Magnetic X-ray Transform

• Magnetic X-ray transform is a mathematical framework that reconstructs 3D magnetization fields from XMCD measurements via line integrals.
• It utilizes linear mapping, regularized least squares, and iterative algorithms (e.g., SIRT) to accurately recover vector field components.
• The method drives advanced magnetic nanotomography by integrating precise experimental setups and robust computational pipelines.

The magnetic X-ray transform is a mathematical and experimental framework for reconstructing the three-dimensional vector magnetization field in magnetic materials, based on line integrals measured via X-ray Magnetic Circular Dichroism (XMCD) with polarized X-ray beams. It generalizes classical scalar tomography to vector field imaging, and is closely connected to the magnetic ray transform of tensor fields found in differential geometry and inverse problems. Magnetic X-ray transforms are foundational for synchrotron-based magnetic nanotomography, underpinning recent advances in vector tomography algorithms and nanomagnetism imaging (Banerjee et al., 2024, Zhou, 2016, Herguedas-Alonso et al., 21 Jan 2025).

1. Mathematical Formulation and Physical Model

The magnetic X-ray transform implements a linear mapping from the local magnetization field $M(\mathbf r) \in \mathbb{R}^3$ to dichroic projection data by way of XMCD. Given a sample exposed to left- ($C-$) and right- ($C+$) circularly polarized hard X-rays, the transmitted intensity for a ray $L$ through position $\mathbf r$ and direction $\hat{\mathbf k}(\theta, \phi)$ is: $$I^{\pm}(\theta, \phi) = I_0^{\pm} \exp\left[-\int_L\big(\mu_0(\mathbf r) \pm \kappa(\mathbf r)\,\hat{\mathbf k}(\theta, \phi) \cdot M(\mathbf r)\big)\,d\ell(\mathbf r)\right]$$ where $\mu_0$ is the nonmagnetic absorption, $\kappa$ is the dichroic coefficient, and incident fluxes $I_0^\pm$ are measured independently. The logarithmic absorbance difference isolates the magnetic signal,

$$D(\theta, \phi) = \ln[I_0^+/I^+(\theta, \phi)] - \ln[I_0^-/I^-(\theta, \phi)] = 2\int_L \kappa(\mathbf r)\,[\hat{\mathbf k}(\theta, \phi)\cdot M(\mathbf r)]\,d\ell$$

This linear transform—here called the Magnetic X-ray Transform (MXT)—is central to vector-field tomography. The measured dichroic projections at multiple tilt angles build up a mildly overdetermined system for the components of $M(\mathbf r)$ (Banerjee et al., 2024). In practical implementations, the forward operator discretizes as a matrix mapping voxel-wise magnetization components to projection data, with weights assigned by beam geometry and attenuation.

2. Inverse Problem: Vector Tomography and Reconstruction

The inversion of the magnetic X-ray transform aims to recover $M(\mathbf r)$ on a three-dimensional grid from dichroic projections acquired at sequential tilt and rotation angles. The canonical approach is regularized least squares, minimizing the data-fidelity functional: $$\min_{M} \sum_{i=1}^P \Bigl\| D_i - \mathcal{A}_i[M] \Bigr\|_2^2 + \lambda\,R[M]$$ where each $\mathcal{A}_i[M]$ represents the linear projection for measurement $i$, $R[M]$ is a regularization (e.g., Tikhonov or total variation) to promote stability or piecewise structure, and $\lambda$ is the regularization weight. The simultaneous iterative reconstruction technique (SIRT) is commonly adopted for unregularized or smoothed inversion, leveraging sequential updates and back-projection. For full-field and scanning transmission X-ray microscopy datasets, the MARTApp software suite implements algebraic reconstruction techniques (ART), gradient descent, and ordered-subset strategies optimized for large data (Herguedas-Alonso et al., 21 Jan 2025).

Alignment of projection images and axis registration are essential, typically accomplished by joint reconstruction–reprojection (JRR), Fourier-based cross-correlation, and rigid-body transforms. Multi-tilt acquisition (e.g., $\phi = 0^\circ, 30^\circ$ or orthogonal axes) is required to reconstruct all three cartesian components of $M(\mathbf r)$ (Banerjee et al., 2024, Herguedas-Alonso et al., 21 Jan 2025).

3. Geometric and Analytic Aspects of the Magnetic Ray Transform

The magnetic ray transform, a generalization of the geodesic X-ray transform, involves integrals of symmetric tensor fields along magnetic geodesics dictated by the underlying Lorentz force on a Riemannian manifold $(M,g,\Omega)$. The defining equation is

$$I_\Omega f(x,v) = \int_0^{\tau(x,v)} f_{i_1\cdots i_m}\bigl(\gamma_{x,v}(t)\bigr)\,\dot\gamma^{i_1}(t)\,\dots\,\dot\gamma^{i_m}(t)\,dt$$

where $\gamma_{x,v}$ is a magnetic geodesic and $\tau(x,v)$ is the exit time. Tensor field order coupling arises from the non-time-reversible character of magnetic flows; for instance, potential fields of the form $f = d_s \beta - E(\beta) + d p$ (with $d_s$ the symmetrized covariant derivative and $E$ the Lorentz force) are invisible to the transform (Zhou, 2016). The kernel, injectivity, and gauge obstructions are precisely understood via microlocal analysis and scattering pseudodifferential calculus. Local invertibility (up to this natural gauge) is guaranteed under strict magnetic convexity of the boundary, and global injectivity holds with suitable foliation and non-trapping conditions.

