Monocular Normal Supervision Techniques

• Monocular normal supervision is a framework that estimates surface normals from a single image using deep learning and contextual geometric cues.
• It combines physics-driven priors, self-supervised consistency, and joint depth-normal optimization to address the inherent ill-posed nature of monocular reconstruction.
• Adaptive fusion and distortion-aware techniques enhance 3D scene understanding and robustness, achieving state-of-the-art results on diverse datasets.

Monocular normal supervision refers to the estimation and exploitation of surface normal maps from single images, typically using deep learning frameworks trained with various forms of supervision—explicit ground truth, self-supervision, or indirect constraints. The paradigm has gained prominence due to its critical role in geometric scene understanding, 3D reconstruction, robotics, and autonomous perception, particularly in settings where dense ground-truth normal data are unavailable or impractical to collect. Contemporary methods span from physics-driven approaches that leverage geometric priors (such as planar regularity) to hybrid architectures fusing local and global cues, and adaptive constraints unifying depth and surface normal predictions.

1. Foundations of Monocular Normal Supervision

Monocular normal supervision encompasses any methodology wherein networks are trained or constrained to estimate surface normals from a single input image, without recourse to additional sensors or ground-truth multi-view 3D data. Unlike stereoscopic or multi-view systems, the monocular case is fundamentally ill-posed: depth and surface orientation must be inferred contextually from image cues—shading, texture, edges, object boundaries, and learned priors.

Historically, dense normal annotation datasets are sparse, often limited to synthetic or indoor scenes, restricting generalization. This necessitates innovative designs such as multi-task learning (predicting both depth and normals), self-supervised consistency strategies (enforcing photometric or geometric coherence), and leveraging complementary signals (e.g., metric depth, geometric context, completed sparse depth). Prominent frameworks include NDDepth (Shao et al., 2023, Shao et al., 2023), Metric3Dv2 (Hu et al., 22 Mar 2024), ASN (Long et al., 8 Feb 2024), PanoNormal (Huang et al., 29 May 2024), and NRE-Net (Liu et al., 4 Aug 2025).

2. Physics-Driven Geometric Regularization

Certain state-of-the-art methods encode explicit geometric priors directly into the learning process. NDDepth (Shao et al., 2023, Shao et al., 2023) demonstrates a two-head architecture: one head predicts pixel-wise surface normal $N(p)$ and plane-to-origin distance $\mathcal{D}(p)$, the other regresses depth directly. For planar regions, depth is computed via

$$D(p) = \frac{\mathcal{D}(p)}{N(p) \cdot K^{-1}\tilde{p}}$$

where $K$ is the camera intrinsic matrix and $\tilde{p}$ the homogeneous coordinate of $p$. Surface normals and distances are regularized within detected planes using a plane-aware consistency constraint:

$$\mathcal{L}_{pc} = \sum_{p} \mathcal{M}(p)|\nabla N(p)| + \sum_{p} \mathcal{M}(p)|\nabla\mathcal{D}(p)|$$

with $\mathcal{M}(p)$ denoting planar region masks. The hybrid design, paired with iterative contrastive refinement (ConvGRU), allows accurate depth and normal prediction across planar and non-planar regions. Experimental results reveal SOTA performance on diverse datasets (NYU-Depth-v2, KITTI, SUN RGB-D), with strengthened generalization via geometry regularization (Shao et al., 2023, Shao et al., 2023).

3. Adaptive and Joint Optimization Frameworks

Recent advances promote adaptive constraints and unified optimization of depth and normals. The Adaptive Surface Normal (ASN) constraint (Long et al., 8 Feb 2024) introduces geometric context features learned from images, guiding both reliable plane selection for normal recovery and region prioritization for normal estimation. For each point, the method samples $K$ candidate neighboring triplets, computing candidate normals as cross-products, weighting their contribution by both geometric similarity

$$\mathcal{L}(P_i, P_j) = \exp\left(-0.5 \|F_{\text{geo}}(P_i)-F_{\text{geo}}(P_j)\|_2\right)$$

and projected triangle area, then aggregating with

$n_i = \frac{\sum_k s_k g_k n_k}{\sum_k s_k g_k}$

where $g_k$ and $s_k$ are the geometric context and area weights, respectively. The global loss couples these recovered normals with directly predicted normals (with multi-scale supervision), leading to sharper edge preservation and improved 3D structure in reconstructions, as shown across scan-based and synthetic benchmarks.

Metric3Dv2 (Hu et al., 22 Mar 2024) innovates with a joint depth-normal optimization module. Depth and normal maps are recurrently refined via ConvGRU:

$H^{(t+1)} = \mathrm{ConvGRU}(D^t, N^t, H^0, H^t)$

$$\Delta D^{(t+1)} = G_d(H^{(t+1)}),\quad \Delta N^{(t+1)} = G_n(H^{(t+1)})$$

$$D^{(t+1)} = D^t + \Delta D^{(t+1)},\quad N^{(t+1)} = N^t + \Delta N^{(t+1)}$$

This design enables normal estimation to benefit implicitly from abundant metric depth labels, even in the absence of explicit normal supervision. Depth-normal consistency loss connects the modalities, and large-scale cross-dataset training produces robust zero-shot generalization.

