### Citations

320 | Introduction to Fourier Optics. McGraw-Hill - Goodman - 1996 |

291 | Phase retrieval algorithms: a comparison
- Fienup
- 1982
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Citation Context ...leaving only four independent equations in six unknowns. Therefore one can, for example, choose values a and b in Fig. 2(c) arbitrarily, and then the values of c, d, e, and f are determined from Eqs. =-=(6)-=- as follows: c = (15-a + 2b)/4, (7a) d = (25 + a-2b)/4, (7b) e = (35-a-2b)/4, (7c) f = (15 - a)/2. (7d) At this point the H-Q algorithm would have been stopped, leaving this ambiguity. An alternative ... |

288 |
A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures
- Gerchberg, Saxton
- 1972
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Citation Context ... problem). r(m, n), the (aperiodic) autocorrelation of f(m, n), is given by' M-1 N-1 r(m, n) = > Zf(j, k)f*(j-m, k-n) (2) j=0 k=O M-1 N-1 = E , f*(J, k)f(j + m, k + n) (3) j=O k=O = i-I[[lF(p, q)l2], =-=(4)-=- where the asterisk denotes a complex conjugate and where it is assumed that f(A, k) = 0 for m outside [0, M - 1] and for n outside [0, N - 1]. Note that, when simulating data, in order to avoid alias... |

161 | Statistical properties of laser speckle patterns. - Goodman - 1984 |

154 |
The Fourier Transform."
- Bracewell
- 1989
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Citation Context ...es to reconstruct f(m, n), an object function, from IF(p, q)J, the modulus of its Fourier transform, where F(p, q) = IF(p, q)lexp[i4(p, q)] = V[f(m, n)] P-i Q-1 = E sE f(m, n)exp[-i22r(mp/P + nq/Q)], =-=(1)-=- m=O n=O wherem,p=0,1,...,P-landn,q=0,1,...,Q-1. The discrete transform is employed here since in practice one deals with sampled data in a computer. The problem of reconstructing the object from its ... |

41 |
Phase-retrieval stagnation problems and solutions,
- Fienup, Wackerman
- 1986
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Citation Context ...rgSaxton algorithms. Here we refer to the original Gerchberg-Saxton algorithm"l" 2 as GS and the accelerated versions as GS1 and GS2,13-15 the latter having the imagedomain operations [combining Eqs. =-=(9)-=- and (10) of Ref. 14] gk+1(X) = g9' + 3[21g(x)l gk() -g(x) - Ig(x)l gk(X) (1) where a is a constant, Ig(x)l is the modulus of the lowresolution image, gk(X) is the input image to the kth iteration, an... |

37 | Reconstruction of a complex-valued object from the modulus of its fourier transform using a support constraint
- Fienup
- 1987
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Citation Context ..."l" 2 as GS and the accelerated versions as GS1 and GS2,13-15 the latter having the imagedomain operations [combining Eqs. (9) and (10) of Ref. 14] gk+1(X) = g9' + 3[21g(x)l gk() -g(x) - Ig(x)l gk(X) =-=(1)-=- where a is a constant, Ig(x)l is the modulus of the lowresolution image, gk(X) is the input image to the kth iteration, and gk'(X) is the output image from the kth iteration. The rates of convergence... |

36 | Fienup, "Reconstruction of an Object from the Modulus of its Fourier Transform - R - 1978 |

36 |
X-Ray Structure Determination.
- Stout, Jensen
- 1972
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Citation Context ...alternative is to continue by adding, to the existing set of underdetermined equations, the equations corresponding to the next row of the autocora = 15 - 2b, and combining Eq. (8b) with Eqs. (7) and =-=(9)-=- yields b2 - lOb + 24 = 0 (9) (lOa) or b = 4 or 6. (lOb) Evaluation of the other unknowns by Eqs. (9) and (7) gives the two solutions shown in Figs. 1(a) and 1(b). The equations for the unused points ... |

31 | On the ambiguity of the image reconstruction problem, - Bruck, Sodin - 1979 |

