Single-distance phase retrieval at large phase shifts

Julian Moosmann; Ralf Hofmann; Tilo Baumbach

doi:10.1364/OE.19.012066

1. Introduction

Third-generation synchrotron light sources produce brilliant photon beams of spatial and temporal coherence properties at the sample which are suitable for routine application of phase-sensitive X-ray imaging methods. Within low-Z samples of low atomic number density (polymers, soft biological tissue, etc.), the attenuation of highly energetic X-rays is weak, thus yielding poor absorption contrast. However, such samples introduce sizable phase shifts to X-ray wave fronts creating intensity contrast [1] downstream of the object due to a free-space propagated transmission function [2–5]. Notice that phase contrast can be 10³ times larger than absorption contrast [6].

The present work reports on two novel and complementary theoretical approaches addressing the nonlinear, noniterative, and single-distance phase-retrieval problem. Their usefulness is demonstrated by numerical experiments: For pure-phase objects, parallel-beam geometry [7], and in Fresnel diffraction theory we consider the regime where the approximation of a linear and local relation between the phase shift ϕ_z ₌₀ in the object-exit plane and intensity I_z at a finite object-detector distance z is inapplicable. This happens for large relative phase shifts and/or large z values (former relevant to phase-sensitive imaging, e.g., of biological organisms, latter occurring, e.g., in measurements with bulky sample environments). That is, the relation between ϕ_z ₌₀ and I_z can be nonlinear and nonlocal because relative phase shifts in the object plane or the value of z, where intensity is measured, or both are large.

2. Review of linear approaches

Let us briefly review two widely used linear relations between ϕ_z ₌₀ and I_z pointing out their strengths and limitations. In the small-z limit the transport-of-intensity equation (TIE) (imaginary part of the paraxial equation [5, 8]) leads to the approximation

g_{z} (\vec{r}) \approx - \frac{z}{k} \nabla_{⊥}^{2} ϕ_{z = 0} (\vec{r}) .

Here

g_{z} (\vec{r}) \equiv \frac{I_{z} (\vec{r})}{I_{z = 0}} - 1

is the intensity contrast, r⃗ a vector in the plane transverse to the optical axis, ∇_⊥ the associated planar nabla operator,

k \equiv \frac{2 π E}{hc}

denotes the wave number of the incident monochromatic X-rays, and h, c are Planck’s quantum of action, speed of light in vacuum, respectively. Moreover, E = hν is the energy of a photon associated with a wave of frequency ν. In the following we refer to phase retrieval based on Eq. (1) as linearized TIE because of the truncation at linear order in z of the Taylor expansion of I(z) in the TIE. (This allows to express ∂_zI(z) at z = 0 linearly in terms of g(z) yielding Eq. (1).) Equation (1) states that out of a given phase map behind the object intensity contrast strengthens with increasing distance z. In the small-relative-phase limit,

| ϕ_{z = 0} (\vec{r} - \frac{π z}{k} \vec{ξ}) - ϕ_{z = 0} (\vec{r} + \frac{π z}{k} \vec{ξ}) | ≪ 1,

where ξ⃗ is the Fourier conjugate to r⃗, an important nonlinear and nonlocal relation between the autocorrelation of the object’s transmission function ψ_z ₌₀(r⃗) and the 2D Fourier transform (ℱI_z)(ξ⃗) [9]

(ℱ I_{z}) (\vec{ξ}) = \int d^{2} r exp (- 2 π i \vec{r} \cdot \vec{ξ}) ψ_{z = 0} (\vec{r} - \frac{π z}{k} \vec{ξ}) ψ_{z = 0}^{*} (\vec{r} + \frac{π z}{k} \vec{ξ})

simplifies to yield the linear contrast-transfer function [10–14]. Notice that relation (3) is tied to Fresnel diffraction theory, the Fraunhofer limit of the paraxial approximation would yield a different relation between ψ_z ₌₀(r⃗) and (ℱ I_z) (ξ⃗). Notice also that condition (2) is required to facilitate the truncation at linear order of the exponential of

i (ϕ_{z = 0} (\vec{r} - \frac{π z}{k} \vec{ξ}) - ϕ_{z = 0} (\vec{r} + \frac{π z}{k} \vec{ξ}))

. This exponential arises upon substituting

ψ_{z = 0} = \sqrt{I_{z = 0}} e^{i ϕ_{z = 0}}

into Eq. (3).

The position-space version of the contrast-transfer function reads

g_{z} (\vec{r}) \approx - 2 [\sum_{i = 1}^{\infty} \frac{1}{(2 i - 1)!} {(\frac{z}{2 k})}^{2 i - 1} {(\nabla_{⊥}^{2})}^{2 i - 1}] ϕ_{z = 0} (\vec{r}) .

Again, relation (2) must hold for all r⃗ and ξ⃗ to guarantee the validity of Eq. (4).

In Fourier space relation (4) reads

(ℱ g_{z}) (\vec{ξ}) = 2 sin (2 \frac{π^{2} z}{k} {\vec{ξ}}^{2}) (ℱ ϕ_{z = 0}) (\vec{ξ}),

and we will refer to phase retrieval based on Eq. (5) simply as CTF. Truncating the power-series expansion of the sine at linear order, yields the Fourier-space version of Eq. (1). Thus Eq. (4) reduces to Eq. (1) in the limit of small values of

\frac{z}{k} {\vec{ξ}}_{max}^{2}

where

{| \vec{ξ} |}_{max} = \frac{1}{2 Δ x}

is the upper cutoff for transverse spatial frequency set by the effective linear pixel size Δx of the detector.

Figures 1(b), 1(c), and 1(d) depict function log |ℱ g_z|(ξ⃗) based on the standard Lena test pattern ϕ_z ₌₀(r⃗) of Fig. 1(a), scaled to a maximal phase shift ϕ _max = 0.01, 1, 6, respectively. The frame-like structure in log |ℱ g_z|(ξ⃗) is due to the Gaussian-blurred transition from the zero-padded region to the actual phase map, see Fig. 1(a), and truncation rods occur (crosses centered at origin). Notice the decrease in visibility of the sinusoidal, radial modulation (rings of minima) from Figs. 1(b) to 1(d). This indicates the failure of CTF at large ϕ _max. Phase retrieval then is affected by quasiperiodic texture superimposed onto the exact phase map, see Figs. 2(b), 2(c): While Eq. (5) expects a sinusoidal modulation of (ℱ g_z)(ξ⃗) this modulation does not occur at large ϕ _max. As a result, the zeros of the sine in Eq. (5) artificially enhance the spectrum of ϕ_z ₌₀ by simple poles in ξ⃗ ². Since |ℱ g_z|(ξ⃗) decays towards large |ξ⃗| the dominating effect comes from the singularities associated with the most central ring of minima. Upon inverse Fourier transformation this yields in transverse position space a 2D quasiperiodic artifact of wavelength $\sqrt{\frac{2 π z}{k}}$ . The amplitude of these quasioscillations rapidly grows with increasing ϕ _max. In contrast to that, quasiperiodic artifacts are absent in linearized TIE, compare Figs. 2(c) and 2(d). For small relative phase shifts Fig. 2(a) demonstrates the superiority of CTF over linearized TIE. When increasing ϕ _max this is no longer true, see Figs. 2(b) and 2(c).

Fig. 1 (a) exact 2D zero-padded phase map (Lena test pattern) as input for intensity computation by free-space propagation; (b), (c), and (d): intensity contrast in Fourier space, log |ℱ g_z|(ξ⃗), with maximal phase shift ϕ _max for E = 10keV and z = 50cm. The colorbar relates to (b), (c), and (d) only. The effective linear pixel size Δξ in Fourier space is given as $Δ ξ = \frac{1}{M Δ x}$ where Δx denotes the effective linear pixel size of the detector, and M is the linear number of pixels. The yellow line in (a) depicts the line cut relating to the presentations in Figs. 2(a) and 2(b). The linear extents of the 2D position-space maps in Figs. 2,3,4, and 5 are all set by the yellow line in (a).

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Fig. 2 Linear phase retrieval from the computed intensity associated with Figs. 1(b), 1(c), see (a), (b), and 1(d), see (c), (d). Since the retrieved phase is undetermined up to an additive constant we subtract mean values to compare exact and retrieved phase. For ϕ _max = 6, see Fig. 2(c), a line-cut presentation of CTF as in Figs. 2(a), 2(b) no longer is adequate since the texture fluctuations have a very large amplitude, compare with 2(d) where linearized TIE was applied yielding reasonable retrieval at reduced resolution.

