Modulation-Based X-ray Imaging

• Modulation-based X-ray imaging is a set of techniques that impose known spatial or temporal modulations to reveal phase, scattering, and spectral contrasts through inversion methods.
• It employs various modulators—such as phase gratings, random diffusers, and coded apertures—to encode subtle attenuation and diffraction effects for improved image reconstruction.
• These methods enable high-resolution, quantitative imaging across applications like CT, crystallography, and dynamic inspections with enhanced sensitivity and dose efficiency.

Modulation-based X-ray imaging designates a class of techniques in which a known spatial or temporal modulation is imposed on the X-ray beam, and the sample is characterized by inverting how that modulation is altered by attenuation, refraction, phase shifts, ultra-small-angle scattering, strain, spectral filtering, or temporal transmission changes. In the cited literature, the modulating element may be a phase grating, a random diffuser, a single membrane, a Talbot array illuminator, a coded aperture, a checkerboard primary modulator, a spectral strip filter, a downstream Bragg modulator, a rotating slat mask, or a rapidly spinning random mask; the measurements may be full-field detector images, phase-stepping sequences, diffraction patterns, detector count-rate profiles, or single-pixel bucket signals (Lautizi et al., 2024, Berujon et al., 2015, Gustschin et al., 2021, Shi et al., 2022, Ulvestad et al., 2018, Zeng et al., 2 Jan 2026).

1. Physical basis of modulation and contrast formation

In transmission and full-field implementations, the central idea is to convert weak phase and scattering effects into measurable changes of a structured reference pattern. The sample attenuation is represented by $A(x,y)=\exp\!\left(-\int \mu(x,y,z)\,dz\right)$, the projected phase shift by $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$, and the local refraction angle by $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$. In speckle-vector tracking, the detector displacement obeys $v=d\,\alpha$, and dark-field arises from visibility loss, for example $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$ (Berujon et al., 2015). In low-coherence random-mask formulations, the sample-modulated intensity is modeled as a shifted and blurred version of a reference modulation, so refraction appears as subpixel displacement while unresolved microstructure appears as a local diffusion or blur term (Magnin et al., 2023).

A tensorial generalization is used when the dark-field signal is anisotropic. In the universal wavefront-modulation formalism, the transmitted intensity in a local analysis window is described by convolution with a Gaussian whose covariance is a 2D or projected 3D scattering tensor, and the Fourier-domain log-magnitude ratio is fitted by a positive-definite bilinear form, $$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$. This places attenuation and directional dark-field in the same energy-conservation model and makes the tensor eigenstructure measurable from omnidirectional modulators such as circular gratings, fractal arrays, or random speckles (Lautizi et al., 2024).

In Bragg geometries, modulation does not encode projected refractive index directly, but the crystalline displacement field. The complex object is written $O(r)=A(r)e^{i\phi(r)}$ with $\phi(r)=Q\cdot u(r)$, so the phase carries lattice displacement information and the strain tensor follows from $\epsilon_{ij}(r)=\partial u_i/\partial x_j$. The unmodulated far-field intensity is $I(q)=\left|\int O(r)e^{-iq\cdot r}\,dr\right|^2$, and a known downstream modulator adds deterministic diversity that makes single-view phase retrieval possible for extended crystals (Ulvestad et al., 2018).

A different branch of the field modulates spectral or temporal content rather than a monochromatic wavefront marker. In spectral modulation, the detector measures a projection of $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$0 or of checkerboard-encoded primary beams in which the modulation is energy dependent, while scatter is treated as a low-frequency additive term. In single-pixel and rotating-mask systems, the encoded quantity is a time series, such as $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$1 for ghost imaging or $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$2 for count-rate modulation, and the image emerges only after inversion of a forward operator (Deng et al., 2022, Shi et al., 2022, Zeng et al., 2 Jan 2026, Budden et al., 2010).

2. Modulator classes and imaging geometries

The literature spans several distinct geometrical realizations. Full-field methods place a wavefront-marking element upstream of the sample: abrasive paper or biological membrane in XSVT, sandpaper in low-coherence MoBI, circular $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$3-shifting gratings, fractal pattern arrays, and single nickel membranes with random, honeycomb, or Vogel-spiral hole topologies. These elements create a reference intensity field at the detector, and sample insertion perturbs that field by local displacement, attenuation scaling, and visibility loss. In these architectures, the field of view is set by the detector and beam footprint, and experimental complexity is shifted from optics to numerical demodulation (Berujon et al., 2015, Magnin et al., 2023, Lautizi et al., 2024, Magnin et al., 14 Apr 2025).

