## Criticality in single-distance phase retrieval |

Optics Express, Vol. 19, Issue 27, pp. 25881-25890 (2011)

http://dx.doi.org/10.1364/OE.19.025881

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### Abstract

We investigate why in free-space propagation single-distance phase retrieval based on a modified contrast-transfer function of linearized Fresnel theory yields good results for moderately strong pure-phase objects. Upscaling phase-variations in the exit plane, the growth of maxima of the modulus of the Fourier transformed intensity contrast dominates the minima. Cutting out small regions around the latter thus keeps information loss due to nonlocal, nonlinear effects negligible. This quasiparticle approach breaks down at a critical upscaling where the positions of the minima start to move rapidly. We apply our results to X-ray data of an early-stage Xenopus (frog) embryo.

© 2011 OSA

*λ*, circular frequency

*ω*, energy

*E*, quantum of action

*h*, speed of light in vacuum

*c*) of a pure-phase object, which does not diminish the (ideal) spatial coherence properties of the incoming wavefront, we consider free-space propagation of the modulated exit wavefront

*ψ*

_{z}_{=0}(

*r⃗*) away from plane

*z*= 0 to generate intensity contrast

*z*> 0 [1

1. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of Xray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. **66**, 5486–5492 (1995). [CrossRef]

7. M. R. Teague, “Deterministic phase retrieval: a Greens function solution,” J. Opt. Soc. Am. **73**, 1434–1441 (1983). [CrossRef]

*I*is the intensity measured in plane

_{z}*z*, and

*I*

_{z}_{=0}≡ const for a pure-phase object. In such a setting, we consider phase retrieval based on a projected version of the

*contrast-transfer function*(CTF) [8], which represents a

*linear*and

*local*[9

9. T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun **259**, 569–580 (2006). [CrossRef]

*g*and the phase shift

_{z}*ϕ*

_{z}_{=0}exiting the object, when

*ϕ*

_{z}_{=0}violates the CTF criterion Here

*ξ⃗*is a transverse-plane wave vector. Such a regularized form of CTF retrieval, named

*projected CTF*, was proposed in [10

10. J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express **19**, 12066–12073 (2011). [CrossRef] [PubMed]

*effective*phase in the spirit of a quasiparticle model [11], remarkably good results for single-distance phase retrieval. The present paper aims at a deeper understanding of this situation.

*ϕ*

_{z}_{=0}→

*α*≡ const. According to Eq. (2), then (

*ℱI*)(

_{z}*ξ⃗*) ≡

*δ*

^{(2)}(

*ξ⃗*)

*I*

_{z}_{=0}, thus (

*ℱg*)(

_{z}*ξ⃗*) ≡ 0, and the right-hand side of Eq. (3) is

*ξ⃗*= 0. (The impossibility of retrieving

*α*from this relation is due to a global U(1) or constant-phase-shift symmetry of Fresnel theory.) When

*ϕ*

_{z}_{=0}starts to vary, infinitely many CTF “vacua” appear at |

*ξ⃗*|

*, and excitations of finite “energy” occur in between these “vacua”. Explicit violations of scaling symmetry are introduced by the nonlinear and nonlocal terms in Eq. (3) starting at*

_{n}*O*((

*ℱϕ*

_{z}_{=0})

^{2}). These cause CTF “vacua” to become “finite-energy” minima of |

*ℱg*|.

_{z}*σ*denotes the Gaussian’s width, and the maximal, relative phase variation is unity. It is straight-forward to derive an expression for the right-hand side of Eq. (3) when evaluated in this model after letting

*π*

^{2}

*σ*

^{2}and

*S*

^{2}. Their ratio is

*λ*= 10

^{−10}m,

*σ*= 10

^{−6}m,

*z*= 1m this yields a value of

^{−2π2σ2ξ⃗2}is

*φ*of the sine function (order

*S*) as induced by the cosine correction (order

*S*

^{2}) in the vicinity of

*S*and with the above parameter values a phase shift

*φ*of the sine function in Eq. (4) given as Thus, for sufficiently small values of

*S*, the shift of the first zero of the sine function as introduced by the quadratic, nonlocal corrections in Eq. (3) is negligible for the Gaussian model considered. We demonstrate below in a full numerical treatment of the Fresnel forward propagation for a generic, that is, realistically complex situation that the position of the first minima |

*ξ⃗*|

_{min,1}does not move away from

*S*values that upscale the regime where linear CTF retrieval is applicable. A critical increase of |

*ξ⃗*|

_{min,1}sets in rather late at a maximal relative phase variation larger than unity. Therefore, the entirety of higher-order corrections to the right-hand side of Eq. (3) actually stabilizes our perturbative, Gaussian-model finding of a slow variation of |

*ξ⃗*|

_{1}for small

*S*to a situation of no variation at all up to

*S*. Moreover, the all-order result changes a polynomial dependence of |

_{c}*ξ⃗*|

_{1}on

*S*, as it is obtained in finite-order perturbation theory, to a fractional power of

*S*–

*S*for

_{c}*ℱg*are not moving for 0 <

_{z}*S*≤

*S*indicates that explicit scaling-symmetry violation is not supplemented by

_{c}*dynamical*breaking all the way up to

*S*. Recall that

_{c}*explicit*symmetry breaking refers to the fact that finite as opposed to vanishing values of the “energy” (

*ℱg*)(

_{z}*ξ⃗*) are no longer invariant under the symmetry. On the other hand, for the symmetry to be broken

*dynamically*the locations, where minimal “energy” is attained, are shifted under the symmetry. For a continuous symmetry such as scaling symmetry the latter situation changes the spectrum drastically: It introduces new degrees of freedom (Goldstone bosons [12

12. J. Goldstone, “Field theories with superconductor solutions,” Nuovo Cimento **19**, 154–164 (1961). [CrossRef]

14. Y. Nambu, “Quasiparticles and gauge invariance in the theory of superconductivity,” Phys. Rev. **117**, 648–663 (1960). [CrossRef]

*S*≥

*S*. (The quasiparticle concept leading to an effective CTF phase then is as useless as the description of an atomic crystal in terms of moderately interacting atoms which, however, applies to the liquid phase.) If condition (1) is sufficiently well satisfied then limited resolution in transverse Fourier space in any discretized formulation does not resolve the small-residue poles of CTF retrieval that appear in

_{c}*ℱϕ*

_{z}_{=0}at |

*ξ⃗*|

*, and numerical Fourier inversion yields satisfactory phase retrieval. We define*

_{n}*ϕ*

_{max}≡ max{

*ϕ*

_{z}_{=0}(

*r⃗*)} with the convention that 0 ≤

*ϕ*

_{z}_{=0}(

*r⃗*). With our pixel resolution of Δ

*x*= 1.1

*μ*m,

*E*= 10keV, and

*z*= 0.5m we are in this CTF scaling regime when setting

*ϕ*

_{max}= 0.01 for the phase map

*ϕ*

_{z}_{=0}in Fig. 1(a) which serves as an input to Fresnel forward propagation. In this case we refer to the phase map as

*ϕ*

_{z}_{=0,CTF}. In Fig. 1(b) we show angular averages

*ξ⃗*| obtained for two inputs

*ϕ*

_{z}_{=0}=

*Sϕ*

_{z}_{=0,CTF}with

*S*= 200 <

*S*= 356 and

_{c}*S*= 450 >

*S*. (

_{c}*S*= 1 corresponds to

*ϕ*

_{z}_{=0,CTF}) While for

*S*= 200 the position of |

*ξ⃗*|

_{min,1}coincides with the CTF “vacuum” |

*ξ⃗*|

_{1}this is not true for

*S*= 450. “Finite-energy” minima at

*S*= 200 introduce poles in Fourier space and thus quasiperiodic artifacts [10

10. J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express **19**, 12066–12073 (2011). [CrossRef] [PubMed]

10. J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express **19**, 12066–12073 (2011). [CrossRef] [PubMed]

*ɛ*is a threshold (0 <

*ɛ*< 1) such that minima are centrally cut about the CTF “vacua”. Applying CTF retrieval to the projected intensity contrast on the right-hand side of replacement (6) yields good results even for very small values of

*ɛ*.

*S*≤

*S*, where no dynamical breaking occurs, we have investigated in Fig. 2 the behavior of the transfer function of the CTF approximation in dependence of

_{c}*S*for the phase map of Fig. 1(a). Observing a smooth sinusoidal shape for

*S*= 1 justifies the above-mentioned consideration of

*ϕ*= 0.01 as a representative of the linear scaling regime. Notice the increasingly dramatic and nervous deviations from this sinusoidal dependence for

_{max}*S*= 100 and

*S*= 200. Therefore, we conclude that even for maximal phase shifts well below

*S*× 0.01 ∼ 3.6 (moderately strong maximal phase variation) the assumed linearity of CTF retrieval fails judging by the behavior of the associated transfer function.

_{c}*S*≤

*S*. Figure 3 shows how the position of the first minimum |

_{c}*ξ⃗*|

_{min,1}changes with increasing

*S*. At

*S*= 356, which corresponds to

_{c}*ϕ*

_{max}∼ 3.6 and thus to a profound violation of condition (1), a critical increase of |

*ξ⃗*|

_{min,}_{1}away from the first CTF “vacuum” |

*ξ⃗*|

_{1}takes place. A fit to

*A*|

*S*–

*S*|

_{c}*+*

^{ν}*B*(

*A*,

*B*,

*ν*real,

*S*>

*S*) of this critical behavior, which resembles a second-order phase transition, yields an exponent

_{c}*ν*∼ 0.15 ± 0.1, the large error being associated with instabilities w.r.t. the length of the fitting interval. (|

*ξ⃗*|

_{min,1}– |

*ξ⃗*|

_{1}is only a pseudo-order parameter for dynamical scaling-symmetry breaking since the latter occurs on top of explicit breaking. For a discrete-symmetry analog, consider an Ising model with magnetic field

*H*. For

*H*= 0 the model is

**Z**invariant, for

_{2}*H*≠ 0 not. For

*T*≤

*T*ferromagnetic ordering occurs, and, given moderate values of

_{c}*H*, it makes sense to consider mean magnetization a pseudo-order parameter for

*dynamical*

**Z**breaking.)

_{2}*R*of

*S*for 1 ≤

*S*≤

*S*= 356. Notice that for all such

_{c}*S*the growth of the first maximum by far outraces that of the first minimum. This can be understood as follows. While, according to Eq. (3), the growth of the minima solely is due to the nonlocal terms at quadratic and higher order in

*ℱϕ*

_{z}_{=0}there is a local component in the growth of the maxima (scaling proportional to

*S*due to the linear and local CTF order in Eq. (3)). For reasonably “nervous”

*ℱϕ*

_{z}_{=0}and for sufficiently moderate

*S*successive

*n*-fold autoconvolutions of

*ℱϕ*

_{z}_{=0}(

*n*≥ 2) tend to homogenize the nonlinear corrections, which are proportional to

*S*, to small values. Thus, in this regime the dominantly linearly and locally driven growth of the maxima outraces the growth of the minima, and little information is lost if for 1 ≤

^{n}*S*≤

*S*thin rings centered at |

_{c}*ξ⃗*|

*are cut out to enable singularity-free phase retrieval [10*

_{n}**19**, 12066–12073 (2011). [CrossRef] [PubMed]

*g*(

_{z}*r⃗*), let

*S*, say,

*S*= 2, Fresnel propagate

*z*to generate the new intensity contrast

*ξ⃗*|

_{min,1}have moved in

*S*= 1,2) are depicted for projected CTF applied to the Xenopus data of Fig. 6(a),(d), and it is obvious that |

*ξ⃗*|

_{min,1}did not move. Thus we conclude that projected CTF retrieval self-consistently operates within its regime of validity for this particular experiment.

*z*under Fresnel propagation from a pure-phase induced exit-plane test map of realistic complexity. On one hand, an investigation was performed of the response of the position of the first minimum |

*ξ⃗*|

_{min,1}of

*S*). For

*S*= 1 phase variations were prepared to lie within the Fresnel scaling regime (symmetry under moderate upscaling, linearity). For a large range 1 ≤

*S*≤

*S*the value of |

_{c}*ξ⃗*|

_{min,1}is observed to be indifferent to upscaling: It stays at the first CTF “vacuum” |

*ξ⃗*|

_{1}. At

*S*= 356, which corresponds to a maximal phase variation of about 3.6,

_{c}*critical behavior*sets in which resembles a second-order like phase transition of critical exponent

*ν*= 0.15±0.1. (At

*S*= 356 explicit breaking is supplemented by a

_{c}*dynamical*breakdown of scaling symmetry, and |

*ξ⃗*|

_{min,1}– |

*ξ⃗*|

_{1}is the associated pseudo-order parameter.) We have not in detail investigated other minima of

*ξ⃗*|

*(*

_{n}*n*= 1, 2,3,...) from the Fourier transform of the intensity contrast, as is done in projected CTF to enable regular phase retrieval at large values of

*S*, works all the way up to

*S*. On the other hand, we have shown that under upscaling the growth of the first maximum of

_{c}*S*≤

*S*. This can be understood by the fact that the growth of maxima is generated linearly in

_{c}*S*and locally in the Fourier transformed phase map

*ℱϕ*

_{z}_{=0}while the growth of minima, albeit subject to higher powers in

*S*, is due to successive autoconvolutions of

*ℱϕ*

_{z}_{=0}which yield small coefficients generically. As a consequence, the omission of thin rings around |

*ξ⃗*|

*(*

_{n}*n*= 1, 2,3,...) from the Fourier transform of the intensity contrast keeps information loss at a low level. Therefore, it seems that below

*S*the use of projected CTF for the retrieval of moderately strong phases is justified. We have applied projected CTF to the phase retrieval from single-distance intensity induced by an early-stage Xenopus embryo under coherent X-ray illumination. Moreover, we have shown self-consistency of projected CTF in this case by a moderate upscaling of the retrieved phase and subsequent Fresnel forward propagation, and we have performed a tomographic reconstruction of the biological sample.

_{c}15. F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part I.,” Physica **9**, 686–698 (1942). [CrossRef]

16. F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part II.,” Physica **9**, 974–986 (1942). [CrossRef]

## Acknowledgments

## References and links

1. | A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of Xray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. |

2. | S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature |

3. | K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. |

4. | P. Cloetens, “Contribution to phase contrast imaging, reconstruction and tomography with hard synchrotron radiation,” PhD dissertation, Vrije Universiteit Brussel (1999). |

5. | P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. |

6. | S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X rays,” Rev. Sci. Instrum. |

7. | M. R. Teague, “Deterministic phase retrieval: a Greens function solution,” J. Opt. Soc. Am. |

8. | J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik |

9. | T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun |

10. | J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express |

11. | L. D. Landau, “The theory of a Fermi liquid,” Sov. Phys. JETP |

12. | J. Goldstone, “Field theories with superconductor solutions,” Nuovo Cimento |

13. | J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Phys. Rev. |

14. | Y. Nambu, “Quasiparticles and gauge invariance in the theory of superconductivity,” Phys. Rev. |

15. | F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part I.,” Physica |

16. | F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part II.,” Physica |

17. | J. Moosmann, V. Altapova, D. Hänschke, R. Hofmann, and T. Baumbach, “Nonlinear, noniterative, single-distance phase retrieval and developmental biology,” submitted to AIP Proceedings, ICXOM 21. |

**OCIS Codes**

(100.2960) Image processing : Image analysis

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

(340.7440) X-ray optics : X-ray imaging

**ToC Category:**

Image Processing

**History**

Original Manuscript: September 16, 2011

Revised Manuscript: October 21, 2011

Manuscript Accepted: October 26, 2011

Published: December 5, 2011

**Virtual Issues**

Vol. 7, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Ralf Hofmann, Julian Moosmann, and Tilo Baumbach, "Criticality in single-distance phase retrieval," Opt. Express **19**, 25881-25890 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-25881

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### References

- A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of Xray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum.66, 5486–5492 (1995). [CrossRef]
- S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996). [CrossRef]
- K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett.77, 2961–2964 (1996). [CrossRef] [PubMed]
- P. Cloetens, “Contribution to phase contrast imaging, reconstruction and tomography with hard synchrotron radiation,” PhD dissertation, Vrije Universiteit Brussel (1999).
- P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys.32, A145–A151 (1999). [CrossRef]
- 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]
- M. R. Teague, “Deterministic phase retrieval: a Greens function solution,” J. Opt. Soc. Am.73, 1434–1441 (1983). [CrossRef]
- J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik49, 121–125 (1977).
- T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun259, 569–580 (2006). [CrossRef]
- J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express19, 12066–12073 (2011). [CrossRef] [PubMed]
- L. D. Landau, “The theory of a Fermi liquid,” Sov. Phys. JETP3, 920–925 (1957).
- J. Goldstone, “Field theories with superconductor solutions,” Nuovo Cimento19, 154–164 (1961). [CrossRef]
- J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Phys. Rev.127, 965–970 (1962). [CrossRef]
- Y. Nambu, “Quasiparticles and gauge invariance in the theory of superconductivity,” Phys. Rev.117, 648–663 (1960). [CrossRef]
- F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part I.,” Physica9, 686–698 (1942). [CrossRef]
- F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part II.,” Physica9, 974–986 (1942). [CrossRef]
- J. Moosmann, V. Altapova, D. Hänschke, R. Hofmann, and T. Baumbach, “Nonlinear, noniterative, single-distance phase retrieval and developmental biology,” submitted to AIP Proceedings, ICXOM 21.

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