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 paper reformulates arbitrary-scale medical slice super-resolution as a constrained inverse problem. By utilizing range-null space decomposition, it achieves a hard constraint for measurement consistency. Within the null space, a Mixture-of-Splines (MoS) is employed to adaptively allocate analytical continuity orders according to local anatomical features, strictly guaranteeing that the acquired slices remain unchanged while generating anatomically plausible interpolation details.

Background & Motivation

Modern volumetric imaging pipelines such as CT and MRI are limited by acquisition times and radiation doses, often yielding anisotropic 3D volumes where the inter-slice spacing is significantly larger than the in-plane resolution. This anisotropy severely hinders downstream tasks such as 3D visualization, medical analysis, and diagnosis, making arbitrary-scale slice super-resolution (reconstructing arbitrary intermediate slices from sparse sampling) a necessary post-processing step. While classic linear interpolation offers deterministic stability, it fails 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 unconstrained residual regression, the networks generate non-zero errors in the observed slice regions and may even hallucinate anatomically implausible details, which is unacceptable in clinical scenarios. The key challenge lies in the trade-off: ensuring the reconstruction results are perfectly consistent with the originally acquired slices (data fidelity) while generating high-fidelity anatomical details in unobserved regions. Existing INR methods lack explicit data fidelity constraints; hence, 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 from basis pursuit theory naturally divides 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 \(\Pi_{\mathcal{N}} = I - \mathcal{U}\mathcal{D}\) in range-null space decomposition as a hard constraint, the network output is forced to map to the null space of the degradation operator, mathematically guaranteeing that all acquired slices remain unchanged. Meanwhile, within this null space, a Mixture-of-Splines (MoS) dynamically allocates different analytical continuity orders, adaptively regulating the rigidity 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 observed slices remain strictly unchanged) and a learnable null-space component (accommodating all high-frequency detail generation). The entire framework consists of three cascaded core modules. First, the pseudo-inverse of the degradation operator is used to perform trilinear upsampling on the low-resolution voxels, obtaining the immutable range-space anchor; concurrently, a 3D encoder extracts low-resolution spatial features. Then, these features enter the Null-Space Estimator (NSE)—first executing content-adaptive geometric interpolation in a continuous coordinate field via the Mixture-of-Splines (MoS), and then injecting local inductive biases using the Local Spatial Consistency Decoder (LSCD) to generate raw null-space prediction voxels. Finally, the Measurement Consistency Projection (MCP) acts as a fixed orthogonal projection operator to map the raw prediction to the null space of the degradation operator, which is then added to the range-space anchor to yield the final super-resolution result.

Since the MCP projection satisfies \(\mathcal{D}\Pi_{\mathcal{N}} = 0\), any component projected into it is "invisible" under the degradation operator. Thus, this term does not affect the acquired slices, guaranteeing data fidelity with strict linear algebra.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Low-Resolution Voxels V_LR"] --> B["Trilinear Upsampling<br/>Range-Space Anchor V_R"]
    A --> C["3D Encoder<br/>Extract Spatial Features"]
    C --> D["Continuous Coordinate Field Query"]
    D --> E["Mixture-of-Splines MoS<br/>Content-Adaptive Geometric Interpolation"]
    E --> F["Local Consistency Decoder LSCD<br/>Inception-style Multi-scale Fusion"]
    F --> G["Raw Null-Space Prediction V̂_N"]
    G --> H["MCP Orthogonal Projection<br/>\Pi_N = I - UD"]
    H --> I["Null-Space Component V_N"]
    B --> J["Super-Resolution Voxels V_SR<br/>= V_R + V_N"]
    I --> J

Key Designs

1. Measurement Consistency Projection (MCP): Hard Fidelity Constraint via Range-Null Space Decomposition

MCP parameterizes the reconstruction problem as an orthogonal bisection based on range-null space decomposition. Given a degradation operator \(\mathcal{D}\) (corresponding to the downsampling process of CT/MRI slice acquisition), its pseudo-inverse \(\mathcal{U}\) (here, trilinear upsampling) is constructed to satisfy \(\mathcal{D}\mathcal{U} = I\). For any super-resolution voxel \(V_{SR}\), the orthogonal decomposition always holds: \(V_{SR} = \mathcal{U}\mathcal{D}(V_{SR}) + (I - \mathcal{U}\mathcal{D})V_{SR}\). The first term \(\mathcal{U}(V_{LR})\) is uniquely determined by the low-resolution input, serving as a deterministic anchor in the range space; the second term is obtained after being applied by the projection \(\Pi_{\mathcal{N}} = I - \mathcal{U}\mathcal{D}\). Since \(\mathcal{D}\Pi_{\mathcal{N}} = 0\), it naturally resides in the null space of the degradation operator and is "invisible" to the acquired slices.

However, \(V_{SR}\) in the second term is the unknown ideal voxel. The approach of DP-NSL is as follows: it uses the Null-Space Estimator (NSE) to output a raw prediction \(\hat{V}_{\mathcal{N}}\), and then forces it into the null space via MCP projection \(\Pi_{\mathcal{N}}(\hat{V}_{\mathcal{N}})\). The final reconstruction is expressed as \(V_{SR} = V_R + \Pi_{\mathcal{N}}(\hat{V}_{\mathcal{N}})\). This linear algebraic constraint guarantees that no matter what the network predicts, the pixel values of the originally acquired slices remain strictly unchanged—providing a stronger, theoretically guaranteed constraint compared to any soft constraint (such as penalty terms in loss functions).

2. Mixture-of-Splines (MoS): Adaptive Continuity Order Allocation per Local Anatomical Features

Biological tissues exhibit heterogeneous smoothness: homogeneous organs (such as liver parenchyma) require high-order continuity to generate smooth transitions, while structural interfaces (such as bone-soft tissue boundaries) require low-order continuity to retain sharp edges. The core idea of MoS is to replace standard MLP black boxes with learnable B-spline bases to model continuous space interpolation, and to adapt to this heterogeneity by combining multiple spline experts with different analytical orders.

Specifically, MoS maintains \(K\) B-spline upsampling experts \(\{\mathcal{U}_{p_k}\}\), where each expert has a distinct analytical continuity order \(p_k\) (experimentally, \(p \in \{2, 3, 4\}\)). For a query coordinate \(\mathbf{q}\), structure-aware scaling coefficients \(\mathbf{c}\), adaptive knot vector \(\mathbf{k}\), and continuous dilation scales \(\mathbf{s}\) are projected from the nearest feature neighborhood using shallow convolutions. Then, each expert computes the B-spline basis response using separable tensor products:

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

where \(\mathbf{b}_d\) is a 1D B-spline basis vector, calculated by substituting the relative projection distance \(\Delta q_d\) modulated by knot and dilation parameters into the B-spline basis function \(\beta^p\) of order \(p\). Meanwhile, a lightweight routing network predicts the mixing weights \(\pi(\mathbf{q}) = \text{Softmax}(R_{\pi}(\tilde{\mathbf{F}}_{LR}^q))\) from the local interpolated features. Ultimately, the high-resolution feature at this coordinate is a weighted mixture of the outputs from each expert: \(\mathbf{F}_{SR}^q = \sum_k \pi_k(\mathbf{q}) \cdot \mathcal{U}_{p_k}(\mathbf{F}_{LR}; \mathbf{q})\). This mechanism allows homogeneous regions to be dominated by low-order experts (smooth transitions) and boundary regions to be dominated by high-order experts (retaining sharpness), enabling spontaneous routing allocation without explicit boundary supervision.

3. Local Spatial Consistency Decoder (LSCD): Supplementing Local Inductive Bias via Multi-Scale Convolutions

While MoS provides a macroscopic continuity prior, it inherently evaluates continuous basis functions independently for each coordinate, thus still lacking consistency constraints in local spatial neighborhoods. The goal of LSCD is to inject local inductive bias into the step of mapping from the high-resolution feature volume to the raw null-space prediction.

LSCD draws inspiration from Inception's split-transform-merge mechanism: it splits the high-resolution features output by MoS into 5 groups along the channel dimension—the identity branch (retaining high-frequency basis signals), a standard 3D convolutional branch, and three depthwise 3D convolutional branches with kernel sizes increasing to \(3 \times 3 \times 3\), \(5 \times 5 \times 5\), and \(7 \times 7 \times 7\), respectively. This design enables the network to capture multi-scale local spatial contexts at a low computational cost: small kernels focus on fine neighborhood consistency, while larger kernels focus on broader structural coordination. After cascading several such Inception-style blocks, the final original null-space prediction \(\hat{V}_{\mathcal{N}}\) is projected. Ablation studies show that LSCD not only improves reconstruction accuracy but also yields superior performance with only 71K parameters (compared to 214K for the MLP decoder) and lower FLOPs.

Loss & Training

The model is optimized using the \(L_1\) reconstruction loss, solely supervising the discrepancy between the final super-resolution voxels \(V_{SR}\) and the ground truth voxels \(V_{HR}\) (without supervising intermediate layers separately or adding extra regularization terms). The optimizer used is Adam (\(\beta_1 = 0.9\), \(\beta_2 = 0.999\)), with an initial learning rate of \(1\times10^{-4}\) halved every 200 epochs, training 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)	\(\times 2\)	PSNR	42.55	41.84 (EDSR3D)	+0.71 dB
Colon (CT)	\(\times 3\)	PSNR	38.43	37.78 (SAINR)	+0.65 dB
Colon (CT)	\(\times 4\)	PSNR	35.95	35.37 (EDSR3D)	+0.58 dB
Liver (CT)	\(\times 2\)	PSNR	42.49	41.42 (HIIF)	+1.07 dB
IXI (MRI)	\(\times 4\)	PSNR	41.92	41.41 (DC2SR)	+0.51 dB
IXI (MRI)	\(\times 5\)	PSNR	40.17	39.55 (SAINR)	+0.62 dB

Ablation Study

Configuration	\(\times 2\) PSNR	\(\times 4\) 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)	Only adding null-space constraint 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)	Two priors are complementary
Full DP-NSL	42.54 (+0.72)	35.93 (+0.63)	Further improvement after adding LSCD

Key Findings

• Ablation studies on MoS spline orders show that among single-order configurations, \(p=3\) is the best. However, the mixture of multiple orders (\(p \in \{2, 3, 4\}\)) significantly outperforms any single-order configuration across all scenarios, indicating that heterogeneous anatomical regions indeed require spatially varying continuity priors.
• Routing weight visualization confirms that low-order experts activate more strongly within homogeneous soft tissues, while high-order experts concentrate on fine vessel branches and organ boundaries. This content-aware allocation is entirely derived from data-driven learning without requiring explicit boundary annotations.
• Slice-level MAE validation of MCP perfectly demonstrates a zero-error pattern: at the originally acquired slice locations (\(s=0, R, 2R, \dots\)), the MAE is exactly zero, whereas all baseline methods exhibit non-negligible errors at these locations. This implies that other methods alter the raw clinical data to varying degrees.
• On unseen extrapolation scales (\(\times 5\), \(\times 6\), \(\times 7\)), the advantages of DP-NSL become even more pronounced (yielding a \(0.62\text{ dB}\) gain on IXI at \(\times 5\)). This is attributed to the inherent continuity of the B-spline basis, which does not depend on sampling density, allowing disciplined extrapolation at unseen resolutions.

Highlights & Insights

• Null-space orthogonal decomposition is the optimal data fidelity strategy: MCP encodes the requirement of "preserving the acquired slices unchanged" as a hard constraint via linear algebra rather than a loss term. This constraint cannot be violated regardless of the network's output—making it cleaner, stronger, and theoretically validated compared to any loss-based soft constraints, representing the optimal problem re-parameterization for this task.
• Exemplifying geometric priors with B-spline basis functions: While most methods relegate continuous coordinate mapping to black-box MLPs, DP-NSL replaces them with analytical B-spline bases, making the "continuity order" of the interpolation behavior a learnable and routable parameter. This design philosophy of "shifting the focus of learning from implicit weights to explicit prior parameters" is worth emulating in other structure-aware tasks.
• Semantic spontaneity of routing allocation: Without anatomical boundary supervision, the routing network of MoS learns spatial allocations highly consistent with CT anatomical structures (low-order = soft tissue interior, high-order = boundaries/vessels). This suggests that the order of geometric continuity priors is a rational design dimension tightly coupled with local image structures.
• Transferable, device-agnostic design: MCP only needs to know the degradation operator to enforce the hard constraint, without depending on any specific degradation model assumptions. This makes it naturally applicable to other medical image inverse problems (such as compressed sensing MRI and low-dose CT denoising).

Limitations & Future Work

• The current MCP projection assumes that the degradation operator \(\mathcal{D}\) is known and fixed (anisotropic downsampling), whereas actual clinical degradation patterns can be far more complex (e.g., non-uniform sampling, mixed motion artifacts). The authors point out that extending MCP to more realistic degradation scenarios is a future direction.
• The three B-spline experts (\(p=2, 3, 4\)) in MoS are manually designated, lacking automated search for the number of orders and values. Whether optimal combinations of orders (e.g., fractional orders) or a larger number of experts could further enhance performance remains unexplored.
• Although optimization with \(L_1\) loss yields competitive results, it lacks guidance on anatomical plausibility in the synthesized regions of the null space. Future work could incorporate adversarial loss or anatomical segmentation-based priors to further improve structural fidelity.
• Regarding computational costs, DP-NSL is slightly more demanding than simpler models (such as ArSSR and CycleINR), requiring about \(1.1\times \text{ to } 1.3\times\) more FLOPs, primarily due to the multi-expert evaluation in MoS and the multi-branch processing in LSCD. Nevertheless, this remains within an acceptable range.

Related Work & Insights

• vs. ArSSR / SAINR / CycleINR / DC2SR (medical arbitrary-scale INRs): These methods model volumes as continuous functions but decode them using coordinate-independent MLPs and lack hard fidelity constraints. The discrepancy of DP-NSL lies in: (1) providing strict measurement consistency guarantees via range-null space decomposition (as opposed to soft constraints); (2) replacing MLPs with a Mixture-of-Splines, turning interpolation continuity into a learnable and explicit prior; (3) injecting local consistency using a convolutional decoder to offset the limitations of point-wise MLPs.
• vs. DDNM (range-null space decomposition in zero-shot inverse problems): DDNM utilizes pre-trained diffusion models to refine null-space details and iteratively projects them. DP-NSL introduces range-null space decomposition to arbitrary-scale medical slice super-resolution for the first time, and utilizes a learnable MoS module (as opposed to a diffusion model) to generate null-space content, making it better suited for end-to-end training scenarios.
• vs. BTC (B-spline coefficient prediction for natural image SR): BTC predicts B-spline coefficients, knots, and dilations from encoded features, which essentially constitutes single-order B-spline interpolation. The MoS of DP-NSL extends this to a multi-order mixture-of-experts, turning the continuity order itself into a routable and spatially adaptive variable—which is particularly critical in medical scenarios with pronounced anatomical heterogeneity.

Rating

• Novelty: ⭐⭐⭐⭐⭐ Formulating medical slice super-resolution via range-null space decomposition and embodying geometric priors with a Mixture-of-Splines is an ingenious approach to problem re-parameterization, backed by sound theoretical completeness.
• Experimental Thoroughness: ⭐⭐⭐⭐⭐ Comprehensive evaluation involving 3 CT and 1 MRI datasets, both in-distribution and out-of-distribution scales, ablation studies, routing visualization, real-world data validation, downstream segmentation tasks, and supplementary computational/convergence analyses.
• Writing Quality: ⭐⭐⭐⭐⭐ Featuring clear motivation, rigorous methodology derivation, and highly convincing visualizations (especially the slice-level MAE validation curve that achieves perfect zero error).
• Value: ⭐⭐⭐⭐⭐ Preserving acquired clinical slices unchanged is an absolute necessity in medical scenarios. MCP offers the current state-of-the-art solution, and the MoS component can be adopted independently from this framework.

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