Grating-based systems use near-field diffraction rather than absorptive speckles. Two-dimensional Talbot array illuminators generate high-visibility, micrometer-scale intensity patterns at fractional Talbot distances, with $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$4 and visibility $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$5. Circular gratings provide omnidirectional directional-dark-field sensitivity in a single shot, while rotated 2D TAIs enable bidirectional phase sensitivity with one-dimensional stepping and UMPA demodulation (Gustschin et al., 2021, Lautizi et al., 2024). The same general category includes integrating-bucket grating interferometry, where G2 is moved continuously instead of by discrete phase stepping (Wali et al., 2017).

Bragg coherent modulation imaging uses a different geometry: a known amplitude/phase mask is inserted between the sample and the detector, $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$6 mm downstream of the sample in the reported experiment, while the far-field detector remains at $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$7 m. The modulator does not merely improve fringe visibility; it deliberately diversifies the diffracted wavefront so that phase retrieval can be performed from a single diffraction pattern at one angular slice (Ulvestad et al., 2018).

Spectral modulators are pre-patient elements that alter the spectrum as a function of ray position. SMFFS uses a stationary stacked two-dimensional arrangement of one-dimensional molybdenum strip filters, producing four effective filter states $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$8 through 0, 0.2, 0.4, and 0.6 mm Mo. SSQI uses a checkerboard primary modulator whose semitransparent cells are copper of 210 $\varphi(x,y)=-(2\pi/\lambda)\int \delta(x,y,z)\,dz$9m thickness, combined with a dual-layer detector that splits the transmitted beam into top and bottom spectral channels (Deng et al., 2022, Shi et al., 2022).

Temporal and single-pixel variants use modulation masks whose state changes in time rather than through detector-resolved pattern analysis. Tabletop X-ray ghost video employs a 500 $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$0m thick brass disk carrying an annular aggregate of random binary codes and a single-pixel detector; rotating-modulator hard-X-ray imaging uses a single rotating mask of parallel slats above an array of non-imaging detectors; rotational CGI uses a single-column striped coding plate rotated through multiple angles to generate a grayscale measurement matrix from area overlaps (Zeng et al., 2 Jan 2026, Budden et al., 2010, Zhou et al., 2023).

3. Forward models and inversion strategies

In transmission MoBI, XSVT, and related single-mask methods, inversion is typically local and overdetermined. XSVT builds per-pixel “speckle vectors” from multiple diffuser positions and estimates the displacement field by maximizing the Pearson correlation coefficient between sample and reference vectors, after which $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$1, $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$2, and dark-field follows from the ratio of standard deviations normalized by means (Berujon et al., 2015). In the low-coherence system formulation, the reference and sample images satisfy $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$3, and multi-position acquisitions solve for $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$4, $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$5, $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$6, and $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$7 per pixel. TAI systems instead use UMPA on stepped structured illumination to estimate local displacement vectors, transmission, and visibility reduction, while integrating-bucket interferometry replaces discrete phase sums by integrals over continuous motion of the analyzer grating (Magnin et al., 14 Apr 2025, Gustschin et al., 2021, Wali et al., 2017).

Bragg coherent modulation imaging uses iterative wave propagation with explicit enforcement of the detector modulus. For a single-view reconstruction, one initializes a complex exit wave $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$8, Fresnel propagates it to the modulator, multiplies by the known mask $$\alpha(x,y)=(\lambda/2\pi)\nabla_{\perp}\varphi(x,y)$$9, Fourier propagates to the detector, replaces amplitudes by $v=d\,\alpha$0, backpropagates, divides out the modulator, and reapplies a finite support at the sample plane. The experimental modulator-recovery step uses a PIE-like update,

$v=d\,\alpha$1

with diversity supplied by the angular slices of a standard BCDI rocking curve rather than by scan overlap (Ulvestad et al., 2018).

Spectral modulation and single-pixel modulation lead to different inverse problems. In SSQI, four sub-measurements $v=d\,\alpha$2 are used to solve for two material line integrals and two scatter images; the forward model integrates the effective spectra and PM transmission with additive scatter, and material decomposition is implemented through calibrated fifth-degree bivariate polynomials plus a per-pixel scatter-consistency optimization (Shi et al., 2022). In SMFFS, residual projections such as

$v=d\,\alpha$3

exploit the hypothesis that closely deflected focal spots share similar scatter, so subtractive residuals are approximately scatter free (Deng et al., 2022). In ghost imaging, the forward model is $v=d\,\alpha$4, and the reported video implementation reconstructs each frame by solving

$v=d\,\alpha$5

using TV regularization and positivity (Zeng et al., 2 Jan 2026). In rotating-modulation astronomy, characteristic count-rate profiles are precomputed analytically to form a system matrix $v=d\,\alpha$6, and reconstruction proceeds by regularized linear inversion, expectation-maximization, correlation methods, or NCAR (Budden et al., 2010, Huo et al., 2015).

