OSA's Digital Library

Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 52, Iss. 36 — Dec. 20, 2013
  • pp: 8627–8633

Phase retrieval of images using Gaussian radial bases

Russell Trahan, III and David Hyland  »View Author Affiliations

Applied Optics, Vol. 52, Issue 36, pp. 8627-8633 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (673 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Here, the possibility of a noniterative solution to the phase retrieval problem is explored. A new look is taken at the phase retrieval problem that reveals that knowledge of a diffraction pattern’s frequency components is enough to recover the image without projective iterations. This occurs when the image is formed using Gaussian bases that give the convenience of a continuous Fourier transform existing in a compact form where square pixels do not. The Gaussian bases are appropriate when circular apertures are used to detect the diffraction pattern because of their optical transfer functions, as discussed briefly. An algorithm is derived that is capable of recovering an image formed by Gaussian bases from only the Fourier transform’s modulus, without background constraints. A practical example is shown.

© 2013 Optical Society of America

OCIS Codes
(070.6020) Fourier optics and signal processing : Continuous optical signal processing
(100.3190) Image processing : Inverse problems
(100.5070) Image processing : Phase retrieval
(110.2990) Imaging systems : Image formation theory
(100.3175) Image processing : Interferometric imaging
(110.3175) Imaging systems : Interferometric imaging

ToC Category:
Image Processing

Original Manuscript: October 4, 2013
Revised Manuscript: November 13, 2013
Manuscript Accepted: November 13, 2013
Published: December 11, 2013

Russell Trahan and David Hyland, "Phase retrieval of images using Gaussian radial bases," Appl. Opt. 52, 8627-8633 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–250 (1972).
  2. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986). [CrossRef]
  3. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef]
  4. J. R. Fienup, “Reconstruction of an object from the modulus,” Opt. Lett. 3, 27–29 (1978). [CrossRef]
  5. J. S. Wu, U. Weierstall, and J. Spence, “Iterative phase retrieval without support,” Opt. Lett. 29, 2737–2739 (2004). [CrossRef]
  6. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003). [CrossRef]
  7. 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]
  8. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007). [CrossRef]
  9. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003). [CrossRef]
  10. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002). [CrossRef]
  11. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection–reflection method for phase retrieval,” J. Opt. Soc. Am. A 20, 1025–2034 (2003). [CrossRef]
  12. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005). [CrossRef]
  13. R. Trahan and D. Hyland, “Mitigating the effect of noise in the hybrid input–output method of phase retrieval,” Appl. Opt. 52, 3031–3037 (2013). [CrossRef]
  14. S. Babaoe-Kafaki, “A quadratic hybridization of Polak–Ribière–Polyak and Fletcher–Reeves conjugate gradient methods,” J. Optim. Theory Appl. 154, 916–932 (2012). [CrossRef]
  15. R. Fletcher and C. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964). [CrossRef]
  16. E. Polak, “Computational methods in optimization; a unified approach,” Math. Program. 3, 131–133 (1972).
  17. E. Polak and G. Ribiére, “Note sur la convergence de méthodes de directions conjuguées,” Rev. Fr. Inf. Rech. Oper. 3, 35–43 (1969).
  18. G. Liu, “Fourier phase retrieval algorithm with noise constraints,” Sig. Process. 21, 339–347 (1990). [CrossRef]
  19. G. Liu, “Object reconstruction from noisy holograms: multiplicative noise model,” Opt. Commun. 79, 402–406 (1990). [CrossRef]
  20. R. Bates and D. Mnyama, “The status of practical Fourier phase retrieval,” Advances in Electronics and Electron Physics (Academic, 1986), Vol. 67, pp. 1–64.
  21. M. Kohl, A. A. Minkevich, and T. Baumback, “Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm,” Opt. Express 20, 17093–17106 (2012). [CrossRef]
  22. G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007). [CrossRef]
  23. F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007). [CrossRef]
  24. W. Chen and X. Chen, “Quantitative phase retrieval of complex-valued specimens based on noninterferometric imaging,” Appl. Opt. 50, 2008–2015 (2011). [CrossRef]
  25. R. Hanbury Brown and R. Q. Twiss, “A new type of interferometer for use in radio astronomy,” Philos. Mag. 45(7), 663–682 (1954).
  26. R. Hanbury Brown and R. Q. Twiss, “A test of a new type of Stellar Interferometer on Sirius,” Nature 178, 1046–1048 (1956). [CrossRef]
  27. R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light,” Proc. R. Soc. B 242, 300–324 (1957). [CrossRef]
  28. H. Altwaijry and D. Hyland, “Detection and characterization of near Earth asteroids using stellar occultation,” in AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, South Carolina, 2013.
  29. E. Gur, V. Sarafis, I. Falat, F. Vacha, M. Vacha, and Z. Zalevsky, “Super-resolution via iterative phase retrieval for blurred and saturated biological images,” Opt. Express 16, 7894–7903 (2008). [CrossRef]
  30. A. R. Smith, “A pixel is not a little square,” Microsoft Technical Memo 6 (Microsoft, 1995).
  31. R. Fernando, GPU Gems: Programming Techniques, Tips and Tricks for Real-Time Graphics (Pearson Higher Education, 2004).
  32. M. Pharr and R. Fernando, GPU Gems 2: Programming Techniques for High-Performance Graphics and General-Purpose Computation (Addison-Wesley, 2005).
  33. P. S. Heckbert, “Survey of texture mapping,” IEEE Comp. Grap. Appl. 6, 56–67 (1986). [CrossRef]
  34. D. Shreiner, G. Sellers, J. M. Kessenish, and B. M. Licea-Kane, OpenGL Programming Guide (Addison-Wesley, 2013).
  35. E. Hartman and J. Keeler, “Layered neural networks with Gaussian hidden units as universal approximations,” Neural Comput. 2, 210–215 (1990). [CrossRef]
  36. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).
  37. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

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.

Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited