## 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

With the advent of 3rd-generation synchrotron light sources, producing highly intense and spatially coherent X-rays, the investigation of materials of low absorption but considerable phase-shifting capability can be and is routinely performed. This opens up the potential for the application of new, nondestructive imaging techniques relevant to in vivo investigation of biological samples. Moreover, satisfactory phase retrieval from a single-distance projection in free-space propagation in combination with the small exposure times due to large photon fluxes at synchrotron beamlines enable time resolved tomographic imaging for the study of evolution processes on the cellular and subcellular level. Therefore, in particular the field of developmental biology should benefit from the method discussed in this paper.

For monochromatic and parallel X-ray illumination (wave number
$k=\frac{2\pi}{\lambda}=\frac{2\pi E}{hc}$, wave length *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 ${g}_{z}\equiv \frac{{I}_{z}-{I}_{z=0}}{{I}_{z=0}}$ at distance

*z*> 0 [1–7]. Here

*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] relation between

*g*and the phase shift

_{z}*ϕ*

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

*ϕ*

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

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

*projected CTF*, was proposed in [10] and yields, via the local retrieval of an

*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.

In Fresnel theory the following important relation holds [8]

*ϕ*

_{z}_{=0}yields upon substitution into Eq. (2) and use of the Fourier convolution theorem

*linear*order in

*ℱϕ*

_{z}_{=0}. Provided that (

*ℱg*)(

_{z}*ξ⃗*) exhibits zeros of the same order as those of the sine function at ${\left|\overrightarrow{\xi}\right|}_{n}\equiv \sqrt{\left(kn\right)/\left(2\pi z\right)}$ (

*n*= 0,1,2, …) CTF retrieval in Fourier space does not produce singularities. In analogy to quantum statistical mechanics, CTF represents a dispersion law between complex “energy” (

*ℱg*)(

_{z}*ξ⃗*) and complex “momentum” (

*ℱϕ*

_{z}_{=0})(

*ξ⃗*), both parameterized by

*ξ⃗*. Being a linear model, the spectrum exhibits scaling symmetry: The ratio $\frac{\mathcal{F}{g}_{z}}{\mathcal{F}{\varphi}_{z=0}}\left(\overrightarrow{\xi}\right)$ is invariant under

*ℱg*,

_{z}*ℱϕ*

_{z}_{=0}→

*Sℱg*,

_{z}*Sℱϕ*

_{z}_{=0}where

*S*is positive and real. The phase

*ϕ*

_{z}_{=0}in position space and thus inverse Fourier transformation is understood as an average related to a “partition function”

*Z*: An

_{r⃗}*r⃗*dependent “Hamiltonian”

*ℋ*≡ 2

_{r⃗}*πξ⃗r⃗*is used to define

*Z*≡ tr exp(

_{r⃗}*iℋ*) where the trace symbol is understood as a sum over all wave-vector states. Notice that changing the value of

_{r⃗}*ϕ*

_{z}_{=0}= tr [(

*ℱϕ*

_{z}_{=0}) exp (

*iℋ*)] by a change of the measure-zero set

_{r⃗}*V*≡ {(

_{n}*ℱϕ*

_{z}_{=0})(|

*ξ⃗*|

*(cos*

_{n}*θ*,sin

*θ*))|0 ≤

*θ*≤ 2

*π*}, which is undetermined in CTF retrieval, possesses vanishing probability.

Scaling symmetry of the spectrum is exact in the trivial limit *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 $\frac{{\delta}^{\left(1\right)}\left({\overrightarrow{\xi}}^{2}\right)}{2\pi}\text{sin}\left(2{\pi}^{2}z{\overrightarrow{\xi}}^{2}/k\right)\left(\alpha +i{\alpha}^{2}+O\left({\alpha}^{3}\right)\right)$. That is, the only state that occurs is the “vacuum” (state of zero “energy”) at

