3D Gaussian Flats in Neural Scene Reconstruction

• 3D Gaussian Flats are specialized anisotropic Gaussian primitives that combine planar and freeform representations to accurately model flat surfaces in 3D neural rendering.
• They integrate 2D planar Gaussians embedded in 3D with unconstrained 3D Gaussians, optimized via a unified photometric and geometric framework.
• This approach achieves state-of-the-art performance in reconstructing low-texture planes and enables robust planar mesh extraction in indoor scenes.

3D Gaussian Flats are specialized anisotropic Gaussian primitives used in hybrid neural scene representations, in which flat (planar) surfaces are modeled with 3D Gaussians whose support is constrained to lie exactly or nearly within a geometric plane. This design addresses fundamental limitations of previous Gaussian-based neural rendering pipelines in reconstructing large, low-texture planar surfaces (e.g., walls, floors, glass) that confound traditional radiance field and splatting models using only freeform 3D Gaussians. The core methodology of 3D Gaussian Flats involves a hybrid parameterization: planar (2D) Gaussians are embedded in 3D and jointly optimized with unconstrained 3D Gaussians within a unified, end-to-end differentiable photometric framework (Taktasheva et al., 19 Sep 2025). This approach achieves state-of-the-art geometric and photometric performance in scene reconstruction and enables reliable planar mesh extraction, outperforming prior art especially in indoor environments with extensive planar structure.

1. Mathematical Foundations and Representation

3D Gaussian Flats employ two classes of Gaussian primitives:

Planar (2D) Gaussians in 3D: Each planar Gaussian $g_k$ is a 2D Gaussian with mean $\mu_k \in \mathbb{R}^2$ and covariance $\Sigma_k \in \mathbb{R}^{2 \times 2}$, supported on a plane $P_p$ defined by origin $o_p \in \mathbb{R}^3$ and normal $n_p \in \mathbb{R}^3$. The embedding into global coordinates uses a rotation $R_p \in SO(3)$ mapping the $z$-axis to $n_p$, and a homogeneous transformation $T_p = [R_p|o_p]$. The 3D mean and covariance are:

• Mean: $\mu_k^3 = T_p \cdot [\mu_k; 0; 1]$
• Covariance: $$\Sigma_k^3 = T_p \cdot \mathrm{diag}(\Sigma_k,0) \cdot T_p^\top$$ This structure enforces zero variance along the third principal axis, so that all density is concentrated within $P_p$ (Taktasheva et al., 19 Sep 2025, Li et al., 25 Mar 2025).

Freeform (3D) Gaussians: For non-planar (generic) regions, fully unconstrained anisotropic 3D Gaussians $\mathcal{N}(\mu_k, \Sigma_k)$ are used. $\mu_k \in \mathbb{R}^3$, $\Sigma_k \in \mathbb{R}^{3\times3}$, $\det \Sigma_k > 0$, enabling flexible modeling of arbitrary non-planar local geometry.

“Flat” Specialization in Hierarchical Schemes: Flat Gaussians can be promoted by clamping the smallest principal axis of the covariance, e.g., $$\Sigma = R \cdot \mathrm{diag}(\sigma_\parallel^2, \sigma_\parallel^2, \varepsilon^2) \cdot R^\top$$ with $\varepsilon \ll \sigma_\parallel$. In GaussianForest (Zhang et al., 2024), a threshold on $s_{i,3}$ distinguishes these flats during the hierarchical compression process.

2. Hybrid 2D/3D Scene Modeling Pipeline

The principal algorithmic framework alternates between identifying and optimizing planar Gaussians and updating freeform 3D Gaussians:

1. Initialization: The process begins with an unconstrained 3D Gaussian splatting warm-up phase (3,000–5,000 iterations) using only freeform 3D Gaussians to capture a rough scene estimate via photometric loss and geometric regularizers.
2. Planar Region Detection and Fitting:
   • Candidate Gaussians for planar assignment are selected via 2D mask projections (from semantic segmentation models, e.g. SAM or PlaneRecNet), opacity, and depth proximity.
   • Planes are fit to candidate mean positions using RANSAC; those with sufficient inliers and low residual error are retained.
   • Inlier Gaussians are “snapped” to the plane by converting their mean/covariance to 2D coordinates, enforcing zero variance out-of-plane and preserving in-plane rotation.
   • Planes are merged if close in normal/origin.
3. Block-Coordinate Optimization:
   • Alternating phases update plane parameters (with Gaussians frozen) and Gaussian parameters (with planes frozen) over iterative cycles until convergence.
4. Planar Densification (Relocation):
   • Low-opacity freeform Gaussians near planes are stochastically converted to planar Gaussians based on their distances $d_\perp, d_\parallel$ to the plane, using a CDF-based probability.
5. Joint Rendering and Supervision:
   • Photometric losses, planar mask supervision, depth variation regularization, and scale/opacity regularizers drive the optimization. Both planar and general primitives are rendered together using volumetric splatting and alpha blending (Taktasheva et al., 19 Sep 2025).

