# DEQ-MPI: A Deep Equilibrium Reconstruction with Learned Consistency for Magnetic Particle Imaging

arXiv: https://arxiv.org/abs/2212.13233 · HTML: https://mlsanity.com/paper/2212.13233/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Dec 2022.

## Built on by (most cited first, 9 of 9)

- [VasTSD: Learning 3D Vascular Tree-state Space Diffusion Model for Angiography Synthesis](https://mlsanity.com/paper/2503.12758/) (2025)
- [UCS: A Universal Model for Curvilinear Structure Segmentation](https://mlsanity.com/paper/2504.04034/) (2025)
- [Fast Trajectory-Independent Model-Based Reconstruction Algorithm for Multi-Dimensional Magnetic Particle Imaging](https://mlsanity.com/paper/2505.22797/) (2025)
- [Equivariant Deep Equilibrium Models for Imaging Inverse Problems](https://mlsanity.com/paper/2511.18667/) (2025)
- [Lipschitz Multiscale Deep Equilibrium Models: A Theoretically Guaranteed and Accelerated Approach](https://mlsanity.com/paper/2602.03297/) (2026)
- [CalibFPA: A Focal Plane Array Imaging System based on Online Deep-Learning Calibration](https://mlsanity.com/paper/2309.11421/) (2023)
- [DUN-SRE: Deep Unrolling Network with Spatiotemporal Rotation Equivariance for Dynamic MRI Reconstruction](https://mlsanity.com/paper/2506.10309/) (2025)
- [High-Resolution Magnetic Particle Imaging System Matrix Recovery Using a Vision Transformer with Residual Feature Network](https://mlsanity.com/paper/2511.02212/) (2025)
- [Deep Unrolled Networks in Representation Space Applied to MRI Reconstruction](https://mlsanity.com/paper/2606.21602/) (2026)

## Abstract (arXiv)

Magnetic particle imaging (MPI) offers unparalleled contrast and resolution for tracing magnetic nanoparticles. A common imaging procedure calibrates a system matrix (SM) that is used to reconstruct data from subsequent scans. The ill-posed reconstruction problem can be solved by simultaneously enforcing data consistency based on the SM and regularizing the solution based on an image prior. Traditional hand-crafted priors cannot capture the complex attributes of MPI images, whereas recent MPI methods based on learned priors can suffer from extensive inference times or limited generalization performance. Here, we introduce a novel physics-driven method for MPI reconstruction based on a deep equilibrium model with learned data consistency (DEQ-MPI). DEQ-MPI reconstructs images by augmenting neural networks into an iterative optimization, as inspired by unrolling methods in deep learning. Yet, conventional unrolling methods are computationally restricted to few iterations resulting in non-convergent solutions, and they use hand-crafted consistency measures that can yield suboptimal capture of the data distribution. DEQ-MPI instead trains an implicit mapping to maximize the quality of a convergent solution, and it incorporates a learned consistency measure to better account for the data distribution. Demonstrations on simulated and experimental data indicate that DEQ-MPI achieves superior image quality and competitive inference time to state-of-the-art MPI reconstruction methods.