Propagation-Based Phase-Contrast Tomography

• Propagation-based phase-contrast tomography is a method that converts phase shifts into measurable intensity variations to reconstruct 3D distributions of refractive and absorptive properties.
• It employs forward models such as TIE, CTF, and multislice approaches to overcome challenges of weak attenuation contrast, noise, and resolution limitations.
• The technique is pivotal for low-dose biomedical imaging and industrial applications, offering significant improvements in SNR, contrast, and spatial resolution.

Propagation-based phase-contrast tomography is a tomographic imaging modality in which free-space propagation converts object-imposed phase shifts into measurable intensity modulations, after which phase retrieval and tomographic reconstruction are used to recover three-dimensional distributions of refractive and absorptive properties. In the X-ray setting it is formulated in the near-field Fresnel regime and is commonly discussed under the projection and paraxial approximations; in contemporary practice it spans single-distance homogeneous Transport of Intensity Equation retrieval, contrast-transfer-function formalisms, nonlinear Fresnel-model inversion, and post-reconstruction three-dimensional filtering. The method is used when attenuation contrast is insufficient, when dose or flux is limiting, and when dynamic, multi-material, or high-resolution imaging makes the choice of forward model and reconstruction strategy decisive (Thompson et al., 2022, Jadick et al., 17 Aug 2025).

1. Physical basis and forward models

The basic material model in propagation-based X-ray phase-contrast imaging uses the complex refractive index

$$n_E(\vec{r}) = 1 - \delta_E(\vec{r}) + i\beta_E(\vec{r}),$$

where $\delta_E$ is the refractive component and $\beta_E$ is the absorptive component. Under the projection approximation, refractive effects within the object are neglected, so the exit wavefield is written as

$$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$

This reduces a three-dimensional scattering problem to a set of two-dimensional projections and underlies much of conventional propagation-based CT processing. Its validity is commonly assessed by the Fresnel number

$N_F = \frac{\Delta^2}{\lambda z_0},$

with $N_F \gg 1$ indicating validity; thick objects, high detector resolution, and lower X-ray energies drive the approximation toward breakdown (Jadick et al., 17 Aug 2025).

In free-space propagation phase-contrast imaging, intensity in the detector plane may be modeled by the Transport of Intensity Equation in the near field as

$$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$

For laboratory systems, finite source size and detector resolution enter through an effective point spread function. One expression given for the system blur is

$$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$

with corresponding optimal magnification

$M_{\text{opt}} = 1 + \frac{\sigma_d}{\sigma_s}.$

Because this geometry uses no optical elements, it makes full use of source flux and is consequently attractive for conventional laboratory sources as well as continuous and time-resolved imaging (Allan et al., 12 Apr 2025).

When the projection approximation is not adequate, the multislice model divides the object into thin slices, modulates the wavefield slice by slice, and propagates between slices with Fresnel diffraction. In Fourier space the propagation kernel is

$$H_R(\nu_x, \nu_y) = \exp\left[ -\frac{i \pi h c }{E} R (\nu_x^2 + \nu_y^2) \right].$$

This explicitly captures within-object refraction and becomes relevant for sub-micron detector resolution or relatively thick samples, where projection-approximate and multislice reconstructions diverge most strongly around fine structures and edges (Jadick et al., 17 Aug 2025).

2. Phase-retrieval formalisms

The dominant classical phase-retrieval approach in propagation-based tomography is Paganin’s method, also denoted TIE-Hom. It assumes a monomorphous object with constant $\delta_E$0, a single propagation distance, and near-field propagation. In one formulation,

$\delta_E$1

and reconstruction applies the inverse operator

$\delta_E$2

The standard choice is $\delta_E$3, but the system PSF already performs some phase retrieval, so an optimized choice is reduced to

$\delta_E$4

This explicitly encodes the core PB-CT trade-off: strong SNR gain through suppression of high-frequency noise at the cost of some edge blurring (Gureyev et al., 25 Jun 2025, Gureyev et al., 12 Jan 2026).