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Condition (2) can also be satisfied for large ϕ _max if $z ≪ \frac{k Δ x}{\sqrt{2} π | \nabla_{⊥} ϕ_{z = 0} |_{max}}$ where the maximum of |∇_⊥ ϕ_z ₌₀| is taken over the entire field of view. Thus Eq. (4) is valid at large phase shifts when using sufficiently small values of z. In practice this may or may not be an option considering constraints imposed by a limited exposure time (imaging of processes, dose restriction, detector dynamics) and/or a bulky sample environment. Small exposure times imply large statistical noise beating the weakly developed intensity contrast at small z. To investigate this we introduce Poisson noise to the computed intensity at each pixel before phase retrieval. We use a mean value of N_c times the computed, noise-free intensity ratio I_z/I_z ₌₀ where N_c is the expected number of photons detected by a pixel. At I_z/I_z ₌₀ ∼ 1 and N_c = 8000 this yields ∼ 1% noise which we employ from now on. In Figs. 3(a) and 3(b) results of CTF for two distinct distances and energies are shown. Compared to Fig. 3(b) no large texture artifact occurs in Fig. 3(a) because CTF acts like a linearized TIE retrieval: $sin (2 \frac{π^{2} z}{k} {\vec{ξ}}^{2}) \sim 2 \frac{π^{2} x}{k} {\vec{ξ}}^{2}$ for all ξ⃗. Notice, however, the importance of noise artifacts due to the small propagation distance. Figure 3(b) depicts CTF at z = 50cm and E = 10keV where noise effects are marginal but a large texture artifact occurs. On the other hand, linearized TIE at z = 50cm and E = 10keV yields reasonable results but suffers from a limited resolution, compare Fig. 3(c) with Fig. 3(a).

Fig. 3 2D phase maps from linear retrieval subject to intensity computed at ϕ _max = 1. Intensity is subject to Poisson noise.

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3. Nonlinear, noniterative approaches

The above discussion indicates the need for nonlinear approaches to the retrieval of large relative phases.

3.1. Quasiparticle approach

One way to work beyond fundamental linearity is to appeal to an effective linear and local model in Fourier space. Namely, we propose a modified CTF retrieval, dubbed projected CTF, in the following sense: Artificial peaks in the spectrum of ϕ_z ₌₀ introduced by CTF are removed by considering an intensity contrast which is filtered in Fourier space, $(\tilde{ℱ g_{z}}) (\vec{ξ}) \equiv Θ (| sin (\frac{2 π^{2} z}{k} {\vec{ξ}}^{2}) | - ɛ) \cdot (ℱ g_{z}) (\vec{ξ})$ . Here $\frac{2 π^{2} z}{k} {\vec{ξ}}^{2} > \frac{π}{2}$ Θ denotes the Heaviside step function, ε is the threshold for this binary filter (0 ≤ ε < 1), and on the left-hand side of Eq. (5) (ℱ g_z)(ξ⃗) is replaced by $(\tilde{ℱ g_{z}}) (\vec{ξ})$ . Let us discuss two physical aspects of projected CTF. First, to assure the absence of singularities in the effective phase ℱ ϕ_z ₌₀ the above binary filtering is used. Second, the use of a local relation between $(\tilde{ℱ g_{z}}) (\vec{ξ})$ and (ℱ ϕ_z ₌₀)(ξ⃗) renders the latter an effective quantity. Namely, nonlinear effects in the forward propagation of the exact phase fundamentally imply a nonlocal relation between the Fourier transform of the latter and ℱ g_z. Exemplarily, this can be checked by truncating at quadratic order the expansion of the exponential of phase after substitution of $ψ_{z = 0} = \sqrt{I_{z = 0}} exp (i ϕ_{z = 0})$ into Eq. (3). The resulting relation between ℱ I_z and ℱ ϕ_z ₌₀ is (by construction) nonlinear and (by the Fourier convolution theorem) nonlocal. As a result, a ξ⃗ dependent rescaling between the Fourier transform of the exact phase and ℱ ϕ_z ₌₀ as retrieved from our modified, local version of Eq. (5) occurs. But for phase retrieval in transverse position space, an inverse Fourier transformation needs to be applied which averages out spurious spectral information contained in ℱ ϕ_z ₌₀. According to our simulations, the result indeed is close to the exact phase. An analogy is the successful quasiparticle concept in the quantum theory of condensed-matter physics and quantum field theory where a free dispersion law is altered by strong interactions yielding a free quasiparticle. In our case, the concept of a quasiparticle translates into the linear and local relation between effective phase and filtered intensity while the average over particle momenta in some normalized derivative of the partition function is analogous to inverse Fourier transformation. Namely, an r⃗ dependent ‘action’ S_r⃗ ≡ 2πξ⃗ · r⃗ is used to define the partition function Z_r⃗ as Z_r⃗ = tr exp(iS_r⃗) where the trace symbol is understood as a sum over all wave-vector states. We stress that projected CTF is nonperturbative. Figure 4(a) depicts $log | \tilde{ℱ g_{z}} |$ at ε = 0.01. Comparison of Figs. 4(b) and 4(c) reveals that an increase of resolution is traded for an increase of the texture artifact when decreasing the threshold ε. Projected CTF essentially improves CTF, compare Fig. 3(b) with Figs. 4(b), 4(c). Notice that in the limit ε → 1 projected CTF yields results that are similar to those of linearized TIE, compare Fig. 4(c) with Fig. 3(c).

