Underwater image enhancement (UIE) has been extensively studied to counteract the severe color shifts and contrast loss caused by light absorption and scattering [8]. According to the classic Jaffe-McGlamery optical model [9], underwater imaging is a depth-dependent process where the degradation is highly correlated with the distance between the camera and the reflective object. Early approaches heavily relied on hand-crafted physical priors, such as the Dark Channel Prior (DCP) [10], to estimate transmission maps. However, these prior-based methods often suffer from limited generalization due to their brittle assumptions.

In recent years, learning-based UIE methods have dominated the field [11–14]. While achieving significant improvements in visual quality, many learning-based UIE methods still formulate restoration mainly as an image-to-image mapping problem. Physical priors, when used, are often introduced as auxiliary inputs, supervision signals, or output-level constraints, rather than being deeply embedded into the feature interaction process.

State Space Models (SSMs) derive from continuous-time systems, and Mamba [7] excels at modeling long-term dependencies. Applying 1D sequence models to 2D vision problems generally requires a spatial scanning strategy, such as the four-directional cross-scan in SS2D [15].

However, existing methods generally treat Mamba as a general-purpose feature extractor and organize 2D feature maps into 1D sequences along predetermined image-plane paths. Such coordinate-defined scanning orders do not explicitly align the state-propagation path of SSMs with the depth-dependent optical degradation in underwater scenes. Consequently, scanning feature maps across different degradation states with a fixed image-plane order may be less aligned with underwater degradation statistics.

Recognizing the critical role of 3D physical geometry in underwater degradation, previous UIE studies have attempted to incorporate depth or physical priors through multi-task depth estimation, feature concatenation, region-level guidance, or physical constraints [16–18]. These methods demonstrate that geometry-related cues are useful for UIE, but the prior usually participates as an auxiliary input, an additional supervision signal, or a constraint on the restored image. In contrast, IG-Mamba uses a depth-correlated potential field to define the token ordering of the SSM sequence itself. Thus, the prior does not merely enrich the input representation; it organizes the state-propagation topology of Mamba through iso-potential routing. This distinction shifts depth guidance from feature injection to geometry-aware sequence organization. To avoid training an underwater depth estimator from scratch, our implementation uses DepthAnythingV2 [19] as a default prior provider, while later experiments further show that the framework is not tied to this specific estimator.

The overall architecture of the proposed Isoline-Guided Evolutionary Mamba (IG-Mamba) is illustrated in Fig. 2. Unlike many UIE networks that rely solely on degraded color images, IG-Mamba adopts a physics-informed dual-input strategy. Given a degraded underwater image Iin∈RB×3×H×W and its corresponding depth-correlated geometry prior Pin∈RB×1×H×W, our objective is to reconstruct the enhanced image Iout. Specifically, Iin is processed by a convolutional stem to extract initial visual features X0∈RB×C×H×W. Concurrently, Pin serves as the initial potential field P0 for the routing and modulation stream. As shown in Fig. 2, the backbone follows a multi-stage encoder-decoder structure: the encoder consists of IGStage modules, while the decoder uses corresponding IGStage_up modules with skip connections from the encoder. Each stage contains stacked IGBlock units for geometry-guided restoration, followed by spatial downsampling or upsampling operations. For notation consistency, IGBlock denotes the basic unit containing IsolineSort-guided Mamba modeling, local detail preservation, potential updating, and transmission-inspired modulation.

For clarity, P0, P, Pq, and Pnew denote different computational states of the same potential stream. P0 is the external initial geometry prior, P is the current potential field received by a block, Pq is the quantized potential used only for isoline sorting, and Pnew=P+ΔP is the evolved potential used for subsequent propagation and physical modulation. The framework does not assume P0 to be metrically accurate depth; it serves as a coarse spatial geometry anchor.

