GSM-GS: Geometry-Constrained Gaussian Splatting

• GSM-GS is a surface reconstruction framework that enhances 3D Gaussian Splatting by integrating explicit geometric supervision into unstructured Gaussian point clouds.
• It employs single-view texture-aware constraints and multi-view normal consistency to reshape Gaussians towards coherent, detailed surfaces.
• Empirical results on DTU, Tanks and Temples, and Mip-NeRF360 demonstrate improved reconstruction metrics and robust rendering quality.

Searching arXiv for papers on GSM-GS and closely related 3D Gaussian Splatting surface reconstruction work. GSM-GS, short for “Geometry-Constrained Single and Multi-view Gaussian Splatting for Surface Reconstruction,” is a reconstruction-oriented extension of 3D Gaussian Splatting that targets the geometric weaknesses of unstructured Gaussian point clouds. It preserves the explicit Gaussian primitive representation and fast rasterization associated with 3DGS, but adds two forms of geometric supervision: single-view, texture-aware constraints that reshape Gaussians toward the local surface, and multi-view, geometry-guided consistency constraints that enforce cross-view structural agreement through 3D point-cloud normal alignment. The method is designed for surface reconstruction rather than merely photorealistic novel-view synthesis, with particular emphasis on texture-rich micro-structures, texture-less regions, and cross-view geometric coherence (Ren et al., 13 Feb 2026).

1. Problem setting and conceptual scope

GSM-GS addresses a limitation of baseline 3D Gaussian Splatting: Gaussians are optimized primarily to minimize photometric error, yet they are not required to lie on a coherent surface. The resulting Gaussian clouds are therefore described as unstructured and irregular, which can produce floating Gaussians, poor surface conformity, noisy normals, and loss of high-frequency geometric detail in complex regions (Ren et al., 13 Feb 2026).

The method is built on planar-based 3DGS and PGSR. Its design is motivated by three failure modes. In texture-rich regions, high-frequency appearance can be explained either by surface normal variation or by covariance changes, which creates ambiguity between geometry and radiance. In texture-less regions, weak image gradients make normal-based optimization unstable and prone to local minima. In multi-view settings, image-level consistency alone does not impose explicit 3D structural coupling, so surfaces can remain misaligned across views. GSM-GS responds by coupling rendered depth, rendered normals, image gradients, and cross-view point-cloud structure in a unified optimization framework (Ren et al., 13 Feb 2026).

A common misconception is to treat GSM-GS as only a rendering refinement. The paper instead frames it as a surface-reconstruction method that uses rendering as one component of a broader geometric objective. This distinction is important because several of its contributions operate directly on depth-derived 3D structure rather than on color alone.

2. Single-view adaptive geometry constraints

The single-view component begins from two rendered depth quantities: a distance-weighted depth map $D$ and an unbiased depth map $\hat D$, with

$\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$

If the current Gaussian geometry is accurate, $D$ and $\hat D$ should agree. GSM-GS converts their discrepancy into a confidence weight

$$W(i, j) = 1 - \frac{\Delta D(i, j)}{\|\Delta D\|_{\infty,\Omega}},$$

where $\Delta D(i,j)=|D(i,j)-\hat D(i,j)|$. A trust region

$$\mathcal{H} = \{ (i, j) \in \Omega \mid W(i, j) \ge \theta \}$$

is then defined with $\theta = 0.8$, so that strong geometry supervision is applied only where the rendered geometry is already sufficiently self-consistent (Ren et al., 13 Feb 2026).

GSM-GS next partitions the image into texture-rich and texture-less sub-regions using Sobel gradients: $$\begin{aligned} G_x(i,j) &= \frac{\partial I(i,j)}{\partial x} * K_x,\ G_y(i,j) &= \frac{\partial I(i,j)}{\partial y} * K_y,\ G(i,j) &= \sqrt{G_x(i,j)^2 + G_y(i,j)^2}. \end{aligned}$$ Let $\hat D$0 be the 75th percentile of $\hat D$1. Pixels with $\hat D$2 belong to $\hat D$3 (texture-rich), and the remainder belong to $\hat D$4 (texture-less). This yields a dual-branch constraint design: geometry is enforced differently in $\hat D$5 and in $\hat D$6 (Ren et al., 13 Feb 2026).

For texture-rich, high-confidence regions, GSM-GS uses a normal-consistency term between the rasterized normal map $\hat D$7 and the normal $\hat D$8 derived from unbiased depth, weighted by $\hat D$9. It also introduces a depth-normal orthogonality term $\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$0, which encourages the depth gradient to remain tangent to the surface and hence orthogonal to the normal. The single-view geometry term is written as

$\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$1

with $\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$2, and within $\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$3 the orthogonality component is weighted by $\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$4 (Ren et al., 13 Feb 2026).

For texture-less regions, the method avoids relying on weak or unstable gradients. Instead it applies a color-aware total-variation regularizer over normals: $\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$5 This encourages smooth normals on homogeneous surfaces while preserving discontinuities near appearance edges. A plausible implication is that GSM-GS treats flat regions not as missing information but as regions requiring a different prior.

