High-Order Degradation Modeling

• High-order degradation modeling is a framework that explicitly represents and disentangles complex, multi-stage degradation processes in physical, chemical, and visual signals.
• It employs progressive cascades and explicit operator coupling to improve inversion methods, using dual priors and state-space networks across various applications.
• Empirical results show significant performance gains in restoration metrics (e.g., PSNR, SSIM) across imaging, remote sensing, and battery modeling domains.

High-order degradation modeling refers to the explicit representation, disentanglement, or inversion of degradation processes whose physical or statistical structure cannot be faithfully captured using simple, single-stage (first-order) models. Such degradation mechanisms often arise as a sequential or compound effect of multiple operators acting on physical, chemical, or visual signals, driving research in computational imaging, remote sensing, universal restoration, and materials modeling. High-order degradation models move beyond classical approaches—typically limited to a single blur kernel or additive noise—by allowing for cascades, spatial heterogeneity, parameter interaction, and path dependence.

1. Definition and Scope of High-Order Degradation Modeling

High-order degradation modeling encompasses systems where the observed signal results from the composition or coupling of multiple degradation mechanisms. In computational imaging and restoration, this typically involves sequences of blur, resampling, noise, compression, occlusion, or non-uniform artifacts, as opposed to scalar or kernel-based first-order models. In the electrochemical sciences, high-order models describe the coupled evolution of several degradation mechanisms—such as SEI growth, lithium plating, mechanical cracking, and loss of active material—whose interplay drives nonlinear, history-dependent behavior in batteries.

Several leading works formalize this perspective:

• In image restoration, high-order degradation is encoded as a multistage pipeline:

$$g_\ell = D_\ell(g_{\ell-1}) = N_\ell \circ S_\ell \circ B_\ell(g_{\ell-1}),\quad \ell=1,...,k,$$

where $B_\ell$ is a blur, $S_\ell$ a resampling operator, and $N_\ell$ additive noise, yielding an observable $g = g_k$ for $k$ sequential stages (Feng et al., 10 Dec 2024).

• In battery physics, degradation is modeled by the explicit dynamical coupling and feedback between SEI-layer growth, lithium plating/stripping, particle fracture, and mechanical loss of active material in an electrode (O'Kane et al., 2021).
• Universal image degradation models separate global (homogeneous) and local (inhomogeneous) degradation features for arbitrary compound processes (Yang et al., 19 May 2025), and advanced restoration pipelines inject fine-grained degradation prompts into all-in-one networks for domain-adaptive denoising, deblurring, and artifact removal (Liu et al., 24 Apr 2025).

2. Mathematical Formulations and Model Architectures

High-order degradation frameworks explicitly parameterize both the composition of degradation operators and the network modules tasked with their inversion or emulation.

Progressive Degradation Cascades for Remote Sensing

In the progressive restoration paradigm (Feng et al., 10 Dec 2024), the observation $g$ reflects a $k$-stage application of cascaded operators: $$g_\ell = D_\ell(g_{\ell-1}) = N_\ell(S_\ell(B_\ell(g_{\ell-1}))),\quad \ell=1,\ldots,k,$$ with inversion cast as a sequence of energy minimizations: $$g_{\ell-1} = \arg\min_g \tfrac{1}{2}\|g_\ell - S_\ell(B_\ell(g))\|_2^2 + \lambda_\ell f_\ell(g),$$ leading to deep networks that separately implement denoising (proximal mapping), super-resolution, and deblurring (truncated Neumann expansion with dual-domain learning).

Explicit Degradation Factorization for Weather Restoration

MODEM (Wang et al., 23 May 2025) represents spatially heterogeneous degradations by decomposing them into:

• Global prior, $\theta_G$ (e.g., haze, fog severity)
• Local/high-order prior, $\theta_L$ (e.g., streaks, small-scale occlusions) and formulates the observation as $x = \mathrm{Degrade}(y ; \theta_G, \theta_L)$, with both priors explicitly estimated and used to condition state-space backbone modules.

