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Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 52, Iss. 17 — Jun. 10, 2013
  • pp: 3977–3986

Nonlinear approaches for the single-distance phase retrieval problem involving regularizations with sparsity constraints

Valentina Davidoiu, Bruno Sixou, Max Langer, and Francoise Peyrin  »View Author Affiliations

Applied Optics, Vol. 52, Issue 17, pp. 3977-3986 (2013)

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The phase retrieval process is a nonlinear ill-posed problem. The Fresnel diffraction patterns obtained with hard x-ray synchrotron beam can be used to retrieve the phase contrast. In this work, we present a convergence comparison of several nonlinear approaches for the phase retrieval problem involving regularizations with sparsity constraints. The phase solution is assumed to have a sparse representation with respect to an orthonormal wavelets basis. One approach uses alternatively a solution of the nonlinear problem based on the Fréchet derivative and a solution of the linear problem in wavelet coordinates with an iterative thresholding. A second method is the one proposed by Ramlau and Teschke which generalizes to a nonlinear problem the classical thresholding algorithm. The algorithms were tested on a 3D Shepp–Logan phantom corrupted by white Gaussian noise. The best simulation results are obtained by the first method for the various noise levels and initializations investigated. The reconstruction errors are significantly decreased with respect to the ones given by the classical linear phase retrieval approaches.

© 2013 Optical Society of America

OCIS Codes
(100.3190) Image processing : Inverse problems
(100.5070) Image processing : Phase retrieval
(110.7440) Imaging systems : X-ray imaging
(180.7460) Microscopy : X-ray microscopy

ToC Category:
Imaging Systems

Original Manuscript: January 31, 2013
Revised Manuscript: May 7, 2013
Manuscript Accepted: May 7, 2013
Published: June 6, 2013

Virtual Issues
Vol. 8, Iss. 7 Virtual Journal for Biomedical Optics

Valentina Davidoiu, Bruno Sixou, Max Langer, and Francoise Peyrin, "Nonlinear approaches for the single-distance phase retrieval problem involving regularizations with sparsity constraints," Appl. Opt. 52, 3977-3986 (2013)

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  1. T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995). [CrossRef]
  2. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997). [CrossRef]
  3. U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965). [CrossRef]
  4. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast x-ray computed tomography for observing biological tissues,” Nat. Med. 2, 473–475 (1996). [CrossRef]
  5. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003). [CrossRef]
  6. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105, 248102 (2010). [CrossRef]
  7. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D 29, 133–146 (1996). [CrossRef]
  8. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996). [CrossRef]
  9. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996). [CrossRef]
  10. T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-rays quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999). [CrossRef]
  11. 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]
  12. D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
  13. S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Instrum. 76, 073705 (2005). [CrossRef]
  14. J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617–1619 (2007). [CrossRef]
  15. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef]
  16. T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003). [CrossRef]
  17. A. Alpers, G. T. Herman, H. Poulsen, and S. Schmidt, “Phase retrieval for superposed signals from multiple binary objects,” J. Opt. Soc. Am. A 27, 1927–1937 (2010). [CrossRef]
  18. V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Non-linear phase retrieval based on Fréchet derivative,” Opt. Express 19, 22809–22819 (2011). [CrossRef]
  19. V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Nonlinear phase retrieval using projection operator and iterative wavelet thresholding,” IEEE Signal Process. Lett. 19, 579–582 (2012). [CrossRef]
  20. H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “CPRL—an extension of compressive sensing to the phase retrieval problem,” Adv. Neural Inf. Process. Syst. 25, 1376–1384 (2012).
  21. E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion” (2011), http://arxiv.org/abs/1109.0573 .
  22. E. J. Candès and X. Li, “Solving quadratic equations via PhaseLift when there are about as many equations as unknowns” (2012), http://arxiv.org/abs/1208.6247 .
  23. I. Waldspurger, A. D’Aspremont, and S. Mallat, “Phase recovery, maxcut, and complex semidefinite programming” (2012), http://arxiv.org/pdf/1206.0102.pdf .
  24. S. Mukherjee and C. S. Seelamantula, “An iterative algorithm for phase retrieval with sparsity constraints: application to frequency domain optical coherence tomography,” In 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 553–556.
  25. T. Gaass, G. Potdevin, P. B. Nol, A. Tapfer, M. Willner, J. Herzen, and A. Haase, “Compressed sensing for phase contrast CT,” AIP Conf. Proc. 1466, 150–154 (2012). [CrossRef]
  26. M. C. Newton, “Compressed sensing for phase retrieval,” Phys. Rev. E 85, 056706 (2012). [CrossRef]
  27. R. Ramlau and G. Teschke, “A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints,” Numer. Math. 104, 177–203 (2006). [CrossRef]
  28. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997).
  29. V. Dicken, “A new approach towards simultaneous activity and attenuation reconstruction in emission tomography,” Inverse Probl. 15, 931–960 (1999). [CrossRef]
  30. M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008). [CrossRef]
  31. M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2428–2436 (2010). [CrossRef]
  32. J. Moosmann, R. Hofmann, A. V. Bronnikov, and T. Baumbach, “Nonlinear phase retrieval from single-distance radiograph,” Opt. Express 18, 25771–25785 (2010). [CrossRef]
  33. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1987).
  34. C. R. Vogel, “Numerical solution of a non-linear ill-posed problem arising in inverse scattering,” Inverse Probl. 1, 393–403 (1985). [CrossRef]
  35. O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variational Methods in Imaging (Springer-Verlag, 2008).
  36. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006). [CrossRef]
  37. I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).
  38. F. Dupe, J. M. Fadili, and J. L. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321(2009). [CrossRef]
  39. G. Teschke and R. Ramlau, “An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector-valued regimes and an application to color image impainting,” Inverse Probl. 23, 1851–1870 (2007). [CrossRef]
  40. R. Ramlau, “A steepest descent algorithm for the global minimization of the Tikhonov functional,” Inverse Probl. 18, 381–403 (2002). [CrossRef]
  41. L. Chaâri, N. Pustelnik, C. Chaux, and J. C. Pesquet, “Solving inverse problems with overcomplete transforms and convex optimization techniques,” Proc. SPIE 7446, 74460U (2009). [CrossRef]

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