Nonlinear phase retrieval from single-distance radiograph

Julian Moosmann; Ralf Hofmann; Andrei V. Bronnikov; Tilo Baumbach

doi:10.1364/OE.18.025771

1. Introduction

With the advent of third-generation synchrotron light sources producing brilliant and partially coherent radiation, whose spectrum contains highly energetic X-ray components, the application of phase sensitive imaging methods became feasible. Within samples, whose atoms are of low Z and of low number density ρ̂ (polymers, soft biological tissue, etc.), the attenuation of highly energetic X-rays is weak yielding a poor absorption contrast. Sizable phase shifts are, however, induced to the coherent X-ray wave fronts. The latter induce intensity contrast that can be used for 2D and 3D imaging.

The motivation of this paper is to elaborate on the optics-free in-line propagation technique where phase contrast in the object plane is retrieved from intensity contrast measured at a single object-detector distance z. Retrieving phase information from intensity at a single distance is a precursor for 2D and 3D in-situ imaging. Specifically, we aim at a higher flexibility in setting z to values beyond the edge-enhancement regime (from typically a few centimeters for X-ray energies above 20 keV to several ten centimeters). This flexibility is required in experiments where the sample is suspended by a large environment not accessible to the detector. As can be argued on the basis of the contrast-transfer function [1–3], at larger z small object scales, such as the fine-structure of edges, increasingly contribute to the fringe pattern discerned by the detector. The promise to resolve such structures at larger z represents another motivation for our work.

The interaction of radiation with matter is effectively described by the complex refractive index

n (x_{1}, x_{2}, x_{3}) = 1 - δ (x_{1}, x_{2}, x_{3}) + i β (x_{1}, x_{2}, x_{3}),

where δ = 1 – Re(n) is the real refractive index decrement introduced by elastic scattering, and β refers to the real absorption index. Assuming the projection approximation to be valid (negligible scattering away from the ray path), the effects induced by the object onto a monchromatic, coherent wave field propagating along, say, the x₂ direction, see Fig. 1, is described by the transmission function ψ_z₌₀(x, y) as

\begin{array}{l} ψ_{z = 0} (x, y) & = & exp (- k \int {d x}_{2} β (x, x_{2}, y)) exp (- i k \int {d x}_{2} δ (x, x_{2}, y)) \\ \equiv & exp (- B (x, y)) exp (i ϕ_{z = 0} (x, y)), \end{array}

where

k \equiv \frac{2 π}{λ}

, λ denotes the wavelength, the integration over x₂ is terminated at the object boundaries, and x,y are coordinates transverse to the ray path along x₂. The 2D phase map ϕ_z₌₀(x, y) is interpreted as the projection of the electron density. Functions B and ϕ are radiographs of the object which contain some information on the object that can be interpreted. Since –B(x,y) + iϕ_z₌₀(x, y) essentially is the Radon transform of n(x₁, x₂, x₃) a stack of such radiographs, obtained by a stepwise rotation of the object about axis x₃, is input for computed tomographic 3D reconstruction of n(x₁, x₂, x₃), see for example [4].

Fig. 1 The experimental set-up and coordinate conventions for single-distance phase-contrast tomography employing the in-line propagation method.

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For the retrieval of the phase-sensitive part of the transmission function from an intensity map several methods have been explored. Interferometric techniques exploit the intensity pattern I generated by the phase-distorted wavefront through interference with an undistorted reference beam [5–9]. Crystal based Bragg-type interferometry requires an extreme stability of the environment to prevent spatial displacements down to the atomic-level within the interferometric set-up. Interferometric phase-contrast imaging employing diffraction gratings is much less sensitive to the positioning and the stability of the experimental setup [10, 11]. Appealing to the relation Δα = −1/k∂_xϕ, the Schlieren technique (or crystal diffraction enhanced imaging) analyses the angular deviation Δα of the transmitted X-rays. The quantitative analysis of the images requires input from dynamical X-ray diffraction theory (finite Darwin width) [12–14].

The instrumentally simple and robust in-line or free-space propagation technique is based on Fresnel diffraction theory [15, 16]: An intensity pattern I_z, which develops due to interference of different, directionally displaced parts of the transmitted and propagated wavefront ψ_z(x, y), orginating by plane-wave illumination of the object, is observed by a detector placed at distance z behind the object. Within any transverse plane z ≥ 0 only a non-homogeneous gradient of the phase (∂_xϕ_z ≠ const and ∂_yϕ_z ≠ const) generates an intensity contrast $g_{z} (x, y) \equiv \frac{I_{z > 0}}{I_{z = 0}} - 1$ [17].

In some free-space propagation techniques [18, 19] one directly appeals to the transport-of-intensity equation (TIE):

- k \partial_{z} I_{z} = - \nabla_{⊥} (I_{z} \nabla_{⊥} ϕ_{z}),

where ∇_⊥ denotes the two-dimensional Nabla operator in the tranverse plane. In the limit of constant absorption (I_z₌₀ ≡ const) Eq. (3) takes the following form

k \partial_{z} g_{z} = - \nabla_{⊥} ((g_{z} + 1) \nabla_{⊥} ϕ_{z}) .