3. Rendering, Losses, and Optimization

Rendering with 3D Gaussian Flats proceeds via volume splatting:

• Per-ray Accumulation: Rays $r(t) = o + td$ are integrated over the sum of all (planar and non-planar) Gaussians using transmittance-weighted alpha blending.
• Photometric Loss: The L1 distance between observed and synthesized pixel colors is minimized.
• Planar Mask Loss: Rendered planar mask images supervise the assignment of Gaussians to planes.
• Regularization:
  • Total variation penalty on local depth gradients ($L_{TV}$)
  • L2 penalty on scale parameters ($L_{scale}$)
  • L1 penalty on opacities ($L_{opacity}$)

For methods incorporating self-supervised or open-surface learning—such as GaussianUDF (Li et al., 25 Mar 2025)—the supporting losses include Chamfer distance terms (for far-field supervision), near-surface regression to UDF values (for local thickness), and normal alignment constraints ensuring the estimated UDF and the plane normal agree exactly on the flat Gaussian supports.

4. Empirical Performance and Applications

3D Gaussian Flats substantially improve reconstruction accuracy, completeness, and mesh extraction fidelity in photometric scene reconstruction benchmarks, especially in environments dominated by flat, textureless surfaces. Key numerical results include:

Method	RMSE (ScanNet++)	PSNR (ScanNet++)	% Planar Primitives
3DGS	0.44	27.09	—
2DGS	0.39	25.56	—
PGSR	0.35	25.78	—
3D Gaussian Flats	0.27	27.01	27.8

Planar mesh extraction accuracy is likewise state-of-the-art, with planar region F1 scores around 50–54%, outperforming AirPlanes and PlanarRecon by a large margin on both DSLR and iPhone acquisition subsets (Taktasheva et al., 19 Sep 2025).

In hybrid renderers, 3D Gaussian Flats facilitate consistent scene interpretation over variable camera geometries and density distributions without overfitting to specific acquisition modalities.

5. Relation to Broader Gaussian Splatting Methods

3D Gaussian Flats are integrated into several modern frameworks:

• GaussianForest (Zhang et al., 2024): Enables planar specialization via hierarchical parameter sharing and pruning—flats occupy minimal variance along one axis; freeform splats handle residual structure. Compression ratios exceeding $10\times$ are achieved for large scenes.
• RaySplats (Byrski et al., 31 Jan 2025): Extends splatting with a ray-tracing backend that supports true 2D “flat” Gaussians as microfacets for advanced lighting and reflection models.
• TR-Gaussians (Liu et al., 17 Nov 2025): Leverage planar-constrained Gaussians to model transmission/reflection, mirroring Gaussians across explicit learned planes, and blending via Fresnel-based weights.
• GaussianUDF (Li et al., 25 Mar 2025): Uses flats for explicit coverage of open surfaces, constraining a distance function's zero set to align with the support of the planar Gaussians.

6. Limitations and Extensions

Despite their strengths, current 3D Gaussian Flats pipelines have recognized limitations:

• Reliance on initial 3DGS “warmup” phases can result in underpopulation of featureless planar regions.
• Segmentation errors from upstream mask generators (SAM, PlaneRecNet) propagate into the plane assignment, limiting geometric precision.
• Plane fitting via RANSAC introduces computational overhead (∼1 hr/scene on A6000-class hardware).
• Spherical harmonics used for appearance generalization are inadequate for extreme view-dependent effects.
• Gradient-based relocation and densification may still inadequately represent extremely large uniform surfaces.

Anticipated research directions include differentiable or learned plane proposal mechanisms, enhancement of segmentation models, and more flexible appearance models for view-dependent rendering.

7. Theoretical and Practical Significance

By combining the low-dimensionality, semantic interpretability, and efficiency of planar representations with the flexibility of unconstrained Gaussians, 3D Gaussian Flats bridge a long-standing gap between high-fidelity photometric neural rendering and robust, accurate geometric scene modeling. This hybridization enables both state-of-the-art performance in depth and mesh recovery and easy integration with scattering, lighting, and physical simulation modules in downstream tasks (Taktasheva et al., 19 Sep 2025, Li et al., 25 Mar 2025, Zhang et al., 2024). The modular design further enables broad applicability in areas such as digital twin construction, mixed-reality environments, compressed rendering, and open-surface modeling.

In summary, 3D Gaussian Flats represent a hybrid 2D/3D Gaussian-based paradigm that robustly captures planar and non-planar geometry in photometric scene representations, with joint, end-to-end optimization delivering best-in-class accuracy, efficiency, and generalization across camera and scene modalities (Taktasheva et al., 19 Sep 2025).