30 | Iterative method applied to image reconstruction and to computer generated holograms - Fienup - 1980 |

18 | Reconstruction of the support of an object from the support of its autocorrelation, - Fienup, Crimmins, et al. - 1982 |

13 | Direct phase retrieval,’’ - Lane, Fright, et al. - 1987 |

10 | Reconstruction and Synthesis Applications of an Iterative Algorithm - Fienup - 1982 |

8 | Phase retrieval for a complex-valued object by using a low-resolution image. - Fienup, Kowalczyk - 1990 |

8 | Enforcing irreducibility for phase retrieval in two dimensions - Fiddy, Brames, et al. - 1983 |

8 | Uniqueness of phase retrieval for functions with sufficiently disconnected support - Crimmins, Fienup - 1983 |

6 |
High Resolution Image Formation Through the Turbulent Atmosphere
- Liu, Lohmann
- 1973
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Citation Context ...(m, n) by the inverse (discrete) Fourier transform (hence the name phase-retrieval problem). r(m, n), the (aperiodic) autocorrelation of f(m, n), is given by' M-1 N-1 r(m, n) = > Zf(j, k)f*(j-m, k-n) =-=(2)-=- j=0 k=O M-1 N-1 = E , f*(J, k)f(j + m, k + n) (3) j=O k=O = i-I[[lF(p, q)l2], (4) where the asterisk denotes a complex conjugate and where it is assumed that f(A, k) = 0 for m outside [0, M - 1] and ... |

6 | Reconstruction of objects having latent reference points - Fienup - 1983 |

5 |
Fourier phase retrieval when the image is complex
- Bates, Tan
- 1985
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Citation Context ...d deviation, a,, of the error of the phase retrieved over the small aperture. As is shown in Appendix A, the expected relationship, if errors in the modulus are ignored, is ABSERR 2 1 - exp(-o-, 2 ). =-=(4)-=- Figure 7 shows examples of algorithm convergence. Twenty iterations of either GS2 or GS were followed by twenty iterations of GS. The optimum value of Q was found to be approximately 1.5 to 2. The al... |

5 | Computer Techniques for Image Processing - unknown authors - 1978 |

5 |
Inferring phase information from modulus information in two-dimensional aperture synthesis,” Astron
- Napier, Bates
- 1974
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Citation Context ...e nonlinear. The first two such equations are (after rearranging the right-hand sides) 52 = 12 + 2a + 2c + d + 3f and 2 3 3 3 3 a b c 2 100 = 12 + 4b + 3c + 2d + 4e + af. Combining Eq. (8a) with Eqs. =-=(7)-=- yields 3 d e f 2 1 2 2 2 1 (C) Fig. 2. Functions (a) and (b), which generate the object shown in Fig. 1(a) by cross correlation and in Fig. 1(b) by convolution. In (c) is the general form of the obje... |

4 | A new direct algorithm for image reconstruction from fourier transform magnitude - Izraelevitz, Lim - 1987 |

4 | The importance of boundary conditions in the phase retrieval problem - Hayes, Quatieri - 1982 |

3 |
Clinthorne, "Effect of tapered illumination and Fourier intensity errors on phase retrieval
- Paxman, Fienup, et al.
- 1987
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Citation Context ...ution image reconstruction from the Fourier modulus, for which the only image-domain constraint is a support constraint, the ODEM is given, instead of by Eq. (2), by EIgk(x)12 ODEM2 = xIS E Igk'(x)12 =-=(5)-=- i.e., the energy outside the support constraint S. 4. IMAGE-RECONSTRUCTION EXAMPLE Figure 11 shows an example of image reconstruction that uses the approach described above. Only the modulus of each ... |

3 | Image reconstruction from partial Fresnel zone information - Rolleston, George - 1986 |

