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REVIEW 3 major objections 5 minor 61 references

Applying the Fokker--Planck equation to grating-based x-ray phase and dark-field imaging

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Pith's one-line read Applying the finite-difference X-ray Fokker–Planck equation to sinusoidal grating illumination yields analytic formulas for dark-field edge effects, phase focusing, and phase-edge cross-talk.

desk verdict A solid theory paper with new analytic expressions for cross-talk artifacts, but an unresolved ambiguity about the diffusion coefficient in the central finite-difference equation. read the letter →

arxiv 1908.01452 v2 pith:MUMMZDP2 submitted 2019-08-05 physics.optics physics.app-ph

Kaye S. Morgan , David M. Paganin This is my paper · ORCID
classification physics.opticsphysics.app-ph
keywords X-rayFokker–Planckequationgrating-basedphasecontrastimagingdark-fieldgratinginterferometryedgeilluminationsmall-anglescatteringtransportofintensityretrieval
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper seeks to show that a single differential equation—the Fokker–Planck equation—can describe both the phase signal and the dark-field signal in grating-based x-ray imaging. When the illuminating pattern is a sinusoidal grid, the drift term of the equation produces the transverse shift of the pattern (the differential-phase signal), and the diffusion term reduces pattern visibility (the dark-field signal). The authors derive closed-form expressions for the extra signals that appear at scattering edges, phase edges, and regions of phase curvature, and show the same expressions carry over to grating interferometry and edge-illumination setups. If correct, this gives a quantitative way to predict and remove cross-talk between attenuation, phase, and dark-field signals, and a starting point for retrieving both phase and dark-field images from grating measurements.

What carries the argument

The load-bearing object is the finite-difference X-ray Fokker–Planck map, Eq. (6), obtained by replacing time with propagation distance $z$ in the classical Fokker–Planck equation: intensity $I$ plays the role of probability density, the drift coefficient is $(1/k)\partial_x\varphi$ (the sample-induced phase gradient), and the diffusion coefficient $D(x,z)$ encodes unresolved small-angle scattering. Inserting sinusoidal illumination $I=a\sin(x/p)+b$ into that map is what makes the analysis explicit: uniform diffusion multiplies the sinusoidal amplitude by $1-D\Delta/p^2$, a spatially varying $D$ generates local bright/dark bands through $\partial^2D/\partial x^2$ and an additional shift through $\partial D/\partial x$, and phase curvature $\partial^2\varphi/\partial x^2$ changes the mean, period, and shift of the pattern. The linearization in $\Delta$ makes drift and diffusion effects add, which is why a single combined formula, Eq. (25), can describe both.

What would settle it

Set up a directly-resolved absorbing or phase grating to illuminate a sharp boundary between two uniform scattering materials with known diffusion coefficients $D_\text{left}$ and $D_\text{right}$, and record intensity profiles for a range of propagation distances $\Delta$ and grating periods $p$. The model predicts a local bright/dark band in the mean intensity whose shape follows $\partial^2 D/\partial x^2$ and whose amplitude scales with $\Delta(D_\text{right}-D_\text{left})/p^2$, plus a visibility overshoot on the weakly scattering side. If the measured edge signature does not follow that shape or scaling, the diffusion model is falsified.

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Extended reading notes

Core claim

The paper's central claim is that the finite-difference form of the X-ray Fokker–Planck equation, $$I(x,z=\$\Delta$)\approx I(x,z=0)+\$\Delta$\left[-\frac{1}{k}\frac{\partial}{\partial x}\left(I\frac{\partial\varphi}{\partial x}\right)+\frac{\$partial^{2}$}{\partial $x^{2}$}(D I)\right],$$ with the illuminating pattern taken as the sinusoid $I=a\sin(x/p)+b$, is a valid forward model for grating-based phase and dark-field x-ray imaging. Under this model the differential-phase signal is the drift term and the dark-field signal is the diffusion term. The paper derives explicit expressions for what happens where the sample changes: a scattering edge produces local extra bright and dark bands and visibility changes (Eqs. 12 and 16); a phase gradient shifts the grid pattern without changing visibility (Eq. 22); phase curvature focuses or defocuses the pattern, changing mean, period, and shift together (Eq. 23); a phase edge adds a bright/dark fringe (Eq. 24); and the two effects add (Eq. 25). The same machinery carries over to two-grating systems: the stepping curve of a grating interferometer has mean $pM/2$, visibility $V/2$, and shift $s$ (Eq. 28), and edge-illumination signals combine in the same way (Eqs. 29–30).

Load-bearing premise

The analysis stands on treating unresolved small-angle x-ray scattering as a genuine diffusion process governed by a single z-dependent coefficient $D(x,z)$ with negligible higher-order Kramers–Moyal terms; if scattering is non-Gaussian or has significant third-order terms, the predicted edge and lensing signatures will not match grating-based measurements.

Editorial extensions

If this is right

  • At scattering edges, the predicted mean-intensity band and visibility change can be recognized as artifacts and subtracted, so dark-field images are not contaminated by edge signals.
  • Strong phase curvature produces simultaneous changes in mean, period, and shift; the analytic expressions quantify how much apparent attenuation and dark-field a lensing object can create.
  • Phase edges add a bright/dark fringe over the structured illumination, providing a grating-based analogue of propagation-based edge contrast that can be modelled rather than mistaken for sample structure.
  • Since drift and diffusion terms add in the finite-difference approximation, Eq. (25) is a single forward model that can serve as the basis for joint phase/dark-field retrieval.
  • For grating interferometry and edge illumination, the directly-resolved-grid results transfer with simple scalings: stepping-curve mean becomes $pM/2$, visibility becomes $V/2$, and the shift remains $s$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension beyond the paper: the same linearized model suggests a direct route to simultaneous phase and dark-field retrieval by inverting Eq. (25) for $\varphi$ and $D$, with edge terms included rather than treated as noise.
  • Extension beyond the paper: because source-size blur, detector blur, and sample scattering would all enter as additive diffusion coefficients, the formalism gives a principled way to separate their contributions from measurements at different distances, grating periods, or source sizes.
  • Extension beyond the paper: the visibility-zero condition $D\Delta=p^2$ (blur width equal to the grid period) implies a design trade-off, suggesting that the optimal grating period for dark-field sensitivity is set by the expected scattering length scale, not merely by resolution.
  • Extension beyond the paper: since arbitrary illumination can be built from sinusoids by Fourier decomposition, the single-sinusoid formulas should transfer to speckle-tracking and non-periodic grid patterns, though the paper only gestures at this connection.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper proposes that the finite-difference form of a Fokker-Planck equation with an added diffusion term can describe the propagation of periodic (sinusoidal) intensity patterns in grating-based and speckle-based x-ray phase and dark-field imaging. The authors start from the X-ray Fokker-Planck equation (Eq. 2), and derive analytic expressions for dark-field visibility loss (Eq. 8), dark-field edge signatures (Eqs. 12 and 14), an integral representation for the large-blur regime (Eq. 16), the prism phase shift (Eq. 22), phase focusing and phase-edge effects (Eqs. 23 and 24), a combined phase/dark-field expression (Eq. 25), and extensions to grating interferometry and edge illumination (Eqs. 27-30). The results are benchmarked against known limiting cases, such as the Lynch et al. dark-field coefficient D/p^2 and the Morgan et al. prism shift. The paper includes Python code for interactive visualization of the derived formulas. No experimental validation is provided, and the central model equation is taken from an in-press companion paper.

Significance. If correct, the paper provides a valuable unifying framework in which phase drift and dark-field diffusion are described by a single partial differential equation, and it yields closed-form expressions for edge and focusing artifacts that are usually discussed only qualitatively in grating-interferometry literature. The benchmarking against independent known expressions and the reproducible code are strengths. However, several key displayed equations contain algebraic errors or ambiguities that affect the quantitative predictions, and the treatment of the z-dependent diffusion coefficient in the finite-difference step is unresolved. These issues must be fixed before the results can be used as a reliable quantitative basis.