4. Information channels recovered by modulation

The most common outputs are attenuation, differential phase, and dark-field. In speckle, MoBI, grating, and TAI systems, attenuation comes from the DC or mean-intensity term, refraction comes from local pattern displacement or local fringe phase, and dark-field comes from visibility loss or a diffusion coefficient. Directional dark-field generalizes this scalar dark-field by estimating a $v=d\,\alpha$7 or $v=d\,\alpha$8 scattering tensor whose eigenstructure describes anisotropy. In tensor tomography, the reconstructed voxel tensor is eigendecomposed to obtain mean scattering $v=d\,\alpha$9, fractional anisotropy, and the eigenvector associated with the smallest eigenvalue, which represents the preferential local fiber orientation. The laboratory random-mask implementation validated DDF orientation against SAXS at two positions with a maximum discrepancy of 2 degrees (Lautizi et al., 2024, Magnin et al., 2023).

Bragg modulation extends the contrast space to crystalline defects and strain. Because $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$0, the displacement field along $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$1 is $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$2, and local strain follows from spatial derivatives. In BCMI simulations of a nanoindented Ni crystal embedded in a larger film, dislocations appeared as phase vortices with $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$3 winding, while phase gradients encoded strain around the defect field. This is a qualitatively different use of modulation: the mask is not used to recover refractive phase but to lift ambiguities in a Bragg diffraction phase problem while retaining defect sensitivity (Ulvestad et al., 2018).

Spectral modulation provides material-specific rather than purely wavefront-specific contrast. SMFFS produces multi-energy blended data and virtual monochromatic images by estimating basis-material coefficients under scatter similarity constraints; SSQI uses a primary modulator plus a dual-layer detector to recover two material-specific images and two scatter images in a single exposure. These methods address beam hardening and scatter jointly, rather than treating phase and dark-field as the primary outputs (Deng et al., 2022, Shi et al., 2022).

Single-pixel modulation accesses yet another regime, in which the desired image is the object transmission map or source-intensity distribution rather than a local wavefront quantity. Tabletop X-ray ghost video reconstructs moving objects from random binary bucket measurements, while rotational CGI reconstructs an $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$4 image using only $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$5 single-stripe masks rotated over $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$6 angles. In hard-X-ray and soft-gamma astronomy, rotating modulation reconstructs sky brightness from time-folded detector count profiles rather than from detector pixels (Zeng et al., 2 Jan 2026, Zhou et al., 2023, Budden et al., 2010, Huo et al., 2015).

5. Representative performance regimes

The reported operating envelope spans synchrotron diffraction microscopy, full-field multimodal CT, laboratory phase contrast, quantitative spectral CBCT, and single-pixel video. In BCMI, the modulator-recovery experiment used a standard BCDI rocking curve with 121 angular slices over $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$7 and acquisition of about 10 minutes, whereas single-view BCMI avoided rocking curves and overlapping scans for the targeted 2D projection and improved temporal resolution to the exposure time of one pattern, typically ~0.5 s; the reported reconstructions used about 500–1000 iterations and showed quantitative agreement at the array center (Ulvestad et al., 2018). Interlaced XSVT reduced sample exposures by a factor 5, from 36,000 to 7,200 for $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$8 projections, while the mixed single-image method used 1,800 sample images; the measured standard deviation of blank wavefront-gradient maps was ~0.35 $DGF(x,y)=-\ln(V/V_0)=\int \Sigma(x,y,z)\,dz$9rad for interlaced XSVT versus ~0.75 $$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$0rad for the mixed method (Berujon et al., 2015).

High-resolution structured-illumination systems reached much finer scales. Two-dimensional TAIs resolved features near 2 $$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$1m in projection imaging and reached ~3 $$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$2m resolution in phase-contrast CT of an unstained murine artery, with bidirectional differential-phase sensitivities $$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$3 nrad and $$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$4 nrad and an estimated dose-efficiency improvement by a factor of 6 relative to a P1000 diffuser (Gustschin et al., 2021). A waveguide-plus-TAI-plus-photon-counting configuration reported measured peak visibilities of 94.8% for CMOS and 93.4% for photon counting, detector QE $$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$5 at 8 keV, and angular sensitivity 1.55 nrad versus 2.15 nrad for CMOS (John et al., 31 Oct 2025). On a low-coherence laboratory Xeuss 3.0 system, ten reference/sample pairs at $$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$6 m were sufficient to retrieve absorption, phase, DF, and DDF with orientation validation against SAXS at 2-degree maximum discrepancy (Magnin et al., 2023).

