Multislice Projection (MP) Imaging

Updated 24 May 2026

Multislice Projection (MP) is a computational modeling technique that partitions a finite-thickness specimen into thin slices, allowing the sequential propagation of a coherent wave to encode 3D structural information.
The method accurately captures cumulative multiple scattering and intra-object refractive effects, enabling superior resolution in X-ray phase-contrast CT and electron ptychography compared to traditional single-projection models.
MP underpins advanced inverse imaging algorithms, supporting robust depth-resolved recovery of material properties and offering enhanced convergence and accuracy under challenging imaging conditions.

Multislice Projection (MP) is a computational modeling technique central to propagation-based imaging modalities, where a coherent wave probes a finite-thickness object and the object’s 3D structure is encoded in the transmission and propagation of the probe wave. MP is widely applied in X-ray phase-contrast computed tomography (CT) and transmission electron microscopy (TEM) ptychography, and is essential for accurate forward and inverse modeling when the projection approximation fails. The model explicitly accounts for cumulative multiple scattering and intra-object refractive effects by partitioning the specimen into thin slices, propagating the probe wave sequentially through each and updating the complex wavefield at each propagation step. This approach enables superior accuracy for high-resolution and thick-object imaging relative to single-step “projection approximation” models and underpins recent developments in atomic-resolution 3D imaging and single-shot volumetric reconstruction.

1. Mathematical Formulation of Multislice Projection

In MP, the incident wave (typically plane or convergent) interacts with an object subdivided into $N$ equi-thickness slices along the beam direction. For X-rays, the object is characterized by a spatially varying complex refractive index $n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ , while in electron microscopy, the interaction is via an electrostatic potential $V(x, y, z)$ . Within slice $k$ , the projected material property (e.g., mean $\delta_k(x, y)$ or $V_k(x, y)$ ) is held constant over the slice thickness $\Delta z$ .

At each slice interface, the propagation alternates between:

Multiplication by the slice transmission operator
- For X-rays: $T_k(x, y) = \exp[-i k_0 \delta_k(x, y) \Delta z - k_0 \beta_k(x, y) \Delta z]$
- For electrons: $t_k(x, y) = \exp[i \sigma V_k(x, y)]$
Free-space (Fresnel or paraxial) propagation over $\Delta z$ , implemented via Fourier methods using a transfer function $n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ 0:

$n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ 1

for X-rays and analogous forms for electrons.

The iterative update at each slice is:

$n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ 2

After $n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ 3 slices, the emergent probe is typically further propagated to the detector, where the intensity is $n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ 4 [ $n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ 5, $n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ 6].

2. Physical Basis and Applicability

The projection approximation neglects multiple and refractive scattering within the object, implicitly assuming local phase or amplitude attenuation without inter-slice coupling. This approximation fails when either (i) the sample thickness $n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ 7 becomes large compared to the coherence length or (ii) the detector resolution $n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ 8 is small enough that the slice-to-slice Fresnel number $n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z)$ 9 approaches unity or less. Under these conditions, the full MP model is required, as it accurately captures the cumulative intra-object phase effects, edge-diffracted Fresnel fringes, and inter-slice mixing that are essential at sub-micron detector resolutions or for thick, high-Z material specimens. The validation range can be assessed via benchmarking PA and MP image outputs as a function of $V(x, y, z)$ 0 and total sample thickness [ $V(x, y, z)$ 1].

3. Inverse Algorithms and Ptychographic Multislice Formulations

MP underpins advanced inverse imaging methods where depth-resolved material properties are recovered from measured intensity datasets. In 4D-STEM electron ptychography, this requires solving for the projected potential $V(x, y, z)$ 2 in each slab from probe-scanned diffraction intensities. Recent developments implement maximum-likelihood inversion (under Poisson noise) with gradient-based updates that exploit the recursive MP forward model:

Slice update (gradient descent): update $V(x, y, z)$ 3 proportional to the real part of the error backpropagated through the sliced model
Mixed-state modeling: updating multiple probe modes to account for partial coherence and instabilities
Depth regularization: Fourier filtering in $V(x, y, z)$ 4 suppresses spurious delocalized solution modes

Two algorithmic approaches are prominent: alternating block-coordinate minimization (layer-wise Amplitude Flow) and sparse-matrix decomposition, the latter enabling explicit diagonal decomposition of each slice transmission and improved unique recovery of atomic layer potentials, especially at lower electron accelerating voltages [ $V(x, y, z)$ 5, $V(x, y, z)$ 6].