Fig. 4 Intensity contrast in Fourier space after binary filtering with threshold ε and retrieved phase maps at ϕ _max = 1, z = 50cm, E = 10keV. Intensity is subject to Poisson noise.

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3.2. Perturbation theory

Let us now consider another nonlinear phase-retrieval algorithm which, in contrast to CTF, is perturbative. Namely, expanding the intensity contrast as $g_{z} (\vec{r}) \equiv \sum_{i = 1}^{N} g^{(i)} (\vec{r}) z^{i}$ , the idea is to determine coefficient g ⁽¹⁾(r⃗) as precisely as possible to solve $g^{(1)} (\vec{r}) = - \frac{1}{2 k} \nabla_{⊥}^{2} ϕ_{z = 0} (\vec{r})$ , see Eq. (1). In [15] g ⁽¹⁾(r⃗) is extracted from a fit to a stack of intensity measurements {Iz_i|i = 1, ⋯ ,N} the better the larger N is. In [16] a theoretical alternative to that approach, which only uses a single-distance intensity map, was proposed: All coefficients g ⁽ ⁱ ⁾(r⃗) are, by virtue of the full paraxial equation, expressed by 2i transverse derivatives, appearing in terms that are linear and nonlinear in ϕ_z ₌₀. Truncating the expansion at N ≥ 2, one obtains a nonlinear partial differential equation (PDE) for ϕ_z ₌₀ subject to the single-distance source term g_z(r⃗) [16]. At N = 2 or next-to-leading order (NLO) the explicit expansion of g_z(r⃗) reads [16]

g_{z} (\vec{r}) \approx - \frac{z}{k} \nabla_{⊥}^{2} ϕ_{z = 0} (\vec{r}) + \frac{z^{2}}{2 k^{2}} [{(\nabla_{⊥}^{2} ϕ_{z = 0})}^{2} + \frac{1}{2} \nabla_{⊥}^{2} {(\nabla_{⊥} ϕ_{z = 0})}^{2} + (\nabla_{⊥} \nabla_{⊥}^{2} ϕ_{z = 0}) \cdot \nabla_{⊥} ϕ_{z = 0}] .

To approximate the solution ϕ_z ₌₀ to the nonlinear PDE (6) perturbatively is to solve it at leading order N = 1, to use this solution in estimating the next-to-leading order (we refer to the result as PNLO), and to finally invert the Laplacian on

- \frac{k}{z} (g_{z} (\vec{r}) - PNLO)

[16]. This type of nonlinear phase retrieval is dubbed linearized TIE + PNLO. As Fig. 5 indicates linearized TIE + PNLO is useful at very large relative phase shifts. Obviously, CTF yields unacceptable results. Comparing Figs. 5(d) and 5(c), linearized TIE + PNLO improves linearized TIE within regions where the phase varies sufficiently slowly, compare also with [16]. This lowers the mean retrieval error per pixel by about 40%. Comparing linearized TIE + PNLO with projected CTF, see Fig. 5(d) and Fig. 5(b), the mean retrieval error per pixel in the former is reduced by about 42% w.r.t. the latter. Linearized TIE + PNLO, however, may artificially hollow regions which are bounded by strong phase slopes (transition across Lena’s eyes). Finally, let us remark that PNLO can be modified to PNLO′ by using CTF to estimate the nonlinear terms in Eq. (6) [17]. In simulations we see that CTF + PNLO′ works whenever CTF already is excellent (typically for ϕ _max < 0.01). Practically, this is irrelevant because of the smallness of the correction PNLO′. For ϕ _max ≥ 0.1 PNLO′ exhibits large texture artifacts.

