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LGGS: Longitudinal MRI Reconstruction via Guided Group Sparsity

Nikola Janjušević, Hersh Chandarana, Yao Wang, Li Feng

arXiv:2610.22643v1eess.IVcs.CV

TL;DR

Serial MRI is clinically important, but long exams constrain access, and reconstruction methods often leave patient-specific information in prior scans unused. LGGS uses feature correspondence and phase-invariant attention to incorporate prior information into reconstruction regularization. The authors report better reconstruction quality than evaluated unguided and longitudinal methods, with substantially fewer learned parameters, while noting limits in the study’s data and validation scope.

  • Problem

    MRI exams can be lengthy, while existing reconstruction methods often treat visits independently despite the patient-specific information available in prior scans and the challenges of inter-visit differences.

    Source: §I · p. 1

  • Method

    LGGS adapts an unrolled reconstruction network’s regularizer using explicit correspondence between current and prior features, with circulant attention designed to accommodate misregistration and phase differences.

    Source: §V · p. 9

  • Results

    LGGS achieved superior reconstruction quality to evaluated unguided methods and L-TGVN while requiring substantially fewer learned parameters.

    Source: §V · p. 9

  • Takeaways & Limitations

    The compact method and tolerance of accelerated prior images support its potential use for serial MRI exams in practice.

    Source: §V · p. 10

    Evaluation was limited to T1-weighted brain MRI and retrospective undersampling; large rotations and anatomical deformations are not explicitly addressed, and prospective validation is warranted.

    Source: §V · p. 10

Abstract

from arXiv · show

Serial MRI is routinely performed in current clinical practice for disease surveillance and applications such as MR-guided radiotherapy. However, long MRI scan times increase costs and limit patient access, making accelerated image reconstruction of broad clinical importance. Previous scans of the same patient contain image information highly correlated with the current exam, yet existing reconstruction methods rarely exploit this information. Major challenges in the effective use of prior scans include inter-visit misregistration, pathological changes, and image phase variation. In this work, we propose Longitudinal Guided Group Sparsity (LGGS), an interpretable deep-learning framework for longitudinally informed MRI reconstruction derived from classical group sparsity. The prior scan is incorporated into the reconstruction regularizer through learned phase-invariant circulant attention that identifies corresponding features across exams. This formulation provides robustness to local in-plane and through-plane misregistration as well as inter-visit phase variation. Across acceleration rates, LGGS achieves state-of-the-art performance on longitudinal T1-weighted multi-coil brain MRI data, outperforming the evaluated unguided and longitudinally informed reconstruction methods, with substantially fewer learned parameters. We further evaluated the contributions of multiple guide slices and phase-invariant similarity and demonstrated that accelerated prior images can also serve as effective guides with minimal degradation in reconstruction quality.

I. INTRODUCTION AND BACKGROUND

Serial MRI provides clinically important longitudinal information, but long exams constrain access and prior scans are difficult to use safely because visits can differ in alignment, pathology, and image phase. LGGS uses prior-feature correspondence to adapt reconstruction regularization, with phase-invariant similarity and local search designed to address these differences.

I. INTRODUCTION AND BACKGROUND

  • Long MRI exams limit clinical throughput and patient tolerance, while prior scans offer patient-specific information that existing reconstruction methods often do not exploit.Prior information must be used carefully to avoid biasing reconstruction toward outdated anatomy.Source: §I · pp. 1–2
  • Inter-visit misregistration makes direct image-difference comparisons unreliable, while disease progression can make a guide misleading and phase variation can compromise feature similarity.Misregistration includes in-plane and slice misalignment; phase differences may not reflect anatomical change.Source: §I · p. 2
  • LGGS incorporates prior-exam information into an unrolled reconstruction network’s regularizer using explicit feature correspondence rather than directly transferring prior-image features.The guided group-sparsity regularizer uses learned, row-normalized associations to adapt current reconstruction regularization.Source: §I · p. 2
  • Phase-invariant circulant attention identifies corresponding features despite inter-visit phase differences, with local search and neighboring guide slices addressing residual misregistration.Feature similarity determines how much prior information influences reconstruction regularization.Source: §I · p. 2
  • The proposed interpretable longitudinal network is reported to outperform evaluated unguided and longitudinal methods across acceleration rates with substantially fewer learned parameters.The authors also report effective reconstruction when prior images are themselves accelerated reconstructions.Source: §I · p. 2

A. Accelerated and Longitudinal MRI

The section introduces MRI reconstruction from undersampled multi-coil k-space and reviews guided reconstruction and learned group sparsity. It motivates LGGS as an explicit way to use prior-exam correspondence in reconstruction regularization.

