Surface Normal Concordance (SNC)

• Surface Normal Concordance is a method that enforces consistent normal predictions across viewpoints to address multi-view inconsistency in neural SDF-based reconstructions.
• It utilizes a view-dependent bias network and a concordance-driven loss combining photometric, normal supervision, and eikonal regularizers to correct normal biases.
• Key applications include improved indoor scene reconstructions with smoother surfaces and sharper local details, as demonstrated by quantitative F-score improvements.

Surface Normal Concordance (SNC) refers to the methodological enforcement and evaluation of agreement between surface normals predicted or reconstructed from multi-view or monocular geometric priors in neural implicit 3D reconstruction frameworks. Originating from challenges in fusing view-inconsistent normal estimates, SNC is critical for global consistency and local detail fidelity in neural signed distance field (SDF)–based indoor scene reconstruction. SNC, as instantiated in NC-SDF, is achieved by view-dependent normal compensation, concordance-driven loss design, geometry-informed pixel sampling, and hybrid geometry modeling, collaboratively addressing the multi-view inconsistency inherent in monocular supervision (Chen et al., 1 May 2024).

1. Definition and Motivation

Surface Normal Concordance denotes the enforced consistency of surface normal directions across multiple viewpoints or priors in neural surface reconstruction. In modern frameworks leveraging neural SDFs for 3D reconstruction, geometric supervision from monocular priors introduces view-dependent biases that manifest as multi-view inconsistency, hampering accurate, locally consistent, and globally smooth reconstructions. SNC addresses this by introducing mechanisms that compensate for such biases, thus promoting consistent normal directions across supervisory signals and views.

The principal motivation lies in empirical observations that state-of-the-art neural implicit surface representations, when supervised with monocular geometric priors (e.g., predicted normal maps), suffer from degraded multi-view concordance, limiting both surface fidelity in detailed regions and smoothness in planar areas.

2. Mathematical Formulation of Normal Compensation

NC-SDF implements SNC by explicitly modeling and correcting view-dependent biases in normal priors. At spatial point $\mathbf{x}_i$, the SDF normal $\mathbf{n}_i^{\mathrm{SDF}}$ is defined as

$$\mathbf{n}_i^{\mathrm{SDF}} = \nabla_{\mathbf{x}} s_i, \quad s_i = f_g(\mathbf{x}_i),$$

where $f_g$ denotes the hybrid geometry network.

To correct for view and geometry–dependent normal bias, a bias network $f_n$—a 4-layer MLP with ReLU activations—predicts three Euler angles $(\gamma, \beta, \theta)$, representing small axis rotations: $$(\gamma, \beta, \theta) = f_n(\mathbf{x}_i, \mathbf{v}, \mathbf{n}_i^{\mathrm{SDF}}, \mathbf{F}_g),$$ where $\mathbf{v} \in \mathbb{R}^3$ is the viewing direction and $\mathbf{F}_g$ is a deep geometry feature.

The compensated normal is then given by applying rotations about each axis: $$\mathbf{n}_i^{\mathrm{comp}} = R_Z(\theta) R_Y(\beta) R_X(\gamma) \mathbf{n}_i^{\mathrm{SDF}},$$ where $R_{*}$ are standard $3 \times 3$ rotation matrices.

Typical input dimensions to $f_n$ comprise 3 (position) + 3 (viewing direction) + 3 (SDF normal) + $\dim \mathbf{F}_g$.

3. Concordance-Driven Loss Functions

SNC is enforced via compound loss terms that align compensated normals with monocular priors:

• The photometric loss ($\mathcal{L}_c$) is the L1 distance between rendered color and ground truth.
• The normal supervision loss ($\mathcal{L}_n$) comprises:
  • L1 normal difference: $$\bigl\| \widehat{\mathbf{N}^{\mathrm{comp}}(\mathbf{r})} - \mathbf{N}(\mathbf{r}) \bigr\|_1$$
  • Cosine alignment: $$\bigl\|1 - \widehat{\mathbf{N}^{\mathrm{comp}}(\mathbf{r})}^\mathsf{T} \mathbf{N}(\mathbf{r})\bigr\|_1$$
• The Eikonal regularizer ($\mathcal{L}_e$) enforces unit gradient norm of the SDF, $\|\nabla s\| = 1$.

The rendered normal map for a camera ray $\mathbf{r}$ is: $$\widehat{\mathbf{N}^{\mathrm{comp}}(\mathbf{r})} = \sum_{i=1}^{N} T_i \alpha_i \mathbf{n}_i^{\mathrm{comp}},$$ with $T_i$ and $\alpha_i$ as ray transmittance and opacity per NeuS.

The total loss is: $$\mathcal{L} = \mathcal{L}_c + \lambda_n \mathcal{L}_n + \lambda_e \mathcal{L}_e,$$ with typical coefficients $\lambda_n = \lambda_e = 0.1$.

A two-stage training regime is adopted: Stage 1 optimizes SDF and radiance ($f_g, f_c$) with $\mathcal{L}_c + \lambda_e \mathcal{L}_e$; Stage 2 introduces $f_n$ and the full SNC-aware loss, allowing the bias MLP to absorb erroneous normal supervision.

4. Pixel Sampling Strategy for Detail-Driven SNC

SNC prioritization in geometrically intricate regions is achieved through an adaptive pixel sampling scheme informed by image texture content:

• A Canny edge detector produces intensity map $I(\mathbf{u})$ for each image.
• "Informative" pixels with $I(\mathbf{u}) \ge l_i$ form $\mathcal{P}_{\mathrm{canny}}$; others are sampled randomly.
• Each training batch samples $N_\mathrm{sample}$ pixels as
  • $r N_\mathrm{sample}$ from $\mathcal{P}_{\mathrm{canny}}$,
  • $(1 - r) N_\mathrm{sample}$ from $\mathcal{P}_{\mathrm{random}}$.
• Parameters $r$ (informative fraction) and $l_i$ (intensity threshold) follow a coarse-to-fine schedule: $r$ increases and $l_i$ decreases during training.

This strategy emphasizes SNC in regions where geometric complexity is highest (e.g., thin structures, object boundaries), reducing multi-view disagreement specifically in the areas where prior inconsistencies are most detrimental.

5. Hybrid Geometry Modeling and Its Role in SNC

The hybrid geometry model fuses MLP and voxel-grid representations:

• The "smooth branch" ($\Phi_{\mathrm{MLP}}$) maps $\mathbf{x}_i$ to $\mathbf{F}_{\mathrm{smooth}}$, promoting global surface smoothness.
• The "grid branch" ($\mathcal{G}(\mathbf{x}_i)$) provides $\mathbf{F}_{\mathrm{grid}}$, capturing high-frequency local detail.
• The concatenated geometry feature $$\mathbf{F}_g = [\mathbf{F}_{\mathrm{smooth}}, \mathbf{F}_{\mathrm{grid}}]$$ stabilizes the compensation MLP ($f_n$).

This architecture permits planar regions to resist overfitting to noisy/bias-prone supervision while simultaneously enabling sharp reconstructions under compensated supervision in fine-detail areas. The stabilized geometry features also enhance disentanglement of bias, a pivotal mechanism for effective SNC.

6. Empirical Evaluation of SNC

Quantitative and qualitative analyses on synthetic and real-world datasets, including ScanNet, validate the impact of SNC-driven techniques:

Method	F-score (%)	Relative Improvement
VolSDF	0.430	—
NeuS	0.320	—
MonoSDF (MLP)	0.748	—
NC-SDF (Ours)	0.781	+4.3% over MonoSDF
Hybrid+IPS (No NC)	0.749	—
+ Normal Compensation	0.781	+4.2% over Hybrid+IPS

Visualization of learned bias correction shows spatially structured patterns, with largest corrections at oblique wall and floor views, corresponding to known prior deficiencies.

Qualitative improvements include:

• Multi-view–consistent, smooth planar regions.
• Sharper reconstructions in slender, detailed objects such as chair legs and handles.
• Improved novel-view renderings and geometric visualizations.

While no explicit mean angular error is reported, the cosine alignment term in $\mathcal{L}_n$ directly optimizes SNC, resulting in visually discernible reductions in flicker and shading artifacts.

7. Implications and Applications

The SNC paradigm substantially improves neural reconstruction quality by reconciling multi-view and prior-induced inconsistencies. Its principled approach—involving view-dependent compensation, concordance-induced loss weighting, spatially informative supervision, and hybrid geometry encoding—provides a substantially more robust foundation for neural SDF frameworks in indoor scene reconstruction. A plausible implication is that SNC techniques can generalize to other neural reconstruction contexts where inconsistent geometric cues are prevalent, including robotics, AR/VR, and multi-sensor fusion.

Further research may explore adaptive SNC enforcement for heterogeneous sensor priors and investigate the extension of SNC-centric inductive biases to broader classes of neural surface representations.