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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 27, Iss. 10 — Oct. 1, 2010
  • pp: 2244–2251

Non-interferometric, non-iterative phase retrieval by Green’s functions

Johannes Frank, Stefan Altmeyer, and Guenther Wernicke  »View Author Affiliations

JOSA A, Vol. 27, Issue 10, pp. 2244-2251 (2010)

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In this paper a non-interferometric, non-iterative method for phase retrieval by Green’s functions is presented. The theory is based on the parabolic wave equation that describes propagation of light in the Fresnel approximation in homogeneous media. Green’s first identity will be used to derive an algorithm for phase retrieval considering different boundary conditions. Finally it will be shown that a commonly used solution of the transport-of-intensity equation can be obtained as a special case of the more general Green’s function formulation derived here.

© 2010 Optical Society of America

OCIS Codes
(100.0100) Image processing : Image processing
(100.3010) Image processing : Image reconstruction techniques
(100.3190) Image processing : Inverse problems
(100.5070) Image processing : Phase retrieval

ToC Category:
Image Processing

Original Manuscript: May 18, 2010
Revised Manuscript: August 4, 2010
Manuscript Accepted: August 24, 2010
Published: September 22, 2010

Johannes Frank, Stefan Altmeyer, and Guenther Wernicke, "Non-interferometric, non-iterative phase retrieval by Green’s functions," J. Opt. Soc. Am. A 27, 2244-2251 (2010)

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  1. G. Wernicke and W. Osten, Holografische Interferometrie (Physik-Verlag, 1982).
  2. T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).
  3. U. Schnars and W. Jueptner, Digital Holography (Springer Verlag, 2005).
  4. G. Wernicke, J. Frank, H. Gruber, M. Duerr, A. Langner, S. Eisebitt, C. Guenther, L. Bouamama, S. Krueger, and A. Hermerschmidt, “Applications of the high-resolution optical reconstruction of digital holograms,” Proc. SPIE 6136, 61360Q (2006). [CrossRef]
  5. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, 1998).
  6. 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).
  7. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
  8. F. Roddier and C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991). [CrossRef] [PubMed]
  9. L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001). [CrossRef]
  10. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983). [CrossRef]
  11. D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586–2589 (1998). [CrossRef]
  12. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998). [CrossRef]
  13. E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002). [CrossRef] [PubMed]
  14. M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004). [CrossRef] [PubMed]
  15. C. Dorrer and J. D. Zuegel, “Optical testing using the transport-of-intensity equation,” Opt. Express 15, 7165–7175 (2007). [CrossRef] [PubMed]
  16. N. Nakajima, “Noniterative two-dimensional phase-retrieval method from two Fourier intensities by use of an exponential filter,” Appl. Opt. 42, 2492–2497 (2003). [CrossRef] [PubMed]
  17. N. Nakajima, “Lensless imaging from diffraction intensity measurements by use of a noniterative phase-retrieval method,” Appl. Opt. 43, 1710–1718 (2004). [CrossRef] [PubMed]
  18. N. Nakajima, “Noniterative phase retrieval from a single diffraction intensity pattern by use of an aperture array,” Phys. Rev. Lett. 98, 223901 (2007). [CrossRef] [PubMed]
  19. N. Nakajima, “Lensless coherent imaging by a deterministic phase retrieval method with an aperture-array filter,” J. Opt. Soc. Am. A 25, 742–750 (2008). [CrossRef]
  20. N. Nakajima, “Phase retrieval from a high-numerical-aperture intensity distribution by use of an aperture-array filter,” J. Opt. Soc. Am. A 26, 2172–2180 (2009). [CrossRef]
  21. N. Nakajima, “Phase retrieval using an aperture-array filter under partially coherent illumination,” Opt. Commun. 282, 2128–2135 (2009). [CrossRef]
  22. J. F. Nye and M. F. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974). [CrossRef]
  23. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1941 (1995). [CrossRef]
  24. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996). [CrossRef]
  25. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).
  26. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  27. G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996). [CrossRef]
  28. A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists (Chapman & Hall, 2002).
  29. P. M. Morse and H. Feshbach, Methods of Theoretical Physics Part 1 (McGraw-Hill, 1953).
  30. E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989). [CrossRef]
  31. H. Maitre and I. Lyuboshenko, “Robust algorithms for phase unwrapping in SAR interferometry,” Proc. SPIE 3217, 176–187 (1997). [CrossRef]
  32. I. Lyuboshenko and H. Maitre, “Phase unwrapping for interferometric synthetic aperture radar by use of Helmholtz equation eigenfunctions and the first Green’s identity,” J. Opt. Soc. Am. A 16, 378–395 (1999). [CrossRef]
  33. B. E. Allman and K. A. Nugent, “Shape imaging in defence operations,” in Proceedings Land Warfare Conference (2006).
  34. V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002). [CrossRef] [PubMed]
  35. A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924 (2006). [CrossRef] [PubMed]
  36. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984). [CrossRef]
  37. T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003). [CrossRef]
  38. National Institute of Standards and Technology, “Green’s functions tutorial, introduction to Green’s functions,” http://www.boulder.nist.gov/div853/greenfn/tutorial.html.

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