A. Accelerated and Longitudinal MRI

  • Multi-coil MRI reconstruction estimates an image from undersampled k-space measurements modeled with coil sensitivities and complex Gaussian noise.The measurements are written as y = Ex + ν, with E combining sampling, Fourier transformation, and coil sensitivities.Source: A. Accelerated and Longitudinal MRI · pp. 2–3

B. Guided Reconstruction

  • LGGS uses prior-image feature correspondence to adapt the current reconstruction’s regularization, without directly transferring prior-image features.Its learned phase-invariant circulant attention is designed to identify corresponding features despite residual misregistration and phase differences.Source: B. Guided Reconstruction · p. 3

C. Learned Group Sparsity

  • GroupCDL forms dynamic soft adjacency weights that encode latent-pixel group membership, and learned group-thresholding approximates the group-sparsity proximal operator.The adjacency is shared across feature channels, and a window size of one reduces the penalty to ℓ1 sparsity.Source: C. Learned Group Sparsity · pp. 3–4
  • LGGS extends learned group sparsity longitudinally, using guide-feature correspondence to regularize the reconstruction rather than adding an intra-image non-local prior.The guide enters the unrolled architecture through its analysis codes and guided group-thresholding; setting those codes to zero recovers LPDSNet [9].Source: C. Learned Group Sparsity · pp. 3–4, Figure 2 · p. 4

D. Learned Primal-Dual Splitting

LPDSNet unrolls an analysis-dictionary sparse-coding objective with an ℓ1 penalty using primal-dual splitting. Its analysis-domain guide codes can be obtained with a single convolution.

D. Learned Primal-Dual Splitting

  • LPDSNet unrolls an analysis-dictionary convolutional sparse-coding objective with an ℓ1 penalty through K primal-dual splitting iterations.Each layer learns analysis and synthesis convolutions, step sizes, and a dual proximal operator implemented as feature-dependent thresholding.Source: D. Learned Primal-Dual Splitting · p. 4
  • The guided unrolling retains the LPDSNet iteration and differs only in its proximal operator.That proximal operator is replaced to incorporate the guide in the proposed method.Source: D. Learned Primal-Dual Splitting · p. 4
  • Guide analysis codes require a single analysis convolution, unlike the separate sparse-coding pass that may be needed for an equivalent synthesis-domain representation.LPDSNet also outperformed CDLNet on supervised and unsupervised MRI joint denoising and reconstruction tasks.Source: D. Learned Primal-Dual Splitting · p. 4

III. PROPOSED METHOD

LGGS extends group sparsity with cross-similarity between current reconstruction and prior-exam features. Its primal-dual unrolling uses guided group-thresholding to incorporate that correspondence into regularization.

A. Longitudinal Guided Group Sparsity (LGGS)

  • The guide’s analysis codes contribute cross-similarity to a row-normalized adjacency matrix, whose off-diagonal entries couple reconstruction and guide features.The regularizer is minimized over the reconstruction codes alone.Source: A. Longitudinal Guided Group Sparsity… · pp. 4–5
  • LGGS places longitudinal reconstruction-to-guide coupling in the group-sparsity regularizer, with no intra-image non-local grouping.The reconstruction self-similarity block is diagonal, while guide similarity is searched across a neighborhood; without a guide, the regularizer becomes ℓ1 sparsity and the unrolling becomes LPDSNet.Source: §III · p. 4, A. Longitudinal Guided Group Sparsity… · p. 5

B. Guided Reconstruction with Learned Primal-Dual Splitting

  • The K-iteration primal-dual solver uses a guided group-energy map and guided group-thresholding as its proximal operator.The unrolling learns per-layer parameters and enforces data consistency through the MRI encoding operator.Source: B. Guided Reconstruction with Learned… · p. 5

C. Circulant Attention for the Adjacency Matrix

LGGS constructs longitudinal adjacency through local circulant similarity between reconstruction and guide features. It uses phase-invariant similarity so relative phase differences do not erase structural agreement.

