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ContiLNN: Mitigating Slice Sampling Discontinuity with Liquid Neural Networks for Medical Image Restoration

Jialei He, Enhe Liu, Sifan Song, Pengfei Jin, Jionglong Su, Hongbin Wang, Zhixiang Lu, Yanhao Huang, Anteng Cai, Zhengyong Jiang, Jiaman Ding, S. Kevin Zhou, Jinfeng Wang

arXiv:2610.12337v1cs.CVcs.AI

TL;DR

Medical image restoration must recover anatomical information from acquisitions constrained by radiation dose, scan time, or signal availability, while independent 2D processing does not explicitly retain ordered slice relationships. ContiLNN adds interval-conditioned bidirectional continuous-time modules to 2D backbones, improving fidelity over Restore-RWKV across CT, MRI, and PET.

  • Problem

    Independent 2D processing does not explicitly retain ordered relationships between slices, although neighboring anatomy provides information for restoration.

    Source: §1 · p. 1

  • Method

    ContiLNN augments 2D backbones with bidirectional slice-axis dynamics and regularizes cross-slice intensity changes and agreement with a frozen 2D backbone.

    Source: §1 · p. 2, §3.3 · p. 6

  • Results

    Mean PSNR increases over Restore-RWKV by 0.1907 dB for CT, 1.0176 dB for MRI, and 1.2482 dB for PET, with higher SSIM and lower RMSE across the three tasks.

    Source: §4.3 · p. 7

  • Takeaways & Limitations

    Experiments support data efficiency and compatibility with a second backbone, while PET ablations support ordered propagation and complementary regularization.

    Source: §6 · p. 14

    The CT test partition contains one patient, and generalization across acquisition protocols has not been evaluated.

    Source: §5.3 · p. 14

Abstract

from arXiv · show

Anatomical continuity provides complementary information for medical image restoration, but its use requires accounting for local anatomy and variations in slice sampling. We introduce ContiLNN, which augments two-dimensional restoration backbones with bidirectional closed-form continuous-time (Bi-CfC) modules for cross-slice modeling while retaining in-plane feature extraction. Slice-index intervals modulate gates determined by local features and hidden states, enabling propagation to respond to sampling variations without numerical ODE integration. Reference-guided consistency aligns first- and second-order cross-slice intensity differences to preserve anatomical variation, while distillation from a frozen backbone helps retain in-plane fidelity. Across five training seeds, ContiLNN improves mean PSNR over Restore-RWKV by 0.1907, 1.0176, and 1.2482 dB for CT denoising, MRI super-resolution, and reduced-count PET restoration, respectively, with lower RMSE in all three tasks. CT results are descriptive for one held-out patient. PET ablations support ordered propagation beyond additional pointwise capacity. Under contiguous training, Bi-CfC achieves higher fidelity than a Bi-GRU with similar parameter counts and arithmetic costs across all tested sampling conditions. Matched seven-slice profiling shows 52.8% lower latency and 57.0% lower peak GPU memory use than Bi-GRU. Mixed-gap training improves sparse and irregular-context performance for both operators, without a uniform ranking across metrics and contexts. Experiments with fewer training patients and a second backbone further support data efficiency and backbone compatibility.

1. Introduction

Medical restoration must recover anatomy from noisy or undersampled acquisitions, while independent 2D processing does not explicitly retain ordered relationships between slices. ContiLNN adds interval-conditioned bidirectional slice modeling to 2D backbones and uses consistency and distillation to guide restoration.

1. Introduction

  • Restoration addresses low-quality inputs produced by acquisition constraints, including increased CT noise, limited MRI resolution, and reduced PET counts.These constraints can make structural detail or local uptake differences harder to recover.Source: §1 · p. 1
  • Independent 2D processing does not explicitly retain ordered through-slice relationships, and available slices may be regularly or irregularly spaced in the source series.Original slice indices distinguish neighboring inputs that need not be consecutive source slices.Source: §1 · p. 1
  • ContiLNN adds bidirectional liquid propagation to 2D restoration backbones, retaining an output for each observed slice.Slice-index intervals condition the updates, and backward states are realigned to the original order.Source: §1 · p. 2, Figure 1 · p. 2
  • Cross-slice consistency preserves first- and second-order intensity changes in the reference sequence, while frozen-backbone distillation retains agreement with in-plane estimates.The method is developed on Restore-RWKV and also applied to DASMamba.Source: §1 · p. 2
  • The evaluation spans CT, MRI, and PET, with analyses of sampling, limited-patient training, backbone compatibility, and ablations.These studies assess restoration quality and the method's behavior across sampling settings.Source: §1 · p. 2

2. Related work

Prior restoration methods provide in-plane feature hierarchies and use through-plane context in several ways. ContiLNN models ordered slice features with interval-conditioned bidirectional dynamics and regularizes both cross-slice variation and agreement with a frozen 2D backbone.

