[Paper Review] DEQ-MPI: A Deep Equilibrium Reconstruction with Learned Consistency for Magnetic Particle Imaging

Alper Güngör, Baris Askin+4 more|arXiv (Cornell University)|Dec 26, 2022

Characterization and Applications of Magnetic Nanoparticles63 references4 citations

TL;DR

DEQ-MPI proposes a physics-driven deep equilibrium model for magnetic particle imaging reconstruction that integrates a learned consistency measure and implicit neural networks into an ADMM-based optimization framework. It achieves superior image quality and competitive inference speed by enforcing data consistency and regularization through a convergent, implicit solution, outperforming both traditional and learning-based MPI reconstruction methods.

ABSTRACT

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.

Motivation & Objective

• Address the limitations of hand-crafted priors in MPI reconstruction, which fail to capture complex image features and require manual tuning of regularization weights.
• Overcome the computational inefficiency and suboptimal convergence of conventional unrolled deep learning methods in MPI by enabling a convergent, implicit solution via deep equilibrium modeling.
• Improve data consistency modeling in MPI by replacing hand-crafted consistency measures with a learned consistency block that better conforms to the true data distribution.
• Enhance generalization and reconstruction fidelity by integrating a physics-based optimization framework (ADMM) with neural network priors and learned consistency, while maintaining competitive inference times.
• Demonstrate the effectiveness of DEQ-MPI on both simulated and experimental MPI data, showing robust performance across varying signal-to-noise and resolution conditions.

Proposed method

• DEQ-MPI employs a deep equilibrium model (DEQ) that implicitly defines a fixed-point solution to an iterative optimization process, avoiding the need for explicit unrolling of layers.
• The method integrates a learned consistency (LC) block into an ADMM-based optimization framework, replacing conventional hand-crafted data consistency terms with a neural network that models the true data distribution.
• A two-component auxiliary variable splitting in ADMM separates the proximal mappings for data consistency and regularization, enabling more stable and accurate convergence in non-convex or non-smooth problems.
• The regularization and learned consistency blocks are trained via implicit differentiation, allowing backpropagation through the fixed-point solution without explicit unrolling.
• The architecture uses densely connected residual connections across convolutional layers to improve feature learning and model performance, differing from standard residual connections in prior DEQ methods.
• Dedicated initialization strategies are applied to both the regularization and learned consistency blocks to improve convergence and final reconstruction quality.

Experimental results

Research questions

• RQ1 Can a deep equilibrium model with learned consistency improve image reconstruction quality in MPI compared to traditional and learning-based methods?
• RQ2 Does integrating a learned consistency block into an ADMM framework enhance data fidelity and generalization compared to hand-crafted consistency measures?
• RQ3 Can implicit differentiation and deep equilibrium modeling achieve high-quality reconstructions with competitive inference times, avoiding the computational burden of unrolled networks?
• RQ4 How does the proposed two-component variable splitting in ADMM improve convergence and stability in non-convex MPI reconstruction problems?
• RQ5 To what extent does the performance of DEQ-MPI generalize across different imaging conditions, including low-signal regions and upsampled system matrices?

Key findings

• DEQ-MPI achieves superior image quality compared to state-of-the-art MPI reconstruction methods on both simulated and experimental data, with enhanced contrast and reduced artifacts.
• The method demonstrates competitive inference time, outperforming iterative methods with similar or better reconstruction fidelity, making it suitable for real-time applications.
• The learned consistency block significantly improves data fidelity by better modeling the complex, correlated noise and frequency-domain characteristics of MPI data.
• The use of implicit differentiation and deep equilibrium modeling enables convergence to a high-quality solution without explicit unrolling, reducing computational overhead.
• The two-component ADMM splitting with dedicated proximal mappings for consistency and regularization leads to more stable and accurate optimization compared to single-variable approaches.
• DEQ-MPI generalizes well to upsampled system matrices, with visual acuity scaling effectively at 2× higher resolution, indicating robustness to SM resolution variations.