major comments (3)
  1. [Sections 2 and 3.1.1, Eqs. (2), (6), (8), (9)] The paper states that D(x,z) is made z-dependent so that the blur width grows linearly with z, but the finite-difference approximation (6) and all subsequent dark-field results use a single constant D multiplied by Delta. If D(z) = D0 z is inserted into Eq. (2), the diffusion term at the initial plane z=0 vanishes, so a forward-Euler step gives no first-order-in-Delta dark-field contribution; evaluating D at z=Delta instead makes the term O(Delta^2). The manuscript never specifies what D in Eq. (6) represents (initial value, final value, or an integrated average), and the factor-of-two remark after Eq. (9) does not resolve this. Because Eqs. (8), (10), (12), (14), and (16) all inherit this step, the quantitative dark-field predictions rest on an unresolved internal inconsistency.
  2. [Section 3.2, Eqs. (21)-(25)] The phase-shift term in Eq. (21), namely tan^{-1}( (dphi/dx) p / (k/Delta - d2phi/dx2) ), does not follow from Eq. (20). Combining the sin and cos terms in Eq. (20) yields a phase shift of tan^{-1}[ (dphi/dx) / (p(k/Delta - d2phi/dx2)) ], so p belongs in the denominator rather than the numerator. Consequently Eq. (21) and the derived Eq. (23) contain an extra factor p^2 in the shift. Setting A=0 (a linear phase ramp) in Eq. (23) gives a shift B p^2 Delta/k instead of B Delta/k, contradicting Eq. (22) and the known prism shift. This error propagates into the phase-edge formula (24) and the combined formula (25), so the phase-related analytic results are not reliable in their current printed form.
  3. [Section 3.1.2, Eq. (16)] The large-blur convolution result is displayed as a complex expression involving imaginary error functions. The mean-intensity term contains 2b e^{(2ipx+(D_l+D_r)Delta)/(2p^2)} (2 - Erf(x/sqrt(2D_l Delta)) + Erf(x/sqrt(2D_r Delta))), whose oscillatory e^{ix/p} factor cannot arise from the constant contribution b of the illumination, and whose limiting value as x to +/- infinity is not b. This suggests a typographical or algebraic error in the displayed formula. Since Eq. (16) underlies the Python-based visualizations of the dark-field edge effect in the large-blur regime, it needs to be rederived and numerically verified.
minor comments (5)
  1. [Section 3.1.2, text below Eq. (13)] The sentence '...D = Dright far to the left of the origin, and D = Dleft to the right of the origin' is opposite to what Eq. (13) actually gives; it should read Dleft on the left and Dright on the right, consistent with Fig. 4.
  2. [Eq. (5) and Eq. (26)] Equation (5) omits the factor 2pi inside sin(x/p), but Eq. (26) reintroduces it; the authors should state explicitly that p in Eqs. (5)-(24) is a dimensionless spatial-frequency parameter and not the physical grating period, to avoid confusion.
  3. [Section 3.4.1, Eqs. (27)-(28)] The integration leading from Eq. (27) to Eq. (28) is not shown; a brief derivation or a statement of the orthogonality relation used would make the result easier to verify.
  4. [General, ref. [24]] The paper relies on an in-press companion paper for the central Fokker-Planck equation; the authors should confirm that the companion is published and include a short summary of the assumptions (paraxiality, projection approximation, etc.) in the present manuscript.
  5. [Section 3.1.1 and 3.2.2] The connection between the blur width L = sqrt(D Delta) and the z-dependent diffusion coefficient is stated in passing; a precise definition of D(z) and a short derivation of the resulting linear growth would help the reader understand the finite-difference treatment.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the assumed Fokker-Planck model is applied, not re-derived from its own outputs.

full rationale

The paper's derivations are self-contained once the stated input model is accepted. The X-ray Fokker-Planck equation (Eq. 2) is imported from the authors' companion paper (ref. 24), which is a self-citation, but it functions as the assumed physical model rather than as a result derived within this paper; the paper's contribution is the application of that model to sinusoidal structured illumination. The subsequent analytic results (Eqs. 8, 12, 14, 16, 21-25, and 28-30) follow by substituting the sinusoidal illumination of Eq. (5) into the finite-difference approximation of Eq. (6) and performing algebraic, trigonometric, and elementary integral manipulations. No parameter is fitted to experimental data, and known limiting cases are recovered from independent literature: the linear dark-field coefficient D/p^2 matches Lynch et al. (Eq. 19), and the prism shift matches Morgan et al. (Eq. 22). These are consistency checks, not circular inputs. The apparent tension between the z-dependent D discussed in Sec. 2 and the use of a single D in the one-step finite-difference Eq. (6) is an internal consistency or correctness concern about the validity of the linearization, not a circular reduction, because the claimed predictions are explicitly linear-in-Delta approximations of the assumed equation rather than a disguised restatement of the model's assumptions.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No data are fitted and no new entities are introduced. The central equation is assumed from the authors' companion paper. The main modeling choices are the z-dependent diffusion coefficient, the sinusoidal illumination, and the Gaussian convolution kernel for diffusive propagation.

