GeoSVR: Explicit Voxel Surface Reconstruction

• GeoSVR is an explicit voxel-based framework that enables geometrically accurate 3D surface reconstruction in radiance field modeling.
• It uses a sparse octree structure paired with uncertainty-driven depth constraints and multi-view regularization to overcome limitations of NeRF and Gaussian Splatting.
• Experimental results demonstrate improved geometric accuracy, detail completeness, and computational efficiency across diverse datasets.

GeoSVR is an explicit voxel-based framework designed for geometrically accurate surface reconstruction in 3D radiance field modeling. It is developed to address representational bottlenecks and geometric ambiguities in dominant approaches such as Gaussian Splatting and NeRF-based methods. GeoSVR leverages sparse voxel representations with novel uncertainty-driven constraints and regularization techniques, achieving detailed, complete, and efficient scene reconstruction across a variety of challenging scenarios (Li et al., 22 Sep 2025).

1. Motivation and Historical Context

Prior to GeoSVR, 3D scene reconstruction was dominated by implicit representation methods (NeRF family, SDF-based models) and Gaussian Splatting approaches. NeRF-style methods excel in photorealistic rendering but often entail high computational cost, limited geometric clarity, and sensitivity to initialization. Gaussian Splatting improves efficiency and coverage but relies heavily on sparse point-cloud initialization (via multi-view geometry) and suffers from blurred or ambiguous geometric boundaries because of the smoothness of the Gaussian primitives.

GeoSVR is conceptualized as a response to these constraints. By deploying explicit sparse voxels defined in an octree hierarchy, GeoSVR inherently preserves geometrical boundaries, coverage completeness, and supports robust convergence even in regions lacking dense initialization. The method draws technical inspiration from SVRaster, extending its principles for surface modeling beyond prior usage.

2. Sparse Voxel Representation and Scene Coverage

GeoSVR utilizes a sparse voxel octree, where the scene is partitioned into hierarchical cubic cells (voxels), each parameterized by trilinear density fields and Spherical Harmonic (SH) coefficients for color. Each voxel is initialized to fully cover the global scene domain, in contrast to approaches that restrict the parameterization to point-cloud regions or rely on adaptive, local fitting.

Voxel Definition

For a voxel indexed by (i, j, k) at octree level $l$, the center, size, and local density/color are defined as:

• $size = R \times 2^{-l}$
• $center = -0.5 \times size + size \times v$
• Trilinear density interpolates the eight corner values.
• SH coefficients enable directional color encoding.

During rendering, GeoSVR employs a volumetric alpha-blending technique analogous to NeRF and Gaussian Splatting:

$$C = \sum_{i=1}^{N} T_{i} \alpha_{i} c_{i}, \quad T_{i} = \prod_{j=1}^{i-1}(1 - \alpha_j)$$

with $\alpha$ computed by exponential mapping of local voxel density.

Explicit voxel boundaries provide geometric clarity, making the method well-adapted for regions infrequently covered by point clouds and reducing ambiguity in reconstructed surfaces.

3. Voxel-Uncertainty Depth Constraint

Sparse voxels historically lack strong scene constraints, leading to potential divergence or incomplete geometry reconstruction. GeoSVR introduces Voxel-Uncertainty Depth Constraint to regulate scene convergence via monocular depth cues, while mitigating the risks associated with noisy or uncertain external estimators.

Uncertainty Modeling

• Base uncertainty $U_{base}(l)$: Inversely related to voxel's octree level and controlled by a scaling factor, typically:

$U_{base}(l) = \frac{C}{\beta \cdot (l + l_0)}$

where $C$ is constant, $l$ is level, and $\beta$ is a scale.

• Voxel-specific uncertainty $U_{geom}(v)$: Modulated by interpolated density, reflecting local constraint confidence.

$$U_{geom}(v) = U_{base}(l) \cdot (1 - \exp(-interp(..., q)))$$

The final uncertainty is transformed into a weight $W$ that regulates the influence of the monocular depth loss:

$$L_{D-unc}(D, \widetilde{D}) = W \cdot L_{D-patch}(D, \widetilde{D})$$

where $L_{D-patch}$ aggregates pixel-wise patch losses from estimated and rendered depth maps.

