Dual-Prior Guided Null-Space Learning with Mixture-of-Splines for Arbitrary Medical Slice Super-Resolution¶

Conference: ECCV2026
arXiv: 2606.26716
Code: https://github.com/DeepMed-Lab-ECNU/Medical-Image-Reconstruction
Area: Medical Image
Keywords: Medical slice super-resolution, Null-space learning, Mixture-of-Splines, Measurement consistency projection, Inverse problems

TL;DR¶

This work reformulates arbitrary-scale medical slice super-resolution as a constrained inverse problem. By utilizing range-null space decomposition, it achieves a hard measurement consistency constraint. Within the null space, a Mixture-of-Splines (MoS) adaptively assigns different continuity orders based on local anatomical features, strictly guaranteeing that the acquired slices remain unchanged while generating anatomically plausible interpolated details.

Background & Motivation¶

Modern volumetric imaging pipelines such as CT and MRI are limited by acquisition time and radiation dosage, typically resulting in anisotropic 3D volumetric data where the inter-slice spacing is much larger than the in-plane resolution. This anisotropy significantly hinders downstream tasks like 3D visualization, medical analysis, and diagnosis. Therefore, arbitrary-scale slice super-resolution (reconstructing arbitrary intermediate slices from sparse scan) becomes a necessary post-processing step. Classic linear interpolation is deterministically stable but struggles to recover high-frequency anatomical details, producing over-smoothed results. Recently, implicit neural representation (INR) methods have modeled volumes as continuous functions, achieving remarkable visual improvements.

However, arbitrary-scale medical slice super-resolution is inherently an ill-posed inverse problem. When treated as an unconstrained residual regression, the networks generate non-zero errors on the observed slices and even hallucinate anatomically implausible details, which is unacceptable in clinical scenarios. The key challenge lies in the trade-off between ensuring that the reconstructed results are perfectly consistent with the originally acquired slices (data fidelity) and generating high-fidelity anatomical details in unseen regions. Existing INR methods lack explicit data fidelity constraints, and pure black-box networks cannot provide the mathematical guarantees required in clinical practice.

The key insight of this paper is that since the range-null space decomposition in basis pursuit theory naturally divides the reconstruction into measurable range-space anchors and learnable null-space components, slice super-resolution can be reformulated from "unconstrained prediction" to "dual-prior constrained reconstruction". Core Idea: By utilizing the orthogonal projection Π_𝒩 = I − UD in range-null space decomposition as a hard constraint, the network output is forcibly mapped back into the null space of the degradation operator, mathematically guaranteeing that all acquired slices remain unchanged. Concurrently, within this null space, a Mixture-of-Splines (MoS) dynamically assigns different analytical continuity orders to adaptively adjust the stiffness of the geometric prior according to local anatomical features.

Method¶

Overall Architecture¶

The core concept of DP-NSL is to decompose medical slice super-resolution into the sum of two orthogonal components: a deterministic range-space anchor (ensuring that the observed slices remain strictly unchanged) and a learnable null-space component (hosting all high-frequency detail generation). The entire framework consists of three cascaded core modules. First, the pseudo-inverse of the degradation operator performs trilinear upsampling on the low-resolution voxels to obtain the immutable range-space anchor; concurrently, a 3D encoder extracts low-resolution spatial features. Then, these features enter the Null-Space Estimator (NSE)—performing content-adaptive geometric interpolation in a continuous coordinate field via a Mixture-of-Splines (MoS), followed by injecting local inductive biases using a Local Spatial Consistency Decoder (LSCD) to generate the raw null-space predicted voxels. Finally, the Measurement Consistency Projection (MCP) acts as a fixed orthogonal projection operator to map the raw predictions onto the null space of the degradation operator. Adding this projection to the range-space anchor yields the final super-resolution result.

