## Algorithms for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data

JOSA A, Vol. 2, Issue 11, pp. 2027-2039 (1985)

http://dx.doi.org/10.1364/JOSAA.2.002027

Enhanced HTML Acrobat PDF (1583 KB)

### Abstract

This paper is a summary of more detailed mathematical work by the authors on recovery of partially known Fourier transforms. These problems of inversion of the finite Fourier transform and of phase retrieval are known to be ill posed. We draw a distinction in the resultant ill conditioning of the problems between global ill conditioning (which is due to the existence of multiple exact solutions) and local ill conditioning (which is due to the existence of large neighborhoods of the true solution, all of whose members are indistinguishable from the true solution if the data are noisy). We then develop extensions of known algorithms that attempt to reduce at least the effects of local ill conditioning on numerical solutions by using the idea of filtered singular-value decomposition and present some numerical examples of the use of those algorithms in the context of optical-diffraction theory.

© 1985 Optical Society of America

**History**

Original Manuscript: January 3, 1985

Manuscript Accepted: July 25, 1985

Published: November 1, 1985

**Citation**

Richard Barakat and Garry Newsam, "Algorithms for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data," J. Opt. Soc. Am. A **2**, 2027-2039 (1985)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-2-11-2027

Sort: Year | Journal | Reset

### References

- R. Gerschberg, W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
- A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975). [CrossRef]
- D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 695–702 (1978).
- J. A. Cadzow, “An extrapolation procedure for bandlimited signals,” IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 4–12 (1979). [CrossRef]
- M. S. Sabri, W. Steenart, “An approach to bandlimited extrapolation: The extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978). [CrossRef]
- R. Burge, M. Fiddy, A. Greenway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976). [CrossRef]
- J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970), pp. 494–500.
- J. R. Fienup, “Space–object imaging through the atmosphere,” Opt. Eng. 18, 529–534 (1979). [CrossRef]
- R. Barakat, G. Newsam, “A numerically stable iterative method for the inversion of wave-front aberrations from measured point-spread-function data,” J. Opt. Soc. Am. 70, 1255–1263 (1980). [CrossRef]
- A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977).
- L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas Propag. AP-39, 381–391 (1981).
- D. L. Misell, “The phase problem in electron microscopy,” in Advances in Optics and Electron Microscopy, Vol. 7, V. E. Coslett, R. Barer, eds. (Academic, New York, 1978).
- H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978). [CrossRef]
- R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data: I. The prototypical linear problem,” J. Integral Eq. 9, 49–76 (1985).
- R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data: II The nonlinear problem of phase retrieval,” J. Integral Eq. Suppl. 9, 77–125 (1985).
- G. Newsam, “Numerical reconstruction of partially known transforms,” Ph.D. dissertation (Harvard University, Cambridge, Mass., 1982).
- C. T. Baker, The Numerical Treatment of Integral Equations (Oxford U. Press, Oxford, 1977).
- G. Newsam, R. Barakat, “Essential dimension as a well-defined number of degrees of freedom of finite-convolution operators appearing in optics,” J. Opt. Soc. Am. A 2, 2040–2045 (1985). [CrossRef]
- R. H. Hanson, “A numerical method for solving Fredholm integral equations of the first kind using singular values.” J. Soc. Ind. Appl. Math. Num. Anal. 8, 616–622 (1971).
- D. Slepian, H. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty: I,” Bell Syst. Tech. J. 40, 43–64 (1961). [CrossRef]
- H. J. Landau, H. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty: II,” Bell Syst. Tech. J. 40, 65–84 (1961). [CrossRef]
- H. J. Landau, H. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty: III,” Bell Syst. Tech. J. 41, 1295–1336 (1962). [CrossRef]
- H. J. Landau, “The eigenvalue behavior of certain convolution equations,” Trans. Am. Math. Soc. 115, 566–569 (1964).
- H. J. Landau, “Necessary density conditions for sampling and interpolation of certain entire functions,” Acta Math. 117, 37–52 (1967). [CrossRef]
- H. J. Landau, H. Widom, “Eigenvalue distribution of time and frequency limiting,” J. Math. Anal. Appl. 77, 469–481 (1980). [CrossRef]
- H. Widom, “Asymptotic behavior of the eigenvalues of certain integral equations: II,” Arch. Rat. Mech. Anal. 17, 215–229 (1964). [CrossRef]
- E. J. Akutowicz, “On the determination of the phase of a Fourier integral—I,” Trans. Am. Math. Soc. 83, 179–192 (1956).
- E. J. Akutowicz, “On the determination of the phase of a Fourier integral—II,” Proc. Am. Math. Soc. 84, 234–238 (1957).
- A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963). [CrossRef]
- E. M. Hofstetter, “Construction of time-limited functions with specified autocorrelation functions,” IEEE Trans. Inf. Theory IT-10, 119–126 (1964). [CrossRef]
- R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. N. Y. 25, 3190–3193 (1984). [CrossRef]
- P. P. Boas, Entire Functions (Academic, New York, 1954).
- P. J. Napier, “The brightness temperature distributions defined by a measured intensity interferogram,” N.Z. J. Sci. 15, 342–355.
- L. Ronkin, Introduction to the Theory of Entire Functions of Several Complex Variables (American Mathematical Society, Providence, R.I., 1974).
- W. Lawton, “Uniqueness results for the phase-retrieval problem for radial functions,” J. Opt. Soc. Am. 71, 1519–1522 (1981). [CrossRef]
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
- L. Gubin, B. Polyak, E. Raik, “The method of projections for finding the common point of convex sets,” USSR Comp. Math. Math. Phys. 7, 1–24 (1967). [CrossRef]
- Y. Censor, G. T. Herman, “Row generation methods for feasibility and optimization problems involving sparse matrices and their applications,” in Sparse Matrix Proceedings 1978, I. Duff, G. Stewart, eds. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1979).
- G. Walsh, Methods of Optimization (Wiley, New York, 1975).
- R. B. Holmes, “Smoothness of certain metric projections on Hilbert spaces,” Trans. Am. Math. Soc. 183, 87–100 (1973). [CrossRef]
- R. Barakat, L. Riseberg, “Diffraction theory of the aberrations of a slit aperture,” J. Opt. Soc. Am. 55, 878–881 (1965). [CrossRef]

## 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.