Speckle-based X-ray Imaging Methods

• Speckle-based X-ray imaging is a family of methods that uses random intensity modulations as probes to extract phase, attenuation, and dark-field information.
• Techniques range from near-field speckle tracking and geometric-flow inversion to ptychographic and optimization-based approaches, balancing sensitivity and dose.
• Recent advances leverage model-based inversion and deep learning to enhance resolution, minimize noise, and enable multimodal imaging.

Searching arXiv for recent and foundational papers on speckle-based X-ray imaging to ground the article. arXiv search: speckle-based X-ray imaging ghost imaging speckle tracking MIST PXST PWF Speckle-based X-ray imaging is a family of X-ray methods that uses spatially random intensity modulations as an imaging carrier, a wavefront marker, or a correlation reference. In the reported implementations, the relevant speckle field may be generated by a random mask in the near field, by the sample itself in a divergent beam, or by the shot noise of synchrotron emission from isolated electron bunches; the measured speckle deformation or inter-beam correlation is then used to recover attenuation, phase shift, dark-field, wavefront aberrations, or, in ghost-imaging configurations, an image formed from photons that never interacted with the sample (Berujon et al., 2015, Pelliccia et al., 2016, Paganin et al., 2018, Pavlov et al., 2019, Lee et al., 26 Mar 2025).

1. Physical basis

The common physical substrate is paraxial X-ray propagation through matter with complex refractive index $n = 1 - \delta + i\beta$. In the projection approximation, the sample induces a phase shift

$$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$

while attenuation is governed by $\beta$. In near-field speckle methods, the resulting transverse phase gradients redirect intensity rather than merely reducing it, so random intensity structure becomes a probe of refraction as well as absorption (Paganin et al., 2018, Berujon et al., 2015).

A central relation, written in several forms across the literature, is that a local speckle displacement is proportional to the phase gradient. In geometric-optics form,

$$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$

or equivalently $\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$ with $\boldsymbol{\theta}=(1/k)\nabla\phi$. This is the basis of differential phase retrieval in near-field speckle tracking, geometric-flow inversion, and ptychographic X-ray speckle tracking (Morgan et al., 2020, Shi et al., 2023).

The same formalism accommodates dark-field. In multimodal intrinsic speckle-tracking, unresolved microstructure is modeled by an effective diffusion coefficient $D_{\mathrm{eff}}(x,y;\Delta)$, which enters a Fokker–Planck-type transport equation alongside the coherent drift term associated with $\nabla\phi$. This formulation treats attenuation, refraction, and small-angle scattering within a single PDE, rather than as three disconnected observables (Pavlov et al., 2019).

In sponge-like or highly porous media, repeated small-angle refractions can themselves generate a granular phase-contrast texture. The reported propagation-based single-shot thickness method for such media models the angular broadening as a random walk and empirically relates local speckle contrast to thickness via

$C(T) = a \ln(bT + 1),$

with inversion by $T(x,y) = [\exp(C(x,y)/a)-1]/b$ after calibration under the stated geometry (Xi et al., 2015).

2. Speckle generation and experimental geometries

One major branch uses an external random modulator. Berujon and Ziegler’s near-field speckle-scanning work employed a sandpaper membrane on a motorized piezo stage, with a 17 keV monochromatic synchrotron beam, diffuser-to-sample distance $$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$0, sample-to-detector distance $$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$1, and effective detector pixel size $$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$2. Speckle contrast was reported as approximately $$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$3 at $$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$4, and the same framework was later extended to tomography with sparse and interlaced acquisition schemes (Berujon et al., 2015, Berujon et al., 2016).

A second branch dispenses with a dedicated diffuser and uses the sample itself as the structured near-field object. In ptychographic X-ray speckle tracking, the specimen is scanned downstream of a high-NA focus, producing a stack of magnified projection holograms. This is specifically designed for highly divergent wavefields such as those formed by wedged multilayer Laue lenses, where a conventional undistorted reference image is not available (Morgan et al., 2020, Morgan et al., 2020).

A third branch uses naturally occurring X-ray speckles. Pelliccia et al. demonstrated hard X-ray ghost imaging at ESRF ID19 using a synchrotron beam split by a $$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$5 thick Si crystal in Laue diffraction geometry, operated at 20 keV. The beamline ran in a four-bunch timing mode with $$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$6 bunch separation, and an ultrafast camera resolved speckle fluctuations associated with shot noise in isolated synchrotron pulses. The sample was a $$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$7 diameter Cu wire (Pelliccia et al., 2016).

Speckle structure also appears at the coherence-function level. For a modern storage-ring undulator source, the cross-spectral density $$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$8 can itself become speckled, with coherence vortices and domain walls that partition the nominal coherence patch into smaller correlation cells. This is not a detector-plane artifact but a property of the two-point coherence function, and it places a statistical constraint on correlation-based imaging and interferometric visibility (Paganin et al., 2019).

3. Principal methodological families

Near-field speckle tracking, speckle vector tracking, and speckle scanning form the classical core of the field. In these methods a reference speckle image or reference stack is compared with a sample-modified speckle image or stack, and the displacement maximizing local correlation is mapped to refraction angle and phase gradient. X-ray Speckle-Vector Tracking extends this idea by building vectors from multiple mask positions, thereby recovering absorption, phase, and dark-field for each projection and enabling multimodal CT with interlaced acquisition (Berujon et al., 2015).

Speckle scanning trades acquisition efficiency for sensitivity and additional structure recovery. In the raster-scan formulation, the sample and reference intensity sequences $$\phi(x,y) = -k \int \delta(x,y,z)\,dz,\qquad k=\frac{2\pi}{\lambda},$$9 and $\beta$0 are compared over diffuser-step coordinates $\beta$1, yielding a displacement $\beta$2 and thus

$\beta$3

The same data can be deconvolved, pixel by pixel, to recover a local ultrasmall-angle scattering distribution $\beta$4 and its moments, rather than only a scalar dark-field image (Berujon et al., 2015).

Single-image geometric-flow speckle tracking recasts the displacement field as an irrotational conserved current. With $\beta$5 the reference image and $\beta$6 the sample image, the linearized brightness-conservation equation

$\beta$7

is solved by introducing a scalar potential $\beta$8 such that $\beta$9, leading to a Poisson equation and an FFT-based inversion. Once $$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$0 is obtained, the phase follows from $$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$1 (Paganin et al., 2018).

Multimodal Intrinsic Speckle-Tracking generalizes optical-flow phase retrieval by retaining a diffusive term in addition to coherent transport. In its Fokker–Planck form,

$$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$2

so phase and local SAXS can be recovered simultaneously from only two sample-present images taken at two mask positions, together with reference images at the same positions. Later variants remove the slowly-varying-dark-field assumption, retain the transverse derivatives of $$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$3, and explicitly address rapidly varying diffusive dark-field (Pavlov et al., 2019, Alloo et al., 2023, Liu et al., 27 Aug 2025).

Ptychographic X-ray speckle tracking is a scan-redundant geometric-flow method for strongly divergent wavefields. Its forward model,

$$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$4

treats each near-field projection as a warped and shifted view of a single virtual reference image, with the warp encoding the detector-plane phase gradient. This formulation recovers both the illumination wavefront and an aberration-free specimen projection (Morgan et al., 2020).

Single-shot quantitative phase retrieval has recently moved beyond shift-only models. Preconditioned Wirtinger flow reconstructs the complex log-field $$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$5 by minimizing the mismatch between measured intensity and a forward model that includes propagation from sample to diffuser and diffuser to detector, multiplication by diffuser transmission, and convolution with an intensity point-spread function derived from measured partial coherence. It is explicitly presented as an assumption-free single-shot quantitative phase method (Lee et al., 26 Mar 2025).

4. Correlation, transport, and inverse formulations

Ghost imaging uses correlations across two speckle-correlated beams rather than local displacement within a single beam. In the hard X-ray realization, the bucket signal was

$$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$6

and the reconstructed ghost image was computed by the mean-subtracted correlation

$$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$7

Pelliccia et al. used 20,000 frames and Fourier filtering around the aliased storage-ring frequency $$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$8 or the half-ring component $$\Delta \mathbf{r}(\mathbf{r}) \approx \frac{z}{k}\,\nabla \phi(\mathbf{r}),$$9, while side windows away from these peaks produced no ghost image (Pelliccia et al., 2016).

Optical-flow and geometric-flow methods replace windowed correlation by PDE inversion. In the geometric-flow formulation, the measured difference $\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$0 is the divergence of a current, and the inverse problem reduces to a Poisson solve with exclusion of the Fourier DC term. This produces a global displacement field without explicit local matching windows, which is why the method is described as maintaining spatial resolution relative to conventional multi-image speckle tracking (Paganin et al., 2018).