3 |
Analogy between holography and interferometric image formation
- Goodman
- 1970
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Citation Context ... (hence the name phase-retrieval problem). r(m, n), the (aperiodic) autocorrelation of f(m, n), is given by' M-1 N-1 r(m, n) = > Zf(j, k)f*(j-m, k-n) (2) j=0 k=O M-1 N-1 = E , f*(J, k)f(j + m, k + n) =-=(3)-=- j=O k=O = i-I[[lF(p, q)l2], (4) where the asterisk denotes a complex conjugate and where it is assumed that f(A, k) = 0 for m outside [0, M - 1] and for n outside [0, N - 1]. Note that, when simulati... |

3 |
Digital Computation of Image Complex Amplitude from Image-and Diffraction-Intensity: an Alternative to Holography,” Optik 44
- Dallas
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Citation Context ...1 3 7 4 4 2 3 6 5 4 2 1 2 2 2 1 (a) 2 3 3 3 1 3 3 6 6 2 3 4 5 6 2 1 2 2 2 1 (b) M-1 = E f*(,O)f(j+m, N-2) j=o M-1 + E f*(i, l)f(i + mn, N -1) j=o M-1 = I a* (j)f(j + m, N-2) j=o M-1 + f*(j, 1)(j + m) =-=(5)-=- j=o form =-M+ 1,...,M-1. Theseare2M-l equations, one for each value of m, in 2M - 4 unknowns, f(j, N - 2) and f(j, 1), forj = 1,2,.. ., M-2. Recall that a(j), 3(j),Jf(0, N2), f(M - 1, N - 2), f(O, 1)... |

3 | Recursive phase retrieval using boundary conditions - Hayes, Quatieri - 1983 |

3 | Image reconstruction for stellar interferoiiietry - Fienup - 1981 |

3 | Experimental evidence of the uniqueness of phase retrieval from intensity data - Fienup - 1984 |

2 | Improved synthesis and computational methods for computer-generated holograms - Fienup - 1975 |

1 |
Wave-front phase estimation from Fourier intensity measurements
- son, Kryskowski
- 1989
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Citation Context ...s in both planes to a Poisson noise process. We chose to simulate the same number of photons in each of the two planes. After the 1.0 LU 0 0 C, U) 0 P a,CE 0E [Igg'() - Ig(x)I] 2 ODEM 2 = X E lg(X)12 =-=(2)-=- which is a measure of how closely the output image modulus agrees with the modulus of the measured low-resolution image and is the criterion by which we judge whether the algorithm has converged. (A ... |

1 |
Improved 7) bounds on object support from autocorrelation support and application to phase retrieval
- Crimmins, Fienup, et al.
- 1990
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Citation Context ... use to evaluate the reconstruction results is the absolute error (ABSERR) of the complex-valued reconstructed image, also a normalized rms error, which is given by E Iag'(x - X0) - g(x)12 ABSERR2 =X =-=(3)-=- E lg(x)12 where a is the complex factor and x0 is the shift that minimizes the ABSERR. It can be shown that x0 is given by the location of the maximum magnitude of rg'g(x), the cross correlation of g... |

1 | Recovery of complex images from Fourier magnitude - Lane - 1987 |

1 |
Reconstructed wavefronts and 1) communication theory
- Leith, Upatnieks
- 1962
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Citation Context ... algorithms. Here we refer to the original Gerchberg-Saxton algorithm"l" 2 as GS and the accelerated versions as GS1 and GS2,13-15 the latter having the imagedomain operations [combining Eqs. (9) and =-=(10)-=- of Ref. 14] gk+1(X) = g9' + 3[21g(x)l gk() -g(x) - Ig(x)l gk(X) (1) where a is a constant, Ig(x)l is the modulus of the lowresolution image, gk(X) is the input image to the kth iteration, and gk'(X) ... |

1 | Phase retrieval from Fourier intensity data - Fienup - 1987 |

1 | Towards a strategy for automatic phase retrieval from noisy Fourier intensities - McCallum, Bates - 1989 |

1 | p. 480. Note that the earlier edition was in error - Gradshteyn, Ryzhik - 1980 |

1 | The solution to the phase retrieval problem using the sampling theorem - Arsenault, Chalasinka-Macukow - 1983 |

1 | Proposal for phase recovery from a single intensity distribution - Greenaway - 1977 |