4. Extensions: Attenuation, Higher-Rank Tensors, and Lorentzian Connections

Generalizations include the attenuated magnetic ray transform, encompassing matrix-valued attenuation operators (connections and Higgs fields) on vector bundles, permitting recovery up to gauge. The transform for a bundle equipped with a unitary connection $A$ and Higgs field $\Phi$, acting on function-plus-1-form sources $(f,\omega)$, is: $$I_{A,\Phi}[f,\omega](x,v) = u_{f,\omega}(x,v)\quad \text{with} \quad (X+\lambda V + \mathcal{A}) u(x,v) = -f(x) - \omega_x(v)$$ Injectivity is modulo the gauge $(f,\omega) \mapsto (-\Phi p, d_A p)$ for $p$ vanishing on the boundary (Ainsworth, 2012). The kernel for higher-rank tensor tomography is characterized similarly via potential pairs.

Recent work connects the magnetic X-ray transform to the light ray transform on stationary Lorentzian manifolds, using projection of null geodesics to magnetic geodesics on the underlying Riemannian manifold with magnetic vector field $F = -d\omega$ (Oksanen et al., 6 Feb 2025). A key technical property for injectivity is finite vertical Fourier degree, ensuring that the magnetic generator of the flow does not disperse data into arbitrarily high modes.

5. Computational and Experimental Implementation

State-of-the-art experimental implementations utilize scanning transmission X-ray microscopy (STXM) combined with XMCD. Acquisition protocols include rotating the sample around the vertical axis at multiple tilt angles, measuring with $C+$ and $C-$ polarizations, and recording dense projection series (e.g., 360 angles in $1^\circ$ increments). Image alignment utilizes JRR, cross-correlation shift estimation, and multi-step volume registration. Absorption and dichroic images are calculated via logarithmic ratios of transmitted intensities for each polarization.

Reconstruction deploys iterative schemes (SIRT, ART), regularization, and multi-tilt merging. MARTApp (Herguedas-Alonso et al., 21 Jan 2025) automates these steps, incorporating alignment, normalization, solving for the vector field components, and GUI-based quality control. The reconstruction fidelity is benchmarked via synthetic test data (e.g., hopfion magnetization profiles), achieving high voxel-wise correlation and low RMSE in normalized units. Table 1 summarizes key practical parameters (see (Banerjee et al., 2024, Herguedas-Alonso et al., 21 Jan 2025)):

Parameter	Value / Method	Source
Beam focus (spatial res.)	120 nm spot size	(Banerjee et al., 2024)
XMCD flipping ratio (SNR)	∼1.1 %	(Banerjee et al., 2024)
Data alignment	JRR, SIRT, Gridrec, CG register	(Banerjee et al., 2024)
Fly-scan step sizes	50/100 nm px	(Banerjee et al., 2024)
Computational pipeline	Python/Matlab, GPU optional	(Herguedas-Alonso et al., 21 Jan 2025)

6. Theoretical Impact and Applications

The magnetic X-ray transform underpins 3D imaging of magnetic domains at the nanometer scale, crucial for elucidating material properties linked to magnetic heterogeneity. Its analytic underpinnings enable rigorous tensor tomography—recovering scalar, vector, and higher-rank objects modulo well-characterized gauge freedoms (Zhou, 2016, Ainsworth, 2012, Ainsworth, 2012, Oksanen et al., 6 Feb 2025). Inverse problems engineered around the magnetic X-ray transform inform foundational work in geometric analysis, including stability estimates, microlocal parametrix constructions, and injectivity theorems. Applications span magnetic nanostructure imaging, entropy production analysis in dynamical systems, and inverse boundary problems in Lorentzian geometry.

Recent software frameworks, notably MARTApp (Herguedas-Alonso et al., 21 Jan 2025), have scaled these methods to experimental data volumes, facilitating routine 3D nanomagnetic vector tomography. The methodology is extensible to multi-energy, phase-contrast, or compressed-sensing tomography. A plausible implication is that further integration with ptychographic and spectroscopic techniques will push the spatial resolution and physical insight of MXT beyond current limitations.

7. Limitations, Extensions, and Outlook

The magnetic X-ray transform is limited by missing wedge artifacts from incomplete angular sampling, anisotropic resolution, and the assumption of single-scattering (straight-ray) propagation. Only the magnetization component parallel to the beam is sensed per projection, requiring multiple tilt axes for full vector recovery. Natural obstructions—gauge freedoms—are inseparable from tensor tomography in magnetic systems.

Extensions include higher-rank tensor tomography, attenuated transforms on vector bundles, dynamic (time-resolved) tomography, and integration with forward models for Lorentzian metric perturbations (Oksanen et al., 6 Feb 2025). Computational developments focus on regularization, sparse sampling, and combined multimodal analysis. The magnetic X-ray transform stands as a rigorously characterized and experimentally indispensable paradigm for 3D vector-field imaging in condensed matter and geometric analysis.