Universal Disentanglement of Homogeneous and Inhomogeneous Factors

A learned universal model (Yang et al., 19 May 2025) separates degradation into global embedding $\mathbf{e}_g$ and local (spatial) embedding $\mathbf{e}_l$, obtained by: $$\hat{\mathbf{y}} = \hat{f}\bigl(\mathbf{x}, \, \mathbf{e}_g,\, \mathbf{e}_l,\, \mathbf{n}\bigr)$$ where:

• $e_g$: dual-branch convolutional encoder for global (homogeneous) degradations
• $e_l$: encoder for inhomogeneous (local) structure
• Losses encourage content–degradation independence via information-theoretic constraints.

Modular Degradation-Aware Prompting in All-in-One Image Restoration

DPMambaIR (Liu et al., 24 Apr 2025) characterizes high-order degradation as the element-wise, sequential, and spatial interaction of multiple photometric and geometric distortions: $$I(x) = Q( ( \alpha(x) J(x) + \beta(x) \gamma(x) ) \otimes K )$$ where $J(x)$ is the clean image; $\alpha(x)$ is a luminance modulator; $\gamma(x)$, $\beta(x)$ are spatial occlusion masks; $K$ the blur kernel; $Q$ a compression operator. Fine-grained degradation cues are encoded into a prompt vector by a pre-trained extractor and dynamically injected into state-space model parameters, allowing adaptive global and local restoration.

3. Mechanisms of Operator Coupling and Interaction

High-order degradation models are distinguished by their explicit treatment of operator coupling—i.e., the way distinct degradation mechanisms influence each other.

Coupled PDE-Based Degradation for Battery Modeling

In lithium-ion battery models (O'Kane et al., 2021), the coupled evolution of SEI thickness $L_{SEI}$, plated Li $c_{Li}$, dead Li $c_{dl}$, crack surface area $a_{cr}$, and active material fraction $\epsilon_{a}$ is governed by nonlinear differential-algebraic equations with feedback:

• SEI thickness modulates the overpotential for plating.
• Crack generation exposes fresh electrode, accelerating SEI formation.
• Dead lithium formation slows SEI decay, producing path-dependent loss of lithium inventory.
• Loss of active area increases local current density, accelerating mechanical degradation.

This network of couplings yields history-dependent, highly nonlinear degradation trajectories, with multiple "end-of-life" pathways depending on cycling and environmental history.

Dynamic Conditioning for Visual Degradation

In universal and all-in-one restoration models (Yang et al., 19 May 2025, Liu et al., 24 Apr 2025, Wang et al., 23 May 2025), conditioning variables—estimated as either global or spatially varying embeddings—control network modules via adaptive modulation or attention:

• Embeddings modulate convolution kernels, SSM state-transition parameters, or feature normalization paths.
• Explicit separation of homogeneous and inhomogeneous mechanisms allows targeted inversion or synthesis.
• Dynamic injection of degradation priors enables robust generalization to unseen compositions and unpredictable mixtures.

A summary of representative interaction mechanisms is provided below.

Domain	Coupled Operators/Factors	Principal Couplings/Interactions
Battery modeling	SEI growth, Li plating, fracture, LAM	Electrochemical, mechanical, geometric; direct feedback; path dependence
Atmospheric imaging	Haze (global), rain/snow (local)	Factorization via priors; dynamic conditioning in State-Space modules
Remote sensing	Cascade of blur, resampling, noise	Sequential inversion; staged operator splitting
Universal synthesis	Homogeneous vs. inhomogeneous embeddings	Content-degradation disjointness enforced by entropy reduction

4. Architectures and Algorithmic Strategies

High-order degradation modeling drives innovations in both representation learning and algorithmic design.

• Deep Unfolding with Physical Interpretability: HDI-PRNet (Feng et al., 10 Dec 2024) unfolds an explicit inverse problem for cascaded degradations, assigning sub-networks to each physical process and providing stage-wise supervision.
• Degradation-Adaptive State-Space Networks: In DPMambaIR (Liu et al., 24 Apr 2025), fine-grained degradation cues drive the parameterization of continuous-discrete SSMs, modulating state-transition and input/output matrices per sample.
• Dual-Prior Conditioning in Adverse Weather Recovery: MODEM (Wang et al., 23 May 2025) introduces a Morton order 2D scan to maintain spatial locality, and a dual-prior injection enables separate global and local adaptation within its backbone.
• Disentangle-by-Compression: The universal model (Yang et al., 19 May 2025) minimizes entropy of degradation embeddings to enforce statistical independence from content, facilitating consistent transfer and adaptation across diverse pipelines.

5. Empirical Evaluation and Impact

High-order modeling demonstrates consistent performance gains and enhanced generalization across imaging and materials domains.

• Remote Sensing Restoration: HDI-PRNet outperforms first-order baselines by 1–1.5 dB PSNR in high-order degradation scenarios and matches or exceeds state-of-the-art performance for ×2/×3/×4 super-resolution under synthetic and real conditions (Feng et al., 10 Dec 2024).
• All-in-One Image Restoration: DPMambaIR achieves PSNR/SSIM gains across all seven major degradation types, exceeding prior methods (e.g., AdaIR, OneRestore) by up to 0.43 dB PSNR in unified settings (Liu et al., 24 Apr 2025).
• Weather Restoration Benchmarks: MODEM's explicit dual-prior architecture reduces PSNR loss by 0.6–0.7 dB versus ablations that remove either local or global guidance, and its feature clusters exhibit more discriminative alignment with physical degradation types (Wang et al., 23 May 2025).
• Universal Degradation Synthesis: The content-degradation disentanglement model (Yang et al., 19 May 2025) delivers MS-SSIM = 0.879 and SSIM = 0.860 on realistic degradation reproduction, and enables parameter-free transfer of degradations in generative restoration and style transfer frameworks.

6. Cross-Domain Extensions: Electrochemical and Physical Systems

The high-order paradigm extends beyond imaging to electrochemical systems, as in the PyBaMM-based lithium-ion model (O'Kane et al., 2021). Here:

• The coupling of four negative-electrode mechanisms is achieved via modular, open-source submodels, each contributing PDEs or DAEs to the full cell simulation.
• Simulation results identify distinct end-of-life "paths" (SEI-dominated fade, crack-accelerated failure, plating-driven fade, runaway pore clogging) as a function of parameter regimes, emphasizing the role of operator interaction and usage history.
• Parameterization remains a challenge due to the ill-posedness and multiparametric nature of the models, motivating joint experimental-theoretical efforts and data-driven calibration.

7. Open Challenges and Outlook

High-order degradation modeling, while advancing transparency and fidelity, faces notable challenges:

• Parameter identification is complicated by latent or unobservable degradation variables, both in imaging (blind restoration) and materials (electrochemical rate constants, mechanical strengths) (O'Kane et al., 2021, Feng et al., 10 Dec 2024).
• Operator disentanglement in universal frameworks requires robust statistical techniques to ensure that embeddings represent only degradation factors and are invariant to source content (Yang et al., 19 May 2025).
• Dynamic adaptation mandates efficient architectures capable of per-instance inference of complex, spatially resolved priors, motivating further research into lightweight meta-learning and interpretable modulation mechanisms (Liu et al., 24 Apr 2025, Wang et al., 23 May 2025).
• Generalization and compositionality remain active areas, as current works aim at scaling from domain-specific to domain-universal degradation spaces while maintaining accuracy and interpretability.

Ongoing work leverages information-theoretic regularizers, staged deep-unfolding, dual-domain learning, and explicit physical coupling to further the rigor and generality of high-order degradation modeling in both computational and physical sciences.