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]

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]

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

[CrossRef]
[PubMed]

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).

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

[CrossRef]

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

[CrossRef]

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).

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]

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

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

[CrossRef]

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]

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).

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).

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]

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]

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

[CrossRef]

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]

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]

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]

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]

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]

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).

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

[CrossRef]

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]

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

[CrossRef]

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

[CrossRef]

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]

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]

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

[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).

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).

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]

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 .