---
title: Randomized HOSVD Approximations
url: https://www.emergentmind.com/topics/randomized-hosvd-approximations
type: topic
---

# Randomized HOSVD Approximations

Multislice Projection (MP) describes a class of forward models—primarily for electron and X-ray imaging—where a wavefield is propagated through a 3D object by recursively slicing the volume into thin slabs. Within each slab, the wave is modulated by local material properties (refractive index or potential), followed by free-space propagation to the next slice. Unlike the projection approximation, MP accurately accounts for intra-object refraction, diffraction, and multiple scattering, making it indispensable for high-resolution phase-contrast tomography, coherent diffractive imaging, and 4D-STEM ptychography. MP is also integral to contemporary inverse methods for quantitative tomography and volumetric imaging at nanometer and atomic scales.

## 1. Theoretical Formulation of the Multislice Model

The MP framework begins with the decomposition of a scattering object of total thickness $z_0$ (along the $z$-axis) into $N$ slices, each of thickness $\Delta z = z_0 / N$. Assume a coherent plane wave $\psi_0(x, y)$ with wavenumber $k_0 = 2\pi/\lambda$ is incident on the object.

For X-rays, the complex refractive index profile is written as:
$$ n(x, y, z) = 1 - \delta(x, y, z) + i\beta(x, y, z), $$
where $\delta$ and $\beta$ describe, respectively, the decrement from unity and the absorption index. Each $k$-th slice is characterized by spatial averages:
$$
\delta_k(x, y) = \frac{1}{\Delta z} \int_{z_k}^{z_{k+1}} \delta(x, y, z)\, dz, \quad
\beta_k(x, y) = \frac{1}{\Delta z} \int_{z_k}^{z_{k+1}} \beta(x, y, z)\, dz.
$$
The wavefunction is updated recursively:
$$
\Psi_{k+1}(x, y) = \mathrm{Prop}_{\Delta z}\left[\Psi_k(x, y) \times \exp\left(-i k_0 \delta_k(x, y)\Delta z - k_0\beta_k(x, y)\Delta z \right)\right],
$$
where $\mathrm{Prop}_{\Delta z}$ denotes free-space propagation over $\Delta z$ (paraxial/fresnel regime), typically implemented via convolution in real space or Fourier multiplication:
$$
H_{\Delta z}(\nu_x, \nu_y) = \exp{-i\pi\lambda\Delta z (\nu_x^2 + \nu_y^2)}.
$$
For electron imaging, the transmission in slice $j$ is
$$
t_j(x, y) = \exp \left[i \sigma V_j(x, y)\right],
$$
with $\sigma$ the electron–matter interaction parameter and $V_j(x, y)$ the projected electrostatic potential. Propagation steps use the equivalent Fresnel propagator or, for electrons, a parabolic approximation.

After $N$ slices, a final propagation yields the exit wave at the detector. The measured intensity is $I(x, y) = |\Psi_\text{det}(x, y)|^2$ [2508.12505][2407.18063][2205.03902].

## 2. Implementation, Pseudocode, and Numerical Aspects

An explicit pseudocode for the X-ray CT forward MP model is as follows [2508.12505]:
```python
# Given: δ(x,y,z), β(x,y,z), N, Δz, R, λ, k0, detector grid
Initialize Ψ ← 1(x, y)
for k in 0..N−1:
    δk ← (1/Δz) ∫_{z_k}^{z_{k+1}} δ(x, y, z) dz
    βk ← (1/Δz) ∫_{z_k}^{z_{k+1}} β(x, y, z) dz
    T ← exp[–i*k0*δk*Δz  –  k0*βk*Δz]
    Ψ ← Ψ * T
    Ψ̂ ← FFT{Ψ}
    Ψ̂ ← Ψ̂ * H_{Δz}
    Ψ ← IFFT{Ψ̂}
Ψ̂ ← FFT{Ψ}
Ψ̂ ← Ψ̂ * H_{R}
Ψ_det ← IFFT{Ψ̂}
I_det(x, y) ← |Ψ_det(x, y)|^2
```
For electron ptychography, a high-level pseudocode captures lateral probe scanning, iterative propagation through slices, direct simulation of diffraction patterns, and gradient-based updates for object and probe parameters [2407.18063][2205.03902].

MP’s computational cost scales as $O(N \times N_\text{pix} \log N_\text{pix})$ per projection due to repeated propagation, compared to a single step for the projection approximation.

