Projection-Based Nano- and Microtomography

• Projection-based nano- and microtomography is a 3D imaging technique that reconstructs internal structures from multiple 2D projections acquired via X-ray or electron imaging.
• It employs advanced reconstruction methods such as filtered backprojection, BST, and machine learning-enhanced algorithms to optimize resolution and suppress imaging artifacts.
• The approach is widely applied in materials science, bioimaging, and energy storage while addressing challenges like limited angular data and nonrigid sample deformations.

Projection-based nano- and microtomography encompasses a suite of computational and experimental methodologies for the three-dimensional imaging of specimens at nanometer to micrometer scale using projection data, typically obtained via X-ray or electron imaging. These approaches transform a set of 2D projection images, acquired at various angles, into a volumetric representation that enables quantitative analysis of internal structures and materials properties. The field has seen rapid development driven by advances in instrumental resolution, detector technologies, computational reconstruction algorithms, and integration of data-driven machine learning.

1. Fundamentals of Projection-Based Nano- and Microtomography

The core principle of projection-based tomography is the acquisition of a series of two-dimensional X-ray (or other radiation) projections as the specimen is rotated over a defined angular range. The projection data are then inverted—most commonly using variants of the Radon transform—to yield a three-dimensional reconstruction of the object's internal density, phase, or composition profile.

Two major spatial regimes are encountered:

• Microtomography typically refers to imaging with voxel sizes from several microns down to ~0.5 μm, suitable for resolving cellular structures, composite materials, and engineered microstructures.
• Nanotomography pushes the spatial resolution further, achieving voxel sizes on the order of hundreds of nanometers or less, allowing for the interrogation of materials at near-subcellular or subgrain scales (Weitkamp et al., 2020, Kovyakh et al., 2021).

Key parameters dictating capability include (1) the spatial and angular sampling density, (2) instrumental aberrations and mechanical precision, (3) contrast modality (absorption, phase, fluorescence), and (4) the mathematical/algorithmic properties of the inversion method.

2. Image Acquisition: Instrumentation, Geometries, and Beamline Configuration

Projection-based nano- and microtomography requires high-stability sample stages, coherent or partially coherent X-ray sources, and detectors tailored for high dynamic range and low noise. Multiple system architectures exist:

• Parallel-beam setups (common in synchrotron beamlines) achieve full-field imaging for larger samples, with tradeoffs in attainable magnification and resolution.
• Cone-beam and geometrically magnified laboratory systems use micro- or nano-focus X-ray tubes with sample-proximal positioning to achieve higher effective pixel resolution (Graetz et al., 2020, Müller et al., 2021).
• Zone-plate or transmission X-ray microscope-based nano-CT allows for ~20 nm effective pixel resolution, as implemented at the ANATOMIX beamline (Weitkamp et al., 2020).

Beamline flexibility, such as variable energy monochromators, multi-scale optics, and large flat illumination fields, enables imaging of extended objects or performing rapid dynamic (in situ) measurements.

Key advances include variable beam sizes (up to 40 mm), rapid scan rotation stages (enabling up to 18 full tomographies per second), and automated data pipelines for high-throughput and real-time feedback (Weitkamp et al., 2020).

3. Calibration, Alignment, and Artifact Suppression

Precise specimen alignment is critical to prevent image degradation due to mechanical errors, miscalibration, and thermal drift. Several robust alignment pipelines are documented:

• Reference marker-based calibration: Small spherical gold particles (e.g., 0.5 μm diameter) provide high X-ray absorption and appear as fiducial markers in projections. Algorithmic localization (gray value barycenter, circle fitting) tracks their trajectory through the projection set, enabling sub-pixel realignment by shifting images so the marker occupies a consistent reference position (Wang et al., 2014).
• ROI and binary masking: Binary conversion and sequential multiplication of segmented projections highlight the marker's path, facilitating robust centroid or morphology-based localization.
• Custom alignment algorithms: Rather than shifting to a simple axis of symmetry, horizontal positions for alignment may follow a cosine law:

$X(k) = D - R\cos\left(\frac{(k-1)\pi}{N}\right)$

where $D$ is image width, $R$ is marker-to-axis distance, $N$ is projection count, and $k$ is projection index, minimizing total pixel shifts and preserving fidelity (Wang et al., 2014).

