OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 16, Iss. 7 — Jul. 1, 1999
  • pp: 1845–1856

Role of support information and zero locations in phase retrieval by a quadratic approach

Tommaso Isernia, Giovanni Leone, Rocco Pierri, and Francesco Soldovieri  »View Author Affiliations


JOSA A, Vol. 16, Issue 7, pp. 1845-1856 (1999)
http://dx.doi.org/10.1364/JOSAA.16.001845


View Full Text Article

Enhanced HTML    Acrobat PDF (462 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A recently introduced approach to phase-retrieval problems is applied to present a unified discussion of support information and zero locations in the reconstruction of a discrete complex image from Fourier-transform phaseless data. The choice of the square-modulus function of the Fourier transform of the unknown as the problem datum results in a quadratic operator that has to be inverted, i.e., a simple nonlinearity. This circumstance makes it possible to consider and to point out some relevant factors that affect the local minima problem that arises in the solution procedure (which amounts to minimizing a quartic functional). Simple modifications of the basic procedure help to explain the role of support information and zeros in the data and to develop suitable strategies for avoiding the local minima problem. All results can be summarized by reference to the ratio between the effective dimensions of the data space and the space of unknowns. Numerical results identify the approach’s considerable robustness against false solutions, starting from completely random first guesses, if the above ratio is larger than 3. The algorithm also ensures robust performance in the presence of noise in the data.

© 1999 Optical Society of America

OCIS Codes
(070.2590) Fourier optics and signal processing : ABCD transforms
(100.3190) Image processing : Inverse problems
(100.5070) Image processing : Phase retrieval

History
Original Manuscript: October 12, 1998
Revised Manuscript: February 24, 1999
Manuscript Accepted: February 24, 1999
Published: July 1, 1999

Citation
Tommaso Isernia, Giovanni Leone, Rocco Pierri, and Francesco Soldovieri, "Role of support information and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A 16, 1845-1856 (1999)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-16-7-1845


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990). [CrossRef]
  2. R. P. Millane, “Image reconstruction from cylindrically averaged diffraction intensities,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 137–143 (1993). [CrossRef]
  3. T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995). [CrossRef]
  4. T. Isernia, G. Leone, R. Pierri, “Radiation pattern evaluation from near field intensities over two planes,” IEEE Trans. Antennas Propag. 44, 701–710 (1996). [CrossRef]
  5. G. Leone, R. Pierri, F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier transform pairs,” J. Opt. Soc. Am. A 13, 1546–1556 (1996). [CrossRef]
  6. I. Sabba Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2160 (1985). [CrossRef]
  7. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979). [CrossRef]
  8. T. R. Crimmins, J. R. Fienup, “Uniqueness of phase retrieval for function with sufficient disconnected support,” J. Opt. Soc. Am. A 73, 218–221 (1983). [CrossRef]
  9. 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]
  10. J. R. Fienup, “Reconstruction of objects having latent reference points,” J. Opt. Soc. Am. A 73, 1421–1426 (1983). [CrossRef]
  11. B. J. Brames, “Unique phase retrieval with explicit support information,” Opt. Lett. 11, 61–63 (1986). [CrossRef] [PubMed]
  12. T. R. Crimmins, “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am. A 4, 124–134 (1987). [CrossRef]
  13. B. Blaschke-Kaltenbaker, H. W. Engl, “Regularization methods for nonlinear ill-posed problems with application to phase reconstruction,” in Inverse Problems in Medical Imaging and Nondestructive Testing, H. W. Engl, ed. (Springer, New York, 1997), pp. 17–35.
  14. M. Nieto-Vesperinas, R. Navarro, F. J. Fuentes, “Performance of a simulated annealing algorithm for phase retrieval,” J. Opt. Soc. Am. A 5, 30–38 (1988). [CrossRef]
  15. D. S. Weile, E. Michielsen, “Genetic algorithm optimization applied to electromagnetics,” IEEE Trans. Antennas Propag. 45, 343–353 (1997). [CrossRef]
  16. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).
  17. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
  18. J. R. Fienup, “Phase retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993). [CrossRef] [PubMed]
  19. 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. Integr. Eq. 9, 77–125 (1985).
  20. J. R. Fienup, “Reconstruction of a complex valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987). [CrossRef]
  21. D. C. Dobson, “Phase reconstruction via nonlinear least squares,” Inverse Probl. 8, 541–557 (1992). [CrossRef]
  22. T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Image reconstruction from Fourier transform magnitude with applications to SAR imaging,” J. Opt. Soc. Am. A 13, 922–934 (1996). [CrossRef]
  23. P. T. Chen, M. A. Fiddy, C. W. Liao, D. A. Pommel, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 13, 1524–1531 (1996). [CrossRef]
  24. C. C. Wakerman, A. E. Yagle, “Phase retrieval and estimation with use of real plane zeros,” J. Opt. Soc. Am. A 11, 2016–2026 (1994). [CrossRef]
  25. Z. Mou-yan, R. Unbehauen, “Methods for reconstruction of 2-D sequences from Fourier transform magnitude,” IEEE Trans. Image Process. 6, 222–233 (1997). [CrossRef]
  26. T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996). [CrossRef]
  27. G. Leone, R. Pierri, F. Soldovieri, “On the performances of two algorithms in phaseless antenna measurements,” in Proceedings of the 10th International Conference on Antennas and Propagation, Conf. Publ. 436 IEE (Institute of Electrical Engineers, London, 1997), pp. 1.136–1.141.
  28. R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1998).
  29. A. E. Taylor, D. L. Kay, Introduction to Functional Analysis, 2nd ed. (Krieger, Malabar, Fla., 1980).
  30. R. Pierri, G. Leone, “The phase retrieval as a quadratic inversion in microwave applications,” presented at the 25th European Microwave Conference, Bologna, Italy, 1995.
  31. R. Barakat, G. Newsam, “Necessary conditions for an unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984). [CrossRef]
  32. M. H. Hayes, J. H. McClellan, “Reducible polynomials in two or more variables,” Proc. IEEE 70, 197–198 (1982). [CrossRef]
  33. J. L. C. Sanz, T. S. Huang, “Unique reconstruction of a band-limited multidimensional signal from its phase or magnitude,” J. Opt. Soc. Am. 73, 1446–1450 (1983). [CrossRef]
  34. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990). [CrossRef]
  35. J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, San Diego, Calif., 1970).
  36. Note that, if we fix, without any loss of generality, the modulus and the phase of a sample of F, the sequence f cannot assume a null value.
  37. D. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1987).
  38. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).
  39. A different kind of support would imply modification also of the essential dimension of data space (see Subsection 4.B).
  40. J. R. Fienup, “Gradient-search phase retrieval algorithm for inverse synthetic-aperture radar,” Opt. Eng. 33, 3237–3242 (1994). [CrossRef]
  41. T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Synthetic aperture radar imaging from phase-corrupted data,” IEE Proc. Radar Sonar Navigation 143, 268–274 (1996). [CrossRef]
  42. B. R. Hunt, T. L. Overman, P. Gough, “Image reconstruction of Fourier-transform magnitude,” Opt. Lett. 23, 1123–1125 (1998). [CrossRef]
  43. J. Miao, D. Sayre, 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]

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.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited