# Hankel Subspace Self-Supervised Learning for Parallel MRI Reconstruction

arXiv: https://arxiv.org/abs/2610.11560 · HTML: https://mlsanity.com/paper/2610.11560/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 38 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Builds on (most cited first, 18 of 18)

- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Noise2Self: Blind Denoising by Self-Supervision](https://mlsanity.com/paper/1901.11365/) (2019)
- [Zero-Shot Self-Supervised Learning for MRI Reconstruction](https://mlsanity.com/paper/2102.07737/) (2021)
- [Deep Generalization of Structured Low-Rank Algorithms (Deep-SLR)](https://mlsanity.com/paper/1912.03433/) (2019)
- [A theoretical framework for self-supervised MR image reconstruction using sub-sampling via variable density Noisier2Noise](https://mlsanity.com/paper/2205.10278/) (2022)
- [Data augmentation for deep learning based accelerated MRI reconstruction with limited data](https://mlsanity.com/paper/2106.14947/) (2021)
- [Accelerated MRI Reconstruction with Separable and Enhanced Low-Rank Hankel Regularization](https://mlsanity.com/paper/2107.11650/) (2021)
- [Multi-Mask Self-Supervised Learning for Physics-Guided Neural Networks in Highly Accelerated MRI](https://mlsanity.com/paper/2008.06029/) (2020)
- [ENSURE: A General Approach for Unsupervised Training of Deep Image Reconstruction Algorithms](https://mlsanity.com/paper/2010.10631/) (2020)
- [One-shot Generative Prior in Hankel-k-space for Parallel Imaging Reconstruction](https://mlsanity.com/paper/2208.07181/) (2022)
- [Three-Dimensional MRI Reconstruction with Gaussian Representations: Tackling the Undersampling Problem](https://mlsanity.com/paper/2502.06510/) (2025)
- [K-band: Self-supervised MRI Reconstruction via Stochastic Gradient Descent over K-space Subsets](https://mlsanity.com/paper/2308.02958/) (2023)
- [Matrix Completion-Informed Deep Unfolded Equilibrium Models for Self-Supervised k-Space Interpolation in MRI](https://mlsanity.com/paper/2309.13571/) (2023)
- [Self-supervised feature learning for cardiac Cine MR image reconstruction](https://mlsanity.com/paper/2505.23408/) (2025)
- [Bilevel Optimized Implicit Neural Representation for Scan-Specific Accelerated MRI Reconstruction](https://mlsanity.com/paper/2502.21292/) (2025)
- [Convolutional Framework for Accelerated Magnetic Resonance Imaging](https://mlsanity.com/paper/2002.03225/) (2020)
- [Towards a Unified Theoretical Framework for Splitting-based Self-Supervised MRI Reconstruction](https://mlsanity.com/paper/2601.04775/) (2026)

## Abstract (arXiv)

Parallel magnetic resonance imaging reconstruction is an ill-posed inverse problem under undersampling. Multi-coil acquisition and Hankel lifting expose complementary repeated information: observations of the same anatomy across coils and repeated local k-space neighborhoods in overlapping windows. These dependencies guide recovery of missing k-space data. However, splitting lifted Hankel entries for self-supervision can place the original sample in both input and target, causing data leakage. We propose Hankel Subspace Self-Supervised Reconstruction (HSSRecon), a scan-specific reconstruction framework for parallel magnetic resonance imaging. HSSRecon partitions data by physical acquisition units before Hankel lifting and applies multiplicity normalization to repeated Hankel copies in overlapping windows. Rather than learning a mapping that directly predicts missing data, the network learns a compact complex-valued Hankel subspace operator. Reconstruction is performed over the original k-space variables using a conjugategradient solver with hard data consistency. This design separates structural learning in the Hankel domain from data consistency in the physical domain: the former exploits multi-coil and local Hankel correlations, while the latter solves over unacquired degrees of freedom. We provide theoretical analyses of physicalgroup splitting and multiplicity normalization, and establish positive definiteness, uniqueness, hard data consistency, and a finite-step conjugate-gradient error bound for the system. On fastMRI brain data with three contrasts and three sampling masks, HSSRecon achieves competitive peak signal-to-noise ratio, structural similarity, and normalized mean squared error across six aggregated conditions.