4. Hybrid Feature Architectures and Distortion-Aware Approaches

Hybrid architectures address spatial domain limitations and leverage diverse cues. NC-SDF (Chen et al., 1 May 2024) applies monocular normal priors from deep networks to neural implicit representations of indoor scenes (signed distance fields/SDFs), incorporating an explicit view-dependent normal compensation model. For each point,

$$n^{\text{comp}} = R_Z(\theta) R_Y(\beta) R_X(\gamma) n^{(\text{SDF})}$$

where $(\gamma,\beta,\theta)$ are predicted rotation angles that align SDF normals to monocular prior normals, correcting view-dependent biases. Informative pixel sampling (based on edge detection) and a hybrid geometry model (MLP + voxel grids) further enhance detail recovery. Experimental results on ScanNet and ICL-NUIM indicate superior reconstruction quality and multi-view consistency, compared to non-compensating baselines.

For panoramic/spherically-distorted images, PanoNormal (Huang et al., 29 May 2024) combines CNN-based local feature extraction with a distortion-aware transformer encoder. The encoder uses tangent projection sampling and trainable token flow (bias) to counteract geometric distortion inherent in ERP images. Multi-level self-attention and multi-scale decoder output enable fine-grained normal estimation across global and local contexts. Benchmarks across 3D60, Stanford2D3D, Matterport3D, and Structured3D demonstrate consistently lower mean/median angular error and improved MSE, validating holistic scene representation and accuracy under challenging projection distortions.

5. Practical Applications and Fusion Frameworks

Monocular normal supervision has direct practical impact in object detection, navigation, and augmented reality. In adverse lighting conditions, NRE-Net (Liu et al., 4 Aug 2025) employs normal maps—predicted from monocular RGB images—as robust geometric cues within a multi-modal detection framework for autonomous vehicles. Normal maps are estimated from dense depth using gradient-based computation:

$$n_{(uv)} = \frac{(\partial D_\text{dense}/\partial u, \partial D_\text{dense}/\partial v, 1)}{\|\ldots\|}$$

and fused with RGB and event data using the Adaptive Dual-stream Fusion Module (ADFM) and Event-modality Aware Fusion Module (EAFM). These enable cross-attention and adaptive weighting of geometric and dynamic features. Significant mAP improvements for object detection under challenging conditions (DSEC-Det-sub, PKU-DAVIS-SOD) support the value of normal maps for disentangling genuine obstacles from surface reflections and suppressing false positives.

6. Challenges, Limitations, and Future Directions

Despite substantial progress, several challenges persist. Monocular normal supervision remains limited by the availability of ground-truth normal annotations—especially in outdoor and complex scenes. Dependency on physics-driven priors (such as planarity) may reduce accuracy in highly curved or cluttered environments, although hybrid and uncertainty-aware fusion alleviate this in part. View-dependent biases, spherical distortion, and domain generalization remain active research areas, driving innovations in compensation algorithms, context extraction, and joint optimization strategies.

Future research is anticipated in several directions: efficient architectures suitable for resource-constrained deployment (Huang et al., 29 May 2024), enhanced loss functions balancing sharpness and robustness, improved spherical priors, and broader adaptation to tasks beyond normal estimation (e.g., semantic segmentation, full 3D reconstruction). The integration of normal supervision with dynamic modalities (events, RGB, depth), uncertainty modeling, and large-scale geometric foundation models (Hu et al., 22 Mar 2024) opens new opportunities for robust geometric perception in unconstrained environments.

7. Notable Mathematical Formulations and Cross-Paper Relationships

Monocular normal supervision research employs a suite of mathematical formulations for geometric estimation, regularization, and fusion. Table 1 summarizes key equations drawn directly from the discussed literature:

Paper/Framework	Key Equation/Formulation	Purpose
NDDepth	$D(p) = \mathcal{D}(p)/[N(p) \cdot K^{-1}\tilde{p}]$	Depth via normal-distance
NDDepth	$$\mathcal{L}_{pc} = \sum \mathcal{M}(p)|\nabla N(p)| + \ldots$$	Planar consistency loss
ASN Constraint	$n_i = (\sum_k s_k g_k n_k)/(\sum_k s_k g_k)$	Weighted normal recovery
Metric3Dv2	$\Delta D^{(t+1)}, \Delta N^{(t+1)}$ via ConvGRU update	Joint depth-normal refinement
NC-SDF	$$n^{\text{comp}} = R_Z(\theta)R_Y(\beta)R_X(\gamma)n^{(\text{SDF})}$$	View-compensated normal
PanoNormal	$$P(f, \hat{s}) = \sum_m W_m [\sum_{q,k} A_{mqk} W'_m f(\hat{s}_{mqk}+\Delta s_{mqk}) ]$$	Distortion-aware attention
NRE-Net	$$n_{(uv)} = (\partial D_\text{dense}/\partial u, \partial D_\text{dense}/\partial v, 1)/\|\ldots\|$$	Normal from depth gradient

These formulations operationalize the bridging of monocular RGB input to accurate surface geometry, leveraging both explicit and latent geometric signals.

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In summary, monocular normal supervision integrates geometric priors, adaptive context, hybrid features, and multi-modal fusion to produce reliable surface orientations from single images. Advances across physics-driven, transformer-based, and fusion-driven frameworks collectively push the field toward comprehensive scene understanding, robust reconstruction, and real-world deployment in challenging environments.