Matrix completion-informed deep unfolded equilibrium models for self-supervised k $k$ -space interpolation in MRI

Background

Self-supervised methods for magnetic resonance imaging (MRI) reconstruction have garnered significant interest due to their ability to address the challenges of slow data acquisition and scarcity of fully sampled labels. Current regularization-based self-supervised techniques merge the theoretical foundations of regularization with the representational strengths of deep learning and enable effective reconstruction under higher acceleration rates, yet often fall short in interpretability, leaving their theoretical underpinnings lacking.

Purpose

In this paper, we introduce a novel self-supervised approach that provides stringent theoretical guarantees and interpretable networks while circumventing the need for fully sampled labels.

Methods

Our method exploits the intrinsic relationship between convolutional neural networks and the null space within structural low-rank models, effectively integrating network parameters into an iterative reconstruction process. Our network learns gradient descent steps of the projected gradient descent algorithm without changing its convergence property, which implements a fully interpretable unfolded model. We design a non-expansive mapping for the network architecture, ensuring convergence to a fixed point. This well-defined framework enables complete reconstruction of missing $k$-space data grounded in matrix completion theory, independent of fully sampled labels.

Results

Qualitative and quantitative experimental results on multi-coil MRI reconstruction demonstrate the efficacy of our self-supervised approach, showing marked improvements over existing self-supervised and traditional regularization methods, achieving results comparable to supervised learning in selected scenarios. Our method surpasses existing self-supervised approaches in reconstruction quality and also delivers competitive performance under supervised settings.

Conclusions

This work not only advances the state-of-the-art in MRI reconstruction but also enhances interpretability in deep learning applications for medical imaging.