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

cs.CVOct 8, 2026

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

Abstract

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 $i_{t}$ identify the $K$ inputs $x_{i_{t}}$ and outputs $\widehat{y}_{i_{t}}$ ; $t$ denotes window position. The direction-specific closed-form continuous-time (CfC) update $\Phi$ propagates states $h_{t}$ using features $e_{t}$ and normalized intervals $\delta_{t}$ derived from gaps $g_{t}$ . The backward traversal state $b_{r}$ 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.

Published reference results: Restore-RWKV study
Method	MRI super-resolution			CT denoising			PET restoration
Method	PSNR $\uparrow$	SSIM $\uparrow$	RMSE $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	RMSE $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	RMSE $\downarrow$
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

\pm

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 $\uparrow$	SSIM $\uparrow$	RMSE $\downarrow$
Gap $g=1$	Bi-GRU	38.5176	0.9538	0.0753
Gap $g=1$	Bi-CfC	38.5868	0.9546	0.0744
Gap $g=2$	Bi-GRU	37.6217	0.9462	0.0826
Gap $g=2$	Bi-CfC	37.7136	0.9476	0.0817
Gap $g=3$	Bi-GRU	36.5295	0.9366	0.0940
Gap $g=3$	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\in\{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 $\uparrow$	SSIM $\uparrow$	RMSE $\downarrow$
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\times 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.

Model	Params. (M)	GMAC/ slice	GMAC/ window	Latency/ slice (ms)	Memory (MB)	$\Delta$ PSNR $\uparrow$ (dB)	$\Delta$ SSIM $\uparrow$	$\Delta$ RMSE $\downarrow$
	Computational cost					PET metric differences
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\times 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

10^{6}

parameters; GMAC denotes

10^{9}

multiply–accumulate operations. Latency is median forward time per slice; memory is peak allocated GPU memory. Each

\Delta

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 $\uparrow$	SSIM $\uparrow$	RMSE $\downarrow$
A	Restore-RWKV	None	$\mathcal{L}_{\mathrm{rec}}$	37.2722	0.9474	0.0860
B	ContiLNN, reconstruction only	Bi-CfC	$\mathcal{L}_{\mathrm{rec}}$	38.4166	0.9533	0.0760
C	ContiLNN, pointwise MLP	Slice-independent MLP	$\mathcal{L}_{\mathrm{full}}$	36.9683	0.9435	0.0891
D	ContiLNN, recurrent alternative	Bi-GRU	$\mathcal{L}_{\mathrm{full}}$	38.5176	0.9538	0.0753
E	ContiLNN, full objective	Bi-CfC	$\mathcal{L}_{\mathrm{full}}$	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.

\mathcal{L}_{\mathrm{full}}

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 $\uparrow$	$\Delta$ PSNR	SSIM $\uparrow$	RMSE $\downarrow$
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;

\Delta

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 $\uparrow$	SSIM $\uparrow$	RMSE $\downarrow$	$\Delta$ 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;

\Delta

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 $\uparrow$	SSIM $\uparrow$	RMSE $\downarrow$
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 $\uparrow$	SSIM $\uparrow$	RMSE $\downarrow$
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	$\Delta$ PSNR	SSIM	$\Delta$ SSIM	RMSE	$\Delta$ 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.

\Delta

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.

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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

\sim

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