## Phase retrieval using two Fourier-transform intensities

JOSA A, Vol. 7, Issue 3, pp. 441-449 (1990)

http://dx.doi.org/10.1364/JOSAA.7.000441

Enhanced HTML Acrobat PDF (1131 KB)

### Abstract

We consider the problem of reconstructing either a one-dimensional or a two-dimensional signal from its Fourier intensity and the Fourier intensity of another signal that is related to the first by the addition of a known reference signal. Several theorems are given that give conditions under which a unique reconstruction is possible, and a recursive algorithm is provided that allows for the reconstruction of the signal from the pair of Fourier intensities.

© 1990 Optical Society of America

**History**

Original Manuscript: July 18, 1989

Manuscript Accepted: November 9, 1989

Published: March 1, 1990

**Citation**

Wooshik Kim and Monson H. Hayes, "Phase retrieval using two Fourier-transform intensities," J. Opt. Soc. Am. A **7**, 441-449 (1990)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-7-3-441

Sort: Year | Journal | Reset

### References

- 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]
- M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982). [CrossRef]
- M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,” J. Opt. Soc. Am. 73, 1427–1433 (1983). [CrossRef]
- P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from signed Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983). [CrossRef]
- R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron microscopy,” Optik 34, 275–284 (1971).
- R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
- D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. test calculations,” J. Phys. D 6, 2200–2216 (1973). [CrossRef]
- D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. sources of error,” J. Phys. D 6, 2217–2225 (1973). [CrossRef]
- R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in 1980 International Optical Computing Conference (Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–141 (1980). [CrossRef]
- R. A. Gonsalves, “Phase retrieval by differential intensity measurements,” J. Opt. Soc. Am. A 4, 166–170 (1987). [CrossRef]
- N. Nakajima, “Phase retrieval from two intensity measurements using the Fourier series expansion,” J. Opt. Soc. Am. A 4, 154–158 (1987). [CrossRef]
- N. Nakajima, “Phase retrieval using the logarithmic Hilbert transform and the Fourier-series expansion,” J. Opt. Soc. Am. A 5, 257–262 (1988). [CrossRef]
- S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983). [CrossRef]
- G. N. Ramachandran, R. Srinivasan, Fourier Methods in Crystallography (Wiley-Interscience, New York, 1970).
- J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979). [CrossRef]
- J. E. Marsden, Basic Complex Analysis (Freeman, San Francisco, Calif., 1973).
- L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas Propag. AP-29, 386–391 (1981). [CrossRef]
- A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
- We say that a sequence x(n) is nonsymmetric if it does not have even or odd symmetry, i.e., if x(n) ≠ x(n0− n) and x(n) ≠ − x(n0− n) for all integer values of n0. This is equivalent to the constraint that x(n) not be a linear phase sequence.
- A. H. Greenaway, “Proposal for phase recovery from a single intensity distribution,” Opt. Lett. 1, 10–12 (1977). [CrossRef] [PubMed]
- C. L. Mehta, “New approach to the phase problem in optical coherence theory,” J. Opt. Soc. Am. 58, 1233–1234 (1968). [CrossRef]
- J. R. Fienup, “Reconstruction of objects having latent reference points,” J. Opt. Soc. Am. 73, 1421–1426 (1983). [CrossRef]
- M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.