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Hankel Subspace Self-Supervised Learning for Parallel MRI Reconstruction

Mingyu Hu, Siquan Zhu, Xijun Zhong, Qiegen Liu

arXiv:2610.11560v1cs.CV

TL;DR

Parallel MRI reconstruction must recover missing k-space while exploiting coil and local Hankel redundancy, but overlapping lifts can leak held-out samples into self-supervised inputs. HSSRecon splits physical acquisition units before lifting, learns a Hankel subspace operator, and reconstructs with a physics-constrained solver. It reports the highest mean PSNR and SSIM across six evaluated R = 6/R = 8 groups, with a contrast-specific qualification.

  • Problem

    Overlapping Hankel lifting duplicates physical coordinates, so splitting lifted entries can expose held-out targets in the input and distort the physical loss.

    Source: §I · p. 2

  • Method

    HSSRecon splits physical acquisition units before lifting, normalizes repeated Hankel copies, learns a compact Hankel subspace operator, and reconstructs physical k-space with a physics-constrained solver.

    Source: §I · p. 2

  • Results

    HSSRecon has the highest mean PSNR and SSIM in all six R = 6/R = 8 groups, though its AXT2 R8 PSNR is slightly below SSDU while its SSIM is higher.

    Source: §IV · p. 8

  • Takeaways & Limitations

    The fastMRI brain results show HSSRecon's learned Hankel operator is useful across the evaluated contrasts and sampling patterns, with a qualified AXT2 R8 comparison.

    Source: §IV · p. 8

    The evaluation covers only two 2D cohorts and does not isolate frequency-specific fidelity.

    Source: §IV · p. 9

Abstract

from arXiv · show

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.

I. INTRODUCTION

Parallel MRI must recover missing k-space coefficients while preserving acquired data and exploiting inter-coil and local Hankel redundancy. HSSRecon addresses leakage from overlapping Hankel lifts through physical-group self-supervision and combines a learned Hankel subspace operator with a physics-constrained solver.

I. INTRODUCTION

  • Classical and structured low-rank MRI methods exploit coil and local Hankel information, but most use calibration, explicit matrix completion, or iterative prior optimization rather than same-scan self-supervised structural learning.Self-supervised methods define inputs and targets in acquisition coordinates, making their combination with overlapping Hankel lifts nontrivial.Source: §I · p. 1
  • Splitting lifted entries can leave a held-out coefficient's duplicate in the input, while an uncorrected lifted loss overweights coordinates appearing in more windows.A valid formulation must exclude targets at the physical-acquisition level and account for repeated coordinates.Source: §I · p. 2
  • HSSRecon splits physical acquisition units before Hankel lifting, corrects repeated copies with multiplicity normalization, and solves for physical k-space variables using a physics-constrained CG solver.This design retains inter-coil and local Hankel redundancy without allowing repeated representations to leak targets or alter the physical loss.Source: §I · p. 2
  • The paper reports competitive reconstruction quality across evaluated multi-coil datasets and sampling masks, alongside theoretical analysis of well-posed reconstruction and finite-step CG bounds.The introduction describes theoretical and empirical validation across the evaluated settings.Source: §I · p. 2

A. Overview

HSSRecon uses a shared learner and constrained Hankel CG loop during training, then freezes the selected learner for reconstruction. Self-supervision splits physical sampling groups before lifting so held-out coefficients cannot reappear through overlapping windows.

B. Hankel Subspace Self-Supervision

  • Self-supervision assigns input, held-out, and validation roles to physical acquisition units before forming overlapping Hankel windows.This prevents held-out coefficients from reappearing through overlapping windows.Source: B. Hankel Subspace Self-Supervision · pp. 2–3
  • Physical groups contain corresponding measurements from all coils; the grouping uses phase-encoding lines for uniform masks, disjoint spoke-shaped groups for radial masks, and acquired spatial coordinates for Poisson masks.The training view splits these groups into input and held-out sets while reserving validation groups.Source: B. Hankel Subspace Self-Supervision · p. 2

A. Overview

  • At deployment, the selected learner uses the full acquired support to construct the subspace operator, and the constrained solver reconstructs missing coefficients before inverse Fourier imaging.During training, the learner is updated with physical held-out prediction loss and selected using validation groups.Source: A. Overview · p. 2, Figure 2 · p. 3
  • Training splits acquired multi-coil data and mask into disjoint physical groups, processes the input through a shared learner, and updates it using physical held-out loss in a constrained Hankel CG loop.The learner produces factorized subspaces; the validation groups support model selection.Source: Figure 2 · p. 3

1) Hankel Multiplicity Normalization:

Overlapping windows create multiple lifted copies of each physical coefficient, so the loss needs a correction that preserves one physical-coordinate contribution.

