SODUN: Segmentation-Oriented Unfolding Network

• The paper demonstrates SODUN's innovative mapping of iterative optimization steps to learnable network layers for enhanced segmentation.
• SODUN incorporates domain-specific image priors, attribution cues, and reversible estimation to refine sparse, multimodal, and COS segmentation tasks.
• Empirical analyses reveal improved AUC, IoU, and error reduction across models like ℓ1DecNet+ and NUN, highlighting efficiency and performance gains.

A Segmentation-Oriented Unfolding Network (SODUN) is a deep architecture in which each network stage is explicitly derived from an iterative optimization algorithm formulated for segmentation tasks. Rather than treating segmentation as a simple supervised mapping, SODUNs incorporate domain-specific image priors, attribution cues, or reversible separation mechanisms, unfolding each step of the underlying model into learnable blocks. This approach is prominent in recent models for sparse feature extraction, multimodal fusion, and robust real-world segmentation, where interpretability, data efficiency, and forward compatibility with mathematical modeling are a priority.

1. Mathematical Formulations of Segmentation-Oriented Unfolding

Core SODUN frameworks are constructed by designing an energy function or constrained optimization objective whose solution embodies the segmentation. For instance, in the context of sparse feature segmentation, the variational decomposition model seeks a decomposition $f = u + v$, where $v$ isolates the sparse target (e.g., vessel/crack) and $u$ is the background, regularized as

$$\min_{u, v} \quad \beta \|v\|_1 + \sum_{m=1}^M \alpha_m \|K_m u\|_1 + \tfrac{1}{2}\|u + v - f\|_2^2.$$

Here, $\{K_m\}$ are learnable transforms and $\beta, \alpha_m$ balance the regularization strengths (Ren et al., 2022). For joint multimodal fusion and segmentation, the objective integrates attribution-weighted fidelity between the fused image $I_f$ and its sources, plus edge preservation and segmentation loss:

$$L_{\rm total} = \alpha L_{\rm fuse} + \beta L_{\rm seg} + \gamma L_{\rm attr},$$

where $L_{\rm fuse}$ uses pixelwise weights from attribution analysis, and $L_{\rm seg}$ supervises the segmentation network $S$ on both source and fused images (Bai et al., 3 Feb 2025). In concealed object segmentation (COS), foreground–background separation is modeled as:

$$\mathcal{L}(M, B) = \tfrac12 \|X - X \odot M - B\|_2^2 + \beta\,\psi(M) + \lambda\,\phi(B),$$

with $M$ as binary mask, $B$ as background, and regularizers $\psi, \phi$ encoding structure priors (He et al., 22 Nov 2025).

2. Unfolding Algorithms as Network Stages

The defining implementation device of SODUN is algorithm-in-network design, wherein each stage (or layer) maps directly to a step in the iterative solver for the segmentation objective.

For ADMM-based sparse decomposition, one iteration is split into:

• Linear-System-Solver (LSS): simultaneous update for $(u, v)$ via closed-form solves.
• Aux-Variable & Multiplier Updater (AVMU): soft-thresholding proximal steps for auxiliary variables, with all thresholds and convolutions learnable and layer-specific (Ren et al., 2022).

In segmentation-oriented multimodal fusion, each iteration comprises a gradient step with attribution-weighted fidelity, a proximal projection with learned operators, and a memory augmentation for feature re-use:

$$I_f^{(k)} = \mathrm{prox}_h\left(I_f^{(k-1)} - \rho_k\,\nabla f\left(I_f^{(k-1)}\right)\right),$$

with inner blocks for attribution-guided attention and memory propagation (Bai et al., 3 Feb 2025).

For COS, each SODUN stage alternates gradient descent and proximal refinement for both mask and background, each realized via shallow CNNs and non-local modules. The process embodies reversible estimation, facilitating supervision at the mask and RGB residual levels (He et al., 22 Nov 2025).

3. Architectural Components and Feature Routing

SODUN implementations typically exhibit a modular two-stage structure:

• Feature Extraction via Unfolding: The initial blocks conduct deep unfolding onto the latent space dictated by the variational/prior model (e.g., from image to $(U, V)$ sparse/dense bands, or source modalities to fused image).
• Segmentation Head: A lightweight, often miniaturized, UNet variant consumes the unfolded features and produces pixelwise output. For example, $\ell_1$DecNet+ stacks $[U, V]$ and any ancillary modalities prior to segmentation, with network size drastically reduced compared to conventional UNets (∼66K params vs. 31M) (Ren et al., 2022).

More sophisticated SODUNs for fusion (UAAFusion) and COS integrate attention mechanisms driven by stagewise attribution maps, long- and short-term memory units via ConvLSTM, and bi-directional information exchanges with restoration modules (e.g., between mask/background and restoration network in NUN) (Bai et al., 3 Feb 2025, He et al., 22 Nov 2025).

4. Loss Functions, Training Protocols, and Optimization Strategies

Losses in SODUN serve both task supervision and alignment with upstream modeling objectives. Binary cross-entropy and weighted IoU/Dice losses are standard for mask segmentation (He et al., 22 Nov 2025). In UAAFusion, segmentation loss is distributed across source modalities and fused output; attribution-fusion loss is incorporated directly by weighting fidelity to source images according to integrated-gradient attribution scores (Bai et al., 3 Feb 2025).

Training recipes feature:

• Alternating learning rates tied to groupings of network parameters (e.g., decomposition-group vs. segmentation-group) (Ren et al., 2022).
• Multi-phase schedules for efficient convergence.
• Adam and SGD optimizers, with standard hyperparameters and decay protocols.
• Data augmentations for increased robustness, including real-world degradations and simulated corruptions in COS applications (He et al., 22 Nov 2025).

5. Empirical Performance and Ablation Analyses

Empirically, SODUN frameworks consistently outperform same-size baselines. For DRIVE retinal vessel segmentation, $\ell_1$DecNet+ achieves AUC=0.9874, ACC=0.9699, MCC=0.8058 with only 0.07M parameters—a ×450 reduction compared to full UNet (Ren et al., 2022). UAAFusion demonstrates increased fusion and segmentation quality attributed to tightly coupled attribution-guided optimization (Bai et al., 3 Feb 2025).

In COS, the NUN framework and SODUN outperform prior DUN-based methods:

• On COD10K with combined degradations, segmentation error $M$ is reduced from 0.125 (RUN) to 0.071 (NUN); $F_\beta$ increases from 0.457 to 0.629.
• For transparent object detection and polyp segmentation, SODUN/NUN raises mIoU and boundary-aware scores by 0.7–1.8% (He et al., 22 Nov 2025).

Ablation studies confirm monotonic performance gains with increasing network depth, marginal effects from width/kernel size, and highlighted importance of non-local reasoning, attribution-based routing, and cross-stage self-consistency. Removal of SODUN modules or replacement by more generic architectures degrades all core segmentation metrics.

Model	Params (M)	AUC/IoU (major dataset)	Key Quantitative Improvement
$\ell_1$DecNet+ (IDmUNet)	0.07	0.9874 (AUC, DRIVE)	×450 smaller, 3× faster than full UNet
UAAFusion SODUN (fusion/segmentation)	~1–3	up to +2% IoU	Attribution-guided fusion for segmentation
SODUN (NUN) for COS	<10	COD10K: Fβ=0.629	20% lower error vs. prior RUN

6. Extensions: 3D Processing and Robustness to Degradations

SODUN architectures are amenable to volumetric extension. In $\ell_1$DecNet+, all convolutional and segmentation layers are replaceable with their 3D analogs, enabling direct application to MRI, CT, and other volume-segmentation domains with comparable efficiency and convergence properties (Ren et al., 2022). NUN leverages a vision-LLM for dynamic degradation inference, achieving significant performance gains under synthetic and real-world corruptions without requiring explicit prior modeling (He et al., 22 Nov 2025).

7. Significance and Context within Segmentation Networks

SODUN embodies a principled shift toward mathematically-guided network design:

• Each layer is interpretable by direct correspondence with a solver step for segmentation-oriented goals.
• Image priors, attribution, or reversible estimation steps are enforced both structurally and through the loss.
• End-to-end training is facilitated, with unfolded parameters optimizable under task loss, ensuring data-driven adaptation.

Unlike standard encoder-decoder or transformer-based networks, SODUN interfaces optimization, prior modeling, and attention mechanisms, achieving robustness, memory efficiency, and granularity unattainable with purely data-driven pipelines. Its deployment in resource-limited environments and fusion contexts suggests broad applicability within medical imaging, quality-fused scene understanding, and robust object segmentation. The full analytical path from model to task results remains inspectable, supporting interpretability and principled improvement.