Relaxing the assumption of constant absorption, the contrast-transfer function (CTF) considers a linearized version of the transmission function of Eq. (2) in B and ϕ_z₌₀. In [20–22] single-distance phase retrieval based on the CTF is considered and successfully applied. In particular, in [20] an interesting discussion of the extended regime of validity of CTF-based retrieval of the thickness function for a homogeneous object as compared to the TIE-based approach was given using a single-distance radiograph. In [21, 22] the usefulness of a combination of the TIE- and CTF-based approach (both approaches employing a linear relation between phase in the object plane and intensity a distance z away from the object) appealing to a multiplicative separation of a slowly varying factor from a small factor in the transmission function was shown. In Sec. 3.3 we point out the differences of TIE- and CTF-based phase retrieval to the nonlinear method proposed in the present paper.

In [23, 24] the CTF is used to overdetermine B and ϕ_z₌₀ by intensity measurements performed at several distances z_i yielding remarkably accurate results with nonlinear effects taken into account iteratively. Judging by the linear approximation only, large spatial object scales are poorly retrieved by the CTF based approach because the Fourier transform of ϕ_z₌₀ has a sine-function coefficient that vanishes at the origin (zeros of the sine function at nonvanishing frequencies are regularized by summation over various values of z). A mixed TIE and CTF based approach is also applied, see [25–27].

For sufficiently small z function g_z varies in an approximately linear way in z, and Eq. (4) simplifies as: [Substituting g_z(x, y) = g₁(x, y) z and ϕ_z(x, y) = ϕ_z₌₀(x, y) + ϕ₁(x, y) z into Eq. (4), consistently ignoring nonvanishing powers in z on the right-hand side, and multiplying by z yields Eq. (5).]

g_{z} = - \frac{λ z}{2 π} \nabla_{⊥}^{2} ϕ_{z = 0} .

In [4] the approximation of Eq. (5) is used to tomographically reconstruct the object function δ(x₁, x₂, x₃) directly in terms of the data g_z;θ where θ refers to an angle at which the object is rotated about the x₃ direction, see Fig. 1. To retrieve ϕ_z₌₀ radiographs need to be collected at a single distance z only which is a prerequisite for the characterization of processes [27–29].

The applicability of Eq. (5) is, however, restricted to the edge-enhancement regime thus preventing the reconstruction of object information on larger spatial scales whose effect emerges in terms of conspicuous fringes in the intensity pattern at larger values of z only. The edge-enhancement regime with increasing distance z emerges first in the intensity contrast due to a large density of interference points introduced by the strongly curved wavefront in the vicinity of boundaries between regions within the object that exhibit a large difference in δ.

In the present work we propose an extension of the phase-retrieval algorithm for pure-phase, nonperiodic objects relying on Eq. (5). This extension maintains the efficiency of that algorithm (single distance z, simple algorithm for phase retrieval) but works systematically beyond the linear approximation of Eq. (5). We demonstrate the potential usefulness of the extended algorithm by analysing simulated phantom data.

The paper is organized as follows: To set conventions we briefly outline the tomographic reconstruction algorithm of Ref. [4] in Sec. 2. A systematic extension is developed in Sec. 3 by appeal to a model where g_z as well as ϕ_z are expanded in powers series in z. The coefficients of the power series expansion are determined by the paraxial wave equation [Eq. (11)]. We give explicit expressions up to the next-to-leading order and discuss how they can be treated perturbatively. In Sec. 4 we perform first tests of our phase-retrieval algorithm applying it to simulated phantom data. Section 5 summarizes our work. An appendix contains the relation between g_z and ϕ_z₌₀ at next-to-next-to leading order.

2. Tomographic reconstruction algorithm

The quantity required for tomographic reconstruction of function δ(x₁, x₂, x₃) by inverse Radon transformation is $\nabla_{⊥}^{2} ϕ_{z = 0; θ}$ in the case of a pure-phase object. The main purpose of our work is to extend the expression of Eq. (5) for $\nabla_{⊥}^{2} ϕ_{z = 0; θ}$ in terms of intensity contrast g_z;θ. Although we do not perform tomographic reconstruction in the present paper we would like to sketch conventions for the tomographic set-up as used in [4] and to point out how $\nabla_{⊥}^{2} ϕ_{z = 0; θ}$ enters in the reconstruction algorithm.

The algorithm of [4] relates the 3D Radon transform δ̂(s,θ,ω) of the object function δ(x₁, x₂, x₃) to the 2D Radon transform $ℛ \nabla_{⊥}^{2} ϕ_{z = 0; θ}$ of $\nabla_{⊥}^{2} ϕ_{z = 0; θ}$ as

\partial_{s}^{2} \hat{δ} (s, θ, ω) = - \frac{λ}{2 π} (ℛ \nabla_{⊥}^{2} ϕ_{z = 0; θ}) (s, ω) .

Here, x₁, x₂ and x₃ denote the Cartesian coordinates in the object frame. The coordinates in the plane transverse to the direction of propagation are x and y. In this plane the 2D Radon transformation is understood as an integration over lines parameterized as xsinω + ycosω = s (see Fig. 1). In order to apply the Radon inversion formula

δ (x_{1}, x_{2}, x_{3}) = - \frac{1}{4 π^{2}} \int_{0}^{π} d ω sin ω \int_{0}^{π} d θ \partial_{s}^{2} \hat{δ} (s, θ, ω) |_{s = (x_{1} cos θ + x_{2} sin θ) sin ω + x_{3} cos ω}

for the reconstruction of object function δ(x₁, x₂, x₃) from the data g_z;θ(x,y), Eqs. (6) and (7) thus requires a one-to-one relation between

\nabla_{⊥}^{2} ϕ_{z = 0; θ}

and g_z;θ. To leading (linear) order in a Taylor expansion of g_z;θ in z this relation is provided by Eq. (5). Notice that Eq. (4) continues to hold approximately for weakly absorbing objects, that is, if |∇_⊥I_z_=0;_θ| is finite but |∇_⊥I_z_=0;_θ| ≪ |∇_⊥I_z_>0;_θ|. In this case two measurements, I_z_=0;_θ and I_z_>0;_θ, are required to determine g_z;θ and hence, by virtue of Eqs. (6), (5), and (7), object function δ(x₁, x₂, x₃).

