ContiLNN: Mitigating Slice Sampling Discontinuity with Liquid Neural Networks for Medical Image Restoration
Authors: Jialei He, Enhe Liu, Sifan Song, Pengfei Jin, Jionglong Su, Hongbin Wang, Zhixiang Lu, Yanhao Huang, +5 more
Organizations: Faculty of Information Engineering and Automation, Kunming University of Science and Technology, Kunming, Yunnan, 650504, China · Academy of Artificial Intelligence and Advanced Technology, Xi’an Jiaotong-Liverpool University, Suzhou, Jiangsu, 215123, China · Department of Radiology, Massachusetts General Hospital and Harvard Medical School, Boston, Massachusetts, 02115, United States · School of BME & Suzhou Institute for Advanced Research, Center for Medical Imaging, Robotics, Analytic Computing & Learning (MIRACLE), University of Science and Technology of China, Suzhou, Jiangsu, 215000, China
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.
Figures & tables
Figure 1: Ordered-slice restoration with ContiLNN. (A) Independent 2D processing. (B) Bidirectional liquid propagation. (C) One restored output per observed slice. Original slice indices it identify the K inputs xit and outputs yit ; t denotes window position. The direction-specific closed-form continuous-time (CfC) update Φ propagates states ht using features et and normalized intervals δt derived from gaps gt . The backward traversal state br is realigned to the original order; interval conventions and alignment are defined in Eqs. 4 and 10 .
Figure 2: ContiLNN architecture based on Restore-RWKV ( Yang et al., 2026a ) . (A) Bi-CfC modules augment selected feature pathways; the paired CT sequences show the same seven observed positions, ordered back to front. (B) Backbone blocks, the bidirectional liquid operator, and gated residual or serial insertion. Reverse traversal preserves feature–interval pairing (Eq. 10 ). (C) Training supervision. High-quality references supervise reconstruction and first- and second-order inter-slice intensity differences. A frozen 2D backbone supplies single-slice predictions for distillation during training; the ContiLNN backbone and Bi-CfC modules remain trainable.
Method
MRI super-resolution
CT denoising
PET restoration
PSNR ↑
SSIM ↑
RMSE ↓
PSNR ↑
SSIM ↑
RMSE ↓
PSNR ↑
SSIM ↑
RMSE ↓
Published reference results: Restore-RWKV study
ARGAN ( Luo et al., 2022 )
30.0800
0.9083
35.7999
32.9200
0.9111
9.2110
36.7500
0.9389
0.0907
DenoMamba ( Öztürk et al., 2026 )
30.3200
0.9091
34.6972
33.1800
0.9115
8.9512
36.8100
0.9367
0.0895
MambaIR ( Guo et al., 2025 )
31.3100
0.9305
31.3150
33.5000
0.9165
8.6345
37.1700
0.9458
0.0864
AirNet ( Li et al., 2022 )
31.3900
0.9316
31.1141
33.6200
0.9176
8.5226
37.1700
0.9451
0.0864
Table 1: Restoration results for MRI super-resolution, CT denoising and reduced-count PET restoration. Published reference results in the upper and lower blocks follow the Restore-RWKV and DASMamba studies, respectively ( Yang et al., 2026a ; Chan et al., 2026 ) ; the upper published block uses joint training across the three modalities. In the lower published block, ∗ marks methods re-implemented by the DASMamba authors. Each block ends with the corresponding backbone reproduced separately for each modality and its ContiLNN counterpart. The reproduced backbones use single-slice (2D) input, whereas ContiLNN uses local ordered-slice (2.5D) input. ContiLNN values are mean ± sample standard deviation across five training seeds. Bold values indicate improvements over the corresponding reproduced backbone within each block. PSNR is reported in dB.
Test context
Slice-axis operator
PSNR ↑
SSIM ↑
RMSE ↓
Gap g=1
Bi-GRU
38.5176
0.9538
0.0753
Bi-CfC
38.5868
0.9546
0.0744
Gap g=2
Bi-GRU
37.6217
0.9462
0.0826
Bi-CfC
37.7136
0.9476
0.0817
Gap g=3
Bi-GRU
36.5295
0.9366
0.0940
Bi-CfC
36.6652
0.9393
0.0927
Table 2: Bi-CfC and Bi-GRU under contiguous training on the 29-case PET test cohort. Both variants use the same insertion sites, objective and training budget. The slice-index gap g specifies the separation between neighboring observations; irregular windows mix g∈{1,2,3} . All settings evaluate the same original target slices with different neighboring context. Bold indicates the better operator within each test condition.
Figure 3: Restoration examples for (A) CT, (B) MRI and (C) PET, with two examples per modality. Columns contain LQ inputs, Restore-RWKV and ContiLNN outputs, HQ references (ground truth, GT), and error-reduction maps. Boxed regions are enlarged below; orange arrows identify corresponding local structures and intensity transitions. Brighter colors indicate larger positive reductions in absolute error relative to Restore-RWKV, measured in normalized display intensity. Error increases are clipped to zero. Each modality shares one color scale; display masking and saturation settings are given in Supplementary Section A.
Patient budget
Training strategy
PSNR ↑
SSIM ↑
RMSE ↓
5%
Restore-RWKV
36.2923
0.9372
0.0973
5%
Budget-matched ContiLNN
36.8837
0.9390
0.0911
5%
Full-refinement ContiLNN
37.2385
0.9343
0.0875
10%
Restore-RWKV
36.5217
0.9389
0.0947
10%
Budget-matched ContiLNN
37.1106
0.9408
0.0886
10%
Full-refinement ContiLNN
37.2121
0.9430
0.0875
Table 3: PET restoration with 5% and 10% of the training patients. Patient subsets are nested, with the validation and test cohorts unchanged. Budget-matched training uses the same number of slice presentations as Restore-RWKV; full refinement adds optimization on the same patient subset. Bold highlights the ContiLNN results.
Figure 4: Intensity profiles across seven ordered CT, MRI and PET slices. P1–P3 mark tissue interfaces or uptake transitions. Each profile averages a 7×7 region centered at the marked location and is normalized using the joint range of the LQ, HQ, Restore-RWKV and ContiLNN profiles at that location. The horizontal axis gives relative slice position; colors and line styles identify the four image sequences.
Figure 5: Spatial error reduction across seven adjacent CT slices: Restore-RWKV relative to LQ (top), and ContiLNN relative to Restore-RWKV (bottom). Colored overlays indicate positive reductions in absolute error against HQ; each row shares one color scale.
Figure 6: Case-level improvements over Restore-RWKV. (A)–(C) PSNR gain, SSIM gain and relative RMSE reduction; positive values favor ContiLNN. Points show all cases. Boxes span the interquartile range (IQR), with median lines and whiskers ending at the most extreme observations within 1.5 IQR of the box. Dashed lines mark zero. (D) Percentage of improved cases. (E) Median PSNR gains with 95% percentile confidence intervals from 10,000 case-bootstrap resamples for MRI and PET. CT has one test case and is descriptive, without a confidence interval.
Computational cost
PET metric differences
Model
Params. (M)
GMAC/ slice
GMAC/ window
Latency/ slice (ms)
Memory (MB)
Δ PSNR ↑ (dB)
Δ SSIM ↑
Δ RMSE ↓
Restore-RWKV
28.37
37.46
262.21
16.72
987.00
0.0000
0.0000
0.0000
ContiLNN-RWKV, MLP
34.92
60.32
422.26
21.23
1835.22
−0.3039
−0.0039
+0.0031
ContiLNN-RWKV, Bi-GRU
34.91
60.50
423.51
54.42
2315.80
+1.2454
+0.0064
−0.0107
ContiLNN-RWKV, Bi-CfC
34.93
60.34
422.39
25.69
995.03
+1.3146
+0.0072
−0.0116
Table 4: Computational cost and PET restoration gains. Profiling uses 128×128 images, 32-bit precision and one NVIDIA A800 GPU: seven independent slices for Restore-RWKV ( B=7 ), or one seven-slice window for each ContiLNN-RWKV variant. MLP and Bi-GRU denote multilayer perceptron and bidirectional gated recurrent unit. M denotes 106 parameters; GMAC denotes 109 multiply–accumulate operations. Latency is median forward time per slice; memory is peak allocated GPU memory. Each Δ is the model’s result minus the reproduced Restore-RWKV result in Table 5 (37.2722 dB PSNR, 0.9474 SSIM, 0.0860 RMSE). Positive PSNR/SSIM and negative RMSE differences indicate improvement. Profiling details are in Supplementary Section A.
