MoS Module: Adaptive Spline Upsampling

• Mixture-of-Splines (MoS) is a content-adaptive, geometric-continuity upsampling operator that replaces black-box methods with analytic B-spline experts.
• It dynamically selects spline orders via a routing network, ensuring spatially-varying interpolation continuity that preserves sharp anatomical boundaries and smooth tissue regions.
• Empirical evaluations within the DP-NSL framework show that MoS enhances PSNR and generalizes better across scales while maintaining measurement consistency.

The Mixture-of-Splines (MoS) module is a content-adaptive, geometric-continuity upsampling operator designed for null-space estimation in the Dual-Prior Null-Space Learning (DP-NSL) framework for arbitrary medical slice super-resolution. MoS replaces black-box parameteric upsamplers (such as MLPs) with a mixture of analytic B-spline experts, each of configurable order, and leverages a dynamic routing mechanism to enable spatially-varying interpolation continuity. This ensures that homogeneous tissue is rendered smoothly while anatomical boundaries and fine structures retain sharpness, all under explicit measurement consistency constraints (Song et al., 25 Jun 2026).

1. Mathematical Foundation of Mixture-of-Splines

At the core of MoS are tensor-product B-spline basis functions of various analytic orders. The zeroth-order (box) B-spline basis is given by

$$\beta^0(x) = \begin{cases} 1, & |x|<\tfrac{1}{2} \ 0, & \text{otherwise} \end{cases}$$

Higher-order B-splines are obtained by convolution: $$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$ with the shifted, p-th order, i-th basis given by $B_{p,i}(x) = \beta^p(x-i)$. In 1D, this yields $C^{p-1}$ continuity.

In the DP-NSL framework, MoS appears within the null-space estimator as: $$V_{SR} = V_{\mathcal R} + V_{\mathcal N} = \mathcal U(V_{LR}) + \Pi_\mathcal N \big( f_\theta(V_{LR}) \big)$$ Here, $\mathcal U(V_{LR})$ is a range-space upsampling anchor reproducing all observed slices and $\Pi_{\mathcal N}$ projects onto the null space of the downsampling operator, ensuring that only unobservable details are filled in.

2. MoS Module Architecture and Regression Mechanism

For each query coordinate $\mathbf q$ in the high-resolution target grid, MoS computes an upsampled feature vector as a weighted mixture: $$\mathrm{MoS}(\mathbf F_{LR},\mathbf q) = \sum_{k=1}^K \pi_k(\mathbf q) \; \mathcal U_{p_k}(\mathbf F_{LR};\mathbf q)$$ where:

• $\{p_k\}_{k=1}^K$ are the selected spline orders (in practice, $$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$0, $$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$1),
• $$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$2 denotes expert $$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$3's separable 3D B-spline interpolation of order $$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$4,
• $$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$5 are adaptive weights parameterized by a routing network: $$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$6 Here, $$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$7 is a locally interpolated feature vector.

Each expert computes

$$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$8

with axis-aligned basis evaluations

$$\beta^p(x) = (\beta^0 * \beta^0 * \cdots * \beta^0)_{(p+1)\text{ times}}(x)$$9

where the geometric parameters $B_{p,i}(x) = \beta^p(x-i)$0 are functions of the local feature neighborhood, typically predicted by shallow convolutional sub-networks.

3. Content-Aware Order Selection and Continuity Control

Rather than fixing a global spline order, MoS dynamically adapts the mixture ratio per coordinate via the routing network $B_{p,i}(x) = \beta^p(x-i)$1. Low-order experts (e.g., $B_{p,i}(x) = \beta^p(x-i)$2) dominate in homogeneous (smooth) tissue, while high-order experts (e.g., $B_{p,i}(x) = \beta^p(x-i)$3) are prioritized near anatomical boundaries and fine details. Empirically, transition zones leverage a blend of orders, producing spatially-adaptive continuity [(Song et al., 25 Jun 2026), Fig. 5].

