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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 12 — Dec. 19, 2012

Noise models for low counting rate coherent diffraction imaging

Pierre Godard, Marc Allain, Virginie Chamard, and John Rodenburg  »View Author Affiliations


Optics Express, Vol. 20, Issue 23, pp. 25914-25934 (2012)
http://dx.doi.org/10.1364/OE.20.025914


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Abstract

Coherent diffraction imaging (CDI) is a lens-less microscopy method that extracts the complex-valued exit field from intensity measurements alone. It is of particular importance for microscopy imaging with diffraction set-ups where high quality lenses are not available. The inversion scheme allowing the phase retrieval is based on the use of an iterative algorithm. In this work, we address the question of the choice of the iterative process in the case of data corrupted by photon or electron shot noise. Several noise models are presented and further used within two inversion strategies, the ordered subset and the scaled gradient. Based on analytical and numerical analysis together with Monte-Carlo studies, we show that any physical interpretations drawn from a CDI iterative technique require a detailed understanding of the relationship between the noise model and the used inversion method. We observe that iterative algorithms often assume implicitly a noise model. For low counting rates, each noise model behaves differently. Moreover, the used optimization strategy introduces its own artefacts. Based on this analysis, we develop a hybrid strategy which works efficiently in the absence of an informed initial guess. Our work emphasises issues which should be considered carefully when inverting experimental data.

© 2012 OSA

OCIS Codes
(030.4280) Coherence and statistical optics : Noise in imaging systems
(100.5070) Image processing : Phase retrieval
(110.0180) Imaging systems : Microscopy

ToC Category:
Imaging Systems

Virtual Issues
Vol. 7, Iss. 12 Virtual Journal for Biomedical Optics

Citation
Pierre Godard, Marc Allain, Virginie Chamard, and John Rodenburg, "Noise models for low counting rate coherent diffraction imaging," Opt. Express 20, 25914-25934 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-23-25914


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References

  1. K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys.59, 1–100 (2010). [CrossRef]
  2. J. M. Rodenburg, “Ptychography and related diffracted imaging methods,” in “Advances in Imaging and Electron Physics,” 150, P. W. Hawkesed. (Elsevier, 2008), 87–184. [CrossRef]
  3. J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy107, 227–231 (2007). [CrossRef]
  4. D. Claus, A. M. Maiden, F. Zhang, F. G. R. Sweeney, M. Humphry, H. Schluesener, and J. M. Rodenburg, “Quantitative phase contrast optimised cancerous cell differentiation via ptychography,” Opt. Express20, 9911– 9918 (2012). [CrossRef] [PubMed]
  5. M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett.107, 208101 (2011). [CrossRef] [PubMed]
  6. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett.98, 34801 (2007). [CrossRef]
  7. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science321, 379–382 (2008). [CrossRef] [PubMed]
  8. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010). [CrossRef] [PubMed]
  9. P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011). [CrossRef]
  10. F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B82, 121415 (2010). [CrossRef]
  11. M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun.3, 730 (2012). [CrossRef] [PubMed]
  12. C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012). [CrossRef]
  13. S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectroholography,” Nature432, 885–888 (2004). [CrossRef] [PubMed]
  14. V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010). [CrossRef] [PubMed]
  15. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109, 1256–1262 (2009). [CrossRef] [PubMed]
  16. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express16, 7264–7276 (2008). [CrossRef] [PubMed]
  17. A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy120, 64–72 (2012). [CrossRef] [PubMed]
  18. P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys.14, 063004 (2012). [CrossRef]
  19. R. A. Fisher, Statistical Methods and Scientific Inference (Oliver & Boyd, 1956).
  20. M. G. Kendall and A. Stuart, The advanced theory of statistics2a (Griffin, 1963).
  21. F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A451, 596–609 (2000). [CrossRef]
  22. C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A576, 109–112 (2007). [CrossRef]
  23. C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006). [CrossRef]
  24. K. Lange and R. Carson, “EM reconstruction algorithm for emission and transmission tomography,” IEEE Trans. Med. Imag.8, 306–316 (1984).
  25. L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag.1, 113–122 (1982). [CrossRef]
  26. L. B. Lucy, “An iterative technique for the rectification of observed distribution,” New Astron. Rev.79, 745–754 (1974).
  27. G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Cryst.A63, 36–42 (2007).
  28. P. Godard, M. Allain, and V. Chamard, “Imaging of highly inhomogeneous strain field in nanocrystals using x-ray Bragg ptychography: A numerical study,” Phys. Rev. B84, 144109 (2011). [CrossRef]
  29. J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express19, 21333–21344 (2011). [CrossRef] [PubMed]
  30. Provided that the fluctuations in one measurement are accurately described by a Poisson PDF, then this PDF is defined by a single (positive) parameter that is the mean and the variance.
  31. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett.85, 4795–4797 (2004). [CrossRef]
  32. M. F. Freeman and J. W. Tuckey, “Transformations related to the angular and the square root,” Ann. Math. Statist.21, 607–611 (1950). [CrossRef]
  33. C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process.5, 480–492 (1996). [CrossRef] [PubMed]
  34. L. Bouchet, “A comparative-study of deconvolution methods for gamma-ray spectra,” Astron. Astrophys.113, 167–183 (1995).
  35. M. Allain and J.-P. Roques, “High resolution techniques for gamma-ray diffuse emission: application to INTEGRAL/SPI,” Astron. Astrophys.43, 1175–1187 (2006). [CrossRef]
  36. By definition, the likelihood is the PDF of the noise model seen as a function of the unknown parameters ρ. In practice, the opposite of the logarithm of the likelihood is rather considered. However, the logarithm function being a monotonic increasing function, the minimiser of the neg-loglikelihood is also the maximiser of the likelihood, i.e., the ML estimator.
  37. Following [33], it is shown that a second order Taylor expansion around hm,j = ym,j of the Poissonian fitting function ℒ𝒫 leads to (10e).
  38. M. Bertero and P. Boccacci, Introduction to inverse problems in imaging (Institute of Physics Publishing, 1998). [CrossRef]
  39. Since the condition hm,j > 0 is enforced if bm,j > 0 [cf. Eq. (1)], an arbitrary small background component can be introduced, hence allowing all the fitting functions and gradients to be well defined.
  40. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A3, 1897–1907 (1986). [CrossRef]
  41. In the optimization literature, OS algorithms are also known as incremental gradient methods or block iterative methods, see for instance [42, Sec. 1.5.2 ] or [43] for details.
  42. D. P. Bertsekas, Nonlinear programming, 2nd ed. (Athena Scientific, 1999).
  43. Y. Censor, D. Gordon, and R. Gordon, “BICAV: a block-iterative parallel algorithm for sparse systems with pixel-related weighting,” IEEE Trans. Med. Imag.20, 1050–1060 (2001). [CrossRef]
  44. H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered-subset of projection data,” IEEE Trans. Med. Imag.13, 601–609 (1994). [CrossRef]
  45. S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subset algorithms,” IEEE Trans. Med. Imag.22, 613–626 (2003). [CrossRef]
  46. The original version of the PIE introduced by Rodenburg and Faulkner in [31] considers another definition for Dj.
  47. A. M. Maiden, Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am.29, 1606–1614 (2012). [CrossRef]
  48. C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv:optics (2011).
  49. Since the three fitting functions ℒ𝒫, ℒ𝒢 and ℒℛ are equivalent w.r.t. a nil data, only the data such that ym,j ≠ 0 should be considered in order to discriminate the noise-models.
  50. This non-monotonic behaviour of the relative error is standard when inverse problems (e.g., image restoration or tomographic reconstruction) are solved with gradient optimization technics, see for instance [38, Chap. 6].
  51. Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuation to Information (Springer, 2004).
  52. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A20, 40–55 (2003). [CrossRef]
  53. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009). [CrossRef] [PubMed]
  54. From [52, p. 339], one notes that the constraint defined by the data set in the DM strategy takes the form of Eq. (12b), suggesting that the data fluctuations are described by the Gaussian model defined in Sec. 2.2.
  55. J. F. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika35, 246–254 (1948).

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