ID
Setting
Slice-axis operator
Objective
PSNR ↑
SSIM ↑
RMSE ↓
A
Restore-RWKV
None
Lrec
37.2722
0.9474
0.0860
B
ContiLNN, reconstruction only
Bi-CfC
Lrec
38.4166
0.9533
0.0760
C
ContiLNN, pointwise MLP
Slice-independent MLP
Lfull
36.9683
0.9435
0.0891
D
ContiLNN, recurrent alternative
Bi-GRU
Lfull
38.5176
0.9538
0.0753
E
ContiLNN, full objective
Bi-CfC
Lfull
38.5868
0.9546
0.0744
Table 5: PET ablation on the 29-case test cohort. ContiLNN variants share the pretrained backbone initialization, insertion locations, training budget and evaluation procedure. The MLP processes slices independently; Bi-GRU and Bi-CfC use the same bidirectional slice context. Lfull denotes the full training objective. Bold indicates the highest PSNR and SSIM and the lowest RMSE.
Figure 7: PET context perturbations. (A) Ordered, reversed, center-repeated and nonlocal same-case inputs, illustrated using Patient 10 at center index 3; every condition retains the evaluated center image. (B) PSNR loss, SSIM loss and RMSE increase for 29 test scans using the same trained checkpoint. Points show all cases; box-plot conventions follow Fig. 6 A–C. Differences are relative to ordered context, whose zero value is marked by dashed lines.
Context
Cases
Centers
PSNR ↑
Δ PSNR
SSIM ↑
RMSE ↓
Ordered (default)
29
290
38.719844
0.000000
0.954441
0.073956
Reversed
29
290
38.463333
-0.256511
0.952564
0.076172
Center-repeated
29
290
36.104738
-2.615106
0.936650
0.096815
Nonlocal same-case
29
290
32.368766
-6.351078
0.879486
0.150989
Table 6: PET restoration under the context perturbations illustrated in Fig. 7 . The same trained ContiLNN-RWKV checkpoint is held fixed across all four conditions. Ten center slices are evaluated in each of the 29 test cases. Metrics are averaged over centers within each case and then across cases; Δ PSNR is relative to ordered context. Bold indicates the best PSNR, SSIM and RMSE across these conditions.
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Group
ID
Configuration
PSNR ↑
SSIM ↑
RMSE ↓
Δ PSNR
Baseline
A-CT
Restore-RWKV, no slice-axis module
33.7642
0.9189
8.3950
0.0000
Direction/core
C1
One-direction CfC
33.7907
0.9193
8.3676
+0.0265
C2
One-direction LTC
33.7643
0.9187
8.3928
+0.0001
C3
Bidirectional CfC
33.8150
0.9193
8.3446
+0.0508
C4
Bidirectional LTC
33.7694
0.9187
8.3889
+0.0052
Bottleneck/ refinement
C5
Strict encoder insertion with bottleneck CfC
33.9058
0.9204
8.2586
+0.1416
Appendix
Table S1: CT architecture ablation. Results use the same Restore-RWKV reference and evaluation protocol; Δ PSNR is relative to this reference. The selected configuration is marked in bold.
Figure S1: CT architecture ablation, expressed as PSNR differences from Restore-RWKV. (A) Propagation direction and cell type. (B) Bottleneck and refinement placement. (C, D) Encoder and decoder depth. (E) Pathway combinations. (F) Skip pathways. H8 denotes the selected topology.