Spectral and quantitative radiographic systems demonstrate a different balance of accuracy and hardware simplicity. For the anthropomorphic Kyoto chest phantom, SMFFS produced a VMI RMSE of 11.8 HU at 70 keV and non-uniformity 14.1 HU, compared with 14.5 HU and 59.4 HU for DKV-CB with scatter correction and 437.6 HU and 184.0 HU without scatter correction (Deng et al., 2022). SSQI reported RMSE 0.13 cm for acrylic and 0.04 mm for copper in slab validation, and reduced RMSE in material-specific images by 38% to 92% in an anthropomorphic chest phantom (Shi et al., 2022).

Single-pixel temporal modulation reached video rates rather than tomographic precision. Tabletop X-ray ghost video demonstrated 200 fps imaging at 225 $$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$7m resolution over a $$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$8 mm$$\frac{1}{2}\,\bm{k_\perp}\Sigma_y\bm{k_\perp}+\mu(y)\Delta y=-\ln\!\big(|\hat I_s(\bm{k_\perp})|/|\hat I_0(\bm{k_\perp})|\big)$$9 field of view, using 1024 random binary patterns per revolution and a continuous 1 MHz single-pixel readout; clear images were recoverable with only 25% or even 12.5% of the bucket data, implying at least an 8× speedup under higher flux and faster detection (Zeng et al., 2 Jan 2026). In rotating hard-X-ray modulation, analytic characteristic profiles required 0.8 s with the standard formula, 2.6 s with the advanced formula without $O(r)=A(r)e^{i\phi(r)}$0, and 5.4 s with $O(r)=A(r)e^{i\phi(r)}$1, whereas Monte Carlo-derived profiles with per-bin SNR $O(r)=A(r)e^{i\phi(r)}$2 required $O(r)=A(r)e^{i\phi(r)}$3 days per detector; NCAR achieved $O(r)=A(r)e^{i\phi(r)}$4 resolution in a laboratory prototype whose nominal geometric limit was about $O(r)=A(r)e^{i\phi(r)}$5 (Budden et al., 2010). For HXMT imaging observations, direct demodulation with 1-fold cross correlation yielded detection efficiencies of about 41% at 1 mCrab and about 92% at 2 mCrab, and was recommended as the default regularization for faint-source detection (Huo et al., 2015).

6. Limitations, optimization, and outlook

The principal limitations are modality specific but structurally similar. BCMI requires a known or calibrated modulator, accurate Fresnel/Fourier propagation, and sampling grids that avoid aliasing; phase modulation is critical, amplitude-only modulators failed to converge, and reconstruction quality degrades outside the probe footprint (Ulvestad et al., 2018). Speckle-based and MoBI methods require sufficient pattern visibility, repeatable membrane positions, and stable mechanical conditions; strong scatterers can challenge single-image dark-field retrieval, and random-mask laboratory systems depend on matching modulation size to detector PSF and geometry (Berujon et al., 2015, Magnin et al., 2023). SMFFS depends on the scatter-similarity hypothesis for millimeter-scale focal-spot deflections, while SSQI assumes scatter is low-frequency and approximately invariant over neighboring PM cells (Deng et al., 2022, Shi et al., 2022).

A recurrent theme is that simpler hardware moves difficulty into calibration, motion design, and inverse algorithms. In single-mask laboratory MoBI, the topological nature of the membrane strongly affects image quality: a spiral topology “seems to be an optimum both in terms of resolution and contrast-to-noise ratio compared to random and regular patterns,” with best performance when the modulation peak-to-peak distance is approximately 3–6 pixels (Magnin et al., 14 Apr 2025). Membrane stepping itself is also a design variable rather than a neutral acquisition choice: optimized movement strategies based on global or local standard deviation improved contrast-to-noise ratio and reduced angular sensitivity relative to regular or random stepping, and honeycomb membranes showed the highest compatibility with the optimization procedure (Perion et al., 18 Nov 2025). In tensor tomography, the analysis window is an explicit mathematical lever that trades spatial resolution against sensitivity (Lautizi et al., 2024).

The applications stated in the cited papers are correspondingly broad. They include fast, local imaging of strain and dislocation dynamics in extended crystals under operando or reactive environments, multimodal CT from absorption, phase, and dark-field projections, quantitative CBCT for image-guided therapy and interventional radiology, materials and fibrous-tissue tensor tomography, real-time and dynamic quantitative imaging, high-speed inspection of moving objects, and hard-X-ray all-sky or pointed imaging (Ulvestad et al., 2018, Berujon et al., 2015, Deng et al., 2022, Lautizi et al., 2024, Zeng et al., 2 Jan 2026, Huo et al., 2015). This suggests that “modulation-based X-ray imaging” is best understood not as a single instrument class, but as an inversion framework in which a designed encoding makes otherwise inaccessible phase, strain, scattering, spectral, or temporal information observable on detectors that need not be intrinsically energy resolving, phase sensitive, or pixellated in the conventional sense.