Abstract

We present an algorithm for phase retrieval based on improvements to the methods developed by Bates [see Optik 61, 247 (1982) ]. Specifically, we have developed a more precise way of calculating phase differences between adjacent actual sampling points. This leads to a reduction in the error buildup in a recursive phase propagation scheme. Our approach has the advantage of having no adjustable parameters. We present a few examples of how this method can lead to improved image reconstructions.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).
  2. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27-29 (1978).
    [CrossRef] [PubMed]
  3. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttgart) 61, 247-262 (1982).
  4. K. L. Garden and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One-dimensional considerations,” Optik (Stuttgart) 62, 131-142 (1982).
  5. W. R. Fright and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik (Stuttgart) 62, 219-230 (1982).
  6. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  7. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40-55 (2003).
    [CrossRef]
  8. Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304-308 (1979).
    [CrossRef]
  9. M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-30, 140-154 (1982).
    [CrossRef]
  10. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662-1669 (1998).
    [CrossRef]
  11. R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vis. Graph. Image Process. 25, 205-217 (1984).
    [CrossRef]
  12. C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
    [CrossRef]
  13. The solution to the phase problem is not unique in 1D. In the most general case (for a complex image), there can be up to 22M−1 different sets of phases compatible with a set of 2M+1 given magnitudes al. This is consistent with the fact that there are two possible choices for the sign of each phase difference (ωl) between adjacent samples .
  14. K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short exposure images,” Astrophys. J. 193, L45-L48 (1974).
    [CrossRef]
  15. R. H. T. Bates and W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358-365 (1983).
    [CrossRef]

2007 (1)

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
[CrossRef]

2003 (1)

1998 (1)

1984 (1)

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vis. Graph. Image Process. 25, 205-217 (1984).
[CrossRef]

1983 (1)

1982 (5)

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-30, 140-154 (1982).
[CrossRef]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttgart) 61, 247-262 (1982).

K. L. Garden and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One-dimensional considerations,” Optik (Stuttgart) 62, 131-142 (1982).

W. R. Fright and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik (Stuttgart) 62, 219-230 (1982).

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

1979 (1)

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304-308 (1979).
[CrossRef]

1978 (1)

1974 (1)

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

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Bates, R. H. T.

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vis. Graph. Image Process. 25, 205-217 (1984).
[CrossRef]

R. H. T. Bates and W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358-365 (1983).
[CrossRef]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttgart) 61, 247-262 (1982).

K. L. Garden and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One-dimensional considerations,” Optik (Stuttgart) 62, 131-142 (1982).

W. R. Fright and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik (Stuttgart) 62, 219-230 (1982).

Bruck, Y. M.

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304-308 (1979).
[CrossRef]

Chapman, H. N.

Chen, C.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
[CrossRef]

Elser, V.

Fienup, J. R.

Fright, W. R.

R. H. T. Bates and W. R. Fright, “Composite two-dimensional phase-restoration procedure,” J. Opt. Soc. Am. 73, 358-365 (1983).
[CrossRef]

W. R. Fright and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik (Stuttgart) 62, 219-230 (1982).

Garden, K. L.

K. L. Garden and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One-dimensional considerations,” Optik (Stuttgart) 62, 131-142 (1982).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Hayes, M. H.

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-30, 140-154 (1982).
[CrossRef]

Ishikawa, T.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
[CrossRef]

Knox, K. T.

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

Kohmura, Y.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
[CrossRef]

Lee, T.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
[CrossRef]

Miao, J.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662-1669 (1998).
[CrossRef]

Nishino, Y.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
[CrossRef]

Ramunno-Johnson, D.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Sayre, D.

Sodin, L. G.

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304-308 (1979).
[CrossRef]

Song, C.

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
[CrossRef]

Thompson, B. J.

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

Appl. Opt. (1)

Astrophys. J. (1)

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

Comput. Vis. Graph. Image Process. (1)

R. H. T. Bates, “Uniqueness of solutions to two-dimensional Fourier phase problems for localized and positive images,” Comput. Vis. Graph. Image Process. 25, 205-217 (1984).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (1)

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-30, 140-154 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304-308 (1979).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (4)

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttgart) 61, 247-262 (1982).

K. L. Garden and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. II: One-dimensional considerations,” Optik (Stuttgart) 62, 131-142 (1982).

W. R. Fright and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions,” Optik (Stuttgart) 62, 219-230 (1982).

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Phys. Rev. B (1)

C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. Chen, T. Lee, and J. Miao, “Phase retrieval from exactly oversampled diffraction intensity through deconvolution,” Phys. Rev. B 75, 012102 (2007).
[CrossRef]

Other (1)

The solution to the phase problem is not unique in 1D. In the most general case (for a complex image), there can be up to 22M−1 different sets of phases compatible with a set of 2M+1 given magnitudes al. This is consistent with the fact that there are two possible choices for the sign of each phase difference (ωl) between adjacent samples .

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Functions sinc ( ε ) (dashed-dotted curve), J ( ε ) (dashed curve), and J sl ( ε ) (solid curve) truncated to zero outside the interval ( 1 , 1 ) .