C. Circulant Attention for the Adjacency Matrix

  • The adjacency uses separate search windows for self- and guide-similarity, restricting its non-local grouping to the longitudinal dimension.The self-similarity window is one pixel, while the guide window is larger; the resulting construction is translation-equivariant and robust to local misalignment.Source: C. Circulant Attention for the… · p. 5

D. Phase-Invariant Similarity

  • A real inner product can suppress or reverse similarity between structurally identical patches when inter-visit phase differs.Its value scales as cos(φ0), vanishing near φ0 = π/2 and changing sign beyond it.Source: D. Phase-Invariant Similarity · p. 5
  • LGGS uses phase-invariant distance similarity for guide cross-similarity, preserving feature structure rather than reducing comparisons to magnitudes alone.The similarity maximizes over a scalar rotation per query–key pair; the magnitude-only alternative could match an edge patch to a constant patch of equal magnitude.Source: D. Phase-Invariant Similarity · p. 5

E. Multiple Guide Images

LGGS extends its circulant-attention guidance to multiple prior slices, weighting them by reconstruction-space similarity. The experiments use serial brain MRI exams, with prior visits supplying guides for target reconstructions.

E. Multiple Guide Images

  • Multiple guide slices contribute cross-similarity terms, and row normalization keeps total group energy independent of guide count.The guide slices are centered on the target slice; similarity weighting lets slices on either side contribute when the target falls between them.Source: E. Multiple Guide Images · pp. 5–6

A. Experimental Setup

  • The dataset comprised 210 fully sampled T1-weighted multi-coil brain volumes from 70 patients, split by patient into 58 training and 12 held-out test patients.Each patient had three serial exams; each target visit was reconstructed using an adjacent visit as its guide.Source: A. Experimental Setup · p. 6
  • Global integer-voxel registration used low-resolution ACS-derived images, while retrospective Cartesian undersampling added the fully sampled ACS region to the sampling pattern.The effective acceleration rate was lower than the nominal rate because ACS lines were added rather than substituted.Source: A. Experimental Setup · p. 6
  • LGGS used three neighboring guide slices by default, with a guide window of three and phase-invariant distance; L-TGVN used the same number of magnitude-image guide slices.LGGS and LPDSNet shared hyperparameters except for guidance.Source: A. Experimental Setup · p. 6

B. Performance Across Acceleration Rates

Across the tested acceleration rates, LGGS outperformed the evaluated unguided and longitudinally guided methods. Ablations indicate that phase-invariant similarity benefits guidance, while multiple guide slices recover additional gains.

B. Performance Across Acceleration Rates

  • LGGS achieved the best NRMSE, SSIM, and LPIPS at every tested rate, with every difference significant.The gap over unguided methods widened with acceleration; against L-TGVN, LGGS improved NRMSE by 1.12%–1.62% across the sweep and provided 25% more acceleration at matched error.Source: B. Performance Across Acceleration Rates · p. 7

C. Guide Representation and Phase-Invariant Similarity

  • At 8× acceleration, all four guide-and-similarity combinations improved on unguided LPDSNet, and phase invariance had the larger effect.A complex guide benefited most when similarity was phase-invariant; L-TGVN produced 10.33% NRMSE with either guide form.Source: C. Guide Representation and… · pp. 7–8, Figure 6 · p. 8

D. Number of Guide Slices

  • At 8× acceleration, one guide slice recovered most of the available gain, three recovered significantly more, and increasing the window to five did not measurably change NRMSE or SSIM.Mismatching guide slices are down-weighted, so a wider-than-useful window adds computational cost rather than reducing accuracy.Source: D. Number of Guide Slices · pp. 8–9, Figure 7 · p. 9

E. Accelerated Guide Images

LGGS uses corresponding prior-exam features to adapt reconstruction regularization, with attention designed to handle misregistration and phase variation. It performs well with accelerated guide images and is reported to outperform evaluated alternatives with substantially fewer parameters.