The multimodal Fokker–Planck lineage emphasizes that dark-field is not merely a secondary statistic but part of the transport law. The most general isotropic formulation in the provided material keeps both $\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$1 and its transverse derivatives in the expanded Laplacian term, so the unknowns are solved from a per-pixel linear system using multiple reference/sample mask positions and Tikhonov-regularized QR decomposition. Gradient-Flow MIST retains this generality while reducing the minimum data requirement to two speckle image pairs and updating transmission through a dark-field-corrected closed-form step (Alloo et al., 2023, Liu et al., 27 Aug 2025).

The inverse problem becomes more subtle when edge-induced and microstructure-induced dark-field coexist. In the devolving Fokker–Planck perspective, the image recorded with sample and membrane is flowed backward to the reference-speckle image obtained when the sample is removed. This leads to a dark-field $\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$2 whose positive part can be interpreted as unresolved-microstructure dark-field and whose negative part localizes sharp-edge-induced dark-field, enabling a separation not available in earlier evolving formulations (Alloo et al., 2024).

Physically explicit optimization methods depart even further from correlation-based approximations. In preconditioned Wirtinger flow, the forward model

$\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$3

is minimized directly, with partial coherence calibrated from the reference speckle via the Siegert relation and an inverse-Laplacian preconditioner applied to the phase part of the gradient. This replaces local shift estimation by gradient-based optimization over the complex transmission (Lee et al., 26 Mar 2025).

5. Reported operating regimes, performance, and constraints

The reported operating regimes span MHz-resolved synchrotron pulse imaging, multi-exposure tomographic scanning, and single-shot phase retrieval. Hard X-ray ghost imaging was demonstrated at 20 keV with a Photron FASTCAM SA-Z, intended frame rate $\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$4, actual frame rate $\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$5, scintillator decay constants of approximately $\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$6 for CsI:Na and $\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$7 for YAG:Ce, and 20,000-frame ensemble averaging. The reconstructed field of view was approximately $\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$8, and the 200 $\mathbf{u}(x,y)=z\,\boldsymbol{\theta}(x,y)$9 wire appeared as an oblique shadow because of the Laue-diffraction geometry (Pelliccia et al., 2016).

Sensitivity figures are highly method- and geometry-dependent. In single-image geometric-flow phase imaging at ESRF ID17, the noise standard deviation in a sample-free region was reported as below $\boldsymbol{\theta}=(1/k)\nabla\phi$0 in monochromatic radiography at 52 keV. In near-field speckle-scanning CT, Berujon and Ziegler reported angular sensitivities of approximately $\boldsymbol{\theta}=(1/k)\nabla\phi$1 with 25 exposures in the sparse-interpolation scheme, approximately $\boldsymbol{\theta}=(1/k)\nabla\phi$2 with 25 exposures in the vector-tracking scheme, and below $\boldsymbol{\theta}=(1/k)\nabla\phi$3 with 36 exposures, compared with $\boldsymbol{\theta}=(1/k)\nabla\phi$4 for a dense $\boldsymbol{\theta}=(1/k)\nabla\phi$5 scan using 625 images (Paganin et al., 2018, Berujon et al., 2016).

Highly divergent-beam PXST reports substantially finer angular sensitivity because of very large magnification. The three MLL-based experiments reported $\boldsymbol{\theta}=(1/k)\nabla\phi$6–$\boldsymbol{\theta}=(1/k)\nabla\phi$7, approximately $\boldsymbol{\theta}=(1/k)\nabla\phi$8, and approximately $\boldsymbol{\theta}=(1/k)\nabla\phi$9 angular sensitivity, with full-period resolution estimates of 70 nm, 259 nm, and 45 nm, respectively. The highest-magnification diatom subregion additionally showed lattice peaks corresponding to a primitive hexagonal constant of about 601 nm (Morgan et al., 2020).

Dose and acquisition burden are recurring constraints. Speckle scanning and vector tracking improve robustness and sensitivity by increasing data redundancy, but this classically increases the number of sample exposures. That trade-off motivated sparse and interlaced XSS CT, modified XSVT with one image per angular projection for absorption and phase, and single-shot algorithms such as geometric flow, CADE, SPINNet, and PWF (Berujon et al., 2015, Berujon et al., 2016, Shi et al., 2023, Qiao et al., 2022, Lee et al., 26 Mar 2025).

