Edge-Guided Depth Regularization Module

• Edge-Guided Depth Regularization modules are defined by their selective smoothness enforcement that preserves boundary details by leveraging edge cues.
• They integrate robust edge extraction techniques, including Canny, Sobel, and learned detectors, to modulate loss functions and enhance depth accuracy.
• Applications span NeRF, monocular depth estimation, and sparse depth upsampling, delivering measurable improvements in PSNR, SSIM, and RMSE benchmarks.

Edge-guided depth regularization modules constitute a design paradigm in depth estimation, 3D reconstruction, and depth refinement pipelines in which local or global smoothness constraints are explicitly modulated by edge information derived from image, depth, or semantic cues. Such modules enforce geometric and photometric consistency selectively—promoting smoothness within homogeneous regions while preserving or even sharpening depth discontinuities coincident with object boundaries or detected edges. This concept is implemented across a spectrum of architectures: Neural Radiance Fields (NeRF), monocular/stereo CNNs, light field solvers, sparse depth upsampling, and advanced 3D representation learning.

1. Principles of Edge-Guided Depth Regularization

The central concept is to prevent over-smoothing of depth/disparity maps across physical boundaries, elevate accuracy near edges, and suppress geometric artifacts. Standard global or local smoothness regularization terms, e.g., $\sum |\nabla D|$, are replaced or augmented with edge-aware forms: $$L_{\text{smooth}} = \sum_{i,j} |\nabla D_{i,j}| \cdot (1 - E_{i,j})$$ where $E_{i,j}$ is a binary or soft edge mask (from RGB, depth, or semantic data). This gating ensures the regularizer enforces consistency only away from discontinuities, thus avoiding boundary blurring. Edge information can be acquired using basic operators (Canny, Sobel), learned detectors (DexiNed), semantic segmentation branches, or hybrid techniques. Certain approaches further distinguish regularization strength across scales, tasks, or modalities, as in progressive or multi-scale architectures.

2. Algorithmic Implementation Patterns

Implementations vary depending on task and architecture but share several core steps, organized around the pipeline integration point:

1. Edge Extraction: Compute edge maps $E$ from input images, predicted depths, or semantic masks. Techniques include Canny edge detection with hysteresis and dilation (Yu et al., 4 Jan 2026, Huang et al., 6 Aug 2025), semantic edge extraction (Li et al., 2020), or hybrid learned edge prediction (Omotara et al., 18 Nov 2025).
2. Selective Regularization:
   • Patchwise or pixelwise masking: Regularization (smoothness constraints, local variance minimization, normal consistency) is masked to exclude or downweight edge regions, as in the weighted losses of EdgeNeRF and DET-GS (Yu et al., 4 Jan 2026, Huang et al., 6 Aug 2025).
   • Edge-aware geometric losses: Include edge-gated differences for depth, normals, or surface planes (Yu et al., 4 Jan 2026, Li et al., 2024, Yang et al., 2017).
   • Edge-aligned fusion or aggregation: Multi-branch or attention-based networks fuse depth features disproportionately or exclusively at edge locations (Dong et al., 2022, Omotara et al., 18 Nov 2025, Li et al., 2020, Li et al., 2024).
3. Plug-and-play architectures: Most modules are designed for seamless integration with standard training pipelines, requiring at most an edge pre-processing step and loss term augmentation (Yu et al., 4 Jan 2026, Huang et al., 6 Aug 2025, Guo et al., 2020).
4. Hyperparameter tuning: Thresholds for edge extraction (e.g., $\tau_e=125$, $\tau_{\text{edge}}=1e-2$); regularization weights to balance depth and edge contributions.

3. Variants Across Major Application Domains

Application	Edge Signal Source	Key Regularization Mechanism
NeRF/sparse 3D reconstruction	RGB edges (Canny)	Patchwise depth/normal smoothing off-edge only (Yu et al., 4 Jan 2026)
3D Gaussian Splatting	RGB edges (Canny)	Neighborhood mean depth consistency for non-edge (Huang et al., 6 Aug 2025)
Mono/stereo depth estimation	RGB/Semantic edges	Edge-aware smoothness & attention/fusion mechanisms (Dong et al., 2022, Omotara et al., 18 Nov 2025, Li et al., 2020)
Light-field depth (EPI)	Superpixel boundaries	Confidence shrinkage and edge-weight reinforcement (Chen et al., 2017)
Depth refinement/upscaling	Depth/RGB edges	Patchwise edge-gradient losses, adaptive fusion (Li et al., 2024, Guo et al., 2020)

NeRF and 3DGS approaches apply edge masking directly to regularization losses. Monocular and stereo depth estimation networks often incorporate spatial or channel attention modules gated by edge cues, or inject edge features into fusion blocks. In sparse data or refinement, the edge signal may be a continuous edge-distance field rather than binary, and is used to modulate convolutional interpolation kernels (Guo et al., 2020).

4. Mathematical Formulations

Edge-guided regularization adapts the loss surface, with the following archetypes:

Patchwise depth variance loss (EdgeNeRF):

$$L_z = \sum_{m=1}^M \sum_{i=1}^4 \max\big(e_{m,i}|z_{m,i}-\bar z_m|-\tau_1, 0\big)$$

with $e_{m,i} = 0$ on edges, $1$ off-edge; $\bar z_m$ is edge-weighted mean.