3. Multi-view spatial structure refinement

The multi-view module operates on pairs of adjacent views. For a current view $\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$6 and a nearest neighbor $\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$7, the method renders $\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$8 and $\hat{D}(p) = \frac{D(p)}{N(p)\, K^{-1}\tilde p}.$9, then reconstructs depth-derived 3D point clouds $D$0 and $D$1 (Ren et al., 13 Feb 2026).

Cross-view confidence is defined by fusing single-view confidence weights: $D$2 with $D$3. The method then forms a geometric validity mask using image bounds and a depth threshold $D$4, where $D$5 m. Candidate positions satisfy both validity and confidence requirements, with threshold

$D$6

From these candidates, the top-$D$7 positions are sampled by descending confidence; a sensitivity analysis selects $D$8 as a favorable accuracy–time operating point (Ren et al., 13 Feb 2026).

To compare geometry across views, the neighbor point cloud is transformed into the current frame using camera poses

$D$9

yielding $\hat D$0. For each sampled location, GSM-GS computes a local $\hat D$1 patch in each cloud and estimates a normal by PCA, taking the eigenvector associated with the smallest covariance eigenvalue. It further defines a curvature-dependent weight

$\hat D$2

where $\hat D$3 are the current-patch eigenvalues. Flat patches thus receive larger weight than highly curved or unstable patches (Ren et al., 13 Feb 2026).

The multi-view geometry loss is

$\hat D$4

This directly penalizes disagreement between locally estimated 3D normals across views. In contrast to purely image-plane consistency terms, this loss operates on reconstructed local surface structure.

4. Objective function and optimization procedure

GSM-GS keeps the standard Gaussian parameterization—centers, covariances, opacities, and spherical-harmonic colors—but changes the optimization target. The paper presents the unified loss

$\hat D$5

with $\hat D$6, and relates it to the PGSR baseline in the ablation discussion (Ren et al., 13 Feb 2026).

Training proceeds for 30k iterations in two phases. From 0–7k iterations, optimization uses photometric and baseline terms only. From 7k–30k iterations, the GSM-GS geometry terms are enabled. Densification is active until 15k iterations, after which the number of Gaussian primitives is frozen and optimization focuses on geometric refinement. Initialization uses COLMAP sparse point clouds and camera poses; implementation is in PyTorch + C++/CUDA on an NVIDIA RTX 4090 (24 GB) (Ren et al., 13 Feb 2026).

This schedule suggests a deliberate separation between coverage and consolidation: early iterations populate the scene, while later iterations regularize those Gaussians into a more surface-aligned configuration.

5. Empirical results

On DTU, evaluated by Chamfer Distance (CD, mm), GSM-GS reports an average 0.51, compared with 0.53 for PGSR, 0.68 for RaDe-GS, 0.76 for 2DGS, 0.81 for GOF, 0.84 for NeuS, 0.86 for VolSDF, and 1.49 for NeRF. Average training time per scene is 0.45 h, compared with 0.28 h for PGSR, 0.15 h for 2DGS, 0.13 h for RaDe-GS, and 0.58 h for GOF (Ren et al., 13 Feb 2026).

On Tanks and Temples, evaluated by F1-Score, GSM-GS reports a mean 0.36, compared with 0.34 for PGSR, 0.31 for GOF, 0.31 for RaDe-GS, and 0.22 for 2DGS. On Mip-NeRF360, GSM-GS reports average PSNR 27.72, SSIM 0.841, and LPIPS 0.175. In the same comparison, 3DGS attains PSNR 28.12, while GSM-GS attains the best LPIPS and ties the best SSIM (Ren et al., 13 Feb 2026).

The ablation study further isolates the contributions of the two geometry modules. On DTU, the PGSR baseline reports 0.53 CD, adding $\hat D$7 yields 0.50, adding $\hat D$8 yields 0.48, and the full method also reports 0.48. On Mip-NeRF360, the baseline reports PSNR 30.22, SSIM 0.928, LPIPS 0.179; the full model reports 30.80, 0.930, 0.172 (Ren et al., 13 Feb 2026). This suggests that the single-view term contributes strongly to rendering quality, whereas the multi-view term contributes more directly to geometric accuracy.

6. Limitations, interpretation, and research context

The paper identifies several limitations. Transparent and highly specular surfaces remain difficult because radiance and geometry are hard to disentangle under complex light transport. Very thin structures can be oversmoothed by PCA-based local normal estimation and TV regularization. The multi-view module also introduces nontrivial overhead because it requires point-cloud construction, SE(3) transformations, and patchwise PCA, which helps explain the increase from 0.28 h for PGSR to 0.45 h for GSM-GS on DTU (Ren et al., 13 Feb 2026).

Within the Gaussian-splatting literature, GSM-GS is best understood as a method that preserves the speed and explicitness of 3DGS while importing stronger geometric structure into the optimization. Its central claim is not that Gaussian splatting should be replaced, but that unstructured Gaussians can be regularized into a more surface-faithful representation by combining confidence-weighted single-view geometry with cross-view 3D normal consistency. That framing explains why the method improves reconstruction metrics while remaining competitive in rendering quality.