• Manual refinement: Modules for stepwise corrections are used in cases where algorithmic detection is insufficient.

Error sources such as mechanical stage backlash and thermal expansion are explicitly corrected in these workflows, with direct evidence of reduction in blur and streaking artifacts and preservation of true internal structures following alignment, as demonstrated at the Beijing Synchrotron Radiation Facility (Wang et al., 2014), and further enhanced by distributed optimization frameworks that incorporate dense optical flow registration (Farneback’s algorithm) to jointly align and unwarp projections and reconstruct objects robustly even under nonrigid motion (Nikitin et al., 2020).

4. Reconstruction Algorithms: Analytical, Fast, and Model-Driven Strategies

Transforming projection data into volumetric images relies on solving inverse problems.

• Inverse Radon/Filtered Backprojection (FBP): The "workhorse" for most systems, notably utilized in LabVIEW-MATLAB integrated platforms where MATLAB's inverse Radon transform is invoked with filter and interpolation schemes (Ram-Lak, Shepp-Logan, Cosine) (Wang et al., 2014).
• Fast backprojection via Backprojection Slice Theorem (BST): The BST formalism simplifies backprojection to a frequency-domain calculation,

$$\widehat{B g}(\sigma \xi_\theta) = \frac{\widehat{g}(\sigma, \theta)}{\sigma}$$

for each radial frequency $\sigma$ and angle $\theta$, allowing high-efficiency $O(N^2\log N)$ implementations, particularly beneficial for datasets with thousands of high-resolution projections (Koshev et al., 2016).

• Log-polar and NFFT methods: Offer alternatives for fast inversion but can incur errors or instability in low-frequency regions. BST circumvents this by direct 1D FFT domain processing for each projection.
• Hybrid machine learning-physics methods: Recent engines like Perception Fused Iterative Tomography Reconstruction Engine (PFITRE) deploy machine-learned priors (e.g., with U-net-like CNNs trained to remove artifacts and hallucinate missing information), embedded in physics-constrained iterative solvers (ADMM plug-and-play), to mitigate missing wedge and sparse data artifacts, yielding physically consistent and visually enhanced reconstructions under extreme data limitations (Zhao et al., 25 Mar 2025).
• Model-driven CT algorithms for phase-contrast nano-tomography: Innovations in projections acquired via Lau interferometers and zone plates are challenged by signal splitting. These effects are mathematically modeled as convolution by a difference kernel (matrix $B$), with an analytical inversion step to restore "unsplit" phase projections followed by penalized weighted least square total variation (PWLS-TV) denoising, enabling accurate reconstruction of low-density/phase samples such as plastic fibers and resolving edge features with high SNR (Cai et al., 2023).

5. Spatial Resolution and Quantitative Characterization

Resolving fine structures is dependent on the modulation transfer function (MTF) and system point spread function (PSF):

• Quantification by test objects: Focused ion-beam milled test objects with patterned sub-micron features are used to empirically determine MTF. In-plane and through-plane resolution can reach 0.8 μm after zoom reconstruction, matching the optics limit, as determined by the 5% MTF crossing criterion (Mizutani et al., 2016).
• Star pattern and edge test objects: For laboratory-based nano-CT, Siemens star test patterns are used for MTF computation and PSF estimation. Effective system PSF may be described by a sum of Gaussian contributions (e.g., 250 nm FWHM for high-resolution, ~1.1 μm for low-frequency), with the system achieving 150 nm spatial resolution as supported by both frequency domain and physical edge-scan validation (Graetz et al., 2020, Müller et al., 2021).
• Fourier Shell Correlation (FSC): For 3D resolution validation, FSC between paired reconstructions demarcates the frequency where the correlation falls below a threshold (1/2-bit), empirically confirming volumetric resolutions of 170–185 nm for integrated circuit and composite samples (Müller et al., 2021).