1) Hankel Multiplicity Normalization:

  • Each physical coefficient appears in d_i overlapping windows, with D = H*H diagonal and D_ii = d_i > 0; the normalized lift and pseudoinverse average lifted copies back to physical coordinates.The projection P_H maps estimates onto the range of the lift.Source: 1) Hankel Multiplicity Normalization · p. 3
  • After projection onto the Hankel range, all d_i copies share the same residual, so dividing by d_i removes repeated counting and 1/r_i preserves pre-draw target-inclusion weighting.For a uniform fixed-size draw of K target groups from G_train, r_i = K/|G_train|.Source: 1) Hankel Multiplicity Normalization · pp. 3–4
  • For a Hankel-consistent estimate, multiplicity correction reduces weighted squared errors across repeated copies to one physical-coordinate contribution, including for the two-dimensional multi-coil lift.The loss is evaluated on physical held-out coordinates.Source: 1) Hankel Multiplicity Normalization · p. 4, 2) Physical Held-Out Objective · p. 4

2) Physical Held-Out Objective:

The held-out objective is evaluated on physical k-space coordinates, while the learner predicts a scan-specific Hankel operator from visible data and mask geometry. Structural and innovation subspaces define a positive-semidefinite operator whose weighting is bounded.

2) Physical Held-Out Objective:

  • The held-out loss is evaluated on physical coordinates, and the two-view training also penalizes disagreement between predicted operators.The common normalization for the reported two-view implementation is 2C|Λ1|.Source: 2) Physical Held-Out Objective · p. 4

C. Hankel Subspace Learner

  • The learner uses visible multi-coil data and mask geometry, but its geometry descriptor contains sampling locations and no held-out values.Complex-valued factors represent structural and sampling-conditioned innovation components.Source: C. Hankel Subspace Learner · p. 4
  • The reported configuration uses 16 structural columns and 8 innovation columns, with innovation weighting capped at 0.1.The operator weights filter directions with residual Hankel energy; it is not a projector onto a reconstructed signal subspace.Source: C. Hankel Subspace Learner · p. 4
  • For full-column-rank factors, QR-based structural and innovation subspaces define the operator, which is Hermitian positive semidefinite and invariant to factor coordinates.This invariance is not asserted for rank-deficient factors.Source: C. Hankel Subspace Learner · p. 4

D. Physics-Constrained Reconstruction Solver

The solver minimizes a learned Hankel penalty with a ridge term over k-space variables subject to acquired-data consistency. It parameterizes only unmeasured coefficients and uses conjugate gradient (CG) to solve the resulting reduced system.

D. Physics-Constrained Reconstruction Solver

  • The constrained quadratic penalizes filter-direction energy and includes a ridge term while retaining all input measurements.λH weights the learned Hankel penalty and λ0 is positive.Source: D. Physics-Constrained Reconstruction… · p. 4
  • The normal map uses multiplicity normalization so coefficients repeated in more windows do not gain weight solely from their copy count..Source: D. Physics-Constrained Reconstruction… · p. 4
  • Writing each feasible reconstruction as the data-consistent k-space plus free-coordinate variables confines CG to unmeasured coefficients.The acquired-data component is restored at every iteration, with an additional write-back to suppress numerical drift.Source: D. Physics-Constrained Reconstruction… · p. 4
  • CG solves the reduced system Avz = cv from an initial residual and returns the reconstruction after at most KCG iterations.The algorithm forms two physical training views before lifting, evaluates held-out loss, selects checkpoints on validation support, then predicts an operator from full acquired support for iterative reconstruction.Source: D. Physics-Constrained Reconstruction… · pp. 4–5

E. Theoretical Analysis

The analysis establishes leakage exclusion and multiplicity-corrected counting for physical splits, then gives fixed-operator guarantees for the constrained solver. Its finite-step error bounds are tied to the fixed quadratic and do not establish signal-subspace recovery or network-training convergence.

E. Theoretical Analysis

  • Physical-group splitting excludes every lifted copy of held-out values, and multiplicity correction counts each physical-coordinate error once after projection onto the lift’s range.The stated result applies to a pure lift that copies each physical entry and has di > 0.Source: E. Theoretical Analysis · p. 5
  • Under full-column-rank QR steps, the operator eigenvalues lie in {0, 1, αv}, giving a spectral norm bound of max{1, αmax}.The bound has no explicit multiplicity factor, unlike the corresponding unnormalized bound.Source: E. Theoretical Analysis · p. 5
  • For fixed view v and Wθ,v ⪰ 0 with λH ≥ 0 and λ0 > 0, Av is Hermitian positive definite, Eq. (12) has a unique minimizer, and every iterate satisfies hard data consistency.For final residual rK = cv − AvzK, the remaining solver error admits a computable bound.Source: E. Theoretical Analysis · p. 5
  • The finite-step bound measures error relative to the fixed quadratic’s exact solution and includes reference mismatch, penalty response, and incomplete solution.The bounds do not establish recovery of the true signal subspace or convergence of network training.Source: E. Theoretical Analysis · p. 5

III. EXPERIMENTS

Experiments use fastMRI brain cases and a held-out Stanford 2D FSE panel, comparing HSSRecon with reconstruction and self-supervised baselines. Both benchmarks use multi-coil measurements and retrospective undersampling, with PSNR, SSIM, and NMSE evaluated on RSS images.

A. Experimental Setup

  • The fastMRI panel contains 270 cases across three contrasts, five subjects per contrast, three slices, three mask families, and accelerations R ∈ {6, 8}.It uses a 640 × 320 working grid and 16 or 20 receiver coils.Source: A. Experimental Setup · p. 5
  • The Stanford panel contains 90 cases from 10 held-out subjects in one 2D FSE Bone setting, with three masks at R = 4.It retains native full-FOV matrices with variable dimensions and uses 16 receiver coils.Source: A. Experimental Setup · pp. 5–6
  • Comparators are zero filling, SAKE, ALOHA, SSDU, BOINR, and HSSRecon, with BOINR and HSSRecon optimized separately for each test scan.The reported BOINR results use acquired-ACS RSS-based phase estimation followed by acquired-data hard consistency.Source: A. Experimental Setup · p. 6
  • Both experiments report PSNR, SSIM, and NMSE on RSS magnitude images normalized by the fully sampled reference maximum.HSSRecon uses fixed per-mask update budgets: 300 and 1000 for fastMRI Poisson/radial and uniform masks, and 300, 600, and 1000 for Stanford Poisson, radial, and uniform masks.Source: A. Experimental Setup · p. 6

B. Experimental Results

HSSRecon performs competitively across the evaluated fastMRI conditions, with strong pooled results and some contrast- and mask-specific exceptions. Ablations show that physical-group splitting prevents leakage through duplicated Hankel entries, while a learned operator improves on a fixed-operator control.

B. Experimental Results

  • HSSRecon leads in PSNR, SSIM, and NMSE across all six fastMRI mask–acceleration conditions at R = 6/8, with an AXT2 R = 8 exception to the broader ordering.For AXT2 R = 8, SSDU has slightly higher PSNR and lower NMSE, while HSSRecon retains higher SSIM; on Stanford, HSSRecon leads on selected metrics but not every metric.Source: B. Experimental Results · p. 6, Table I · p. 7, Table II · p. 7
  • Qualitative fastMRI examples at R = 6 show reduced structured aliasing and local boundary errors with HSSRecon.Fig. 4 compares three contrasts and sampling patterns; its rows share an error scale within each row.Source: B. Experimental Results · p. 6, Figure 4 · p. 7

C. Ablation Study

  • In the nine-slice ablation, the learned operator improves PSNR by 6.23 dB over the fixed operator.The comparison assesses learning and view conditioning jointly.Source: C. Ablation Study · pp. 6–7
  • Splitting after Hankel lifting left 99.996% of target coordinates accessible through duplicates, producing near-zero training loss but higher held-out loss and 4.73 dB lower reconstruction PSNR.The physical split removes duplicate target copies from the input.Source: C. Ablation Study · p. 7

IV. DISCUSSION

The discussion interprets HSSRecon’s results across fastMRI and Stanford, relates performance to the available structural evidence, and distinguishes the learned Hankel operator from the constrained solver. It also identifies limits in the current evaluation and summarizes the method’s findings.

