[Paper Review] Reconstruction of Current Densities from Magnetic Images by Bayesian Inference

Colin B. Clement, James P. Sethna+1 more|arXiv (Cornell University)|Oct 28, 2019

Advanced X-ray and CT Imaging4 citations

TL;DR

This paper presents a Bayesian inference framework for reconstructing 2D current densities from magnetic images using flexible, non-traditional priors—such as the Frobenius norm of the Hessian and total variation—beyond analytically tractable Tikhonov regularization. The method employs ADMM optimization with empirical prior strength selection, enabling robust, physically plausible reconstructions even with boundary currents and noisy data.

ABSTRACT

Electronic transport is at the heart of many phenomena in condensed matter physics and material science. Magnetic imaging is a non-invasive tool for detecting electric current in materials and devices. A two-dimensional current density can be reconstructed from an image of a single component of the magnetic field produced by the current. In this work, we approach the reconstruction problem in the framework of Bayesian inference, i.e. we solve for the most likely current density given an image obtained by a magnetic probe. To enforce a sensible current density priors are used to associate a cost with unphysical features such as pixel-to-pixel oscillations or current outside the device boundary. Beyond previous work, our approach does not require analytically tractable priors and therefore creates flexibility to use priors that have not been explored in the context of current reconstruction. Here, we implement several such priors that have desirable properties. A challenging aspect of imposing a prior is choosing the optimal strength. We describe an empirical way to determine the appropriate strength of the prior. We test our approach on numerically generated examples. Our code is released in an open-source exttt{python} package called exttt{pysquid}.

Motivation & Objective

• Address the ill-posed inverse problem of reconstructing current density from noisy, blurred magnetic field images in condensed matter physics.
• Overcome limitations of traditional Tikhonov regularization by enabling non-analytically tractable priors such as total variation and Hessian-based regularization.
• Incorporate physical constraints like current conservation and finite device boundaries without requiring mirror symmetry or analytical tractability.
• Develop a practical, open-source implementation (pysquid) for reproducible and extensible current reconstruction in experimental physics.

Proposed method

• Formulate current reconstruction as a Bayesian inference problem using a generative model linking magnetic field measurements to underlying current density.
• Implement priors based on the Frobenius norm of the Hessian matrix to penalize second-order spatial variations, offering better rotational invariance than Laplacian-based priors.
• Apply total variation regularization via a non-smooth, non-quadratic penalty on second derivatives to preserve sharp current boundaries.
• Use Alternating Direction Method of Multipliers (ADMM) to solve the optimization problem, enabling efficient and stable minimization of the log-posterior.
• Introduce empirical methods to determine optimal prior strength, avoiding reliance on cross-validation or theoretical assumptions.
• Integrate finite support and external model priors by modifying the ADMM formulation to constrain current density to physical regions or known current patterns.

Experimental results

Research questions

• RQ1 Can non-traditional, non-analytically tractable priors such as total variation and Hessian-based regularization improve current density reconstruction beyond Tikhonov regularization?
• RQ2 How can rotational invariance and physical consistency be better enforced in current reconstruction compared to standard Laplacian priors?
• RQ3 What is an effective empirical method to select the regularization strength without relying on cross-validation or analytical approximations?
• RQ4 How can boundary currents and non-symmetric device geometries be handled in a physically consistent way within a Bayesian reconstruction framework?

Key findings

• The Frobenius norm of the Hessian provides a more physically motivated prior than the Laplacian, due to its rotational invariance and superior handling of spatial gradients.
• Total variation regularization effectively suppresses high-frequency noise while preserving sharp current boundaries, outperforming Tikhonov in edge retention.
• The proposed ADMM-based optimization framework enables stable and efficient solution of the reconstruction problem with complex, non-smooth priors.
• Empirical selection of prior strength yields robust reconstructions across diverse synthetic test cases, reducing dependence on cross-validation.
• The method successfully handles current injection/exit at image boundaries without requiring mirror symmetry or artificial boundary conditions.
• The open-source Python package pysquid enables reproducible, extensible, and community-driven development of current reconstruction tools for experimental physics.