The most explicit quantitative speed comparisons in the provided material come from ML-based trackers. CADE reported mean displacement RMSE $D_{\mathrm{eff}}(x,y;\Delta)$0 pixels versus $D_{\mathrm{eff}}(x,y;\Delta)$1 for UMPA and $D_{\mathrm{eff}}(x,y;\Delta)$2 for ZNCC, mean bias $D_{\mathrm{eff}}(x,y;\Delta)$3 pixels versus $D_{\mathrm{eff}}(x,y;\Delta)$4 and $D_{\mathrm{eff}}(x,y;\Delta)$5, and runtime of 18 s at about $D_{\mathrm{eff}}(x,y;\Delta)$6 pixels compared with 53 s for ZNCC and 181 s for UMPA (Shi et al., 2023). SPINNet reported simultaneous absorption and phase reconstruction on the order of 100 ms, approximately 0.16 s per $D_{\mathrm{eff}}(x,y;\Delta)$7 image pair on a single NVIDIA A100 GPU, and at least two orders of magnitude speedup over correlation-based analysis (Qiao et al., 2022).

Several limitations recur across the literature. Mechanical instabilities of beam splitters or optics generate low-frequency drift; detector afterglow and finite scintillator persistence mix neighboring pulses; large displacements or caustics violate linearized transport models; division by $D_{\mathrm{eff}}(x,y;\Delta)$8 amplifies noise in low-intensity regions; and the neglect of rotational components in current decompositions can bias reconstructions when the flow is not effectively irrotational (Pelliccia et al., 2016, Paganin et al., 2018, Pavlov et al., 2019).

6. Interpretation, controversies, and current directions

A persistent interpretive issue is the origin of dark-field contrast. Earlier multimodal speckle formalisms treated dark-field primarily as an effective diffusion associated with unresolved microstructure. The devolving Fokker–Planck perspective shows that resolvable sharp edges can generate a distinct dark-field channel, separable from unresolved-microstructure dark-field by the sign structure of the recovered $D_{\mathrm{eff}}(x,y;\Delta)$9. This indicates that “dark-field” is not a single mechanism but a composite observable whose decomposition depends on the forward model (Alloo et al., 2024).

Another frequent simplification is that speckle methods have “low” or “relaxed” coherence requirements. The provided studies consistently support that claim relative to interferometric techniques, but they also show that coherence remains structurally important: it sets speckle contrast, sampling requirements, and, at the two-point level, the topology of the cross-spectral density. The appearance of coherence vortices and domain walls in partially coherent undulator radiation implies that correlation-based observables can be locally suppressed even when the average beam appears highly usable (Paganin et al., 2019).

Current development is split between two complementary tendencies. One tendency seeks more complete physical forward models: PWF incorporates Fresnel propagation, measured diffuser transmission magnitude, and partial coherence calibrated from the reference speckle; Gradient-Flow MIST preserves the full multimodal Fokker–Planck structure while lowering data requirements; and devolving MIST separates edge and microstructure contributions to XDF (Lee et al., 26 Mar 2025, Liu et al., 27 Aug 2025, Alloo et al., 2024). A plausible implication is that the field is moving away from purely kinematic shift estimation toward model-based inversion in which speckle transport, blur, and coherence are jointly estimated.

The other tendency is computational acceleration. CNN-based single-shot trackers and SPINNet replace explicit windowed correlation by learned dense displacement estimation, while still embedding the near-field warp relation between reference and sample images. These approaches are presented not as new imaging physics but as new inverse solvers for the same speckle-mediated observables, with direct implications for in-situ, in-operando, and tomographic workflows (Shi et al., 2023, Qiao et al., 2022).

Taken together, the literature defines speckle-based X-ray imaging not as a single technique but as a methodological class. Its unifying feature is the use of random or pseudo-random X-ray structure as an information carrier. Within that class, correlation imaging, transport-equation inversion, scan-redundant wavefront metrology, and physically explicit optimization now coexist, and the choice among them is determined by the required balance between sensitivity, dose, acquisition burden, coherence tolerance, and the specific contrast channel—attenuation, phase, dark-field, or coherence—that is to be isolated.