## 3. Quantitative Metrics, Resolution, and Validity Criteria

Evaluation of MP's necessity is governed by the slice-to-slice (or overall) Fresnel number:
$$
N_F = \Delta^2 / (\lambda z_0),
$$
with $\Delta$ the lateral sampling (detector pixel size) and $z_0$ the object thickness. For $N_F \gg 1$, intra-object refraction is negligible and the projection approximation is valid; for $N_F \lesssim 1$, MP is required. Simulation results [2508.12505] show, for a 5-mm sample at 20 keV X-rays:
- $\Delta = 2\,\mu$m $\Rightarrow N_F \approx 13$, projection approximation holds.
- $\Delta = 0.5\,\mu$m $\Rightarrow N_F \approx 0.8$, substantial MP-projection differences arise, especially near fine structure.

Metrics in phase contrast CT and ptychography include absolute difference maps, sinogram intensity differences, RMSE in exit-wave intensity, and resolution figures such as depth full-width at 80% max (FW80M) and sub-angstrom lateral resolution [2508.12505][2407.18063].

## 4. Multislice Projection in Tomographic and Ptychographic Inversion

For inversion, MP couples naturally to maximum likelihood (ML), gradient-based, and convex optimization schemes. In electron ptychography, the data likelihood under Poisson statistics is
$$
C = \sum_r \sum_{q} \left[ |\Psi_\text{calc}(q; r)|^2 - I_\text{exp}(q; r)\ln|\Psi_\text{calc}(q; r)|^2 \right],
$$
where $\Psi_\text{calc}$ is the forward-simulated exit wave, and updates are computed via block-coordinate descent alternating between object slice potentials and probe functions [2407.18063].

“Layer-wise” and “sparse matrix decomposition” optimization strategies enable direct recovery of object slice transmissions, probe functions, or the entire thick-object transfer matrix [2205.03902]. MP models are also invertible via algebraic and iterative techniques, including amplitude-flow phase retrieval and proximal gradient descent, with numerical stability enhanced by depth-regularization in Fourier space [2407.18063][2205.03902].

## 5. Applications and Performance Benchmarks

MP is foundational in:
- Propagation-based phase-contrast micro-CT with high-resolution detectors, where it corrects for intra-object refraction and Fresnel-edge fringes [2508.12505].
- 4D-STEM ptychography for 3D atomic imaging, achieving experimental depth resolution $\sim$2.6 nm and sub-angstrom lateral resolution; single-atom sectioning is demonstrated via depth profiles and 3D Fourier analyses [2407.18063].
- Algorithmic advances in thick specimen inversion, where layer-wise and sparse-matrix approaches allow robust recovery at lower acceleration voltages and thicker samples, with simulation results confirming superior atomic-plane separation and stable probe/object recovery in MoS$_2$, SrTiO$_3$, and GaAs [2205.03902].
- Multi-projection X-ray imaging (XMPI), which enables volumetric single-shot readouts without mechanical rotation, reconstructing 3D structure from simultaneous projections via inverse Radon transformation [1808.05434].

## 6. Practical Guidelines and Limitations

The necessity for MP over the projection approximation is dictated by the lateral pixel size, object thickness, and X-ray/electron wavelength:
- For $N_F > 5$–$10$, the projection approximation suffices in CT applications.
- For sub-micron or atomic-scale detectors and/or thick samples ($N_F \lesssim 1$), MP is required to prevent artifacts and loss of resolution [2508.12505][2407.18063].
- In ptychography, recovery of slice structure is enhanced at lower electron energies (longer wavelengths), where Fresnel propagation phase is more pronounced; ambiguity increases at high energies due to near-identity of the propagator [2205.03902].

Computational cost and memory constraints remain limiting factors for large-volume or high-angle datasets, with GPU-acceleration or block-wise reconstruction being practical requirements for contemporary experiments [2407.18063].

## 7. Summary Table: MP Implementation in Contemporary Modalities

| Imaging Modality         | Forward Model Type       | Application Domain                          |
|-------------------------|-------------------------|---------------------------------------------|
| X-ray phase-contrast CT | Refraction & attenuation, Fresnel propagation | Sub-micron phase-contrast tomography        |
| Electron ptychography   | Projected potential, Fresnel propagation      | 3D atomic imaging, defect localization      |
| XMPI (multi-projection) | Geometric Radon projections aggregated in single shot | Single-shot volumetric X-ray imaging |

MP is a critical theoretical and practical tool underpinning high-fidelity 3D reconstruction in modern wave-based imaging, bridging phase-contrast CT, electron ptychography, and single-shot tomographic methods across photon and electron modalities [2508.12505][2407.18063][2205.03902][1808.05434].

Source: https://www.emergentmind.com/topics/randomized-hosvd-approximations