Setting
PSNR ↑
SSIM ↑
RMSE ↓
Full objective
33.954120
0.921013
8.213643
Without distillation
33.923577
0.920509
8.241692
Without cross-slice consistency
33.874285
0.919479
8.290462
Without second-order consistency
33.910046
0.920077
8.256208
Appendix
Table S2: CT loss-component ablation under the selected architecture, seed 42. Components are removed from the full objective one at a time. Bold indicates the best value in each metric column.
T
Val. epoch 3
Val. epoch 5
Val. epoch 10
Best val. (epoch)
Test PSNR
Test SSIM
Test RMSE
5
31.7210
31.7450
31.7619
31.7619 (10)
33.8825
0.9202
8.2806
7
31.7079
31.7542
31.7705
31.7719 (8)
33.8941
0.9201
8.2697
9
31.7047
31.7393
31.7574
31.7574 (10)
33.8760
0.9200
8.2865
11
31.7032
31.7311
31.7490
31.7528 (9)
33.8528
0.9194
8.3079
Appendix
Table S3: CT sequence-length analysis with training seed 42 and a 10-epoch budget. Validation and test each use one case. Bold marks the selected seven-slice window.
Figure S2: CT sequence-length analysis. (A) Validation PSNR at epochs 3, 5, and 10. (B) Highest validation PSNR, with the corresponding epoch annotated. Shading identifies the selected seven-slice window.
Perturbation
PSNR loss (dB)
SSIM loss
RMSE increase
Reversed
0.2565 [0.2160, 0.2962]
0.0019 [0.0015, 0.0022]
0.0022 [0.0017, 0.0027]
Center-repeated
2.6151 [2.4952, 2.7333]
0.0178 [0.0160, 0.0197]
0.0229 [0.0211, 0.0246]
Nonlocal same-case
6.3511 [6.0307, 6.6710]
0.0750 [0.0681, 0.0820]
0.0770 [0.0725, 0.0816]
Appendix
Table S4: PET case-level changes relative to the common ordered-context reference. The same trained checkpoint is used in all conditions. Values are means with 95% percentile confidence intervals from 10,000 case-bootstrap resamples. Positive values denote PSNR or SSIM loss, or RMSE increase. Per-case absolute metrics and signed changes are reported in Table S6 .
Operator
Test context
Training
PSNR ↑
SSIM ↑
RMSE ↓
Bi-GRU
g=1
Contiguous
38.5176
0.9538
0.0753
Mixed-gap
38.4541
0.9522
0.0757
g=2
Contiguous
37.6217
0.9462
0.0826
Mixed-gap
38.2747
0.9517
0.0772
g=3
Contiguous
36.5295
0.9366
0.0940
Mixed-gap
37.9980
0.9504
0.0796
Appendix
Table S5: Additional PET sampling experiments on the 29-case test cohort. Both operators are evaluated after contiguous and mixed-gap training at uniform gaps g=1,2,3 and irregular combinations of these gaps. Mixed-gap training uses a constant gap within each window and varies the gap across windows. Bold indicates the better training strategy for each operator and test context.
Case
Context
PSNR
Δ PSNR
SSIM
Δ SSIM
RMSE
Δ RMSE
Patient 10
Ordered (default)
36.761593
0.000000
0.948692
0.000000
0.073840
0.000000
Patient 10
Reversed
36.449061
-0.312532
0.946832
-0.001860
0.077593
0.003753
Patient 10
Center-repeated
34.391667
-2.369926
0.928067
-0.020624
0.094301
0.020461
Patient 10
Nonlocal same-case
31.008852
-5.752742
0.859689
-0.089003
0.142358
0.068518
Patient 101
Ordered (default)
38.906556
0.000000
0.952976
0.000000
0.062351
0.000000
Patient 101
Reversed
38.850965
-0.055590
0.952068
-0.000908
0.062559
0.000207
Appendix
Table S6: PET context-perturbation results for all 29 test cases under four conditions (116 records). Each metric is the mean over the same ten evaluated center slices within a case. Δ denotes the condition result minus the ordered-context result for the same case; negative PSNR or SSIM changes and positive RMSE changes indicate deterioration. PSNR and its change are in dB; SSIM is dimensionless and RMSE retains the original evaluation units. Metrics and differences are rounded independently to six decimal places.