The system adaptively intensifies depth supervision in regions with high uncertainty while down-weighing confident regions, thereby maintaining geometric accuracy and avoiding overfitting to noisy depth cues.

4. Sparse Voxel Surface Regularization

To achieve detailed and sharp surface geometry, GeoSVR employs specific regularization strategies targeting tiny voxels and their local neighborhoods.

Multi-View Regularization with Voxel Dropout

• Homography-based Patch Warping: Multi-view consistency is enforced by warping image patches according to geometry and camera poses.
• Voxel Dropout: To counteract local overfitting, a fraction of voxels is randomly dropped during regularization, which increases the effective area each remaining voxel is responsible for, improving patch-based consistency and mitigating redundancy.

Surface Rectification and Scaling Penalty

• Surface rectification: Compare “enter” ($p_e$) and “exit” ($p_o$) densities in surface voxels ($V_s$) to align the densest region to the actual surface:

$$R_{rec} = w \cdot \mathbb{I}(v \in V_s) \cdot [interp(\cdot, p_e) - interp(\cdot, p_o)]$$

with $w = T_\alpha$ and $\mathbb{I}$ indicating surface voxel inclusion.

• Scaling penalty: Voxels with excessively large effective sampling distances (implying mislocalized or oversmoothed geometry) are penalized logarithmically, normalized by global minimum voxel size:

$$R_{sp} = w \cdot interp(\cdot, q_c) \cdot \max(0, \log_2(\frac{voxelEffectiveSize}{minVoxelSize}))$$

Combined Loss Function

The total training loss blends photometric loss ($L_{photo}$), uncertainty-modulated depth loss ($\eta L_{D-unc}$), normalized cross correlation for multi-view regularization ($\tau L_{NCC}$), and voxel regularization terms ($\mu_1 R_{rec}$, $\mu_2 R_{sp}$):

$$L = L_{photo} + \eta L_{D-unc} + \tau L_{NCC} + \mu_1 R_{rec} + \mu_2 R_{sp}$$

Empirical hyperparameters are: $$\eta = 0.1, \tau = 0.01, \mu_1 = 10^{-5}, \mu_2 = 10^{-6}$$.

5. Experimental Performance and Comparative Analysis

GeoSVR is extensively evaluated on DTU, Tanks and Temples (TnT), and Mip-NeRF 360 datasets. Results indicate:

• Geometric Accuracy: Lower Chamfer distances and higher F1-scores than NeRF, SDF, Gaussian Splatting, and mesh-based alternatives.
• Detail and Completeness: Superior surface detail and coverage in regions prone to under-constraint or ambiguous geometry.
• Efficiency: Rapid training and inference, without dependency on heavy point cloud initialization.
• Robustness: Improved handling of reflective, textureless, and incomplete coverage areas compared with Gaussian-based techniques.

The approach stands out by eschewing the need for point cloud pre-initializations, replacing ambiguous primitives with explicit voxels, and by carefully modulating depth cue reliance through uncertainty modeling.

6. Implications and Future Directions

GeoSVR's framework provides a robust solution to explicit sparse voxel surface modeling, paving the way for further research in:

• Handling Scene Ambiguity: Expanding voxel "globality" to accommodate lighting variations, textureless surfaces, and highly ambiguous scenarios.
• Integration of New Cues: Investigating the incorporation of richer external priors and more efficient ray tracing to escape local minima.
• Generalization Across Modalities: Potential fusion with hierarchical grid and multi-scale spherical encoding systems (see Sphere2Vec (Mai et al., 2023)) for broader geospatial regression, localization, or analysis.

GeoSVR's contributions—explicit sparse voxel representation, adaptive uncertainty-based constraints, and tailored regularization—demonstrate clear advances over preceding methodologies in geometric surface reconstruction. The framework presents a scalable, accurate, and computationally efficient solution for 3D scene modeling, with applications extending across computer vision, geospatial analysis, and robotics.