2.1. Medical image restoration

  • Prior restoration methods provide in-plane feature hierarchies, while ContiLNN targets cross-slice context (Yang et al., 2026a; Chan et al., 2026).The paper distinguishes this focus from FedFTN's work on institutional heterogeneity in low-count PET denoising (Zhou et al., 2023).Source: §2.1 · p. 3

2.2. Slice-axis context

  • Through-plane context can come from neighboring-slice convolution, volumetric features, supervision, or multi-view inference; ContiLNN instead maintains an ordered feature sequence and restores every supplied slice.Its interval inputs derive from original index gaps between selected observations.Source: §2.2 · p. 3

2.3. Liquid neural dynamics

  • CfCs retain explicit dependence on observation intervals, allowing ContiLNN's bidirectional updates to distinguish larger index gaps from consecutive slices.Bi-CfC is evaluated on sparse and irregular contexts after contiguous training; mixed-gap training is investigated separately.Source: §2.3 · p. 3

2.4. Consistency and distillation

  • The method matches first- and second-order intensity differences between restored and reference sequences, while a separate distillation term encourages agreement with a frozen pretrained 2D backbone.The cross-slice constraint accommodates anatomical boundaries and uptake patterns that vary across slice positions.Source: §2.4 · p. 3

3. Method

ContiLNN restores ordered slice windows by adding bidirectional, interval-conditioned feature propagation to 2D backbones. Reference-guided cross-slice consistency and frozen-backbone distillation complement reconstruction supervision.

3.1. Ordered-slice restoration

  • Each local window maps ordered low-quality slices to position-matched restored outputs, retaining original slice-index gaps to represent the spacing between observations.The model directly supervises each output with its corresponding high-quality reference slice.Source: §3.1 · pp. 3–4

3.2. Bidirectional liquid dynamics

  • Bi-CfC propagates feature states in both directions, conditioning closed-form updates on slice intervals and local features without numerical ODE integration.The interval input is normalized by the number of observations in the window, and both traversal directions preserve feature–interval pairing.Source: §3.2 · p. 4

3.3. Backbone augmentation

  • ContiLNN retains Restore-RWKV’s in-plane blocks and adds Bi-CfC to selected pathways, using serial insertion at the bottleneck and gated residual insertion elsewhere.The same insertion roles are applied to DASMamba to examine backbone compatibility.Source: §3.3 · p. 6

3.4. Training objective

  • Training combines reconstruction, cross-slice consistency, and distillation from a frozen 2D backbone to preserve reference intensity changes and in-plane fidelity.Consistency matches first- and second-order inter-slice differences; gradients update the ContiLNN backbone and Bi-CfC modules, not the frozen backbone.Source: §3.4 · p. 6

3.5. Inference

Inference covers observed slices with local, case-specific windows, including short terminal windows. Overlapping window predictions are fused and returned in the original slice order.

3.5. Inference

  • Inference uses local, case-specific windows to cover the observed slices, retaining short terminal windows.The window length and tiling offsets are specified in Supplementary Section A.Source: §3.5 · p. 7
  • For each slice, inference fuses predictions from every overlapping window that contains it.The fused output averages those window predictions.Source: §3.5 · p. 7
  • Fused outputs return to original slice order, while sparse and irregular contexts retain position-aligned outputs as contextual support changes.Each hidden state is initialized within its own window.Source: §3.5 · p. 7

4. Experiments

ContiLNN is evaluated on CT denoising, MRI super-resolution, and reduced-count PET restoration, with additional tests of sampling robustness, data efficiency, and architectural choices. Across the three tasks, it improves restoration fidelity over Restore-RWKV, while CT results describe one held-out patient.