A B-spline of order $B_{p,i}(x) = \beta^p(x-i)$4 is $B_{p,i}(x) = \beta^p(x-i)$5 continuous, providing the ability to specify locally the needed degree of smoothness. The composite MoS field remains globally continuous and regionally attuned, mitigating both the over-smoothing effect of high-order interpolation and ringing artifacts of low-order kernels, while preserving anatomical fidelity.

4. Algorithmic Workflow

The MoS upsampling process for a single coordinate is as follows:

$$V_{SR} = V_{\mathcal R} + V_{\mathcal N} = \mathcal U(V_{LR}) + \Pi_\mathcal N \big( f_\theta(V_{LR}) \big)$$2 Parameters $B_{p,i}(x) = \beta^p(x-i)$6 allow for local modulation of the spline basis. Parallelization is applied over coordinate batches in practical implementations.

5. Implementation Considerations

• Expert Bank: $B_{p,i}(x) = \beta^p(x-i)$7 B-splines ($B_{p,i}(x) = \beta^p(x-i)$8) are used, with analytic forms in the appendix of (Song et al., 25 Jun 2026).
• Geometric Parameter Net: A shallow 3D CNN (e.g., $B_{p,i}(x) = \beta^p(x-i)$9 and $C^{p-1}$0 convolutions) maps the local $C^{p-1}$1 neighborhood to interpolation parameters.
• Routing Net: Either one or two fully-connected layers, or $C^{p-1}$2 convolutions, produce the mixture logits from a pointwise interpolated feature vector.
• Loss Function: The DP-NSL framework is supervised end-to-end using an $C^{p-1}$3 loss on the reconstructed volume, $C^{p-1}$4, without separate regularization for MoS.
• Computational Characteristics: MoS increases FLOPs by approximately $C^{p-1}$5–$C^{p-1}$6 compared to trilinear upsampling. It remains computationally cheaper than heavy spatial-attention modules and provides significant PSNR improvements [(Song et al., 25 Jun 2026), Table 9].

6. Empirical Effect and Ablation Analysis

Ablation studies in [(Song et al., 25 Jun 2026), Sec. 4.4, Tables 3–4, Fig. 5] demonstrate the unique contribution of MoS:

• Replacing MoS with trilinear upsampling in the null-space estimator reduces $C^{p-1}$7 PSNR by $C^{p-1}$8 dB.
• The combination of MoS and measurement-consistent projection (MCP) obtains the highest fidelity, with an additional $C^{p-1}$9 dB over MoS without MCP.
• Single-order splines (any fixed $$V_{SR} = V_{\mathcal R} + V_{\mathcal N} = \mathcal U(V_{LR}) + \Pi_\mathcal N \big( f_\theta(V_{LR}) \big)$$0) have comparable in-scale performance, but the multi-order MoS generalizes better to scales outside the training distribution.
• Expert-weight visualizations show that MoS order selection is semantically meaningful, aligning with anatomical structure boundaries and tissue homogeneity.

7. Comparative Positioning and Functionality

MoS provides a principled, analytic alternative to black-box MLP or fixed-order spline upsamplers within constrained super-resolution frameworks. Its per-location adaption via a lightweight routing mechanism delivers explicit $$V_{SR} = V_{\mathcal R} + V_{\mathcal N} = \mathcal U(V_{LR}) + \Pi_\mathcal N \big( f_\theta(V_{LR}) \big)$$1 continuity control consistent with anatomical topology in medical imaging. No auxiliary losses or supervision are introduced for MoS itself; end-to-end learning in the context of DP-NSL is sufficient for optimal spatial upsampling.

A plausible implication is that MoS can be generalized as a modular upsampling unit where explicit, content-adaptive continuity is desired, and black-box interpolation mechanisms are ill-suited for measurement-consistent inverse problems. It also suggests that further exploration of spline order ensembles and dynamic weighting strategies could extend to other geometric or scientific domains (Song et al., 25 Jun 2026).