Since the MCP projection satisfies 𝒟Π_𝒩 = 0, any component passing through the projection is "invisible" to the degradation operator. Thus, it does not affect the acquired slices, establishing data fidelity with rigorous linear algebraic guarantees.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["低分辨率体素 V_LR"] --> B["三线性上采样<br/>值空间锚点 V_R"]
    A --> C["3D编码器<br/>提取空间特征"]
    C --> D["连续坐标场 Query"]
    D --> E["混合B样条专家 MoS<br/>内容自适应几何插值"]
    E --> F["局部一致性解码器 LSCD<br/>Inception式多尺度融合"]
    F --> G["原始零空间预测 V̂_N"]
    G --> H["MCP正交投影<br/>Π_𝒩 = I − UD"]
    H --> I["零空间分量 V_N"]
    B --> J["超分辨率体素 V_SR<br/>= V_R + V_N"]
    I --> J

Key Designs¶

1. Measurement Consistency Projection MCP: Hard Fidelity Constraints via Range-Null Space Decomposition

MCP parameterizes the reconstruction problem into an orthogonal dichotomy based on range-null space decomposition. Given a degradation operator 𝒟 (corresponding to the downsampling process of CT/MRI slice acquisition), its pseudo-inverse 𝒰 (trilinear upsampling here) is constructed to satisfy 𝒟𝒰 = I. For any super-resolution voxel V_SR, the orthogonal decomposition V_SR = 𝒰𝒟(V_SR) + (I − 𝒰𝒟)V_SR always holds. The first term 𝒰(V_LR) is uniquely determined by the low-resolution input, serving as a deterministic anchor in the range space. The second term is obtained by applying the projection Π_𝒩 = I − 𝒰𝒟. Since 𝒟Π_𝒩 = 0, it naturally lies in the null space of the degradation operator and is "invisible" to the acquired slices.

However, the V_SR in the second term is the unknown ground-truth voxel. The strategy of DP-NSL is to use the null-space estimator NSE to output a raw prediction V̂_𝒩 and then force it into the null space through MCP projection Π_𝒩(V̂_𝒩). The final reconstruction is V_SR = V_R + Π_𝒩(V̂_𝒩). This linear algebraic constraint guarantees that no matter what the network predicts, the pixel values at the acquired slice positions remain strictly unchanged—providing a stronger, theoretically guaranteed constraint than any loss-based soft constraint.

2. Mixture-of-Splines MoS: Adaptive Continuity Order Assignment Based on Local Anatomical Features

Biological tissues exhibit heterogeneous smoothness: uniform organs (such as liver parenchyma) require high-order continuity to produce smooth transitions, whereas structural interfaces (such as bone-soft tissue boundaries) require low-order continuity to preserve sharp boundaries. The core idea of MoS is to model continuous spatial interpolation using learnable B-spline bases instead of black-box MLPs, and to adapt to this heterogeneity by mixing multiple spline experts of different analytical orders.

Specifically, MoS maintains K B-spline upsampling experts {𝒰_{p_k}}, each having a different analytical continuity order p_k (experimentally chosen as p∈{2,3,4}). For a query coordinate q, structure-aware scaling coefficients c, adaptive knots k, and continuous dilation scales s are projected from the nearest neighbor feature neighborhood through shallow convolutions. Each expert then calculates the B-spline basis response using a separable tensor product:

\[\mathcal{U}_p(\mathbf{F}_{LR};\mathbf{q}) = \mathbf{c} \odot (\mathbf{b}_x \otimes \mathbf{b}_y \otimes \mathbf{b}_z)\]

where b_d is a 1D B-spline basis vector, computed by inserting the relative projection distance Δq_d (modulated by knots and dilation parameters) into the B-spline basis function β^p of order p. Meanwhile, a lightweight routing network predicts the mixture weights π(q) = Softmax(R_π(F̃LR^q)) from local interpolated features. The high-resolution feature at this coordinate is the weighted mixture of each expert's output: F_SR^q = Σ_k π_k(q) · 𝒰(F_LR; q). This mechanism allows uniform areas to be dominated by low-order experts (smooth transition) and boundary areas by high-order experts (preserving sharpness), achieving automatic routing distribution without explicit boundary supervision.

3. Local Spatial Consistency Decoder LSCD: Reintroducing Local Inductive Bias with Multi-scale Convolutions

MoS provides a macroscopic continuity prior, but because MoS essentially evaluates continuous basis functions independently for each coordinate, it still lacks spatial neighborhood consistency constraints. The goal of LSCD is to inject local inductive bias into the mapping step from high-resolution features back to raw null-space predictions.