To maintain a stable scale space for the potential field across hierarchical levels, the potential stream employs average pooling for spatial downsampling and bilinear interpolation for upsampling. During skip connections, element-wise mean fusion is applied (Pout=Pup+Pskip2) instead of standard addition to prevent value inflation beyond a stable numerical range.

Optical Triangle and Radiative-Transfer-Motivated State Propagation. In underwater environments, light propagation follows the optical triangle principle: light travels from the source to the object (distance d1), reflects, and returns to the camera (distance d2). Based on the Jaffe-McGlamery model [9], visual deterioration in a stationary state is primarily governed by the camera-to-object distance. Recently, most Vision Mamba models have converted two-dimensional spatial information into one-dimensional sequences using fixed scanning pathways, such as rasterization. However, spatial-agnostic scanning may connect pixels with different depth-related degradation states in the SSM sequence, thereby weakening local structural integrity. To resolve this issue, we propose the Topology-Preserving Isoline Scanning scheme based on the one-dimensional line-of-sight Radiative Transfer Equation (RTE) theory. In underwater optics, the propagation of light at a distance z obeys the following law:(1)dLdz=−c⋅L(z)+S(z),

Here, z denotes the physical optical-path coordinate. Notably, the continuous-time SSM equation has a similar recursive form:(2)h′(t)=Ah(t)+Bx(t).

In IG-Mamba, the learned potential P is not treated as the physical optical-path coordinate. Instead, it is used as a depth-correlated ordering variable to group pixels into near-to-far iso-potential strata. By grouping pixels into near-to-far iso-potential strata, IsolineSort constructs geometry-aware token sequences in which later hidden states can aggregate information from preceding tokens with related optical paths or degradation states. This provides a physics-inspired inductive bias analogous to radiative transfer, where radiance accumulates along an optical path through attenuation and scattering.

Orthogonal Isoline Routing. To approximate this ordered accumulation over discrete tensors while better preserving local two-dimensional structure, we quantize the potential field P into discrete ordinal iso-potential indices with a bin granularity controlled by K. Specifically, Quantize(⋅,K) maps potential values into discrete ordinal indices, where pixels assigned to the same index are treated as belonging to the same iso-potential stratum. To address potential directional bias, the input features are decomposed by channel into horizontal (Xh) and vertical (Xv) branches, respectively.

The resulting operation, termed IsolineSort, applies stable sorting according to the quantized potential rather than exact metric depth. Stability ensures that pixels within the same iso-potential stratum retain their original row-wise and column-wise local orders in the Xh and Xv branches, respectively. These visual tokens are then flattened to form one-dimensional, isoline-ordered sequences. After independent state-space evolution by MambaH and MambaV, the sequences are mapped back to their original 2D spatial grids by inverse sorting and then concatenated. The detailed dual-route algorithm is shown in Algorithm 1. Here, (⋅)T denotes transposition of the two spatial dimensions H and W.

While the isoline-guided SSM captures macro-level optical dependencies, it might inherently overlook localized, high-frequency textures that are orthogonal to the depth gradient (e.g., coral textures). We introduce a parallel Local Detail Branch utilizing a 3×3 depth-wise convolution and Channel Attention:(3)Xlocal=ℋconv(X)⊙σ(ℋgate(GAP(ℋconv(X)))),

The aggregated feature is fused via a learnable scaling parameter α: Xfused=FFN(Yroute+Xlocal⋅α)+X. This dual-branch design supports both global geometry-aware modeling and local visual fidelity.

Although pre-trained foundation models can provide useful geometric priors, a non-negligible domain gap remains between in-air geometric depth and underwater optical degradation, which is affected by turbidity, backscattering, and wavelength-dependent absorption. The same depth-dependent degradation principle also motivates a complementary modulation path: while geometry-aware routing uses the potential to define the state-propagation order, the Jaffe-McGlamery transmission term t(x)=e−βd(x) suggests that depth-correlated geometry can also provide an attenuation-related signal for feature reweighting.