4.1. Datasets

  • The benchmarks cover paired CT, MRI, and PET restoration tasks, with five-seed estimates and case-disjoint partitions; the CT test partition contains one patient.Tasks are CT denoising, MRI super-resolution, and reduced-count PET restoration.Source: §4.1 · p. 7, §4.2 · p. 7

4.2. Experimental setup

  • ContiLNN jointly trains the augmented backbone and Bi-CfC modules, using a frozen 2D backbone for distillation; principal results use contiguous slice windows.Metrics are averaged within patient or scan, then across cases with equal weight.Source: §4.2 · p. 7

4.3. Quantitative results

  • 0.1907, 1.0176, and 1.2482 dB mean PSNR gains over Restore-RWKV on CT, MRI, and PET accompany higher SSIM and lower RMSE.The results are means across five training seeds; CT is evaluated on one held-out patient.Source: §4.3 · p. 7, §4.2 · p. 7, Table 1 · p. 8

4.4. Sampling robustness

  • Under contiguous training, Bi-CfC exceeds Bi-GRU in PSNR, SSIM, and RMSE across four PET test contexts; mixed-gap training benefits both without a uniform ranking.The Bi-CfC PSNR advantage over Bi-GRU grows from 0.0693 dB at g = 1 to 0.1357 dB at g = 3.Source: §4.4 · pp. 8–9, Table 2 · p. 8

4.5. Data efficiency

  • With matched target-slice exposure, ContiLNN improves PET PSNR by 0.5914 and 0.5889 dB using 5% and 10% of training patients.Full refinement adds optimization on the same subsets, with further PSNR and RMSE gains but different SSIM effects at the two fractions.Source: §4.5 · p. 9, Table 3 · p. 11

4.6. Qualitative results

  • Across CT, MRI, and PET examples, ContiLNN better retains local structures and reference intensity transitions; selected MRI orthogonal views also follow HQ boundaries more closely.For the displayed CT P2, MRI P2, and PET P3 profiles, mean absolute deviations fall from 0.290 to 0.135, 0.150 to 0.066, and 0.257 to 0.085.Source: §4.6 · p. 9, Figure 4 · p. 10

4.7. Case-level analysis

  • All 114 MRI subjects and 29 PET scans improve in PSNR, SSIM, and RMSE; median PSNR gains are 1.0443 dB and 1.1711 dB, respectively.All six Holm-adjusted p values are below 1.2 × 10^-8.Source: §4.7 · p. 9, Figure 6 · p. 12

4.8. Ablation studies

  • With the full objective, Bi-CfC exceeds the slice-independent MLP by 1.6185 dB PSNR and 0.0111 SSIM, and adds 0.1702 dB PSNR over reconstruction-only Bi-CfC, supporting propagation beyond pointwise capacity.Under the same trained ContiLNN-RWKV checkpoint and ten center positions per scan across 29 test scans, reversal, center repetition, and nonlocal same-case sampling reduce PSNR by 0.2565, 2.6151, and 6.3511 dB, respectively.Source: §4.8 · pp. 12–13

5. Discussion

ContiLNN combines ordered cross-slice propagation with supervision of anatomical intensity changes and in-plane fidelity. The discussion reports benefits across modalities and sampling conditions, while noting limits on what the evaluation establishes.

5.1. Anatomical context and restoration

  • PET context perturbations show larger losses when distinct local neighbors are removed or replaced than when their order is reversed.This pattern supports local anatomical content as an important information source, with ordering making an additional contribution to fidelity.Source: §5.1 · p. 13
  • Reference-based differences guide predictions to follow anatomical intensity changes, while frozen-backbone distillation provides an in-plane reference as neighboring observations are incorporated.The discussion describes the two forms of supervision as complementary.Source: §5.1 · pp. 13–14
  • Across CT, MRI, and PET, the combined design improves fidelity under different degradation processes, with case-level analyses extending across the evaluated MRI and PET cohorts.Examples link gains to anatomical interfaces and uptake transitions.Source: §5.1 · p. 14

5.2. Sampling and e

  • Under contiguous training, Bi-CfC maintains higher restoration fidelity than Bi-GRU across tested sparse and irregular contexts, with mixed-gap training benefiting both operators.Bi-CfC also has lower measured latency and peak allocated memory than Bi-GRU in the evaluated setting.Source: §5.2 · p. 14
  • Matched seven-slice profiling on an NVIDIA A800 found 52.8% lower median window latency and 57.0% lower peak allocated memory for Bi-CfC than Bi-GRU.The operators had closely matched parameter counts and arithmetic costs under these profiling conditions.Source: §5.2 · p. 14

5.3. Limitations

  • The evidence uses fixed restoration benchmarks and local slice windows, and the preprocessed data lack physical slice-spacing metadata for testing adaptation to calibrated anatomical distances.Window length and index gaps jointly determine the anatomical extent of available context; the CT test partition also contains only one patient, so seed variation does not establish between-patient robustness.Source: §5.3 · p. 14

5.4. Future directions

  • Future work could use DICOM-derived physical coordinates, adaptive context lengths, and selective feature-level propagation, alongside evaluation across centers, protocols, downstream tasks, and reader studies.The authors also propose comparisons with slice-axis convolution and attention under matched budgets.Source: §5.4 · p. 14

6. Conclusion

ContiLNN adds bidirectional liquid propagation over ordered slice positions to two-dimensional restoration backbones, conditioning updates on slice-index intervals. The conclusion reports improved fidelity across three modalities and supports Bi-CfC as the preferred operator under the evaluated conditions.