6. Practical Applications and Imaging Modalities

Projection-based nano- and microtomography drives advances across numerous fields:

Application Area	Example/Impact	Reference
Materials Science	Metallization in ICs; electrode aging; composite fillers distribution	(Müller et al., 2021)
Life Sciences/Bioimaging	Cerebral tissue analysis with sub-cellular resolution	(Mizutani et al., 2016)
Cultural Heritage	Microstructure and water ingress in ancient Roman concrete	(Xu et al., 2020)
Energy Storage	3D mapping of lithium battery electrode degradation	(Müller et al., 2021)
Metamaterials Testing	In situ loading and reconstruction of polymeric lattices	(Debastiani et al., 2023)
Nano-structure Analysis	PDF-based mapping of 2D combinatorial arrays in thin films	(Kovyakh et al., 2021)

Advanced contrasts including phase-contrast (often with zone-plate or Lau interferometer) extend the field to weakly absorbing materials. Neutron radiography and machine learning-driven regularizers further extend capability for correlating porosity, fracture, and absorption dynamics.

7. Current Challenges and Research Directions

Key challenges and innovative approaches include:

• Limited angular data (missing wedge): Mechanical constraints (e.g., in in situ devices) lead to incomplete angular coverage, severely degrading standard filtered backprojection. Sample rotation (by 90°, for example) to optimize the angular illumination coverage, and the application of deep learning–regularized reconstructions (PFITRE) effectively recover missing data regions and dramatically reduce artifacts even with angular losses exceeding 100° (Debastiani et al., 2023, Zhao et al., 25 Mar 2025).
• Nonrigid and time-dependent sample deformation: Distributed optimization—integrating dense optical flow via Farneback’s algorithm within ADMM—enables recovery of sharp object boundaries despite projection misalignment and warping (Nikitin et al., 2020).
• Automation and user workflow: Platforms with graphical user interfaces (LabVIEW frontends), automated alignment schemes, and MATLAB-LabVIEW integration accelerate on-the-fly experimental handling (Wang et al., 2014).
• Feature-based trajectory refinement: Treating angle correction as a SLAM problem, with keypoint extraction (SURF, RANSAC), and factor-graph optimization, allows for real-time pose estimation and rotation axis correction, yielding substantial improvement in root-mean-square error and potential instrument cost reduction (Griguletskii et al., 2021).
• Integration of mathematical optimization and data-driven priors: Embedding a DNN trained for local edge detection into a mixed-integer optimization model (e.g., MIP) for global tomographic reconstruction encourages solutions with homogeneous material phases separated by crisp edges, substantially enhancing material contrast and boundary sharpness, outperforming conventional regularizers (e.g., total variation) in suppressing streak and blur artifacts (Mishra et al., 7 Sep 2025).
• Speed and scalability: Fast backprojection engines (BST) and GPU-accelerated iterative schemes (HolotomocuPy) achieve sub-minute reconstruction times for gigavoxel volumes, essential for high-throughput, in situ, or dynamic imaging (Koshev et al., 2016, Nikitin et al., 29 Jul 2024).

References to Key Methodological Papers

• Image alignment and LabVIEW workflow: (Wang et al., 2014)
• Fast backprojection and BST formalism: (Koshev et al., 2016)
• Experimental resolution validation and MTF: (Mizutani et al., 2016, Graetz et al., 2020, Müller et al., 2021)
• Advanced machine learning methods for limited angle problems: (Zhao et al., 25 Mar 2025)
• DNN+MIP edge-aware optimization: (Mishra et al., 7 Sep 2025)
• Nonrigid alignment and optical flow correction: (Nikitin et al., 2020)
• In situ missing wedge strategies and experimental protocols: (Debastiani et al., 2023)
• Holotomography probe retrieval and GPU acceleration: (Nikitin et al., 29 Jul 2024)
• SLAM-inspired rotation refinement: (Griguletskii et al., 2021)

The synthesis of algorithmic advances, high-throughput instrumentation, and application-driven imaging protocols continues to expand the reach and reliability of projection-based nano- and microtomography within the physical, life, and engineering sciences.