E. Accelerated Guide Images

  • At 16× nominal acceleration, 4× and 6× guides cost 0.09% and 0.44% NRMSE against the fully sampled prior, versus a 6.89% penalty without a guide.At 12×, the 4× guide was not significantly different from the fully sampled guide in NRMSE.Source: E. Accelerated Guide Images · p. 9

V. DISCUSSION AND CONCLUSION

  • LGGS achieved superior reconstruction quality to two unguided reconstruction methods and L-TGVN across acceleration rates, while requiring substantially fewer learned parameters.The discussion reports the overall comparison; a separate comparison gives 1.08M learned parameters for LGGS and 43.6M for L-TGVN under the stated experimental conditions.Source: §V · pp. 9–10
  • Prior features that correspond to current features adapt group regularization, allowing strongly corresponding features to receive less suppression and potentially preserving anatomical detail.Unlike GroupCDL’s within-image associations, LGGS forms non-local group associations across exams.Source: §V · p. 9
  • Circulant attention searches locally for corresponding features, and neighboring guide slices improve robustness to through-plane misregistration.Similarity-based weights reduce the contribution of prior features that do not match the current reconstruction, including when anatomical or pathological changes occur.Source: §V · p. 9
  • Phase-invariant similarity accounts for inter-exam phase differences while retaining relative complex-feature information; ablations report gains over magnitude-only guidance.The method removes the common phase difference between compared feature vectors rather than discarding all phase information.Source: §V · p. 9
  • LGGS uses 1.08M learned parameters and reconstructs better than L-TGVN’s 43.6M at every evaluated rate, at comparable wall-clock cost.The comparison is under the stated experimental conditions, not against L-TGVN’s full-scale form.Source: §V · p. 10
  • The evaluation covers only T1-weighted brain MRI and retrospective undersampling, and does not explicitly address large rotations or large anatomical deformation.The authors call for prospective validation and systematic evaluation of lesion preservation and true interval pathological changes.Source: §V · p. 10

APPENDIX

The appendix describes visit alignment and input preparation, then motivates phase-invariant similarity by showing how phase variation can undermine naïve comparisons. It also outlines derivations for similarity functions and their backpropagation.

A. Longitudinal Data Preparation

  • Inter-visit registration estimates an integer-voxel 3D translation by maximizing correlation between low-frequency, coil-combined references, then applies the shift to volumes, sensitivity maps, and support masks.The low-frequency reference is intended to make displacement estimation insensitive to contrast change and pathology; rotation is not corrected.Source: A. Longitudinal Data Preparation · p. 11
  • Visit intensities are normalized using the foreground median of a low-frequency ACS reference, while the support mask zeros the reconstruction target background and windows qualitative displays.Sensitivity maps are estimated from the full multi-coil volume, but the forward operator uses maps from the undersampled k-space ACS band.Source: A. Longitudinal Data Preparation · p. 11

B. On Mean-Subtraction

  • The reconstruction input uses a geometry-aware mean-subtracted zero-filled image, subtracting the encoded mean through EHE rather than simply subtracting the mean from EHy.The guide is centered separately by its own spatial mean, jointly over the guide slices; magnitude guides use mean magnitude.Source: B. On Mean-Subtraction · p. 11

C. Derivation of Phase-Invariant Similarity and Backpropagation Rules

  • The appendix derives closed-form phase-invariant dot-product and half-squared-distance similarities and the backpropagation rules needed to train LGGS through them.The derivations support the similarity functions introduced in the main text.Source: C. Derivation of Phase-Invariant… · p. 11

1) Motivation: Failure Modes of Naïve Similarities:

  • Naïve real-inner-product similarity can fail to recognize similar magnitude patches when their latent codes differ by phase near π/2, and can invert similarity near π.This motivates a differentiable phase-invariant latent similarity metric.Source: 1) Motivation: Failure Modes of Na¨ıve… · pp. 11–12
  • The dot product of code magnitudes removes phase but can falsely treat distinct edge and constant-region codes as equally similar as self-matches.For the example vectors, the standard and phase-invariant dot products are zero, while the magnitude-code dot product is 2.Source: 1) Motivation: Failure Modes of Na¨ıve… · p. 12

2) Closed-Form Phase-Invariant Similarities:

This appendix derives phase-optimal similarity forms and the gradients used to train LGGS through similarity operators.