6. Conclusion

  • ContiLNN improves fidelity across CT, MRI, and PET, with limited-patient PET experiments and a second backbone supporting data efficiency and architectural compatibility.Under contiguous training, Bi-CfC has higher fidelity than Bi-GRU across tested sampling conditions, with lower measured inference latency and peak allocated memory.Source: §6 · p. 14

Data availability; Code availability; Ethics statement; Funding; Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

A.1 Slice windows and liquid modules

  • Ordered windows contain at most seven slices, and the interval supplied to the liquid cell is the constant slice-index gap divided by the actual window length.Original slice indices are retained so intervals encode gaps in the source sequence.Source: §A.1 · p. 18

A.2 Restore-RWKV training

  • The Restore-RWKV baseline initializes from its highest-validation-PSNR checkpoint, and a frozen copy supplies predictions for distillation during joint ContiLNN training.Training details include a 300,000-update baseline schedule and five seeds for the main Table 1 results.Source: §A.2 · p. 18

A.3 DASMamba integration

  • ContiLNN-DASMamba follows the Restore-RWKV integration workflow while retaining DASMamba’s native spatial and Fourier-domain reconstruction terms.Cross-slice consistency and distillation objectives are also used.Source: §A.3 · p. 19

A.4 Sampling robustness

The supplementary protocols specify PET sampling comparisons, data-efficiency budgets, intensity processing, computational profiling, and evaluation procedures. The profiling compares Bi-CfC and Bi-GRU under matched seven-slice conditions.

A.4 Sampling robustness

  • PET sampling tests compare Bi-CfC and Bi-GRU after contiguous or mixed-gap training, evaluating the same original target slices across consecutive, sparse, and irregular contexts.Mixed-gap windows use constant gaps of 1, 2, or 3, with approximately equal numbers of windows at each gap and the same total windows and optimizer updates per epoch as contiguous training.Source: §A.4 · p. 19

A.5 Data efficiency

  • The PET data-efficiency experiments use nested subsets of 6 and 12 training patients, with validation and test cohorts fixed.Budget-matched ContiLNN allocates 0.6 million slice presentations to the 2D pathway and a further 0.6 million to sequence training; full refinement adds sequence training after the ratio-matched baseline schedule.Source: §A.5 · p. 19

A.6 Intensity processing

  • Training and evaluation use modality-specific intensity ranges, with random 128 × 128 crops, rotations, and flips applied consistently across slices in each training window.Evaluation includes inverse normalization and modality-specific truncation ranges.Source: §A.6 · p. 19

A.7 Computational profiling

  • Bi-CfC’s median seven-slice window latency is 179.85 ms, compared with 380.92 ms for Bi-GRU, whose models have 34.50 million and 34.52 million trainable parameters, respectively.Profiling uses an NVIDIA A800 GPU, 32-bit precision, and 128 × 128 images; these are model forward-pass timings and exclude data loading, window assembly, and full-volume tiling fusion.Source: §A.7 · pp. 19–20

A.8 Evaluation and statistical settings

  • Slice-level metrics are averaged within each case before aggregation across cases, and multi-seed estimates report means and sample standard deviations across five seeds.Paired case analyses use specified single seeds for each modality; the CT test cohort contains one case, so its measurements describe that case.Source: §A.8 · p. 20

B Architecture ablation

The supplementary ablations favor selective, gated cross-slice insertion and complementary objective terms, with seven slices selected in the CT sequence-length study. PET context perturbations compare the contribution of local anatomical content and its ordering.