To implement this idea while reducing the geometry-optics domain gap, IG-Mamba proposes the Potential Field Evolution (ΔP) mechanism (Algorithm 2). During the forward pass, the coarse geometric prior is refined by a learned residual correction ΔP to obtain the updated optical potential field Pnew∈RB×1×H×W. Here, Pnew should be interpreted as a restoration-oriented, degradation-aware optical potential rather than a metrically accurate depth map. A lightweight convolutional mapping then expands Pnew into a spatial-channel transmission-inspired gate Wmod∈[0,1]B×C×H×W for feature reweighting.

The aggregated visual features are then modulated through element-wise multiplication, Xfused⊙Wmod. This differentiable branch also plays an optimization role beyond its physical motivation. Since the argsort operations in our isoline routing are non-differentiable and prevent direct gradient propagation through sorting indices, Wmod provides an additional differentiable optimization path. Consequently, improper modulation, such as over-suppressing a bright foreground region, increases the reconstruction loss, which backpropagates through Wmod to adaptively calibrate the potential field. Radiometric compensation is mainly handled by the main restoration stream and reconstruction head, while the modulation branch provides transmission-inspired reweighting for scattering-dominant feature responses and encourages the potential stream to retain its association with depth-correlated geometry.

Finally, this design forms a restoration-driven correction loop: the routing path uses the current potential to organize state propagation, while the modulation path allows the reconstruction objective to adapt the potential through a continuous feature-reweighting branch. This encourages ΔP to adjust the raw geometric prior toward a restoration-oriented optical potential without requiring auxiliary depth supervision.

We train IG-Mamba end-to-end with a reconstruction-driven potential evolution strategy. The external prior P0 generated by DepthAnythingV2 [19] is used only to initialize the potential stream as a coarse geometry anchor. Through IsolineSort-guided routing and the differentiable transmission-inspired modulation branch, the image restoration objective optimizes the potential stream and encourages ΔP to adapt the raw geometric prior into a restoration-oriented optical potential that is more consistent with underwater optical degradation.

The overall loss is entirely specified as the L1 reconstruction loss of the image:(4)ℒtotal=ℒ1=1N‖Ipred−Igt‖1where N is the total number of pixels in Ipred.

This depth-supervision-free design allows the potential field to be optimized for image restoration rather than being forced to fit potentially inaccurate metric depth labels.

Datasets. Training and in-domain validation are conducted on UIEB [11]: 800 paired images are used for training (UIEB-T800), 90 paired images for full-reference validation (UIEB-V90), and 60 challenging images without references for UIEB-C60 evaluation. Cross-domain non-reference evaluation is further conducted on U45 [20], UCCS [21], MABLs [22], UFO120 [23], and Z700 [24], covering different color casts, turbidity levels, and real underwater scenes.

Implementation Details. IG-Mamba is implemented in PyTorch and trained end-to-end on an NVIDIA vGPU (32 GB). We use Adam with β1=0.9, β2=0.999, an initial learning rate of 1×10−4, and cosine annealing to 1×10−6. The network is trained for 600 epochs with a batch size of 4. Images are randomly cropped to 256×256 and augmented by random horizontal/vertical flips. The external prior is provided as an input guidance field, and the model is optimized by image reconstruction loss.

The default external prior is generated by DepthAnythingV2 and precomputed before restoration experiments. It is normalized as a depth-correlated potential input and is not used as a supervised depth label. In the degraded-prior robustness study, we keep the trained IG-Mamba checkpoint fixed and only replace or corrupt P0 at inference time, so the results evaluate test-time tolerance to imperfect prior providers rather than robustness obtained from prior augmentation.

For competing methods, we use official results or released models when available; otherwise, reproduced outputs are evaluated with the same metric scripts and testing splits. This keeps the full-reference and non-reference comparisons under a consistent protocol.

Evaluation Metrics. PSNR [25] and SSIM [26] are used for full-reference evaluation, and RMSE is additionally reported on UIEB-V90. UCIQE [27] and URanker [28] are used for non-reference evaluation. Higher values indicate better performance for PSNR, SSIM, UCIQE, and URanker, while lower values indicate better performance for RMSE.

We compare IG-Mamba with classic prior-/CNN-based methods (WaterNet [11], UWCNN [12], Ucolor [13], SPDF [29]), Transformer/advanced restoration methods (U-shape [14], AST [30], VQCNIR [31]), and recent competitive models including FMambaIR [32], UDNet [33], and CDF-UIE [34].

Quantitative Evaluation. Tables 1 and 2 report the quantitative comparison results. Benefiting from geometry-aware routing and global state-space modeling, IG-Mamba achieves consistently strong performance across the evaluated datasets. Compared with SSM-based methods that rely on generic image-plane scanning paths, such as FMambaIR [32], IG-Mamba obtains higher PSNR on UIEB-V90, suggesting that organizing tokens by a depth-correlated potential field provides a useful inductive bias for depth-dependent underwater degradation.

In addition to strong restoration performance, the core enhancement network of IG-Mamba maintains an efficient structure, with 5.32M parameters and 13.02G FLOPs at a resolution of 256×256. The default external prior in our experiments is precomputed, and the framework is not tied to one specific depth estimator. In practice, the coarse geometric anchor may be provided by a lightweight learned proxy, low-resolution prior, or on-board AUV sensing sources such as stereo cameras and sonars. This deployment flexibility is further analyzed in Section 4.4.

Qualitative Evaluation. Figs. 3–5 provide visual comparisons on UIEB-V90 and cross-domain datasets. IG-Mamba reduces color casts and haze-like scattering while preserving foreground structures and object details, showing robust restoration under diverse underwater conditions.

To address the concern that IG-Mamba may rely on an accurate external depth estimator, we conduct a controlled robustness study on UIEB-V90 by deliberately degrading the input geometry prior P0. Downsampled and blurred priors preserve coarse but spatially aligned geometry; noisy and masked priors simulate local estimation errors or missing regions; a constant prior represents the no-geometry fallback; shuffled and random priors represent spatially inconsistent or fully corrupted guidance. Table 3 reports the robustness results under these degraded-prior settings.

The results show that IG-Mamba does not require metrically accurate depth. Moderately degraded but spatially aligned priors, such as downsampled, blurred, noisy, and masked priors, do not cause performance collapse. This suggests that a coarse relative geometry anchor is sufficient for stable iso-potential grouping. In particular, coarse or smoothed priors can remain effective because they preserve the near-to-far spatial layout while reducing high-frequency artifacts in monocular depth estimates. When the spatial layout of the prior is removed or disrupted, as in constant, shuffled, and random priors, the performance drops more noticeably, suggesting that IG-Mamba relies on spatially meaningful geometry anchors rather than metrically precise depth. Together with the subsequent Static Prior ablation study, these results indicate that potential evolution helps calibrate imperfect geometry priors into restoration-oriented optical potentials.

Table 4 reports the efficiency and prior-provider comparison. To address practical deployment concerns, we further evaluate the computational cost of different prior providers at 256×256 resolution. The external clean prior achieves the best restoration quality, but its online prior-generation cost is much higher when the DepthAnythingV2 inference time is included. In contrast, the lightweight learned prior introduces only 0.017M parameters and 0.193G FLOPs, reducing the total latency from 180.06 to 35.26 ms while maintaining competitive restoration quality. This result indicates that IG-Mamba is not tied to a heavy monocular depth foundation model; a fast task-oriented prior proxy can also provide an effective coarse geometry anchor for deployment.