*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}Let us now exemplarily investigate the effect in Eq. (3) of the quadratic, nonlocal correction to the CTF “dispersion law”. To do so, we appeal to a 2D isotropic Gaussian model (GM) of the exit phase map,
${\varphi}_{z=0}^{\text{GM}}\left(\overrightarrow{r}\right)={\text{e}}^{-\frac{{\overrightarrow{r}}^{2}}{2{\sigma}^{2}}}$, where *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
${\varphi}_{z=0}^{\text{GM}}\to S{\varphi}_{z=0}^{\text{GM}}$ (or
$\mathcal{F}{\varphi}_{z=0}^{\text{GM}}\to S\mathcal{F}{\varphi}_{z=0}^{\text{GM}}$):

*π*

^{2}

*σ*

^{2}and ${\pi}^{2}\frac{{z}^{2}}{{k}^{2}{\sigma}^{2}}$ in the two exponentials appearing at order

*S*

^{2}. Their ratio is $\frac{{\sigma}^{4}{k}^{2}}{{z}^{2}}$. For

*λ*= 10

^{−10}m,

*σ*= 10

^{−6}m,

*z*= 1m this yields a value of $\frac{{\pi}^{2}}{2500}\ll 1$. Thus the exponential in the round brackets can be neglected. On the other hand, the ratio of the rates of change of the argument of the sine (and cosine) and the exponential factor e

^{−2π2σ2ξ⃗2}is $\frac{z}{k{\sigma}^{2}}$. For the above-assumed parameter values this yields a value of $\frac{50}{\pi}\gg 1$. Thus the sine and cosine factors vary much faster than the exponential factors, and we can treat the latter as constants as far as the investigation of a phase shift

*φ*of the sine function (order

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

*S*

^{2}) in the vicinity of ${\left|\overrightarrow{\xi}\right|}_{1}^{2}={\overrightarrow{\xi}}_{1}^{2}=\frac{k}{2\pi z}$ is concerned. Using $a\text{sin}x+b\text{cos}x=\sqrt{{a}^{2}+{b}^{2}}\text{sin}\left(x+\phi \right)$, where $\phi =\text{arcsin}\frac{b}{\sqrt{{a}^{2}+{b}^{2}}}$, we have at ${\overrightarrow{\xi}}_{1}^{2}$ to linear order in

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

*φ*of the sine function in Eq. (4) given as

*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 ${\left|\overrightarrow{\xi}\right|}_{1}\equiv \sqrt{k/\left(2\pi z\right)}$ at all for a wide range of

*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 $0<\frac{S-{S}_{c}}{{S}_{c}}\ll 1$.

_{c}The fact that the minima of the modulus of *ℱg _{z}* are not moving for 0 <

*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 [12–14]), and the description in terms of the old spectrum is lost. In our case, this happens for

*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 $\overline{\mathcal{F}{g}_{z}}\left(\left|\overrightarrow{\xi}\right|\right)\equiv \frac{1}{2\pi}{\int}_{0}^{2\pi}d\theta \left|\mathcal{F}{g}_{z}\right|\left(\left|\overrightarrow{\xi}\right|\left(\text{cos}\theta ,\text{sin}\theta \right)\right)$ (modulo a suitable treatment of truncation rods) as functions of |

*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] into position-space CTF retrieval. To cope with this, the following projection is applied [10]

*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*.

To show how scaling symmetry is increasingly broken in an explicit way within the window 1 ≤ *S* ≤ *S _{c}*, where no dynamical breaking occurs, we have investigated in Fig. 2 the behavior of the transfer function of the CTF approximation in dependence of

*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}Let us now spell out the reasons for why projected CTF retrieval is good within the window 1 ≤ *S* ≤ *S _{c}*. Figure 3 shows how the position of the first minimum |

*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}Figure 4 depicts the ratio *R* of
${v}_{\mathit{max},1}\equiv \frac{d\overline{\mathcal{F}{g}_{z}}\left({\left|\overrightarrow{\xi}\right|}_{\mathit{max},1}\right)}{dS}$ and
${v}_{\mathit{min},1}\equiv \frac{d\overline{\mathcal{F}{g}_{z}}\left({\left|\overrightarrow{\xi}\right|}_{\mathit{min},1}\right)}{dS}$ as a function of *S* for 1 ≤ *S* ≤ *S _{c}* = 356. Notice that for all such

*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], see Eq. (6).*

_{n}Figures 5 and 6 show the results of an analysis of experimental data for phase contrast from the four-cell stage of a Xenopus embryo. Figure 5 indicates the uselessness of CTF retrieval in view of the considerable phase variations introduced by the object, and Fig. 6 points out the higher resolving power of projected CTF versus retrieval using the linearized transport-of-intensity equation (yolk particles clearly can be tracked in former case).

The following self-consistency test for projected CTF can be devised in applications to nearly pure-phase objects. Retrieve the phase
${\varphi}_{z=0}^{\text{pCTF}}\left(\overrightarrow{r}\right)$ according to projected CTF (including a subtraction of large-scale variations arising from small absorption effects) from the measured intensity contrast *g _{z}*(

*r⃗*), let ${\varphi}_{z=0}^{\text{pCTF}}\to S{\varphi}_{z=0}^{\text{pCTF}}$ with a moderate value of

*S*, say,

*S*= 2, Fresnel propagate $S{\varphi}_{z=0}^{\text{pCTF}}$ to

*z*to generate the new intensity contrast ${g}_{z}^{\text{S}}\left(\overrightarrow{r}\right)$ and investigate whether, compared to $\overline{\mathcal{F}{g}_{z}}\left(\left|\overrightarrow{\xi}\right|\right)=\overline{\mathcal{F}{g}_{z}^{\text{S}=1}}\left(\left|\overrightarrow{\xi}\right|\right)$, the minima |

*ξ⃗*|

_{min,1}have moved in $\overline{\mathcal{F}{g}_{z}^{\text{S}}}\left(\left|\overrightarrow{\xi}\right|\right)$. In Fig. 7 $\overline{\mathcal{F}{g}_{z}^{\text{S}}}\left(\left|\overrightarrow{\xi}\right|\right)$ (

*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.

To summarize, we have in a quite generic way shown why the local (quasiparticle) approach to single-distance phase retrieval yields robust and good results. Specifically, we have considered the behavior of the angular averaged modulus of the Fourier transform of the intensity contrast
$\overline{\mathcal{F}{g}_{z}}\left(\left|\overrightarrow{\xi}\right|\right)$ which emerges at distance *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
$\overline{\mathcal{F}{g}_{z}}\left(\left|\overrightarrow{\xi}\right|\right)$ to upscaling of the test map (scale factor *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 _{c}* the value of |

*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 $\overline{\mathcal{F}{g}_{z}}\left(\left|\overrightarrow{\xi}\right|\right)$, but, qualitatively, we see similar behavior. Therefore, cutting out thin rings around |

*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 $\overline{\mathcal{F}{g}_{z}}\left(\left|\overrightarrow{\xi}\right|\right)$ outraces the growth of the first minimum for 1 ≤

_{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}Notice that in philosophy projected CTF is similar to Zernike phase contrast where a bias on the spectrum of the wave field is introduced at locations in Fourier space with no relevant information content [15, 16]. In Zernike phase-contrast microscopy this gives rise to useful intensity contrast. As we have shown in the present work, to retrieve phase in a local way in Fourier space from a single-distance intensity-contrast map, projected CTF may introduce a bias on the spectrum of the latter at fixed locations because the associated information loss is minimal.

Since projected CTF is single-distance and applicable to a wide range of relative phase variations it should be useful for real-time tomographic in vivo or in vitro phase-contrast imaging of compact developmental stages of optically opaque biological model systems such as Xenopus embryos [17].

## Acknowledgments

We acknowledge the European Synchrotron Radiation Facility for provision of synchrotron radiation facilities, and we would like to thank Lukas Helfen for assistance in using beamline ID19. We are grateful to L. Waller and H. Suhonen for useful conversations and to V. Altapova and D. Hänschke for the measurement of the Xenopus data. Finally, we would like to thank an anonymous Referee for constructive suggestions, leading, among other changes, to the implementation of Eqs. (4, 5) and Figs. 2, 7. We believe that theses suggestions have improved the paper substantially.

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