Fig. 2
Fig. 2

Propagation scheme in 2D within a single Nyquist cell. The phase is propagated at each actual cell (any square formed by four adjacent black dots) from the lower-left corner to the upper-right corner of the Nyquist cell. The values of the phase at the edges (marked with squares) need to be updated at each subsequent propagation that includes the correction term. The propagation in each Nyquist cell is iterated until a steady solution is reached. In the 1D case the propagation is done within a single line (left to right).

Fig. 3
Fig. 3

General scheme for phase retrieval.

Fig. 4
Fig. 4

Sample eight-component images (solid curves) and reconstructed images (dashed curves). (a), (c) and (e) were obtained using the method described in Section 3 with the correction term from Eq. (33). (b), (d) and (f) were obtained using Bates’s method with relation (19). The (optimal) values of γ used for Bates’s reconstructions are, respectively, γ = 0.509 , γ = 0.497 , and γ = 0.503 . Different linear oversampling ratios were used for each reconstruction: N = 4 for (a) and (b), N = 6 for (c) and (d), and N = 8 (e) and (f).

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

I ( k ) ψ ̂ ( k ) 2 ,
F ( u ) = l F l samp l ( u p l ) ,
samp l ( 0 ) = 1
samp m ( p n p m ) = 0 .
F ( u ) = l F l j = 1 d sinc [ ( u j p l , j ) α j ] ,
B l = 1 ζ [ A l + A l + 1 + 2 a l a l + l cos ( ω l ) ] .
f ( x ) = n = M M C n e 2 π i n x 2 L
C n = 1 2 L L L f ( x ) e 2 π i n x 2 L d x .
F ( u ) = n = M M F n Ny sinc ( 2 L u n ) ,
F ( u ) = L L f ( x ) e 2 π i u x d x .
F n Ny = F ( n 2 L ) = 2 L C n .
F ( u ) = n = M M F n sinc ( 2 L u n ) ,
F ( n 2 L ) = F ( N n 2 L ) ,
F ( [ n + 1 2 ] 2 L ) γ { F n + F n + 1 } ,
F n = a n e i θ n ,
B n = b n 2 = F ( [ n + 1 2 ] 2 L ) 2 ,
E n = cos ( ω n true ) cos ( ω n est ) ,
cos ( ω n true ) = F n F n + 1 * + F n * F n + 1 2 a n a n + 1 ,
cos ( ω n est ) = ζ B n A n A n + 1 2 a n a n + 1 .
A n = j = M M k = M M a j Ny a k Ny c j , k Ny sinc ( ε j , n ϵ ) sinc ( ε k , n ϵ ) ,
A n + 1 = j = M M k = M M a j Ny a k Ny c j , k Ny sinc ( ε j , n + ϵ ) sinc ( ε k , n + ϵ ) ,
B n = j = M M k = M M a j Ny a k Ny c j , k Ny sinc ( ε j , n ) sinc ( ε k , n ) ,
cos ( ω n true ) = 1 a n a n + 1 j = M M k = M M a j Ny a k Ny c j , k Ny sinc ( ε j , n + ε ) sinc ( ε k , n ϵ ) .
c j , k Ny = cos ( θ k Ny θ j Ny ) = cos ( ω j , k Ny ) .
ε j , n = ( n + 1 2 ) N j .
E n = 1 a n a n + 1 [ j = M M k = M M a j Ny a k Ny c j , k Ny Ω ( ε j , n , ε k , n , ϵ ) + ( 2 ζ 2 ) B n ] ,
Ω ( ε 1 , ε 2 , ϵ ) = 1 2 [ sinc ( ε 1 + ϵ ) + sinc ( ε 1 ϵ ) ] × [ sinc ( ε 2 + ϵ ) + sinc ( ε 2 ϵ ) ] + 2 sinc ( ε 1 ) sinc ( ε 2 ) .
Ω ( ε 1 , ε 2 , ϵ ) ϵ 2 ( sinc ( ε 1 ) { π 2 sinc ( ε 2 ) + ( 2 ε 2 2 ) [ cos ( π ε 2 ) sinc ( ε 2 ) ] } + sinc ( ε 2 ) { π 2 sinc ( ε 1 ) + ( 2 ε 1 2 ) [ cos ( π ε 1 ) sinc ( ε 1 ) ] } ) .
E n = 1 a n a n + 1 [ 1 2 N 2 j = M M k = M M a j Ny a k Ny c j , k Ny × sinc ( ε j , n ) J ( ε k , n ) + ( 2 ζ 2 π 2 2 N 2 ) B n ] ,
J ( ε ) = ( 2 ε 2 ) [ sinc ( ε ) cos ( π ε ) ] .
ζ = 4 π 2 N 2 .
J sl ( ε ) = { J ( ε ) for ε 1 0 for ε > 1 } .
cos ( ω n corr ) = cos ( ω n lin ) + b n 2 N 2 a n a n + 1 [ a j Ny c j , n J sl ( ε j , n ) + a j + 1 Ny c j + 1 , n J sl ( ε j , n 1 ) ] .
c j , n = cos { θ [ ( n + 1 2 ) 2 L ] θ j Ny } .
θ ( n + 1 2 2 L ) phase { F n + F n + 1 1 4 N 2 [ F j Ny J sl ( ε j , n ) + F j + 1 Ny J sl ( ε j , n 1 ) ] } .
f ( x ) = n = M M C n e 2 π i n x ,

Metrics