Fig. 5 2D maps of the modulus of the difference between exact and retrieved phase. Parameter values are ϕ _max = 6, z = 50cm, E = 10keV, and intensity is subject to Poisson noise. The colorbar applies to all images.

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4. Summary

To summarize, we have proposed two nonlinear, noniterative single-distance phase-retrieval algorithms. In analogy to the quasiparticle approach in strongly coupled systems [18], an effective phase in Fourier space, satisfying certain minimal requirements, assures at moderately large phase shifts an essential improvement over fundamentally linear methods. Alternatively, we have considered a perturbative approach [16] to the power-series expansion of intensity contrast [15] which works well for very large phase shifts. Our results could be useful for static phase-contrast imaging, but they also are applicable to the imaging of structure dynamics in nonabsorbing materials for which single-distance retrieval is essential. This extends to the case of weak and quasistatic absorption: An absorption image I_z ₌₀ would be taken to define intensity contrasts {g_z} for a sequence of images recorded at a single distance z > 0, think of the quick motion of nonabsorbing air bubbles inside a quasistatic, absorbing frame containing a nonabsorbing fluid. Another way of including absorption effects in single-distance phase retrieval is to make the assumption that ϕ_z ₌₀ = αB_z ₌₀ in $ψ_{z = 0} = \sqrt{I_{z = 0}} exp (i ϕ_{z = 0} - B_{z = 0})$ , compare with Eq. (3). Here α is a real constant that is known a priori, see for example [19].

Let us briefly compare our noniterative, single-distance, nonlinear phase-retrieval algorithms with iterative nonlinear methods, see [20]. First, iterative methods employ a sequence of forward-backward propagations. Backward propagation is simply implemented in case of the full Helmholtz theory (elementary wave fronts are spherical) and the Fraunhofer far-field approximation of the paraxial theory (elementary wave fronts are scaled 2D planes). In Fresnel theory the backward-propagation part is problematic since the theory is not symmetric under z → –z (elementary wave fronts are paraboloids). We insist on Fresnel theory because in Fraunhofer theory the Fourier transform of intensity at z, ℱ I_z, is related to the integral (and not the Fourier transform) of the autocorrelation of the object function ψ ₀ over the object plane. This implies that, unlike in Fresnel theory, the small-relative-phase condition (2) does not yield a linear relation between ℱ I_z and ℱ ϕ_z ₌₀. Namely, the lowest-order relation between ϕ_z ₌₀ and ℱ I_z is

\frac{(ℱ I_{z}) (\vec{ξ}) - I_{z = 0} δ^{(2)} (\vec{ξ})}{I_{z = 0}} = \int d^{2} r ϕ_{z = 0} (\vec{r} - \frac{π z}{k} \vec{ξ}) ϕ_{z = 0} (\vec{r} + \frac{π z}{k} \vec{ξ}) .

This nonlinear relation is not local which makes it rather useless for noniterative, numerically inexpensive algorithms. In the full Helmholtz theory the situation is even worse. Second, for single-distance, iterative methods a certain amount of priori knowledge on the object is required which we would not like to rely on. Usually, the sparser this information is the slower is the convergence process in single-distance iterative phase retrieval [20]. Also, the slow convergence of iterative algorithms may turn out to be fatal for real-time imaging. Because projected CTF and linearized TIE + PNLO are computationally inexpensive and do not require a priori information they represent promising alternatives.

The present work should be relevant for soft-matter and life sciences. In fact, preliminary experimental results on the propagation phase contrast induced by Xenopus (African Clawed Frog) embryos show that CTF is not applicable and indicate that projected CTF indeed yields a substantial improvement over linearized TIE [21] in terms of spatial resolution.

Acknowledgments

The authors acknowledge useful conversations with V. Altapova, A. Bronnikov, D. Hänschke, H. Suhonen, and L. Waller.

References and links

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Single-distance phase retrieval at large phase shifts

Abstract

1. Introduction

2. Review of linear approaches

3. Nonlinear, noniterative approaches

3.1. Quasiparticle approach

3.2. Perturbation theory

4. Summary

Acknowledgments

References and links

Cited By

Figures (5)

Equations (7)

Optics Express