Normal consistency (also in EdgeNeRF):

$$L_n = \sum_{m=1}^M \sum_{i=1}^4 \max\big(e_{m,i}\|\mathbf{n}_{m,i} - \bar{\mathbf{n}}_m\|^2 - \tau_2, 0\big)$$

Image-space masking (DET-GS):

$$\mathcal{L}_{\text{edge}} = \frac{1}{P} \sum_{x_i} m(x_i)\,|\,\mathcal{D}(x_i) - \overline{\mathcal{D}}(x_i)|^2$$

where $m(x_i)=1$ if $x_i$ is not on an edge.

Edge-aware fusion regularization (SDDR):

$$\mathcal{L}_{\mathrm{grad}} = \frac{1}{N_g} \sum_{n=1}^{N_g} \| \beta_1 G_0[P_n] + \beta_0 - G_S[P_n]\|_1$$

over high-gradient patches $P_n$.

Edge-attention and regularization (EGSA-PT):

$$L_{\mathrm{reg}} = \lambda_{\text{edge}} \sum_{i,j} |\nabla D_{i,j}|(1 - E_{i,j})$$

and optionally,

$$L_{\text{edgeAlign}} = \lambda_{\text{align}} \sum_{i,j} E_{i,j}|\nabla D_{i,j}|$$

Implementations may also include edge-aware normalization in convolutional layers, edge-aware WLS solvers, or hybrid loss compositions with auxiliary edge-detection branches.

5. Empirical Effects and Benchmark Results

Across all domains, edge-guided regularization substantially improves depth accuracy near discontinuities and spatially thin structures, as well as global metrics:

• Sparse-view NeRF: EdgeNeRF's module improves PSNR/SSIM by enforcing local depth consistency in non-edge regions and preserving sharp boundaries; a +0.57–0.59 dB gain versus standard depth regularization (Yu et al., 4 Jan 2026).
• Monocular depth: EGD-Net's transformer-based edge-context fusion achieves a +3% absolute gain in $\delta_1$ accuracy over baselines while maintaining run-time efficiency (Dong et al., 2022).
• KITTI/NYUv2 monocular estimation: Edge-aware variants reduce AbsRel, SqRel, and RMSE on state benchmarks (around 5–20% relative depending on baseline) (Talker et al., 2022, Li et al., 2020).
• 3D Gaussian Splatting: Edge-aware regularization in DET-GS yields state-of-the-art geometric and visual results in sparse-view synthesis (Huang et al., 6 Aug 2025).
• Depth refinement: In SDDR, the introduction of edge-guided gradient and adaptive fusion losses reduces ORD from 0.313 to 0.305 and improves robustness to input noise (Li et al., 2024).
• Sparse upsampling: Edge-guided CNNs reduce RMSE/MAE by 6–18% against normalized-convolution-only baselines and prominently recover sharp occlusion boundaries (Guo et al., 2020).

6. Variations, Extensions, and Domain-Specific Adaptations

• Edge signal sources: Some approaches use only RGB-based cues, while others combine semantic/instance boundaries or learned edge detectors for increased robustness in low-contrast or ill-posed regions (Li et al., 2020, Yang et al., 2019).
• Progressive and multi-modal strategies: EGSA-PT phases from RGB-derived edge maps to depth-derived edges as training progresses, improving performance especially on transparent or ambiguous materials (Omotara et al., 18 Nov 2025).
• Hybrid spatial–channel attention: EGD-Net and EGSA-PT implement cross-attention between edge and context features, modulating both spatial and semantic fusion as a function of edge geometry (Dong et al., 2022, Omotara et al., 18 Nov 2025).
• Optimization scope: Some modules (e.g., SDDR, (Li et al., 2024)) target only local high-gradient regions, while others operate globally but mask or weight by edge proximity.
• Edge-aware confidence or weighting: In weighted least squares, superpixel regularization, or CNN normalization, pixelwise weights are contracted or expanded according to the presence of partial occlusions or material boundaries (Chen et al., 2017, Guo et al., 2020).

7. Integration and Computational Considerations

Edge-guided depth regularization modules are generally lightweight and require negligible overhead compared to base models: e.g., EdgeNeRF increases iteration time by $\sim0.3\%$ for depth-only regularization (Yu et al., 4 Jan 2026), with integration amounting to precomputing edge masks and summing additional loss terms. In real-time and lightweight networks such as EGD-Net, explicit edge branches and attention mechanisms are implemented in parallel with minimal parameter increase (Dong et al., 2022). Modular design enables plug-and-play integration into NeRF, 3DGS, monocular/stereo CNNs, and refinement cascades without significant architectural disruption.

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In conclusion, edge-guided depth regularization is a unifying principle underlying many of the recent advances in high-fidelity depth map estimation and 3D scene reconstruction. By embedding edge context into the loss surface or network structure, these modules mitigate boundary over-smoothing, preserve geometric detail, and improve both local and global reconstruction accuracy in challenging, data-sparse, or ambiguous settings (Yu et al., 4 Jan 2026, Huang et al., 6 Aug 2025, Dong et al., 2022, Li et al., 2020, Omotara et al., 18 Nov 2025, Talker et al., 2022, Chen et al., 2017, Li et al., 2024, Guo et al., 2020, Yang et al., 2017).