B Architecture ablation

  • Intermediate encoder and decoder pathways are more effective insertion sites than extending liquid modules across all feature levels, and the selected topology combines serial bottleneck processing with gated feature updates.Among skip-path variants, gated insertion at L2 and L3 gives the highest PSNR and lowest RMSE, supporting H8.Source: B Architecture ablation · pp. 20–21, Table S1 · p. 20, Figure S1 · p. 21

C Objective decomposition

  • Removing cross-slice consistency causes the largest PSNR reduction, while the second-order term and 2D backbone distillation each provide further improvement.The ablation removes one term at a time under the selected architecture while preserving the other terms, initialization, and training setting.Source: C Objective decomposition · pp. 21–22, Table S2 · p. 22

D Sequence length

E Context perturbations

  • PET context perturbations compare ordered context with reversed local order, center repetition, and nonlocal same-case sampling using the same trained checkpoint and fixed center positions.Table S4 reports mean changes relative to ordered context with 95% case-bootstrap confidence intervals; deterioration is indicated by PSNR or SSIM loss or RMSE increase.Source: E Context perturbations · p. 22, Table S4 · p. 23

F PET Stage-A trajectory

The PET Restore-RWKV validation trajectory peaked at update 87,000, and MRI orthogonal-view figures show restoration across three index-space planes for two cases.

F PET Stage-A trajectory

  • 38.1207 dB PSNR was the PET Restore-RWKV validation maximum, reached at update 87,000 with SSIM 0.9541 and RMSE 0.0813.The checkpoint was retained; the remaining 213 validation evaluations did not exceed that PSNR.Source: F PET Stage-A trajectory · p. 23
  • Figure S3 displays all 300 recorded PSNR, SSIM, and RMSE validation values across 300,000 optimizer updates.The dashed line marks the highest validation PSNR at update 87,000.Source: Figure S3 · p. 23, F PET Stage-A trajectory · p. 23

G MRI orthogonal views

  • Figures S4 and S5 show LQ, Restore-RWKV, ContiLNN-RWKV, and HQ images in three index-space planes for two MRI cases.Matched red-boxed regions are enlarged below; the 100 processed slices are retained along the D axis, and aspect ratios follow array indices rather than physical spacing.Source: Figure S4 · p. 24, Figure S5 · p. 25

H Additional mixed-gap training study

Mixed-gap training improves sparse and irregular-context results for both operators, but their relative metric rankings vary with the sampling condition.

Mixed-gap training results

  • Bi-CfC gains 0.5525, 1.3300, and 0.4032 dB in PSNR at g = 2, g = 3, and irregular sampling; Bi-GRU gains 0.6530, 1.4685, and 0.4535 dB.Both operators also improve in SSIM and RMSE in those three settings relative to contiguous training.Source: H Additional mixed-gap training study · p. 26
  • After mixed-gap training, Bi-GRU has marginally higher PSNR at the three uniform gaps, while Bi-CfC has higher SSIM across all four contexts and wins all three metrics under irregular sampling.At g = 1, contiguous training retains PSNR advantages of 0.1568 dB for Bi-CfC and 0.0635 dB for Bi-GRU.Source: H Additional mixed-gap training study · p. 26

Mixed-gap training setup

  • Table S5 evaluates both operators after contiguous and mixed-gap training at g = 1, 2, 3, and irregular gap combinations on 29 PET test cases.Mixed-gap training keeps each window's gap constant while varying it across windows.Source: Table S5 · p. 26

I Contiguous-training performance and inference cost

Figure S6 compares Bi-CfC and Bi-GRU across PET sampling conditions and reports seven-slice latency and peak allocated GPU memory under the profiling conditions.

PET performance comparison

  • Figure S6A compares mean PET PSNR across 29 test cases under four sampling conditions, with annotations showing Bi-CfC minus Bi-GRU.Lines connect categorical conditions for visual comparison; the figure caption does not state the comparison's outcome.Source: Figure S6 · p. 27

Inference-cost profiling

  • Figure S6B reports seven-slice window latency and peak allocated GPU memory, normalized to Bi-GRU, with absolute values and relative reductions labeled.The caption identifies these as the main Table 4 profiling conditions without providing the values.Source: Figure S6 · p. 27

Performance and cost comparison

  • The figure juxtaposes PET fidelity across sampling conditions with inference-cost measurements for a seven-slice window.The performance panel uses the main Table 2 conditions, while the profiling panel uses the main Table 4 conditions.Source: Figure S6 · p. 27

J Case-level PET context results

Table S6 reports PET context-perturbation metrics and within-case changes for four conditions across the 29-case test cohort.

Case-level PET context results

  • Table S6 covers 116 records across four conditions for all 29 PET test cases, averaging each metric over the same ten center slices per case.Metrics are rounded independently to six decimal places.Source: Table S6 · p. 28
  • Table S6 defines Δ as each condition's result minus the ordered-context result for the same case.Negative PSNR or SSIM changes and positive RMSE changes indicate deterioration.Source: Table S6 · p. 28
  • PSNR and its change are reported in dB, SSIM is dimensionless, and RMSE retains the original evaluation units.The caption provides metric conventions but no outcome values.Source: Table S6 · p. 28

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