3. Intensity contrast beyond linearity in z

3.1. Powers in z, paraxial wave equation, and perturbation theory

To characterize larger object scales than achieved within the edge-enhancement regime the object-detector distance z should be increased when applying the single-distance in-line propagation technique: As discussed in the Introduction, the intensity pattern I_z;θ then contains discernable information on the exiting phase ϕ_z_=0;_θ (x,y) and thus on the object function δ(x₁, x₂, x₃). The model of Eq. (5), assuming a linear z dependence of g_z;θ (x,y), however, breaks down at larger z, and thus a more general ansatz is needed:

g_{z; θ} (x, y) = \sum_{l = 1}^{l_{max}} g_{l; θ} (x, y) z^{n} .

Also, function ϕ_z;θ(x,y) is expanded in powers of z:

ϕ_{z; θ} (x, y) = \sum_{m = 0}^{m_{max}} ϕ_{m; θ} (x, y) z^{m} .

Note that {g_l;θ} = {g_1;θ, g_2;θ,...} and {ϕ_m;θ} = {ϕ_0;θ, ϕ_1;θ,..} denote the z-independent coefficients of the expansion of the z-dependent functions g_z;θ and ϕ_z;θ, respectively. Substituting ansatz (8) and ansatz (9) into Eq. (4) and comparing coefficients up to linear order in z yields

\begin{array}{l} g_{1; θ} & = & - \frac{1}{k} \nabla_{⊥}^{2} ϕ_{0; θ}, \\ g_{2; θ} & = & \frac{1}{2 k} [\frac{1}{k} ((\nabla_{⊥} \nabla_{⊥}^{2} ϕ_{0; θ}) \cdot \nabla_{⊥} ϕ_{0; θ} + {(\nabla_{⊥}^{2} ϕ_{0; θ})}^{2}) - \nabla_{⊥}^{2} ϕ_{1; θ}] . \end{array}

Equation (4) follows from the imaginary part of the paraxial equation

(2 i k \partial_{z} + \nabla_{⊥}^{2}) ψ_{z; θ} = 0

for the wavefield

ψ_{z; θ} = \sqrt{I_{z; θ}} exp (i ϕ_{z; θ})

. Again considering the absorption-free limit I_z_=0;θ ≡ const, the real part of Eq. (11) is given as

\partial_{z} ϕ_{z; θ} = \frac{1}{2 k} (\frac{\nabla_{⊥}^{2} \sqrt{g_{z; θ} + 1}}{\sqrt{g_{z; θ} + 1}} - {(\nabla_{⊥} ϕ_{z; θ})}^{2}) .

In the limit z → 0 we have from Eq. (12) for the coefficient ϕ_1;θ in Eq. (9)

ϕ_{1; θ} = lim_{z \to 0} \partial_{z} ϕ_{z; θ} = - \frac{1}{2 k} {(\nabla_{⊥} ϕ_{0; θ})}^{2} .

Combining Eqs. (8), (9), (10), and (13) (truncating at l_max = 2 and m_max = 1, next-to-leading-order) finally yields

\begin{array}{l} \nabla_{⊥}^{2} ϕ_{z = 0; θ} & \equiv & \nabla_{⊥}^{2} ϕ_{0; θ} = - \frac{k}{z} g_{z; θ} + \\ \frac{z}{2 k} [(\nabla_{⊥} \nabla_{⊥}^{2} ϕ_{0; θ}) \cdot \nabla_{⊥} ϕ_{0; θ} + {(\nabla_{⊥}^{2} ϕ_{0; θ})}^{2} + \frac{1}{2} \nabla_{⊥}^{2} {(\nabla_{⊥} ϕ_{0; θ})}^{2}] . \end{array}

The result in Eq. (14) can also be obtained along the lines of [4] by computing intensity I_z from the wavefield ψ_z, Fresnel-propagated from z = 0, and by neglecting variations in x and y of the attenuation factor. The counting in powers of z then is due to a Taylor expansion of the Fourier transform of the Fresnel propagator. In such a derivation of Eq. (14) it is, however, impossible to keep track of how the nonlinear corrections arise. This becomes clear when directly appealing to Eq. (10). Namely, the first two terms in the square brackets arise from the unpropagated phase while the third term is due to phase-propagation.

The formal expansion of $\nabla_{⊥}^{2} ϕ_{z = 0; θ}$ in powers of $\frac{z}{k}$ can be carried to higher orders, see the Appendix for an explicit listing of the next-to-next-to-leading-order correction. In doing so, Eq. (4) and l_max – 2 derivatives of Eq. (12) with respect to z are needed to express the coefficients in the expansions of Eqs. (8) and (9) in terms of polynomials of transverse derivatives acting on ϕ_0;θ. For later discussion we notice that the nonlinear correction in Eq. (14) represents a sum of total derivatives in x and y. The expansion of $\nabla_{⊥}^{2} ϕ_{z = 0; θ}$ on the right-hand side of Eq. (14) formally is in ‘powers’ of derivatives of ϕ_0;θ. We thus are required to demand a weak variation of ϕ_0;θ in dependence of x,y. (Otherwise the coefficients of powers in z would be unacceptably large when increasing the order of the expansion in powers of z.)