Several extensions retain the single-distance workflow while modifying the spectral response. A discrete reformulation of phase retrieval replaces the continuous Laplacian by its discrete Fourier representation,

$\delta_E$5

leading to a generalized Paganin method that preserves more high spatial frequency information on pixelated data. Closely related work using periodic boundary conditions derives a Fourier-space filter that is less suppressive at high spatial frequencies than the original Lorentzian and improves the reconstructed contrast of very fine sample features. Experimental CT measurements reported spatial-resolution improvements of up to $\delta_E$6 when the detector PSF is effectively a single pixel (Pollock et al., 2021, Paganin et al., 2020).

A separate line of development replaces two-dimensional projection-space phase retrieval by three-dimensional post-reconstruction filtering. In the CTF formulation, the central identity

$\delta_E$7

permits phase retrieval to be applied directly to the reconstructed volume. For monomorphous samples, the 3D CTF-Hom relation is

$\delta_E$8

Simulations showed fast and stable post-reconstruction CTF retrieval producing results equivalent to conventional pre-reconstruction 2D CTF phase retrieval, with the additional practical benefit of highly localized application to isolated regions of interest (Thompson et al., 2022).

Nonlinear methods abandon the linearization assumptions of TIE-Hom and CTF. A maximum-likelihood non-linear phase-retrieval framework uses the Fresnel forward model

$\delta_E$9

and solves

$\beta_E$0

optionally with explicit material constraints via

$\beta_E$1

This avoids the weak-absorption, single-material, or small-distance approximations of linear retrieval and was reported to reduce blur, artifacts, and quantitative inaccuracies relative to linear baselines (Mohan et al., 2023).

At long propagation distances and in the presence of strong phase gradients, even linearized single-distance filters may fail. Eikonal phase retrieval instead models photon transport through gradient-induced displacements,

$\beta_E$2

and reconstructs by iterative inversion of an eikonal forward model. This was introduced specifically to exploit fourth-generation synchrotron conditions, where long sample-to-detector distances reveal low-contrast features but also break the small-gradient assumptions underlying Paganin-type filters (Mirone et al., 2023).

Learning-based retrieval constitutes a further departure from analytic inversion. Under laboratory conditions with partial spatial coherence, polychromaticity, and high noise, an end-to-end U-Net-based method was trained on simulated data using physically modeled spectra, blur, and noise, then tested on experimental measurements. The method was reported to be stable under the studied variations, though its successful deployment depended on preprocessing choices, network training considerations, and system modeling (Deshpande et al., 2022).

3. Tomographic inversion and reconstruction architectures

The simplest PB-CT pipeline performs phase retrieval on each projection and then reconstructs with filtered back-projection. In many cases this remains the default because FBP assumes that the input sinograms consist of line integrals. However, when the forward model assumptions are violated, the resulting reconstructions can inherit propagation artifacts and quantitative errors, which motivates coupled or iterative alternatives (Jadick et al., 17 Aug 2025).

One important post-reconstruction alternative is three-dimensional TIE-Hom filtering on the reconstructed volume. In this setting the filter in Fourier space is

$\beta_E$3

and it can be applied only to a localized region of interest. This reduces the need for repeated global 2D filtering and multiple CT reconstructions in multi-material objects, and simulation results indicated a modest improvement in noise suppression over pre-reconstruction 2D TIE-Hom while potentially yielding large computational gains (Thompson et al., 2018).

Iterative reconstruction becomes central when data are noisy, undersampled, or dynamic. In laboratory dynamic micro-CT, the inverse problem is posed as $\beta_E$4, and the reconstruction is regularized by a structure-based prior from a high-quality reference scan: $\beta_E$5 with

$\beta_E$6

The optimization is solved with the Primal-Dual Hybrid Gradient algorithm, and the prior exploits mutual information by aligning image gradients rather than pixel intensities (Allan et al., 12 Apr 2025).