We also study the sensitivity to the number of iso-potential bins K, as reported in Table 5. The sensitivity analysis shows that the no-stratification reference (K=1) is clearly inferior to valid iso-potential grouping, confirming the necessity of geometry-aware sequence organization. The best performance is obtained at K=4, suggesting that underwater restoration benefits from coarse but effective iso-potential strata. When K becomes larger, PSNR decreases, likely because overly fine quantization fragments spatially coherent degradation regions and amplifies high-frequency artifacts in monocular priors. This result is consistent with the degraded-prior robustness analysis: IG-Mamba requires spatially meaningful geometry guidance, but does not rely on metrically precise depth.

We conduct ablation studies on UIEB to validate the main components of IG-Mamba. Table 6 analyzes routing, stable sorting, potential evolution, and modulation through a series of variants from the depth-free baseline to the full model. Here, Model A (Baseline) disables all geometry-related components and achieves 23.552 dB PSNR, providing the reference for evaluating the remaining variants.

1. Effectiveness of Topology-Preserving Isoline Routing. To test the routing algorithm, we separately examine the effects of hard-quantized iso-potential bins and stable sorting.

First, Model B (Continuous Potential) removes the potential quantization process and orders continuous floating-point potentials directly. Although Model B improves PSNR over the baseline from 23.552 to 23.957 dB, its SSIM decreases from 0.902 to 0.898. Since there are usually no identical potential values, the resulting sequence can frequently switch across distant spatial locations. This micro-fragmentation weakens local structure; therefore, discrete iso-potential layers are useful for preserving macroscopically continuous structures.

Second, Model C (Unstable Sort) uses isoline bins but employs a non-stable sorting method. Compared with the full model, PSNR decreases from 24.521 to 23.991 dB, and SSIM decreases from 0.907 to 0.904. Without stable sorting to preserve the local order within iso-potential strata, the resulting token sequence becomes more fragmented, forcing the network to compensate for more complex spatial discontinuities during reconstruction, thereby increasing the difficulty of image enhancement.

2. Synergy of Potential Evolution and Physical Modulation. We introduce two coupled physics-inspired mechanisms in our architecture: Potential Field Evolution (ΔP) for potential calibration and Transmission-Inspired Modulation (Wmod) for attenuation-aware feature reweighting and gradient propagation, as illustrated in Fig. 6. To fully separate their responsibilities, we examine a series of progressive variants.

First, Model D (Routing Only) uses only a static geometric prior (P0) for Mamba sequencing. Using only the routing path, its PSNR reaches 23.899 dB, outperforming Model A. However, directly modulating features using this raw prior, as in Model E (Static Prior), reduces PSNR to 23.682 dB. This negative transfer suggests that uncalibrated geometric distance should not be directly treated as underwater optical attenuation. Geometric depth and underwater optical degradation are correlated but not identical, and direct modulation may suppress useful foreground responses. Finally, Model F (Full Model) achieves the highest PSNR of 24.521 dB. This indicates that dynamic potential calibration is needed to reduce the domain gap of Model E. The Wmod gate serves two functions: it provides transmission-inspired spatial reweighting and establishes a continuous, differentiable optimization path around the non-differentiable argsort operation.

3. Representation Capacity under Deep Physical Constraints. A concern with potential-guided modulation is that explicit spatial constraints may over-restrict deep feature representation. We therefore monitor the gate statistics of Wmod across hierarchical levels after training. If the modulation were harmful, the network could weaken this branch by driving the gate toward an identity-like response, i.e., Wmod≈1. Empirically, however, Wmod does not collapse to such an identity-like gate. Its mean gate values range from 0.428 to 0.521 across hierarchical levels, indicating that the modulation branch remains effective from shallow encoders to deep decoders and that underwater degradation remains relevant across multiple semantic scales rather than only at low-level layers.

This observation supports the division of roles in IG-Mamba. The routing branch organizes long-range dependencies along geometry-aware strata, the potential updater adapts the external prior to underwater restoration, and the modulation branch suppresses scattering-dominant feature responses without replacing the main restoration stream. Therefore, the performance drop of the Static Prior variant mainly arises from using an uncalibrated prior for modulation, rather than from insufficient representation capacity under physical guidance.