Equation (14) represents a nonlinear partial differential equation for ϕ_z_=0;θ linking the latter to the data g_z;θ. Let us now discuss how the solution to Eq. (14) can be approached. In principle, one could try to solve Eq. (14) numerically subjecting it to appropriate boundary conditions. Notice that due to the presence of additional powers of ∇_⊥ϕ_0;θ in Eq. (14) as compared to Eq. (5) an ambiguity ∇_⊥ϕ_0;θ → ∇_⊥ϕ_0;θ + v⃗, v⃗ a constant 2D vector no longer is present. Also, a reduction of the influence of optical vortices [18, 19] should occur in ϕ_0;θ: Phase retrieval is not only constrained by the transport-of-intensity equation (imaginary part of the paraxial wave equation representing Fresnel diffraction theory incompletely) but also by the real part of the paraxial wave equation. Thus, expanding beyond leading order increasingly accounts for the constraints of the full Fresnel theory.

An alternative to a brute-force treatment of Eq. (14) is to approach the right-hand side by appeal to the leading-order situation (setting l_max = 1 and m_max = 0 in Eqs. (8) and (9), respectively) to find a useful approximation. The nonlinear terms in square brackets are then approximated in terms of the solution ϕ_0;θ of Eq. (5) easily obtained by Fourier transformation (perturbation theory).

In general, the right-hand side of Eq. (14) is expressed in terms of the following power series

\nabla_{⊥}^{2} ϕ_{0; θ} = \sum_{l = 0}^{l_{max} + 1} c_{l} (x, y) z^{l} .

Ideally, all coefficients other than c₀(x, y) vanish in Eq. (15). For example, setting l_max = 2, as assumed in deriving Eq. (14), the term

- \frac{k}{z} g_{z; θ}

generates a contribution –k(g_1;θ (x,y) + g_2;θ (x,y)z) while the terms in square brackets is by definition linear in z. Ideally, this linear term cancels the linear term arising in

- \frac{k}{z} g_{z; θ}

. When increasing l_max = 2 → l_max = 3 the quadratic term in

- \frac{k}{z} g_{z; θ}

is cancelled by the quadratic correction on the right-hand side of Eq. (34) in the Appendix and so on.

Perturbation theory respects this counting in powers of z since by inverting the Laplacian in a truncation of the right-hand side of Eq. (14) at the leading order one, indeed, ends up with an estimate for ϕ_0;θ that is independent of z. Due to its perturbative nature, this estimate, when substituted in the corrections to next-to-leading and next-to-next-to-leading order of Eq. (34), does, however, not completely cancel the respective powers in z in the term $- \frac{k}{z} g_{z; θ}$ . Still, it reduces the modulus of their coefficients.

3.2. Numerical implementation

Technically, it is advantageous to rescale coordinates x,y dimensionless as $x, y \to \tilde{x} \equiv \sqrt{\frac{k}{z}} x, \tilde{y} \equiv \sqrt{\frac{k}{z}} y$ . In these variables, the right-hand side of the equivalent of Eq. (14) does not exhibit prefactors $\frac{k}{z}$ or $\frac{z}{k}$ of the leading order (LO) and the next-to-leading order (NLO), respectively. Since g_z;θ is dimensionless its z dependence is determined by the dimensionless variables kz and $α = \frac{z}{a}$ , $β = \frac{z}{b}$ , ··· where a, b, ··· are object scales.

Let us now abbreviate the NLO terms as

\begin{array}{l} {NLO}_{1} (x, y) & \equiv & (\nabla_{⊥} \nabla_{⊥}^{2} ϕ_{0; θ}) \cdot \nabla_{⊥} ϕ_{0; θ} \\ {NLO}_{2} (x, y) & \equiv & {(\nabla_{⊥}^{2} ϕ_{0; θ})}^{2} \\ {NLO}_{3} (x, y) & \equiv & \frac{1}{2} \nabla_{⊥}^{2} {(\nabla_{⊥} ϕ_{0; θ})}^{2} . \end{array}

To express the leading-order estimate for ϕ_0;θ in terms of the data we appeal to Fourier transformation. To leading order we have g_1;θ = g_z/z. Thus in perturbation theory NLO₁ reads as follows:

(\nabla_{⊥} \nabla_{⊥}^{2} ϕ_{0; θ}) \nabla_{⊥} ϕ_{0; θ} = k^{2} ℱ^{- 1} (ξ_{i} ℱ g_{1; θ}) ℱ^{- 1} (\frac{ξ_{i}}{{\vec{ξ}}^{2}} ℱ g_{1; θ}),

where ℱ (ℱ⁻¹) denotes (inverse) Fourier transformation, ξ₁, ξ₂ are the Fourier conjugates to x,y, respectively, and a summation convention for repeated indices is understood (i = 1,2). NLO₂ simply is

{(\nabla_{⊥}^{2} ϕ_{0; θ})}^{2} = k^{2} g_{1; θ}^{2},

and NLO₃ is given as

\frac{1}{2} \nabla_{⊥}^{2} {(\nabla_{⊥} ϕ_{0; θ})}^{2} = \frac{1}{2} k^{2} ℱ^{- 1} {{\vec{ξ}}^{2} ℱ [ℱ^{- 1} (\frac{ξ_{i}}{{\vec{ξ}}^{2}} ℱ g_{1; θ}) ℱ^{- 1} (\frac{ξ_{i}}{{\vec{ξ}}^{2}} ℱ g_{1; θ})]} .