Adapting image restoration models to a stream of new tasks without revisiting past data remains challenging due to catastrophic forgetting. In this work, we propose Restoring without Forgetting (RwF), a filter-level continual adaptation framework for image restoration built upon a critical observation: task-specific knowledge is centered in a small subset of filters and can be separated from those reconstructing general content. RwF first performs parameter-space integrated gradients attribution to localize degradation-critical filters in a coarse-to-fine manner. It then adapts to new tasks by generating task-specific filters from a filter bank using compact factorized low-rank transformations, further augmented with cross-task attention and prototypical contrastive learning, and lastly assembles them back only at localized positions. Experiments on six restoration tasks show that RwF effectively avoids forgetting and achieves competitive restoration quality against all-in-one methods that have full data access, and outperforms LoRA-style adaptation with ∼10× fewer additional parameters. Code is available at https://github.com/funkdub/Restoring-without-Forgetting.
Xin Feng, Jin Zhao, Yizhen Zhang +3
School of Informatics, University of Edinburgh · Baidu Inc. · Tsinghua University +1
Arbitrary slice super-resolution reconstructs isotropic volumes from anisotropic clinical acquisitions by synthesizing intermediate slices at arbitrary scales. However, treating this ill-posed inverse problem as unconstrained residual-based regression risks hallucinating anatomically implausible structures or altering the originally observed data. To address both concerns, this paper presents the Dual-Prior Null-space Learning (DP-NSL) framework, which reformulates the task as a constrained recovery process guided by two complementary priors. A Measurement-Consistent Projection (MCP) enforces a Deterministic Observation Prior: the reconstruction undergoes an exact orthogonal projection that reproduces every acquired slice with zero error, confining all learned details to the unobservable null space. Within this null space, a Mixture-of-Splines (MoS) module imposes a Geometric Continuity Prior by dynamically mixing B-spline experts of different analytic orders, allowing each anatomical region to be modeled with a content-aware level of continuity. To promote spatial coherence, a Local Spatial Consistency Decoder (LSCD) further injects local inductive bias. Experiments on three CT and one MRI benchmark show that DP-NSL outperforms existing approaches while strictly preserving measurement consistency. Code is available at https://github.com/DeepMed-Lab-ECNU/Medical-Image-Reconstruction.
Haofei Song, Siyuan Xu, Xintian Mao +3
Shanghai Key Laboratory of Multidimensional Information Processing, East China Normal University, Shanghai 200241, China
Neighboring-slice self-supervised denoising is attractive for volumetric medical imaging, yet inter-slice misalignment breaks anatomical correspondence and often yields ghosting and blurred margins when adjacent slices are used naively as targets. We propose Neighbor-Guided Patch Sampling (NGPS), a lightweight framework that constructs neighboring supervision under local inter-slice misalignment without explicit registration. To avoid learning from misleading targets, prior methods commonly mask discrepant regions, but this stabilizes training at the cost of leaving a non-trivial portion of neighboring evidence unexploited, particularly around high-frequency anatomical boundaries. NGPS addresses this by decoupling structure matching from signal retrieval: for each masked location, it searches a local neighborhood for structurally similar candidate patches using a simple guide image (e.g., fast bilateral filtering), while retrieving the supervision signal directly from the raw noisy neighbor at the matched coordinates. By matching on a noise-attenuated guide while retrieving raw values from neighboring slices, NGPS constructs local pseudo targets without a learned registration module. Across the evaluated CT and synthetic-Rician MRI settings, NGPS improves fidelity and structure-sensitive metrics. Code is available at https://github.com/cv-cho/NGPS .
Jaehyun Cho, YoungJoon Yoo
Department of Artificial Intelligence, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul, Republic of Korea · SNUAILAB, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea