OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 8 — Apr. 21, 2014
  • pp: 9220–9244

Boundary-artifact-free phase retrieval with the transport of intensity equation: fast solution with use of discrete cosine transform

Chao Zuo, Qian Chen, and Anand Asundi  »View Author Affiliations

Optics Express, Vol. 22, Issue 8, pp. 9220-9244 (2014)

View Full Text Article

Enhanced HTML    Acrobat PDF (2088 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The transport of intensity equation (TIE) is a two-dimensional second order elliptic partial differential equation that must be solved under appropriate boundary conditions. However, the boundary conditions are difficult to obtain in practice. The fast Fourier transform (FFT) based TIE solutions are widely adopted for its speed and simplicity. However, it implies periodic boundary conditions, which lead to significant boundary artifacts when the imposed assumption is violated. In this work, TIE phase retrieval is considered as an inhomogeneous Neumann boundary value problem with the boundary values experimentally measurable around a hard-edged aperture, without any assumption or prior knowledge about the test object and the setup. The analytic integral solution via Green’s function is given, as well as a fast numerical implementation for a rectangular region using the discrete cosine transform. This approach is applicable for the case of non-uniform intensity distribution with no extra effort to extract the boundary values from the intensity derivative signals. Its efficiency and robustness have been verified by several numerical simulations even when the objects are complex and the intensity measurements are noisy. This method promises to be an effective fast TIE solver for quantitative phase imaging applications.

© 2014 Optical Society of America

OCIS Codes
(100.3010) Image processing : Image reconstruction techniques
(100.5070) Image processing : Phase retrieval
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Image Processing

Original Manuscript: December 18, 2013
Revised Manuscript: February 27, 2014
Manuscript Accepted: March 3, 2014
Published: April 9, 2014

Chao Zuo, Qian Chen, and Anand Asundi, "Boundary-artifact-free phase retrieval with the transport of intensity equation: fast solution with use of discrete cosine transform," Opt. Express 22, 9220-9244 (2014)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Reed Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]
  2. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27(7), 1223–1225 (1988). [CrossRef] [PubMed]
  3. K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996). [CrossRef] [PubMed]
  4. K. Ishizuka, B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54(3), 191–197 (2005). [CrossRef] [PubMed]
  5. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984). [CrossRef]
  6. C. Zuo, Q. Chen, W. Qu, A. Asundi, “Noninterferometric single-shot quantitative phase microscopy,” Opt. Lett. 38(18), 3538–3541 (2013). [CrossRef] [PubMed]
  7. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995). [CrossRef]
  8. S. C. Woods, A. H. Greenaway, “Wave-front sensing by use of a Green’s function solution to the intensity transport equation,” J. Opt. Soc. Am. A 20(3), 508–512 (2003). [CrossRef] [PubMed]
  9. L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001). [CrossRef]
  10. S. V. Pinhasi, R. Alimi, L. Perelmutter, S. Eliezer, “Topography retrieval using different solutions of the transport intensity equation,” J. Opt. Soc. Am. A 27(10), 2285–2292 (2010). [CrossRef] [PubMed]
  11. T. Gureyev, A. Roberts, K. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12(9), 1932–1942 (1995). [CrossRef]
  12. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996). [CrossRef]
  13. T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1–6), 339–346 (1997). [CrossRef]
  14. D. Paganin, K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998). [CrossRef]
  15. V. V. Volkov, Y. Zhu, M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33(5), 411–416 (2002). [CrossRef] [PubMed]
  16. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29(10), 1402–1403 (1990). [CrossRef] [PubMed]
  17. I. W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232–1237 (1995). [CrossRef]
  18. A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998). [CrossRef] [PubMed]
  19. E. D. Barone-Nugent, A. Barty, K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002). [CrossRef] [PubMed]
  20. S. S. Kou, L. Waller, G. Barbastathis, C. J. R. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. 35(3), 447–449 (2010). [CrossRef] [PubMed]
  21. L. Waller, L. Tian, G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010). [CrossRef] [PubMed]
  22. C. Zuo, Q. Chen, Y. Yu, A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter--theory and applications,” Opt. Express 21(5), 5346–5362 (2013). [CrossRef] [PubMed]
  23. C. Zuo, Q. Chen, W. Qu, A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express 21(20), 24060–24075 (2013). [CrossRef] [PubMed]
  24. J. Martinez-Carranza, K. Falaggis, T. Kozacki, M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
  25. J. Frank, S. Altmeyer, G. Wernicke, “Non-interferometric, non-iterative phase retrieval by Green’s functions,” J. Opt. Soc. Am. A 27(10), 2244–2251 (2010). [CrossRef] [PubMed]
  26. D. A. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, 2001), Vol. 224.
  27. K. Bhamra, Partial Differential Equations: An Introductory Treatment With Applications (PHI, 2010).
  28. R. Courant and D. Hilbert, “Potential theory and elliptic differential equations,” in Methods of Mathematical Physics (Wiley-VCH Verlag GmbH, 2008), pp. 240–406.
  29. C. Zuo, Q. Chen, A. Asundi, “Light field moment imaging: comment,” Opt. Lett. 39(3), 654 (2014). [CrossRef] [PubMed]
  30. D. W. Trim, Applied Partial Differential Equations (PWS-Kent, 1990).
  31. A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?” in Computer Vision–ECCV 2006 (Springer, 2006), pp. 578–591.
  32. A. Talmi, E. N. Ribak, “Wavefront reconstruction from its gradients,” J. Opt. Soc. Am. A 23(2), 288–297 (2006). [CrossRef] [PubMed]
  33. F. Morse, Methods of Theoretical Physics (1981), Vols. 1 and 2.
  34. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing (Cambridge University, 1992).
  35. D. Paganin, A. Barty, P. J. McMahon, K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004). [CrossRef] [PubMed]
  36. C. Dorrer, J. D. Zuegel, “Optical testing using the transport-of-intensity equation,” Opt. Express 15(12), 7165–7175 (2007). [CrossRef] [PubMed]
  37. L. Tian, J. C. Petruccelli, G. Barbastathis, “Nonlinear diffusion regularization for transport of intensity phase imaging,” Opt. Lett. 37(19), 4131–4133 (2012). [CrossRef] [PubMed]
  38. C. Campbell, “Wave-front sensing by use of a Green’s function solution to the intensity transport equation: comment,” J. Opt. Soc. Am. A 24(8), 2480–2482 (2007). [CrossRef] [PubMed]
  39. S. C. Woods, H. I. Campbell, A. H. Greenaway, “Wave-front sensing by use of a Green's function solution to the intensity transport equation: reply to comment,” J. Opt. Soc. Am. A 24(8), 2482–2484 (2007). [CrossRef]
  40. J. A. Schmalz, T. E. Gureyev, D. M. Paganin, K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011). [CrossRef]
  41. V. V. Volkov, Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98(2-4), 271–281 (2004). [CrossRef] [PubMed]
  42. N. Ahmed, T. Natarajan, K. R. Rao, “Discrete cosine transform,” IEEE Trans. Comput. C-23(1), 90–93 (1974). [CrossRef]
  43. E. Feig, S. Winograd, “Fast algorithms for the discrete cosine transform,” IEEE Trans. Signal Process. 40(9), 2174–2193 (1992). [CrossRef]
  44. J. Tribolet, R. E. Crochiere, “Frequency domain coding of speech,” IEEE Trans. Acoust. Speech Signal Process. 27(5), 512–530 (1979). [CrossRef]
  45. R. Reeves, K. Kubik, “Shift, scaling and derivative properties for the discrete cosine transform,” Signal Process. 86(7), 1597–1603 (2006). [CrossRef]
  46. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1986), Vol. 3.

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.

CrossCheck Deposited