Let us now discuss the expressions NLO₁(x, y), NLO₂(x, y), and NLO₃(x, y) in more detail. Up to the factor ∇_⊥ϕ_0;θ in NLO₁, which relates the Laplacian of ϕ_0;θ to the data g_1;θ in a nonlocal way, the right-hand side of Eq. (17) is local in g_1;θ. Loosely speaking, ∇_⊥ϕ_0;θ is obtained by integrating the leading-order equation

\nabla_{⊥}^{2} ϕ_{z = 0; θ} = \nabla_{⊥}^{2} ϕ_{0; θ} = - \frac{k}{z} g_{z; θ}

once. Moreover, NLO₂ is completely local in g_1;θ while NLO₃ is nonlocal. Each of the terms NLO₁(x, y), NLO ₂(x, y), and NLO₃(x, y), can be decomposed as

{NLO}_{i} (x, y) = 〈 {NLO}_{i} 〉 + δ {NLO}_{i} (x, y), (i = 1, 2, 3),

where 〈NLO_i〉 ≡ ∫dxdyNLO_i(x, y). Since NLO₁(x, y) + NLO₂(x, y) and NLO₃(x, y) separately are sums of partial derivatives we have

\sum_{i = 1}^{3} 〈 {NLO}_{i} 〉 = 0 .

Since NLO₃ is nonlocal in g_z;θ, this term smeares g_1;θ and averages out large amplitudes.

A pragmatic approach to inverting the Laplacian appealing to discrete Fast Fourier Transforms is to simply let

\frac{1}{{\vec{ξ}}^{2}} \to \frac{1}{{\vec{ξ}}^{2} + α^{2}},

where α is a real constant. Requiring that the retrieved phase is insensitive to a variation of α fixes α’s value. It is straightforward to show that using the regularization of Eq. (21) in applying

\nabla_{⊥}^{- 2}

to a function F(x,y) one has

〈 \nabla_{⊥, α}^{- 2} F (x, y) 〉 = α^{- 2} 〈 F 〉,

where

\nabla_{⊥, α}^{- 2}

refers to a version of

\nabla_{⊥}^{- 2}

regularized by virtue of (21).

From Eqs. (22), (20) and the fact that

〈 g_{z; θ} 〉 = 0

(energy conservation for pure-phase objects) it follows that the phase retrieved by applying

\nabla_{⊥, α}^{- 2}

to the right-hand side of Eq. (14) has zero mean (〈NLO₁〉 and 〈NLO₂〉 individually can be large but their sum is nil.). Deviations from this situation are introduced by numerical artifacts introduced through discrete Fourier transformations and thus should be subtracted. The exact phase, which is expressed by a line integral over the object function δ(x₁, x₂, x₃), however has a nonvanishing mean (δ(x₁, x₂, x₃) is sign definite). This ambiguity is due to the fact that an arbitrary constant can be added to

ϕ_{0; θ} (x, y) \equiv 〈 ϕ_{0; θ} 〉 + δ ϕ_{0; θ} (x, y)

without changing Eq. (14) for ϕ_0;θ. To compare exact with retrieved phase we thus subtract means in both cases.

3.3. Comparison with CTF-based single-distance retrieval

An important result due to Guigay [30] is an expression linking the autoproduct of the object transmission ψ₀,_θ to the Fourier transform of intensity ℱI_z,θ at distance z:

ℱ I_{z, θ} (\vec{ξ}) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} dx dy exp (- 2 π i \vec{r} \cdot \vec{ξ}) ψ_{0, θ}^{*} (\vec{r} + \frac{π}{k} z \vec{ξ}) ψ_{0, θ} (\vec{r} - \frac{π}{k} z \vec{ξ}),

where r⃗ ≡ (x,y). Assuming a pure-phase object, one has

ψ_{0, θ} = \sqrt{I_{0, θ}} exp (i ϕ_{0; θ})

where I₀,_θ is homogeneous. In exploiting Eq. (25) one makes the assumption that

ϕ_{0; θ} (\vec{r} + \frac{π}{k} z \vec{ξ}) - ϕ_{0; θ} (\vec{r} - \frac{π}{k} z \vec{ξ}) ≪ 1

for all r⃗ and for all ξ⃗ at fixed z and fixed k. If this assumption is met then one may write

ψ_{0; θ} = \sqrt{I_{0, θ}} (1 + i ϕ_{0; θ}) .

Substituting Eq. (26) into Eq. (25), one has [30]

ℱ I_{z, θ} (\vec{ξ}) = I_{0, θ} (δ (\vec{ξ}) - 2 sin (\frac{2 π^{2}}{k} z {\vec{ξ}}^{2}) ℱ ϕ_{0; θ}),

Eq. (27) provides for an algebraic solution of the phase-retrieval problem in Fourier space provided the zeros of the sine function are regularized in a natural way. Notice that going beyond Eq. (26) in taking higher powers of ϕ_0;θ into account algebraic inversion in the sense of Eq. (27) no longer is possible. This is because in Fourier space quadratic and higher orders enter in Eq. (25) in a highly nonlocal way.

Since

sin (\frac{2 π^{2}}{k} z {\vec{ξ}}^{2}) = \sum_{i = 1}^{\infty} \frac{{(- 1)}^{i - 1}}{(2 i - 1)!} {(\frac{2 π^{2} z}{k} {\vec{ξ}}^{2})}^{2 i - 1}

(with an infinite radius of convergence of the series on the right-hand side) we deduce, upon letting ℱ⁻¹ act on Eq. (27), that

\begin{array}{l} g_{z; θ} (\vec{r}) & = & \frac{I_{z, θ} (\vec{r}) - I_{0, θ}}{I_{0, θ}} \\ = & - 2 \sum_{i = 1}^{\infty} \frac{1}{(2 i - 1)!} {(\frac{z}{2 k})}^{2 i - 1} {(\nabla_{⊥}^{2})}^{2 i - 1} ϕ_{0; θ} (\vec{r}) . \end{array}

Therefore, by assuming the validity of Eq. (26) the inversion of Eq. (27) corresponds to an infinite summation of powers of

\nabla_{⊥}^{2}

acting on ϕ_0;θ(r⃗). We stress that our expansion (8) contains these powers of

\nabla_{⊥}^{2}

in a truncated way, see left-hand side of z× Eq. (14) at order z and the term

\propto {(\nabla_{⊥}^{2})}^{3} ϕ_{0; θ}

in the sixth line of z× Eq. (34) at order z³. Notice that in addition to these terms nonlinear corrections in ϕ_0;θ enter at these orders. This is the reason why the above assumption can increasingly be relaxed taking increasing powers of z into account in Eq. (8). The series Eq. (8) expands in powers of transverse derivatives including nonlinearities in ϕ_0;θ whereas expansion (29) takes into account all powers of derivatives at linear order in ϕ_0;θ only. For weak phase objects expansion (29) is an excellent approximation in treating the case where all frequencies contribute to the overall phase shift such that the latter is smaller than unity but breaks down as soon as a part of the spectrum contributes relative phase shifts comparable or larger than unity. Since the nonlinear expansion of Eq. (8) for pragmatic reasons, needs to be truncated at a finite order in z large frequencies are retrieved in a worse way than they are by inverting Eq. (27) within that equation’s regime of validity.

4. Phase retrieval for simulated phantom data

Let us now apply the phase retrieval algorithm of Sec. 3 to simulated phantom data. We use the regularization of Eq. (21) to invert the Laplacian in Eqs. (14), both in the perturbative estimate and the final expression for ϕ_0;θ.

To see how sensitive the retrieved phase depends on the regularization parameter α we define a function Φ(θ,α) as follows

Φ (θ, α) \equiv \sum_{x, y} | ϕ_{0; θ} (x, y) |,

where ϕ_0;θ(x,y) is the retrieved phase at regularization parameter value α. The value of α is chosen such that Φ(θ,α) is least sensitive to a variation in α. For the phantoms considered below there occurs a wide region (several orders of magnitude with central value α_c ∼ 10⁻⁶) of insensitivity for Φ(θ,α).

To measure the improvement achieved by the inclusion of the perturbatively evaluated correction terms in Eq. (14) compared to the leading-order linear term we define a quantity Ψ_θ,αc as follows:

Ψ_{θ} \equiv | ϕ_{0; θ} - ϕ_{0; θ}^{exact} |,

where ϕ_0;θ either is the leading-order (LO) or includes the next-to-leading-order (LO + NLO) result. The quantity 〈Ψ_θ〉 is defined as the mean of Ψ_θ over all pixels.

Let us now test the algorithm for a pure-phase phantom (Siemens star) with phase retrieval indicated in Fig. 2. All simulations are performed for a central projection (θ = 0) at an X-ray energy of E = 30keV, at δ = 10⁻⁷ on a δ = 0 background, and for thickness d = 0.256mm. A central disk with constant phase shift was introduced to avoid numerical problems arising from unresolved spokes segments. To avoid extreme phase jumps we have applied a normalized Gaussian blurring to the Siemens star to yield an exact phase map from which intensity is computed by free-space propagation. As Figs. 2(e) and 2(f) indicates, function Ψ_θ is smaller when taking next-to-leading corrections into account. To investigate the dependence on distance and resolution of the retrieved phase we perform line cuts through the phase map as indicated in Fig. 2(a). The results are shown in Fig. 3. Keeping the standard deviation of the Gaussian blurring constant at 1.5 pixels, results do not change substantially for smaller values of δ than 10⁻⁷. (This means that at a higher resolution the averaged-over length scale becomes shorter.) For δ > 10⁻⁶, however, strong deviations of retrieved from exact phase maps occur both in the leading-order and the next-to-leading-order result suggesting that a violation of our assumption on weak phase variation takes place. Recall that strong phase variations yield large coefficients in the power series of Eq. (15) and thus worsen the convergence properties of the expansion.

Fig. 2 Phase and intensity maps for the Siemens star in central projection at a resolution of 2048 × 2048. In all cases mean values are subtracted. (a) exact phase map at z = 0, (b) intensity contrast map at z = 0.3m generated from (a) by Fresnel propagation, (c) ϕ^LO for leading-order retrieval based on map (b), (d) the correction ϕ^NLO at next-to-leading-order retrieval, (e) Ψ^LO (see Eq. (31) for definition), and (f) Ψ^LO+NLO. A Gaussian blurring with standard deviation of 1.5 pixels was applied to the Siemens star before propagation. The green line in (a) indicates the trajectory along which cuts of the phase maps are presented in Fig. 3.