A more general all-at-once formulation is given by regularized Newton methods. For near-field phase contrast the forward model is

$\beta_E$7

and the iteratively regularized Gauss-Newton step solves

$\beta_E$8

The same framework extends to all-at-once tomography,

$\beta_E$9

so that phase retrieval and tomographic inversion are carried out simultaneously rather than sequentially (Maretzke et al., 2015).

Material-resolved inversion can also be fully coupled to the propagation model. In dual-energy propagation-based phase-contrast CT, a one-step iterative material-decomposition method reconstructs basis-material concentrations directly from intensities at different energies using Fresnel diffraction in the forward model. This avoids the two-step separation of phase retrieval, reconstruction, and material decomposition and was reported to be superior in accurate material decomposition and noise reduction (Liao et al., 2023).

Multi-material robustness has additionally been addressed with non-iterative compositing in the reconstructed volume. A 3D masked phase-retrieval method first segments high-$$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$0 regions, applies strong 3D phase retrieval to the low-$$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$1 masked volume, and then reinserts interface-tuned reconstructions at the boundaries. This was proposed specifically to avoid the over-blurring that occurs when a single phase-retrieval filter is tuned to one boundary in a multi-material object (Pollock et al., 2023).

4. Image quality, resolution, and dose

The quantitative appeal of propagation-based tomography is usually framed through signal-to-noise, contrast, spatial resolution, and dose. One theoretical relation used for PB-CT is the noise-resolution uncertainty relation

$$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$2

with $$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$3 in tomography. For PB-CT of breast tissue using TIE-Hom retrieval, intrinsic and biomedical image-quality characteristics were introduced,

$$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$4

and experiments on full human mastectomy samples found 2D SNR gains of $$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$5 for an energy-integrating detector and $$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$6 for a photon-counting detector, while 3D tomographic SNR gains were $$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$7 and $$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$8, respectively (Gureyev et al., 25 Jun 2025).

Laboratory dynamic microtomography provides a concrete demonstration of compounded gains from propagation contrast and iterative reconstruction. In a laboratory-based multi-contrast micro-CT system using a Rigaku MicroMax 007-HF source, analytic attenuation tomography gave $$\psi_{z_0} = \psi_0 \cdot \exp\left[ -k \int_0^{z_0} (\beta(\vec{r}) + i\,\delta(\vec{r}))\,dz \right].$$9, analytic phase-retrieved tomography gave $N_F = \frac{\Delta^2}{\lambda z_0},$0, and regularized phase-retrieved tomography gave $N_F = \frac{\Delta^2}{\lambda z_0},$1. These values correspond to a $N_F = \frac{\Delta^2}{\lambda z_0},$2 improvement from free-space propagation phase contrast alone and a $N_F = \frac{\Delta^2}{\lambda z_0},$3 improvement relative to the conventional reconstruction when combined with structure-based prior regularization. The same study reported fully dynamic imaging with temporal resolution of $N_F = \frac{\Delta^2}{\lambda z_0},$4 at a voxel size of $N_F = \frac{\Delta^2}{\lambda z_0},$5, static-region spatial resolution of $N_F = \frac{\Delta^2}{\lambda z_0},$6 FWHM, and up to $N_F = \frac{\Delta^2}{\lambda z_0},$7 degradation in spatial resolution at rapidly changing features (Allan et al., 12 Apr 2025).