assumptions (8)
  • domain assumption The X-ray Fokker-Planck equation, Eq. (2), governs the paraxial evolution of x-ray intensity through samples with phase and unresolved scattering.
    Section 2 replaces time with propagation distance z and probability density with intensity, citing companion paper ref 24 for the derivation; the present paper does not derive or experimentally validate this equation.
  • domain assumption The intensity of the x-ray beam can be treated as a probability density function undergoing drift and diffusion.
    Section 2 states this explicitly; it underlies the whole Fokker-Planck analogy.
  • ad hoc to paper The diffusion coefficient D(x,z) is z-dependent so that the diffusive spread grows linearly with propagation distance.
    Section 2 notes that a constant D would give non-linear spread inconsistent with free-space propagation; D(z)=D0Δ is introduced to force linear growth, with justification deferred to companion paper ref 24.
  • standard math Truncation of the Kramers-Moyal expansion at second order is valid (Pawula theorem).
    Section 2 cites ref 32 to justify omitting third-order and higher terms; the theorem states that if any term above second order vanishes, all above second order vanish.
  • domain assumption A sinusoidal illumination, Eq. (5), is a sufficiently general model because any periodic or speckle illumination can be expanded in a Fourier series.
    Section 3 states this choice and the Fourier-series justification; all analytic results in the paper use this single-harmonic illumination.
  • standard math The finite-difference approximation in Eq. (6) is valid for small propagation distance Δ, i.e., DΔ < p^2.
    Section 3.1.1 explicitly restricts validity to DΔ < p^2; Eq. (8) can produce negative amplitudes outside this regime, which the authors acknowledge.
  • domain assumption The convolution form in Eq. (9) with a Gaussian kernel of variance D(x')Δ describes the diffusive propagation as a position-dependent point spread function.
    Section 3.1.1 introduces this linear integral transform as an alternative to the finite-difference solution; the Gaussian form is an approximation that matches Eq. (8) only to lowest order.
  • ad hoc to paper The error-function transition profile in Eq. (13) models a smooth edge between two scattering regions and enables closed-form evaluation of edge effects.
    Section 3.1.2 uses this specific profile to derive Eq. (14) and Eq. (16); the quantitative shape of the predicted edge signature depends on this choice, though the qualitative behavior is general.

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Cite this review

Pith. "Pith review of Applying the Fokker--Planck equation to grating-based x-ray phase and dark-field imaging." pith.science (2026). https://pith.science/paper/MUMMZDP2

@misc{pith2026190801452,
  author       = {Pith},
  title        = {Pith review of: Applying the Fokker--Planck equation to grating-based x-ray phase and dark-field imaging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MUMMZDP2}},
  note         = {Machine review of arXiv:1908.01452}
}
read the original abstract

X-ray imaging has conventionally relied upon attenuation to provide contrast. In recent years, two complementary modalities have been added; phase contrast and dark-field x-ray imaging, capturing weakly attenuating and sub-pixel sample structures respectively. These three modalities can be accessed using a crystal analyser, a grating interferometer or by looking at a directly-resolved grid, grating or speckle pattern. Grating and grid-based methods extract a differential phase signal by measuring how far a feature in the illumination has been shifted transversely due to the presence of a sample. The dark-field signal is extracted by measuring how the visibility of the structured illumination is decreased, typically due to the presence of sub-pixel structures in a sample. The strength of the dark-field signal may depend on the grating period, the pixel size and the set-up distances, and additional dark-field signal contributions may be seen as a result of strong phase effects or other factors. In this paper we show that the finite-difference form of the Fokker--Planck Equation can be applied to describe the drift (phase signal) and diffusion (dark-field signal) of the periodic or structured illumination used in phase contrast x-ray imaging with gratings, in order to better understand any cross-talk between attenuation, phase and dark-field x-ray signals. In future work, this mathematical description could be used as a basis for new approaches to the inverse problem of recovering both phase and dark-field information.

Figures

Figures reproduced from arXiv: 1908.01452 by the authors.

Figure 1
Figure 1. A photon or x-ray beamlet may encounter: a) a phase gradient in the sample, transversely shifting the position of the detected light at a downstream detector, b) small angle scattering from the sample, spreading out the beamlet, reducing the visibility, and broadening the probability distribution that describes where the photon will land at a downstream detector, or c) both effects. measured by looking at a reductio… view at source ↗
Figure 2
Figure 2. a) The Kramers–Moyal equation and its truncated equivalent, the Fokker–Planck equation, describe the diffusion and shift in a probability function. b) This same equation can be applied to study the x-ray dark-field and phase shift measured using periodic x-ray illumination. an array of beamlets, with a transverse shift of each beamlet seen at z = ∆ when the beamlet passes through a phase-shifting sample ( [PITH_FUL… view at source ↗
Figure 3
Figure 3. a) A diffusion of the illumination is the basis for a dark-field signal, shown here for the simple case where the diffusion D is constant. The reduction in the visibility of the periodic illumination is described by Eqn. (8). b) When the diffusion D is not constant, any sudden spatial change in the scattering properties (i.e. ∂ 2D/∂ x 2 is significant) results in an effect that would be interpreted as variations in … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: a) We can examine the effects seen where the diffusion or dark-field/scattering property D changes spatially. The test sample here is weakly scattering on the left and strongly scattering on the right, as modelled by Eqn. (13), meaning that ∂ 2D/∂ x 2 is non-zero durin…
Figure 5
Figure 5. Figure 5: a) A shift of the illumination is the basis for a phase shift signal, shown here for the simple case where the second derivative of the phase gradient is not significant. The shift in the transverse position of the periodic illumination is described by Eqn. (22). b) Wh…
Figure 6
Figure 6. Figure 6: Phase variations incurred by the sample result in changes to the sinusoidal illumination in a number of ways, shown here (left to right) for a linear projected phase ( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

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