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Fig. 3 Line cuts as indicated in Fig. 2 (a) through phase maps for the Siemens star with 32 spokes in dependence of distance z (top to bottom) and pixel resolution (left to right). Black (blue, red) lines refer to the exact (leading-order, including next-to-leading-order) result for phase retrieval.

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The inclusion of the next-to-leading order result retrieves the exact phase closely at a low pixel resolution (left column) which corresponds to a large physical blurring scale. This is true for all indicated distances z. At high pixel resolution (right column) we still observe an improved phase retrieval but subject to edge-related artifacts induced by a stronger varying phase (smaller physical blurring scale). On the other hand, by increasing the pixel resolution at a fixed physical blurring scale we observe quantitatively stable results. In Fig. 4 we have investigated a more complex situation by increasing the numbers of spokes in the Siemens star and by adding a large central disk. This introduces a scale hierarchy to the object (typical diameter of the spoke to diameter of the disk). Comparing the first line of Fig. 3 with Fig. 4, we notice that the leading-order result deviates much stronger in Fig. 4 from the exact phase than in Fig. 3 and that the inclusion of the next-to-leading order correction yields a significantly improved phase retrieval (making up a ∼ 10% deviation of the leading-order retrieval). Notice also that in Fig. 4 the tendency of the leading-order retrieval in over-estimating the phase shift introduced by the spokes is reversed for the central region such that the mean value remains at zero at a slightly overestimated background (leading order) and a well retrieved background (including next-to-leading order corrections). The proper retrieval induced by the corrections is due to the nonlinearity of the latter (maximal difference in phase shifts ∼ 4). We suspect that the inclusion of all powers of the Laplacian (CTF) but exclusion of nonlinear corrections would only have improved on the edge-related artefacts.

Fig. 4 Phase maps for Siemens star with 256 spokes in central projection at a resolution of 2048 × 2048. (a) exact phase map at z = 0, (b) line cut through exact phase map (black), leading-order result for phase retrieval (blue), and for phase retrieval including next-to-leading order result at z = 0.3m. One has 〈Ψ^LO〉 = 0.1535 and 〈Ψ^LO+NLO〉 = 0.0347, respectively.

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5. Summary and outlook

We have presented a systematic extension of a linear phase retrieval algorithm for the in-line propagation technique which is applicable to pure phase objects (soft tissue, polymers, etc.). This algorithm could be of advantage for computed tomography applied to processes because it appeals to a stack of radiographs taken at a single object-detector distance z only.

Specifically, we have proposed a model where intensity and phase contrast are expanded in powers of z. In Fresnel theory the expansion coefficients are determined in terms of increasing numbers of transverse derivatives of the phase map in the object plane. The resulting partial differential equation starts to be nonlinear at next-to-leading order and can be approached in a perturbative way. Using simulated phantom data, we have demonstrated the promise of this method.

Because the method is applicable to the retrieval of larger relative phase-shifts over the entire projection (nonlinearity) it should yield more satisfactory results for thick objects (ratio between thickness and linear size of field of view close to unity). Moreover, since the accumulated phase shift along a ray path through a thicker sample is less prone to small-scale variations of the projection in dependence of transverse coordinates (averaging out large variations in δ) the requirement of small values of derivatives (convergence of the expansion in Eq. (8)) is better met in this case (less nervousness on small transverse scales). Therefore, we expect the present approach to have applications in the tomography of samples introducing large phase shifts.

It is conceivable that the use of different basis sets than just powers in z (Eq. (8)) yield even better convergence properties of the expansion.

Finally, due to the linear relation (5) between phase and intensity contrast in the leading-order estimate the perturbative treatment of the nonlinearities in the next-to-leading and in the next-to-next-to-leading order corrections of the right-hand sides of Eqs. (14) and (34) extents consistently to the case of polychromatic irradiation: At leading order intensity contrast and effective phase both are understood as an integral over the spectrum involving their monochromatic components. Since in perturbation theory it is the thus defined leading-order estimate for the effective phase that enters into the evaluation of the nonlinearities of Eqs. (14) and (34) one arrives at a consistent generalization of the concept of an effective phase for the polychromatic case when invoking nonlinear corrections.

Appendix: Next-to-next-to leading corrections to $\nabla_{⊥}^{2} ϕ_{0; θ}$

Here we give the next-to-next-to-leading order in the expansion $\nabla_{x, y}^{2} ϕ_{z = 0; θ}$ . To obtain these corrections ansatz Eqs. (8) and ansatz (9) are truncated at l_max = 3 and m_max = 2, respectively. Inserting this truncation into Eq. (4) and comparing coefficients up to quadratic order in z yields

\begin{array}{l} g_{1; θ} & = & - \frac{1}{k} \nabla_{⊥}^{2} ϕ_{0; θ} \\ g_{2; θ} & = & - \frac{1}{2 k} [\nabla_{⊥} g_{1; θ} \cdot \nabla_{⊥} ϕ_{1; θ} + \nabla_{⊥}^{2} ϕ_{1; θ} + g_{1; θ} \nabla_{⊥}^{2} ϕ_{0; θ}] \\ g_{3; θ} & = & - \frac{1}{3 k} [\nabla_{⊥} g_{1; θ} \cdot \nabla_{⊥} ϕ_{1; θ} + \nabla_{⊥} g_{2; θ} \cdot \nabla_{⊥} ϕ_{0; θ} + \nabla_{⊥}^{2} ϕ_{2; θ} + g_{1; θ} \nabla_{⊥}^{2} ϕ_{1; θ} + g_{2; θ} \nabla_{⊥}^{2} ϕ_{0; θ}] . \end{array}

