Towards a Unified Theoretical Framework for Self-Supervised MRI Reconstruction

TL;DR

MRI reconstruction benefits from self-supervised learning, but existing SSL methods are fragmented and empirically driven. UNITS provides a unified theoretical framework that proves SSL can match supervised performance in expectation and introduces sampling stochasticity and cross-consistency to boost generalization and stability. Across a 2D cardiac Cine MRI dataset, UNITS-Base and UNITS-Cross approach or exceed supervised performance, with cross-consistency enabling faster convergence and reduced variance. This work offers a principled, generalizable path toward clinically applicable SSL-based MRI reconstruction and a standardized benchmark for method comparison.

Abstract

The demand for high-resolution, non-invasive imaging continues to drive innovation in magnetic resonance imaging (MRI), yet prolonged acquisition times hinder accessibility and real-time applications. While deep learning-based reconstruction methods have accelerated MRI, their predominant supervised paradigm depends on fully-sampled reference data that are challenging to acquire. Recently, self-supervised learning (SSL) approaches have emerged as promising alternatives, but most are empirically designed and fragmented. Therefore, we introduce UNITS (Unified Theory for Self-supervision), a general framework for self-supervised MRI reconstruction. UNITS unifies prior SSL strategies within a common formalism, enabling consistent interpretation and systematic benchmarking. We prove that SSL can achieve the same expected performance as supervised learning. Under this theoretical guarantee, we introduce sampling stochasticity and flexible data utilization, which improve network generalization under out-of-domain distributions and stabilize training. Together, these contributions establish UNITS as a theoretical foundation and a practical paradigm for interpretable, generalizable, and clinically applicable self-supervised MRI reconstruction.

Figures (10)

• Figure 1: Overview of the proposed UNITS (unified theory for self-supervision) framework. The framework defines a general self-supervised learning paradigm for MRI reconstruction: (a) Initial undersampling: an undersampling mask $M_{y}$ is applied to acquire k-space $y$ for training. (b) Training: at each step, $y$ undergoes re-undersampling with multiple random masks $M_{1},\dots,M_{L}(L\ge 2)$ to generate subsets $y_{1},\dots,y_{L}$, which are flexibly assigned as inputs or supervision signals. Input subsets are processed by the reconstruction network and compared in loss calculation with measured entries in the supervision subsets that differ from the input. Solid arrows denote the main training pathways that are mandatory for learning, while dashed arrows indicate auxiliary pathways that support framework generality and extensibility. (c) Inference: the trained network directly reconstructs images from undersampled data. The bottom panel summarizes the core advantages of the UNITS framework.
• Figure 2: Reconstructions in spatial (x-y) and spatiotemporal (y-t) plane of the proposed UNITS-Base. Each column shows the results for the acceleration rates $R=3,6,9,12,15,18$. The first row presents the undersampled zero-filled input images, the second row shows the reconstructed full images, with enlarged cardiac regions (yellow box) displayed in the third row. The bottom row presents the corresponding $2\times$ scaled relative error maps between the reconstructed and the fully-sampled reference. The dynamic performance in the y-t plane corresponds to the blue line in the reference x-y plane image.
• Figure 3: Comparison between representative self-supervised reconstruction methods within the UNITS framework and supervised learning. Reconstructions in spatial (x-y) and spatiotemporal (y-t) planes are shown for zero-filling, Noisier2Noisemillard2023theoretical, SSDUyaman2020ssdu, supervised learning, UNITS-Base, and UNITS-Cross. All methods were implemented within the UNITS framework using the same network backbone. Both the initially undersampled k-space ($R=8$, top) and the re-undersampled k-space with ratio $0.4$ (effective acceleration $R=20$, bottom) are evaluated as inference inputs. The dynamic performance in the y-t plane corresponds to the blue line in the reference x-y plane image. The error plots present the corresponding $5\times$ scaled relative error maps between the reconstructed images and the fully-sampled reference.
• Figure 4: Ablation results on sampling stochasticity. Quantitative comparison of the five experimental variants summarized in Table \ref{['tab:randomness']}, evaluated using structural similarity index (SSIM) across all slices of all test subjects under three inference conditions: (a) in-distribution (ID): the input is re-undersampled from an initially undersampled k-space ($R=8$) with ratio $0.4$, yielding an effective acceleration of $R=20$ (matching the training setup of UNITS-Fix). (b,c) out-of-distribution (OOD): the input is the initially undersampled k-space with acceleration (b)$R=8$ and (c)$R=12$, without further re-undersampling. Violin plots depict the SSIM distribution, with vertical dashed lines indicating the median and interquartile ranges. Asterisks denote statistically significant differences assessed by the Wilcoxon signed-rank test across subjects (*: $p < 0.05$; *: $p < 0.01$; *: $p < 0.001$; n.s.: not significant).
• Figure 5: Illustration of the case analysis in the proof of Theorem \ref{['thm:equivalence']} In the k-space plots ($y_{0}$, $y$, $y_{1}$), black dots indicate sampled k-space locations and white dots indicate unsampled points. In the mask plots ($M_{y}$ and $M_{1}$), "1" and "0" denote whether a location is sampled or not in the binary mask. Three representative k-space locations are highlighted: Case 1 (yellow): the location is sampled in both initial and re-undersampling stages; Case 2.1 (green): the location is not sampled in the initial undersampling; Case 2.2 (blue): the location is sampled in the initial undersampling but not sampled in the re-undersampling stage.

Theorems & Definitions (2)

• Theorem 1: Equivalence of Self-Supervised and Supervised MRI Reconstruction
• Proposition 1: Variance Reduction via Cross-consistency Loss