IV. DISCUSSION

  • The split-after-lifting control had near-zero training loss but higher held-out loss and poorer reconstruction; multiplicity correction prevents overlap from reweighting physical coefficients.Nearly every held-out physical coordinate remained accessible through duplicate entries after lifting.Source: §IV · p. 7
  • On fastMRI, HSSRecon has the highest mean PSNR and SSIM in all six R = 6/R = 8 groups, with a contrast-specific exception at AXT2 R = 8.The Stanford panel favors HSSRecon under Poisson and radial sampling, while uniform results favor other methods on individual metrics.Source: §IV · p. 8
  • The authors suggest HSSRecon’s benefit depends on sampling geometry and identifiable local multi-coil relations; fastMRI and Stanford also differ in grid, coil, and acceleration settings.These differences mean the two panels do not have the same reconstruction difficulty or improvement headroom.Source: §IV · pp. 8–9
  • The encoder learns factor matrices whose QR projectors define the Hankel penalty, while constrained CG solves for physical k-space values and preserves acquired samples.The theoretical result guarantees a well-posed quadratic problem and a finite-step CG bound under stated conditions.Source: §IV · p. 9
  • The evaluation covers only two 2D cohorts and does not isolate frequency-specific fidelity.The authors propose larger subject-matched 3D and dynamic MRI validation and frequency-resolved metrics as future work.Source: §IV · p. 9

V. CONCLUSION

  • The conclusion reports favorable pooled fastMRI results with an AXT2 R = 8 exception, and selected metric leads on Stanford rather than uniform dominance.It also summarizes the ablation evidence that physical splitting prevents duplicate-entry leakage and the learned operator improves over a fixed control.Source: §V · p. 9

APPENDIX

The appendix proves that splitting before Hankel lifting excludes duplicate held-out entries and that multiplicity normalization recovers the physical-coordinate loss. It also establishes positive definiteness, uniqueness, measurement preservation, and error bounds for the reduced system.

A. Proof of Proposition 1

  • Splitting before lifting excludes every lifted copy of held-out coordinates, while multiplicity normalization cancels copy counts to give the physical-coordinate loss.For each physical coordinate, the repeated terms cancel the factor equal to its number of copies.Source: A. Proof of Proposition 1 · p. 9

B. Proof of Theorem 1

  • The reduced system is Hermitian positive definite and has a unique solution, with its minimum eigenvalue bounded below by λ0.The structural, innovation, and orthogonal-complement subspaces are invariant, with eigenvalues 1, αv, and 0 for Wθ,v.Source: B. Proof of Theorem 1 · p. 9
  • Every iterate retains the input measurements, and conjugate-gradient error is bounded by the reduced-system residual divided by λ0.The proof also gives an energy-norm error bound for the Kth iterate.Source: B. Proof of Theorem 1 · p. 9

C. Proof of Corollary 1

The corollary proof chooses a comparison point that matches the input measurements and expresses the solution’s free-space difference through a first-order condition. Hard data consistency then leaves the input measurement discrepancy unavoidable.

C. Proof of Corollary 1

  • The comparison point kref matches the input measurements, and its difference from k⋆ is the projected measurement discrepancy.The proof writes this difference as PΘv(y − k⋆).Source: C. Proof of Corollary 1 · p. 9
  • The free-space first-order condition determines the solution difference and separates its Hankel and ridge contributions.The proof first expresses the difference from kref in the null space of the input-measurement projector.Source: C. Proof of Corollary 1 · p. 9
  • Hard data consistency makes the input measurement discrepancy unavoidable in the comparison with k⋆.The projected difference equals PΘv(y − k⋆), and coordinate projection cannot increase the norm.Source: C. Proof of Corollary 1 · p. 9
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