For large biological specimens, low-dose operation is a major driver. In infant-sized lungs imaged at the Australian Synchrotron with a DECTRIS Eiger 2M-W photon-counting detector, image quality normalized against dose was optimized as a function of energy and propagation distance. The optimal propagation distance was $N_F = \frac{\Delta^2}{\lambda z_0},$8 meters, the intrinsic image quality was maximal at $N_F = \frac{\Delta^2}{\lambda z_0},$9 keV, and the lowest tested mean absorbed dose was $N_F \gg 1$0 mGy, corresponding to an effective dose of approximately $N_F \gg 1$1. At a voxel size of $N_F \gg 1$2, phase retrieval enabled clear visualization of minor lung airways at doses up to $N_F \gg 1$3 times lower than conventional CT reconstruction, with Fourier Ring Correlation limits of $N_F \gg 1$4 in the best case and $N_F \gg 1$5 at $N_F \gg 1$6 mGy (Pollock et al., 2024).

The same logic extends beyond X-rays. Propagation-based neutron phase-contrast tomography using the ISIS pulsed spallation source reported an SNR boost of $N_F \gg 1$7 compared to attenuation-based tomography and an effective brilliance gain of $N_F \gg 1$8, implying over two orders of magnitude increase. The study emphasized that the gain may be spent either on reduced acquisition time or on improved contrast at fixed exposure (Paganin et al., 2019).

Image quality is not reducible to SNR alone. In proximity to highly attenuating objects, standard strong phase retrieval can greatly blur interfaces. A 3D masked phase-retrieval method was reported to provide a $N_F \gg 1$9-fold improvement in the SNR of rabbit brain tissue relative to standard phase retrieval, and in an aluminium-water phantom it provided a $$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$0-fold SNR boost while preserving the boundary resolution at $$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$1, compared to $$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$2 in conventional phase retrieval (Pollock et al., 2023).

Finally, the choice of discrete rather than continuous mathematics affects the measured resolution when high frequencies survive the detector PSF. Experimental CT validation showed that discrete phase retrieval can yield negligible improvement with broad indirect-detector PSFs, but up to $$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$3 improvement with direct photon-counting detectors characterized by a single-pixel PSF (Pollock et al., 2021).

5. Applications and domain extensions

Propagation-based phase-contrast tomography is particularly effective when attenuation contrast is weak, when interfaces are abundant, or when structural changes are localized. A representative laboratory demonstration measured the movement of a waterfront in the fine vessels of a birch wood skewer, using a high-quality reference scan followed by dynamic scans at $$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$4 with $$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$5 ms exposures and $$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$6 projections per reconstruction. The study identified broader application to fluid flow in porous media, vascular dynamics in plants or biological tissues, pharmacological studies of drug delivery, and any scenario where structural changes are local and the majority of features remain (Allan et al., 12 Apr 2025).

Biomedical imaging has been a major application domain. In breast tissue PB-CT, the quantitative framework of SNR, contrast, and dose was developed specifically for synchrotron imaging with prospective clinical transfer. A related line of work showed that dark-field signal extraction can be performed in propagation-based imaging from a single image in projection imaging and from one image per projection angle in tomography; a preliminary example demonstrated improved visualization of microcalcifications in propagation-based X-ray breast cancer imaging without increasing radiation dose (Gureyev et al., 25 Jun 2025, Gureyev et al., 2020).

Lung imaging is especially well matched to propagation-based methods because lung-air interfaces generate strong phase gradients. Low-dose, high-resolution CT of lamb lungs as a model for paediatric patients demonstrated superior resolution to existing high-resolution CT systems while remaining below current Australian guidelines for infant chest CT exposure of $$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$7 effective dose (Pollock et al., 2024).

Material-resolved imaging has expanded the scope of PB-CT. Dual-energy propagation-based phase-contrast CT has been formulated as a one-step iterative material-decomposition problem in which phase retrieval, reconstruction, and material decomposition are performed directly from intensity data at different energies, rather than as separate pre- or post-reconstruction stages (Liao et al., 2023). A complementary single-distance method for samples composed of unknown materials uses deliberately incorrect Paganin-type phase retrieval, then fits an error-function-based model to residual interface artifacts in the reconstructed volume to recover true $$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$8 and $$I(x,y,z_d) = \frac{I(x,y,z_o)}{M^2} \left(1 + \frac{R_2 \lambda}{2\pi M} \nabla_\perp^2 \phi(x,y;\lambda)\right).$$9 values for distinct materials. Applied to breast tissue, it recovered the refraction component $$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$0 with $$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$1 accuracy compared to theoretical values (Alloo et al., 2021).

Non-X-ray extensions are established rather than merely speculative. With polychromatic neutron beams, propagation-based phase retrieval significantly improved SNR in weakly absorbing samples, and in a bone sample the retrieved phase enabled separation of bone and $$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$2, which is important for in situ flow experiments. This exploits deuteration contrast without chemical contrast enhancement and positions neutron imaging as a complementary method to X-ray imaging of bone (Østergaard et al., 2022).

At high-brilliance fourth-generation synchrotron sources, the method has also been pushed toward long propagation distances in mixed-attenuation biological specimens. Eikonal phase retrieval was introduced for this regime specifically because strong phase gradients and low-contrast features coexist, and standard single-distance linearized filters can generate halos, streaks, and loss of quantitative accuracy (Mirone et al., 2023).

6. Mathematical guarantees, validity limits, and common misconceptions

A recurring misconception is that single-distance propagation-based phase retrieval is intrinsically non-unique for arbitrary complex objects. Under the projection and paraxial approximations, a uniqueness theorem proves injectivity for compactly supported objects in the near-field regime when a known, non-vanishing reference wave is present. The theorem applies to both the nonlinear forward operator and its linearization on any open detector set, and a tomographic corollary states that any compactly supported object $$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$3 with $$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$4 is uniquely determined by single-distance tomographic intensity data for any open set of rotation angles. For tomography, the no-phase-wrapping condition is essential because uniqueness of the logarithm otherwise fails (Maretzke, 2014).

A second misconception is that the projection approximation is harmless at arbitrarily high resolution. Simulated synchrotron micro-CT images of a zebrafish phantom at $$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$5 keV showed that at $$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$6 detector resolution with $$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$7, projection-approximate and multislice images were nearly identical after Paganin retrieval and FBP, whereas at $$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$8 with $$\sigma = \sqrt{\left(1 - \frac{1}{M}\right)^2 \sigma_s^2 + \frac{\sigma_d^2}{M^2}},$$9 the two forward models differed noticeably, especially around fine structures and edges. The practical conclusion was that phase retrieval for such high-resolution data likely requires more detailed forward modeling that avoids the projection approximation (Jadick et al., 17 Aug 2025).

A third misconception is that Paganin’s method is a parameter-free optimum. The method is broadly useful, but its standard parameter choice maximizes noise suppression at the cost of spatial resolution, and optimization depends on the system PSF and on the image-quality metric being targeted. Analytical expressions were derived for the regularization parameter, for deblurring by defocus, and for Tikhonov-regularized deconvolution of the PSF. The same study emphasized that no-reference metrics such as $M_{\text{opt}} = 1 + \frac{\sigma_d}{\sigma_s}.$0 and full-reference metrics such as RMSE and SSIM may favor different operating points, particularly when deconvolution boosts high frequencies (Gureyev et al., 12 Jan 2026).

Finally, tomography itself remains ill-posed even when phase retrieval is successful. In breast PB-CT, the biomedical image-quality characteristic remained well below one despite strong SNR gains, because CT reconstruction amplifies noise at high spatial frequencies. This places a structural limit on how much of the phase-contrast advantage can be translated into volumetric image quality and explains why current research repeatedly returns to coupled inversion, nonlinear forward models, and priors rather than relying on phase retrieval as an isolated preprocessing step (Gureyev et al., 25 Jun 2025).

In practice, propagation-based phase-contrast tomography is therefore not a single algorithm but a family of physically related inverse problems. The central technical question is not whether phase contrast is present, but which forward model, retrieval formalism, and reconstruction architecture remain valid for the object class, detector, coherence, dose regime, and temporal scale at hand.