LSCD adopts an Inception-style split-transform-merge mechanism: the high-resolution features output by MoS are split along the channel dimension into 5 groups—an identity branch (retaining high-frequency base signals), a standard 3D convolutional branch, and three depthwise 3D convolutional branches with kernel sizes of 3×3×3, 5×5×5, and 7×7×7, respectively. This design allows the network to capture multi-scale local spatial contexts at a low computational cost: small kernels focus on fine neighborhood consistency, while large kernels target broader contextual coordination. By cascading multiple such Inception blocks, the final raw null-space prediction V̂_𝒩 is projected. Ablation studies show that LSCD not only improves reconstruction accuracy but also achieves superior performance with only 71K parameters (compared to 214K for MLP decoders) and lower FLOPs.

Loss & Training¶

The model is optimized using L1 reconstruction loss, directly supervising the discrepancy between the final super-resolution voxels V_SR and the GT voxels V_HR (no separate supervision for intermediate layers or extra regularization is used). The optimizer is Adam (β1=0.9, β2=0.999) with an initial learning rate of 1e-4 halved every 200 epochs, for a total of 1000 epochs with a batch size of 8.

Key Experimental Results¶

Main Results¶

Dataset	Scale	Metric	DP-NSL	Prev. SOTA	Gain
Colon (CT)	×2	PSNR	42.55	41.84 (EDSR3D)	+0.71 dB
Colon (CT)	×3	PSNR	38.43	37.78 (SAINR)	+0.65 dB
Colon (CT)	×4	PSNR	35.95	35.37 (EDSR3D)	+0.58 dB
Liver (CT)	×2	PSNR	42.49	41.42 (HIIF)	+1.07 dB
IXI (MRI)	×4	PSNR	41.92	41.41 (DC2SR)	+0.51 dB
IXI (MRI)	×5	PSNR	40.17	39.55 (SAINR)	+0.62 dB

Ablation Study¶

Configuration	×2 PSNR	×4 PSNR	Description
Baseline (w/o MCP/w/o MoS/w/o LSCD)	41.82	35.30	Pure EDSR + trilinear upsampling
+MCP	42.13 (+0.31)	35.52 (+0.22)	Null-space constraint alone is effective
+MoS	42.45 (+0.63)	35.82 (+0.52)	Geometric prior yields the largest improvement
+MCP+MoS	42.52 (+0.70)	35.89 (+0.59)	Complementary dual priors
Full DP-NSL	42.54 (+0.72)	35.93 (+0.63)	Further improvement with LSCD

Key Findings¶

Ablation on MoS spline orders shows that among single-order configurations, p=3 performs best. However, the mixture of multiple orders (p∈{2,3,4}) significantly outperforms any single-order configuration under all scenarios, showing that heterogeneous anatomical regions indeed require spatially varying continuity priors.
Visualization of routing weights confirms semantic spontaneity: low-order experts are strongly activated inside uniform soft tissues, whereas high-order experts focus on fine blood vessel branches and organ boundaries. This content-aware distribution is entirely data-driven, without relying on explicit boundary annotations.
Slice-level MAE validation for MCP perfectly demonstrates the zero-error paradigm: at the positions of originally acquired slices (s=0, R, 2R, ...), the MAE is exactly zero, whereas all baseline methods show non-negligible errors at these positions—implying they alter the original clinical data to various degrees.
On unseen extrapolation scales (×5, ×6, ×7), the advantage of DP-NSL becomes even more pronounced (boosting PSNR by 0.62 dB on IXI at ×5). This is attributed to the inherent mathematical continuity of B-spline bases, which does not depend on sampling density, enabling principal extrapolation under unseen resolutions.

Highlights & Insights¶

Range-null space orthogonal decomposition is the optimal data-fidelity strategy: MCP encodes the requirement of "preserving originally acquired slices" as a hard constraint via linear algebra rather than a loss term. This remains inviolable regardless of the network outputs—rendering it cleaner, more powerful, and mathematically more rigorous than any loss-based soft constraints, thus serving as the optimal problem re-parameterization for this task.
Materializing geometric priors using B-spline basis functions: While most methods delegate continuous coordinate mapping to black-box MLPs, DP-NSL replaces them with analytical B-spline bases, making the "continuity order" of interpolation a learnable, routable parameter. This design style of "shifting the learning focus from implicit weights to explicit prior parameters" is highly generalizable to other structure-aware tasks.
Spontaneous semantic routing: Without structural boundary supervision, the routing network of MoS inherently learns spatial allocations that highly align with CT anatomy (low-order = organ interior, high-order = boundaries/vessels). This suggests that the order of geometric continuity is a highly rational design dimension tightly coupled with local image structures.
Device-agnostic and transferable design: MCP only requires knowledge of the degradation operator to enforce the hard constraint, making it independent of any specific degradation model assumption. Consequently, it is naturally applicable to other medical image inverse problems (e.g., compressed sensing MRI, low-dose CT denoising).

Limitations & Future Work¶

The current MCP projection assumes a known and fixed degradation operator 𝒟 (anisotropic downsampling). However, in actual clinical workflows, degradation patterns can be far more complex (e.g., non-uniform sampling, combined motion artifacts). The authors point out that extending MCP to more realistic degradation scenarios is an important future direction.
The three B-spline experts (p=2,3,4) in MoS are hand-crafted, lacking automated search for the optimal number and selection of orders. Whether better order combinations (such as fractional orders) or more experts can further improve performance remains unexplored.
Although L1 loss optimization works well, it lacks anatomical plausibility guidance in the synthesized null-space regions. Future work could incorporate adversarial losses or anatomical segmentation-based priors to further enhance structural fidelity.
Regarding computational overhead, DP-NSL is slightly heavier than simpler models like ArSSR or CycleINR (about 1.1-1.3× FLOPs), mainly arising from evaluating multiple experts in MoS and the multi-branch design in LSCD, although it remains within an acceptable range.

vs ArSSR / SAINR / CycleINR / DC2SR (Medical Arbitrary-Scale INRs): These methods model the volume as a continuous function but rely on MLPs with independent coordinate queries and lack hard data-fidelity constraints. DP-NSL distinguishes itself by: ① introducing range-null space decomposition to provide strict measurement consistency (rather than soft constraints); ② replacing MLPs with a Mixture-of-Splines to make the interpolation continuity order a learnable, explicit prior; ③ leveraging a convolutional decoder to inject local consistency (mitigating the limitations of pointwise MLPs).
vs DDNM (Range-null space decomposition in zero-shot inverse problems): DDNM utilizes pre-trained diffusion models to refine null-space details and performs iterative projections. DP-NSL introduces range-null space decomposition to medical arbitrary-scale super-resolution for the first time and utilizes a learnable MoS module (instead of diffusion models) to generate null-space content, making it more suitable for end-to-end training.
vs BTC (B-spline coefficient prediction for natural image SR): BTC predicts B-spline coefficients, knots, and dilations from encoded features, essentially operating as a single-order B-spline interpolation. DP-NSL extends this to a multi-order Mixture-of-Splines, enabling the continuity order itself to be a routable, spatially adaptive variable—which is highly crucial in medical scenarios characterized by prominent anatomical heterogeneity.

Rating¶

Novelty: ⭐⭐⭐⭐⭐ Formulating medical slice super-resolution via range-null space decomposition and leveraging Mixture-of-Splines to materialize geometric priors is highly elegant, well-motivated, and theoretically sound.
Experimental Thoroughness: ⭐⭐⭐⭐⭐ Includes validation on 3 CT and 1 MRI datasets, in-distribution and out-of-distribution scales, extensive ablations, routing visualizations, clinical slice validation, and downstream segmentation tasks. Computational complexity and convergence analyses are also provided in the appendix.
Writing Quality: ⭐⭐⭐⭐⭐ The motivation is clear, the methodological derivation is rigorous, and the visualizations are highly convincing (especially the slice-level MAE curves perfectly dropping to zero).
Value: ⭐⭐⭐⭐⭐ Preserving acquired slices unchanged is a critical clinical requirement. MCP offers a solid mathematical solution, and the MoS component can be easily decoupled and integrated into other frameworks.