4. Pseudocode and Implementation

Discrete MP modeling involves:

Initializing the incident wave (plane or focused probe)
Recursively for $V(x, y, z)$ $V (x, y, z)$ 7:
- Compute $V(x, y, z)$ 8, $V(x, y, z)$ 9 or $k$ 0 by integrating through slice $k$ 1
- Compute transmission $k$ 2 or $k$ 3
- Multiply incident wave by slice transmission
- Propagate wavefield over $k$ 4 using Fourier transform and apply transfer function
Final propagation from exit face to detector plane
Measure intensity $k$ 5

Example pseudocode traceable to multislice X-ray CT:

$V_k(x, y)$ 5 [ $k$ 6]

5. Quantitative Performance and Resolution Metrics

MP rigorously models wave coupling, enabling quantitative analysis of finite-depth and sub-nanometer features inaccessible to PA methods. In X-ray CT, performance is quantified by root-mean-square error (RMSE) and maximum absolute exit-wave differences between PA and MP, sinogram intensity deviation, and reconstructed fine-structure contrast. Systematic studies reveal that for Fresnel numbers $k$ 7, the PA produces artifact-laden reconstructions, while MP resolves refractive and edge features accurately.

In electron ptychography, MP enables both deep sub-Å lateral and nanometric axial resolution. For example, experimentally measured full-width at 80% maximum (FW80M) axial resolution for single-atom contrast in interstitial Tm was $k$ 82.7 nm, with lateral resolution $k$ 90.67 Å (Fourier analysis). Algorithmic convergence is robust, with cost functions typically dropping by $\delta_k(x, y)$ 0 over $\delta_k(x, y)$ 1 iterations, and visually resolved depth sectioning appears within $\delta_k(x, y)$ 2 iterations for realistic datasets [ $\delta_k(x, y)$ 3, $\delta_k(x, y)$ 4].

6. Practical Criteria for Model Selection and Limitations

Model selection is dictated by the interplay between detector pixel size, wavelength, and object thickness. The projection approximation is valid when $\delta_k(x, y)$ 5 (empirically, $\delta_k(x, y)$ 6), i.e., when the intra-object refractive coupling is weak. For high-resolution (sub-micron pixel, millimeter-scale sample) imaging or when single-distance phase retrieval is employed under conditions violating this criterion, the MP model is essential to avoid spurious edge fringes and mis-reconstructions.

Sampling requirements: MP accuracy demands $\delta_k(x, y)$ 7 to be small enough that the corresponding transfer function $\delta_k(x, y)$ 8 introduces appropriately sampled phases; undersampling leads to aliasing or numerical artifacts. Computational costs scale linearly with the number of slices $\delta_k(x, y)$ 9 per projection, typically 10–100 $V_k(x, y)$ 0 higher than PA models for thick objects.

7. Impact and Applications Across Modalities

MP is decisive for imaging scenarios that demand quantitative depth-resolved information, such as:

Propagation-based X-ray phase-contrast CT with sub-micron detectors, mesoscopic or thick biological samples, and multi-material specimens [ $V_k(x, y)$ 1]
Multislice electron ptychography, enabling direct 3D localization of individual impurity atoms within host lattices at nanometric depth precision and deep sub-Å in-plane sensitivity, as applied to garnet oxide heterostructures and simulated perovskites [ $V_k(x, y)$ 2, $V_k(x, y)$ 3]
Single-shot multi-projection hard X-ray imaging (XMPI), where multiple projections are recorded from a single XFEL pulse using Laue-diffracted beams, providing volumetric information of irreversible dynamical processes without sample rotation [ $V_k(x, y)$ 4]

The approach is generalizable to any paraxial coherent imaging modality where multiple-scattering, defocus, and refraction within finite-thickness specimens are non-negligible. For accelerating the MP calculations at scale, recent studies utilize GPU-accelerated pipelines, block-coordinate inversion, and proximal optimization exploiting the sparsity and diagonality in the sliced object model.