CrossRef] [PubMed]
  4. K. T. Knox, “Image retrieval from astronomical speckle patterns,” J. Opt. Soc. Am. 66, 1236–1239 (1976).
    [CrossRef]
  5. M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef] [PubMed]
  6. 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]
  7. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  8. J. R. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  9. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  10. 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]
  11. J. Herrmann, “Least-squares wave-front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  12. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  13. R. Cubalchini, “Modal wave-front estimation from phase derivative meaurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  14. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. 71, 989–992 (1981).
    [CrossRef]
  15. Ref. 6, Eq. (15).
  16. B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, New York, 1983), Chaps. 14.4 and 14.8.2.
    [CrossRef]
  17. Ref. 8, Eq. (13).

1984 (1)

1981 (1)

1980 (2)

1979 (3)

1978 (2)

J. R. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
[CrossRef]

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1977 (3)

1976 (1)

1974 (1)

Baxter, B. S.

Cubalchini, R.

Drollinger, B.

Fried, D. L.

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, New York, 1983), Chaps. 14.4 and 14.8.2.
[CrossRef]

Frost, R. L.

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
[CrossRef]

Herrmann, J.

Hudgin, R. H.

Hunt, B. R.

Knox, K. T.

Koliopoulos, C. L.

Lefebvre, J. E.

Makosch, G.

Noll, J. R.

Rimmer, M. P.

Rushforth, C. K.

Southwell, W. H.

Appl. Opt. (3)

J. Opt. Soc. Am. (10)

Proc. IEEE (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Other (3)

Ref. 6, Eq. (15).

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, New York, 1983), Chaps. 14.4 and 14.8.2.
[CrossRef]

Ref. 8, Eq. (13).

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Figures (5)

Fig. 1
Fig. 1

Difference-sampling geometries. ○, wave-front points W; → difference data Sx, Sy.

Fig. 2
Fig. 2

Point-spread functions for configuration A, 64 by 64 array. The source consists of an x difference of value one. (a) Source at (33, 1). Difference peak to valley (P-V) = 0.6737. (b) Source at (1, 1). P-V = 0.6366. (c) Source at (17, 15). P-V = 0.5007. (d) Source at (1, 33). P-V = 0.5003.

Fig. 3
Fig. 3

Point-spread functions for configuration B, 64 by 64 array. The source consists of an x difference of value one. (a) Source at (33, 1). P-V = 0.7505. (b) Source at (1, 1). P-V = 1.0. (c) Source at (17, 15). P-V = 0.5042. (d) Source at (1, 33). P-V = 0.7505.

Fig. 4
Fig. 4

Spatial distribution of the variance of the reconstruction error, 64 by 64 array, (a) Configuration A. P-V = 9.3648. (b) Configuration B. P-V = 15.3356.

Fig. 5
Fig. 5

Mean variance of the reconstruction error. Curve 1: configuration A. Curve 2: configuration A, periodic wave front. Curve 3: configuration A. Curve 4: configuration B. Curve 5: configuration B, periodic wave front. Curve 6: configuration B.

Equations (64)

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S ( m , n ) = W ( m , n ) + n ( m , n ) ,
S = ( S x S y ) , n = ( n x n y ) ,
x W ( m , n ) = W ( m , n + 1 ) W ( m , n )
y W ( m , n ) = W ( m + 1 , n ) W ( m , n ) ;
x W ( m , n ) = W ( m , n + 1 2 ) W ( m , n 1 2 ) ,
y W ( m , n ) = W ( m + 1 2 , n ) W ( m 1 2 , n ) ;
x W ( m , n ) = W ( m + 1 , n + 1 ) W ( m , n ) ,
y W ( m , n ) = W ( m + 1 , n ) W ( m , n + 1 ) ;
x W ( m , n ) = W ( m + 1 2 , n + 1 ) W ( m + 1 2 , n )
y W ( m , n ) = W ( m + 1 , n + 1 2 ) W ( m , n + 1 2 ) .
W ( m , n ) = p , q a p q Z p q ( m , n ) .
Z p q = Z p q = ( Z x , p q Z y , p q )
W ( m , n ) = p , q a p q Z p q ( m , n ) .
F = m , n = 1 N [ S ( m , n ) W ˜ ( m , n ) ] 2
F = m , n = 1 N [ S ( m , n ) p , q a ˜ p q Z p q ( m , n ) ] 2 .
Z p q ( m , n ) = 1 N exp [ 2 π i N ( p m + q n ) ] , 1 m , n N , 0 p , q N 1.
p , q = 0 N 1 Z p q ( m , n ) Z * p q ( m , n ) = δ ( m , m ) δ ( n , n ) ,
m , n = 1 N Z p q ( m , n ) Z * p q ( m , n ) = δ ( p , p ) δ ( q , q ) ,
W ( m , n ) = DF 1 { a p q } = 1 N p , q = 0 N 1 a p q exp [ 2 π i N ( p m + q n ) ] ,
a p q = DF { W ( m , n ) } = 1 N m , n = 1 N W ( m , n ) × exp [ 2 π i N ( p m + q n ) ] .
Z p q ( m , n ) = ( [ exp ( 2 π i N q ) 1 ] [ exp ( 2 π i N p ) 1 ] ) Z p q ( m , n ) .
a ˜ p q F = 0
a ˜ p q = { 0 , p = q = 0 1 4 [ sin 2 ( π N p ) + sin 2 ( π N q ) ] × { [ exp ( 2 π i q N ) 1 ] DF { S x } + [ exp ( 2 π i p N ) 1 ] DF { S y } } , otherwise .
4 [ sin 2 ( π N p ) + sin 2 ( π N q ) ] = [ exp ( 2 π i N p ) 1 ] × [ exp ( 2 π i N p ) 1 ] + [ exp ( 2 π i N q ) 1 ] × [ exp ( 2 π i N q ) 1 ] .
W ( m , N + 1 ) W ( m , 1 ) = n = 1 N S x ( m , n ) ,
W ( m , N + 1 ) = W ( m , 1 ) ,
S x ( m , N ) = n = 1 N 1 S x ( m , n ) , 1 m N
S y ( N , n ) = m = 1 N 1 S y ( m , n ) , 1 n N .
Z p q ( m , n ) = ( [ exp [ 2 π i ( p + q ) N ] 1 ] [ exp ( 2 π i p N ) exp ( 2 π i q N ) ] ) Z p q ( m , n ) .
a = { 0 , p = q = 0 , p = q = N / 2 1 4 [ 1 cos ( 2 π p N ) cos ( 2 π q N ) ] × { [ exp ( 2 π i ( p + q ) N ) 1 ] DF { S x } + [ exp ( 2 π i p N ) exp ( 2 π i q N ) ] DF { S y } } , otherwise .
S x ( m , N ) = n = 1 n odd N 1 S y ( m , n ) n = 1 n even N 1 S x ( m , n ) ,
S y ( m , N ) = n = 1 n odd N 1 S x ( m , n ) n = 1 n even N 1 S y ( m , n )
S x ( N , n ) = m = 1 m odd N 1 S y ( m , n ) m = 1 m even N 1 S x ( m , n ) ,
S y ( N , n ) = m = 1 m odd N 1 S x ( m , n ) m = 1 m even N 1 S x ( m , n )
S x ( N , N ) = m = 1 N 1 S x ( m , m ) ,
S y ( N , N ) = m = 1 N 1 S y ( m , N m )
S ˆ x ( m , n ) = 1 2 [ S x ( m , n ) + S x ( m , n + 1 ) ]
S ˆ y ( m , n ) = 1 2 [ S y ( m , n ) + S y ( m + 1 , n ) ]
S ˆ x ( m , n ) = 1 2 [ W ( m , n + 3 2 ) W ( m , n 1 2 ) ] .
S ˆ x ( m , n ) = S y ( m , n ) + S x ( m , n ) ,
S ˆ y ( m , n ) = S y ( m , n ) S x ( m , n ) ,
σ w 2 ( m , n ) = | W ˜ ( m , n ) W ( m , n ) | 2 ,
σ w 2 ¯ = 1 N 2 m , n = 1 N σ w 2 ( m , n )
C p q p q A = ( a ˜ p q a p q ) ( a ˜ * p q a * p q ) .
σ w 2 ( m , n ) = 1 N DF 1 [ p , q = 0 N 1 C p q ( p p ) ( q q ) A ] ,
σ w 2 ¯ = 1 N 2 p , q = 0 N 1 C p q p q A .
C m n m n S i j = n i ( m , n ) n j ( m , n ) = σ n 2 δ ( i , j ) δ ( m , m ) δ ( n , n ) ,
σ w 2 ¯ / σ n 2 = a + b ln ( N ) ,
b = 1 / π .
σ w 2 ¯ / σ n 2 = a + b ln ( N 1 ) ,
b = 1.5 / π .
b = 3 / π .
C A = ( D D T ) 1 D C S D T ( D D T ) 1 ,
C S = ( C S x x C S x y C S y x C S y y ) .
D = ( D x D y ) ,
C m n m n S x x = C m n m n S y y = σ n 2 δ ( m , m ) δ ( n , n ) , 1 m , m , n , n N 1 , C m N m n S x x = C N m n m S y y = σ n 2 δ ( m , m ) , 1 m , m N , 1 n N 1 , C m n m N S x x = C n m N m S y y = σ n 2 δ ( m , m ) , 1 m , m N , 1 n N 1 , C m N m N S x x = C n m N m S y y = ( N 1 ) σ n 2 δ ( m , m ) , 1 m , m N , C m n m n S x y = C m n m n S y x = 0 , 1 m , m n , n N .
C p q p q A = { 0 , p = q = 0 , p = q = 0 , p = q = 0 , p = q = 0 σ n 2 { δ ( p , p ) δ ( q , q ) 4 [ sin 2 ( π N p ) + sin 2 ( π N q ) ] + [ exp ( 2 π i p N ) 1 ] [ exp ( 2 π i p N ) 1 ] δ ( q , q ) 16 [ sin 2 ( π N p ) + sin 2 ( π N q ) ] [ sin 2 ( π N p ) + sin 2 ( π N q ) ] + [ exp ( 2 π i q N ) 1 ] [ exp ( 2 π i q N ) 1 ] δ ( p , p ) 16 [ sin 2 ( π N p ) + sin 2 ( π N q ) ] [ sin 2 ( π N p ) + sin 2 ( π N q ) ] } , otherwise ,
C p q p q A = { 0 , p = q = 0 σ n 2 2 [ sin 2 ( π N p ) + sin 2 ( π N q ) ] , otherwise .
C m n m n S x x = C m n m n S y y = σ n 2 δ ( m , m ) δ ( n , n ) , 1 m , m , n , n N , C m n m n S x y = C m n m n S y x = 0 , 1 m , m n , n N .
C p q p q A = { 0 , p = q = 0 , p = q = 0 , p = q = 0 , p = q = 0 σ n 2 { [ sin 2 ( π N p ) cos 2 ( π N p ) + sin 2 ( π N q ) cos 2 ( π N q ) ] δ ( p , p ' ) δ ( q , q ' ) 4 [ sin 2 ( π N p ) + sin 2 ( π N q ) ] 2 + [ exp ( 2 π i p N ) 1 ] [ exp ( 2 π i p N ) 1 ] δ ( q , q ) 16 [ sin 2 ( π N p ) + sin 2 ( π N q ) ] [ sin 2 ( π N p ) + sin 2 ( π N q ) ] + [ exp ( 2 π i q N ) 1 ] [ exp ( 2 π i q N ) 1 ] δ ( p , p ) 16 [ sin 2 ( π N p ) + sin 2 ( π N q ) ] [ sin 2 ( π N p ) + sin 2 ( π N q ) ] } , otherwise ,
C p q p q A = { 0 , p = q = 0 σ n 2 sin 2 ( π N p ) [ 1 + cos 2 ( π N p ) ] + sin 2 ( π N q ) [ 1 + cos 2 ( π N q ) ] 4 [ sin 2 ( π N p ) + sin 2 ( π N q ) ] 2 , otherwise .
C p q p q A = { 0 , p = q = 0 , p = q = 0 , p = q = N / 2 , p = q = N / 2 σ n 2 16 [ 1 cos ( 2 π N p ) cos ( 2 π N q ) ] [ 1 cos ( 2 π N p ) cos ( 2 π N q ) ] × ( { [ exp ( 2 π i ( p + q ) N ) 1 ] [ exp ( 2 π i ( p + q ) N ) 1 ] + [ exp ( 2 π i p N ) exp ( 2 π i q N ) ] [ exp ( 2 π i p N ) exp ( 2 π i q n ) ] } × { δ ( p , p ) δ ( q , q ) 1 N 2 N 2 + δ ( p , p ) + δ ( q , q ) 1 2 [ δ ( p , p ) 1 N ] [ δ ( q , 0 ) + δ ( q , N 2 ) + δ ( q , 0 ) + δ ( q , N 2 ) ] 1 2 [ δ ( q , q ) 1 N ] [ δ ( p , 0 ) + δ ( p , N 2 ) + δ ( p , 0 ) + δ ( p , N 2 ) ] + 1 2 [ δ ( p , 0 ) δ ( q , N 2 ) + δ ( p , 0 ) δ ( q , N 2 ) + δ ( q , 0 ) δ ( p , N 2 ) + δ ( q , 0 ) δ ( p , N 2 ) ] } + Re { [ exp ( 2 π i ( p + q ) N ) 1 ] [ exp ( 2 π i p N ) exp ( 2 π i p N ) ] } × { [ δ ( p , p ) + 1 N ] [ δ ( q , 0 ) δ ( q , N 2 ) + δ ( q , 0 ) δ ( q , N 2 ) ] + [ δ ( q , q ) + 1 N ] [ δ ( p , 0 ) δ ( p , N 2 ) + δ ( p , 0 ) δ ( p , N 2 ) ] + [ δ ( p , 0 ) δ ( q , N 2 ) + δ ( p , 0 ) δ ( q , N 2 ) + δ ( q , 0 ) δ ( p , N 2 ) + δ ( q , 0 ) δ ( p , N 2 ) ] } ) , otherwise ,
C p q p q A = { 0 , p = q = 0 , p = q = N / 2 , σ n 2 4 { 3 1 N 2 N 2 1 cos ( 2 π p N ) cos ( 2 π q N ) + 2 N [ δ ( q , 0 ) 1 cos ( 2 π p N ) + δ ( q , N 2 ) 1 + cos ( 2 π p N ) + δ ( p , 0 ) 1 cos ( 2 π q N ) + δ ( p , N 2 ) 1 + cos ( 2 π q N ) ] } , otherwise .
C p q p q A = { 0 , p = q = 0 , p = q = 0 , p = q = N / 2 , p = q = N / 2 σ n 2 δ ( p , p ) δ ( q , q ) 4 [ 1 cos ( 2 π p N ) cos ( 2 π q N ) ] , otherwise .

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