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Neural Vector Tomography for Reconstructing a Magnetization Vector Field

Giorgi Butbaia, Jiadong Zang

TL;DR

This work tackles the problem of reconstructing a magnetization vector field from vector tomography data in the presence of noise and limited measurements. It introduces a neural field representation that models the vector field as a smooth continuous function and trains it to match probed ray-transform measurements, augmented by a smoothing regularizer. The method supports high-resolution reconstructions and can exploit symmetries through equivariant networks, with substantial robustness gains over discretized approaches demonstrated on Bloch points and Hopfion-like fields. The approach offers a scalable, stable framework for 3D nanomagnetism imaging where traditional voxel-based reconstructions struggle under noise and incomplete data.

Abstract

Discretized techniques for vector tomographic reconstructions are prone to producing artifacts in the reconstructions. The quality of these reconstructions may further deteriorate as the amount of noise increases. In this work, we instead model the underlying vector fields using smooth neural fields. Owing to the fact that the activation functions in the neural network may be chosen to be smooth and the domain is no longer pixelated, the model results in high-quality reconstructions, even under presence of noise. In the case where we have underlying global continuous symmetry, we find that the neural network substantially improves the accuracy of the reconstruction over the existing techniques.

Neural Vector Tomography for Reconstructing a Magnetization Vector Field

TL;DR

This work tackles the problem of reconstructing a magnetization vector field from vector tomography data in the presence of noise and limited measurements. It introduces a neural field representation that models the vector field as a smooth continuous function and trains it to match probed ray-transform measurements, augmented by a smoothing regularizer. The method supports high-resolution reconstructions and can exploit symmetries through equivariant networks, with substantial robustness gains over discretized approaches demonstrated on Bloch points and Hopfion-like fields. The approach offers a scalable, stable framework for 3D nanomagnetism imaging where traditional voxel-based reconstructions struggle under noise and incomplete data.

Abstract

Discretized techniques for vector tomographic reconstructions are prone to producing artifacts in the reconstructions. The quality of these reconstructions may further deteriorate as the amount of noise increases. In this work, we instead model the underlying vector fields using smooth neural fields. Owing to the fact that the activation functions in the neural network may be chosen to be smooth and the domain is no longer pixelated, the model results in high-quality reconstructions, even under presence of noise. In the case where we have underlying global continuous symmetry, we find that the neural network substantially improves the accuracy of the reconstruction over the existing techniques.

Paper Structure

This paper contains 8 sections, 1 theorem, 21 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Vector Tomography and ray transform
  3. Neural Reconstruction
  4. Sampling Rays
  5. Approximating $\mathcal{R}_{\mathcal{P}}(\tilde{F}_\theta)(L)$
  6. Symmetries and equivariant neural tomography
  7. Numerical Reconstructions
  8. Conclusion

Key Result

Theorem 1

The mapping $\mathcal{R}_p$ on the space of vector fields has kernel: where $\nabla$ is the usual gradient operator on vector fields.

Figures (5)

  • Figure 1: Outline of the neural field reconstruction of the magnetization vector field.
  • Figure 2: Slices of reconstructions at different levels of noise $\epsilon \sim \mathcal{N}(0,\sigma)$. (a) Reconstruction using neural vector tomography (ours), (b) reconstruction using the method described in Donnelly_2018 . (a') and (b') show the errors (defined as the absolute value of the difference between reconstruction and the ground truth) for (a) and (b), respectively.
  • Figure 3: Comparison of the reconstructed and ground truth isosurfaces $M_z = 0$.
  • Figure 4: Comparison of (a) isosurfaces $M_z = 0$ and (b) slices of reconstructions using different reconstruction techniques.
  • Figure 5: SSIM and MSE metrics of the reconstructions using neural and discrete Donnelly_2018 techniques under different levels of noise.

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1