From Eq. (12) and its derivative w.r.t. z we obtain for z → 0:

\begin{array}{l} ϕ_{1; θ} & = & - \frac{1}{2 k} {(\nabla_{⊥} ϕ_{0; θ})}^{2} \\ ϕ_{2; θ} & = & \frac{1}{4 k} [\frac{1}{2} \nabla_{⊥}^{2} g_{1; θ} - 2 \nabla_{⊥} ϕ_{0; θ} \cdot \nabla_{⊥} ϕ_{1; θ}] \\ = & \frac{1}{4 k^{2}} [- \frac{1}{2} \nabla_{⊥}^{4} ϕ_{0; θ} + \nabla_{⊥} ϕ_{0; θ} \cdot \nabla_{⊥} {(\nabla_{⊥} ϕ_{0; θ})}^{2}] . \end{array}

Including the next-to-next-to-leading order, we thus have

\begin{array}{l} \nabla_{⊥}^{2} ϕ_{0; θ} & = & - {(\frac{z}{k})}^{- 1} g_{z; θ} - \\ \frac{1}{2} (\frac{z}{k}) [(\nabla_{⊥} \nabla_{⊥}^{2} ϕ_{0; θ}) \cdot \nabla_{⊥} ϕ_{0; θ} + {(\nabla_{⊥}^{2} ϕ_{0; θ})}^{2} - \frac{1}{2} \nabla_{⊥}^{2} {(\nabla_{⊥} ϕ_{0; θ})}^{2}] + \\ \frac{1}{12} {(\frac{z}{k})}^{2} [2 \nabla_{⊥} (\nabla_{⊥}^{2} ϕ_{0; θ}) \cdot \nabla_{⊥} {(\nabla_{⊥} ϕ_{0; θ})}^{2}] + \\ 2 \nabla_{⊥} ϕ_{0; θ} \cdot \nabla_{⊥} (\nabla_{⊥} ϕ_{0; θ} \cdot \nabla_{⊥} (\nabla_{⊥}^{2} ϕ_{0; θ})) + \\ 2 \nabla_{⊥} ϕ_{0; θ} \cdot \nabla_{⊥} {(\nabla_{⊥}^{2} ϕ_{0; θ})}^{2} + \nabla_{⊥} ϕ_{0; θ} \cdot \nabla_{⊥} (\nabla_{⊥}^{2} {(\nabla_{⊥} ϕ_{0; θ})}^{2}) - \\ \frac{1}{2} {(\nabla_{⊥}^{2})}^{3} ϕ_{0; θ} + \nabla_{⊥}^{2} (\nabla_{⊥} ϕ_{0; θ} \cdot \nabla_{⊥} {(\nabla_{⊥} ϕ_{0; θ})}^{2}) + 3 \nabla_{⊥}^{2} ϕ_{0; θ} \nabla_{⊥}^{2} {(\nabla_{⊥} ϕ_{0; θ})}^{2} + \\ 2 \nabla_{⊥}^{2} ϕ_{0; θ} \nabla_{⊥} ϕ_{0; θ} \cdot \nabla_{⊥} (\nabla_{x, y}^{2} ϕ_{0; θ}) + 2 {(\nabla_{⊥}^{2} ϕ_{0; θ})}^{2}] . \end{array}

Acknowledgments

JM and RH have benefited from the discussion of available techniques in phase-contrast imaging provided by Peter Cloeten’s and Max Langer’s doctoral thesis [24, 31]. In addition, the queries of a Reviewer, which have led to the inclusion of Sec. 3.3., were perceived to be very helpful and have led to an improvement of the manuscript. Useful conversations with Daniel Hänschke, Lukas Helfen, Tomy dos Santos Rolo, Heikki Suhonen, and Feng Xu are gratefully acknowledged.

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Nonlinear phase retrieval from single-distance radiograph

Abstract

1. Introduction

2. Tomographic reconstruction algorithm

3. Intensity contrast beyond linearity in z

3.1. Powers in z, paraxial wave equation, and perturbation theory

3.2. Numerical implementation

3.3. Comparison with CTF-based single-distance retrieval

4. Phase retrieval for simulated phantom data

5. Summary and outlook

Appendix: Next-to-next-to leading corrections to $\nabla_{⊥}^{2} ϕ_{0; θ}$

Acknowledgments

References and links

Cited By

Figures (4)

Equations (35)

Optics Express

Abstract

1. Introduction

2. Tomographic reconstruction algorithm

3. Intensity contrast beyond linearity in z

3.1. Powers in z, paraxial wave equation, and perturbation theory

3.2. Numerical implementation

3.3. Comparison with CTF-based single-distance retrieval

4. Phase retrieval for simulated phantom data

5. Summary and outlook

Appendix: Next-to-next-to leading corrections to ∇⊥2ϕ0;θ

Acknowledgments

References and links

Cited By

Figures (4)

Equations (35)

Optics Express

Appendix: Next-to-next-to leading corrections to $\nabla_{⊥}^{2} ϕ_{0; θ}$