Randomized HOSVD Approximations

• Randomized HOSVD Approximations are advanced techniques for tensor decomposition that reduce computational complexity while maintaining accuracy.
• The methods leverage random sampling to approximate high-order singular value decompositions, speeding up analyses for large-scale data.
• These approximations are practical for signal processing, image analysis, and big data applications, offering scalable solutions for complex datasets.

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:

$z$0

The wavefunction is updated recursively:

$z$1

where $z$2 denotes free-space propagation over $z$3 (paraxial/fresnel regime), typically implemented via convolution in real space or Fourier multiplication:

$z$4

For electron imaging, the transmission in slice $z$5 is

$z$6

with $z$7 the electron–matter interaction parameter and $z$8 the projected electrostatic potential. Propagation steps use the equivalent Fresnel propagator or, for electrons, a parabolic approximation.

After $z$9 slices, a final propagation yields the exit wave at the detector. The measured intensity is $N$0 (Jadick et al., 17 Aug 2025, Chen et al., 2024, Bangun et al., 2022).

2. Implementation, Pseudocode, and Numerical Aspects

An explicit pseudocode for the X-ray CT forward MP model is as follows (Jadick et al., 17 Aug 2025): $\Delta z = z_0 / N$9 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 (Chen et al., 2024, Bangun et al., 2022).

MP’s computational cost scales as $N$1 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$2

with $N$3 the lateral sampling (detector pixel size) and $N$4 the object thickness. For $N$5, intra-object refraction is negligible and the projection approximation is valid; for $N$6, MP is required. Simulation results (Jadick et al., 17 Aug 2025) show, for a 5-mm sample at 20 keV X-rays:

• $N$7m $N$8, projection approximation holds.
• $N$9m $\Delta z = z_0 / N$0, 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 (Jadick et al., 17 Aug 2025, Chen et al., 2024).

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

$\Delta z = z_0 / N$1

where $\Delta z = z_0 / N$2 is the forward-simulated exit wave, and updates are computed via block-coordinate descent alternating between object slice potentials and probe functions (Chen et al., 2024).

“Layer-wise” and “sparse matrix decomposition” optimization strategies enable direct recovery of object slice transmissions, probe functions, or the entire thick-object transfer matrix (Bangun et al., 2022). 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 (Chen et al., 2024, Bangun et al., 2022).

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 (Jadick et al., 17 Aug 2025).
• 4D-STEM ptychography for 3D atomic imaging, achieving experimental depth resolution $\Delta z = z_0 / N$32.6 nm and sub-angstrom lateral resolution; single-atom sectioning is demonstrated via depth profiles and 3D Fourier analyses (Chen et al., 2024).
• Algorithmic advances in thick specimen inversion, where layer-wise and sparse-matrix approaches allow robust recovery at lower acceleration voltages and thicker samples, with simulation results confirming superior atomic-plane separation and stable probe/object recovery in MoS$\Delta z = z_0 / N$4, SrTiO$\Delta z = z_0 / N$5, and GaAs (Bangun et al., 2022).
• 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 (Villanueva-Perez et al., 2018).

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 $\Delta z = z_0 / N$6–$\Delta z = z_0 / N$7, the projection approximation suffices in CT applications.
• For sub-micron or atomic-scale detectors and/or thick samples ($\Delta z = z_0 / N$8), MP is required to prevent artifacts and loss of resolution (Jadick et al., 17 Aug 2025, Chen et al., 2024).
• 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 (Bangun et al., 2022).

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 (Chen et al., 2024).

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 (Jadick et al., 17 Aug 2025, Chen et al., 2024, Bangun et al., 2022, Villanueva-Perez et al., 2018).