2) Closed-Form Phase-Invariant Similarities:

  • Phase-invariant similarity is obtained by phase-aligning code vectors before computing their similarity.The passage introduces this procedure as the definition of phase-invariant similarity.Source: 2) Closed-Form Phase-Invariant… · p. 12
  • The phase-optimal dot-product similarity is the magnitude of the complex dot product between the vectors.The passage gives dotsimπ(u, v) = |vHu|.Source: 2) Closed-Form Phase-Invariant… · p. 12
  • The appendix also formulates a phase-optimal distance-similarity.The supplied passage introduces this form but does not state its resulting expression.Source: 2) Closed-Form Phase-Invariant… · p. 12

3) Back-Propagation Rules:

  • For end-to-end training, the phase-invariant dot-product similarity of latent matrices is written as an absolute-value operation applied to their matrix product with conjugate transpose.The formulation is S = f(g(X, Y)), with f(Z)ij = |Zij| and g(X, Y) = XYH.Source: 3) Back-Propagation Rules · p. 12
  • The derivation gives the backpropagation rule for the absolute-value operation on a complex scalar.It states dz = sign(z) ds.Source: 3) Back-Propagation Rules · p. 12
  • The appendix derives gradients for the matrix-product operation by differentiating with respect to both latent matrices.The displayed identities express the differential in terms of changes to X and Y.Source: 3) Back-Propagation Rules · p. 12
  • For phase-invariant distance-similarity, the backpropagation rules additionally use derivatives of the quadratic norm terms.The passage names the row-wise norms of X and Y in these terms.Source: 3) Back-Propagation Rules · p. 12

D. Supplementary Visual Comparisons

The supplementary visuals compare guide and similarity choices and show how complex guides retain phase-derived structure that magnitude guides discard. Guide-slice windows help in severe misregistration, while three slices suffice in the mild example.

E. Complex and Magnitude Guides

  • Phase-invariant similarity removes one constant phase per comparison, whereas a magnitude guide discards phase across the filter support; the complex guide is preferred even with phase-invariant similarity.The residual b captures within-support phase deviation that the single rotation cannot absorb.Source: E. Complex and Magnitude Guides · p. 12

D. Supplementary Visual Comparisons

  • Figure S1 compares four combinations of complex or magnitude guides with distance or phase-invariant similarity across two cases at 8× nominal acceleration.The visual panels report NRMSE, SSIM, and LPIPS ×100; complex-valued panels encode phase by hue.Source: D. Supplementary Visual Comparisons · p. 12, Figure S1 · p. 13

E. Complex and Magnitude Guides

  • For severe through-plane misregistration at 8×, widening the guide window from Ng = 3 to Ng = 5 lowers NRMSE from 9.65% to 8.80%.In the mild case, Ng = 3 suffices; Ng = 5 adds nothing on NRMSE or SSIM and improves only LPIPS.Source: Figure S2 · p. 13, Figure S3 · p. 13
  • The discarded phase-variation channel contains anatomy-related structure and correlates more strongly between visits of one patient than between different patients.The reported correlations are 0.73 between visits of one patient and 0.23 between two different patients.Source: E. Complex and Magnitude Guides · p. 13

F. Masked and Unmasked SSIM

The supplementary material contrasts masked and whole-frame SSIM and documents architecture hyperparameters. Whole-frame averaging raises and compresses SSIM comparisons, while the accompanying tables describe the compared network families.

F. Masked and Unmasked SSIM

  • Between 4.7× and 11.6×, LGGS SSIM falls 13.5% on the brain-support mask but only 5.0% over the frame; at 11.6×, the four-method spread narrows from 15.0% to 5.6%.The background covers the remainder of the frame and scores above 99% throughout, so unmasked comparisons show a third of the difference.Source: F. Masked and Unmasked SSIM · p. 13
  • Figure S5 separates local variation into a magnitude-retained channel and a discarded phase-derived channel, pooled over the 7 × 7 code-formation support on one color key.The discarded channel contains structure not present in the retained channel and repeats between visits.Source: Figure S5 · p. 14

G. Architecture Hyperparameters

  • Tables S1 and S2 list hyperparameters for unrolled sparse-coding and variational-net architectures, with L-TGVN adding a trust-region projection solved by conjugate gradient.Parameter counts treat complex weights as two real parameters and tied weights as one, enabling a common counting basis across architectures.Source: Table S1 · p. 14, Table S2 · p. 14, G. Architecture Hyperparameters · p. 14

F. Masked and Unmasked SSIM

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