Radius-based Spatial Thinning & Merge Scheme

1. Technical Overview

To address the severe spatial redundancy of 3D Gaussian Splatting (3DGS) data in high-density regions and the grid artifacts and structural aliasing commonly introduced by traditional voxel-based thinning methods, we propose a radius-based spatial thinning and merge scheme.

This method uses a spatial distance constraint as its core criterion. By defining a minimum spatial separation (radius threshold), Gaussian primitives are processed through point-wise selection and merging. The algorithm prioritizes Gaussians with higher visual contribution and suppresses redundant points within their spatial neighborhoods, thereby achieving a more uniform spatial distribution.

To ensure computational efficiency for large-scale datasets, a hash grid is introduced as a spatial acceleration structure. The 3D space is partitioned into regular cells whose edge length equals the radius threshold. Neighbor searches are restricted to the current cell and its adjacent cells only, reducing the computational complexity to near-linear while guaranteeing completeness.

If a representative Gaussian already exists within the radius neighborhood of a given Gaussian, its geometric and attribute parameters are merged into that representative using weighted aggregation. Otherwise, the Gaussian is retained as a new representative. After merging, an upper bound is enforced on the Gaussian scale to prevent abnormal inflation caused by excessive fusion, ensuring rendering stability.

This scheme does not rely on regular voxel boundaries, multi-level LOD hierarchies, or information-theoretic distance metrics. It is well suited for offline thinning and engineering-level optimization of large-scale 3DGS datasets.

2. Algorithm Workflow

Input:

• Gaussian set
• Radius threshold (r)

Output：

• Thinned and merged Gaussian set

Steps：

1. Importance Sorting
   Gaussians are sorted according to opacity and local scale to prioritize primitives with higher visual contribution.

2. Spatial Hash Construction
   The 3D space is divided into grid cells with edge length (r), and a hash mapping is built as a spatial index.

3. Radius Neighborhood Query
   For each Gaussian, neighbor search is performed only within its own grid cell and the surrounding 27 neighboring cells.。

4. Thinning and Merging

   • If no representative exists within radius(r), the Gaussian is retained as a new representative.
   • If a representative exists, the Gaussian is merged into it using weighted aggregation.

5. Scale Constraint and Parameter Update
   A scale upper bound is applied to merged Gaussians, and the final Gaussian set is output.

3. Mathematical Formulation

3.1 Gaussian Representation

Let the input Gaussian set be defined as:

$$\mathcal{G} = { g_i \mid g_i = (\mathbf{x}_i, {\Sigma}_i, \mathbf{f}_i, \alpha_i) }$$

where:

• $(\mathbf{x}_i \in \mathbb{R}^3 )$ is the Gaussian center
• $({\Sigma}_i)$represents scale or covariance approximation
• $( \mathbf{f}_i)$denotes color or spherical harmonics (SH) features
• $(\alpha_i)$ is the opacity

3.2 Radius Constraint

For a Gaussian $(g_i)$，if there exists a representative $(g_j )$ such that:

$$|\mathbf{x}_i - \mathbf{x}_j|_2 \le r$$

then the two Gaussians are considered spatially redundant and should be merged. Otherwise, $(g_i)$ is preserved as a new representative.

3.3 Weight Definition

The merge weight is defined as:

$$ w_i = \alpha_i $$

（In practice, this can be extended to $( w_i = \alpha_i \cdot s_i )，where (s_i)$ is a local scale factor.）

3.4 Position and Attribute Merge

In radius-constrained spatial aggregation, multiple Gaussian splats falling within the same aggregation cluster are merged into a representative Gaussian in order to reduce data size while preserving overall visual appearance. The merge operation is performed across multiple dimensions, including position, features, opacity, scale, and rotation.

3.4.1 Position Merge

The position of the merged Gaussian is computed as a weighted centroid:

$$ \mathbf{x}_{\text{new}} = \frac{\sum_i w_i \mathbf{x}_i}{\sum_i w_i} $$

where:

• $\mathbf{x}_i$ denotes the center position of the $i$-th Gaussian,
• $w_i$ is the corresponding weight, typically related to opacity or importance.

3.4.2 Feature Merge (Color / Spherical Harmonics)

Color and higher-order spherical harmonics (SH) features of Gaussians are defined in a linear space and can therefore be merged using weighted averaging:

$$ \mathbf{f}_{\text{new}} = \frac{\sum_i w_i \mathbf{f}_i}{\sum_i w_i} $$

where $\mathbf{f}_i$ includes the DC component and higher-order spherical harmonics coefficients. Since spherical harmonics are defined in a linear space, weighted averaging preserves the expected radiance.

3.4.3 Opacity Merge

To correctly model the occlusion relationship among multiple Gaussians, opacity is merged using cumulative occlusion probability:

$$ \alpha_{\text{new}} = 1 - \prod_i (1 - \alpha_i) $$

This formulation is equivalent to accumulating $\log(1 - \alpha)$ in log space and avoids the underestimation caused by linear averaging.

3.4.4 Scale Merge

The spatial scale is first determined by the dispersion of Gaussian centers, whose statistical variance is given by:

$$ \sigma^2_{\text{pos}} = \mathbb{E}[\mathbf{x}^2] - \mathbf{x}_{\text{new}}^2 $$

In practice, this expectation is computed using weighted summation and combined with the original Gaussian scales to obtain the merged scale:

$$ \sigma_{\text{new}}^2 = \frac{\sum_i w_i \lVert \mathbf{x}_i - \mathbf{x}_{\text{new}} \rVert^2}{\sum_i w_i} + \frac{\sum_i w_i \sigma_i^2}{\sum_i w_i} $$

To prevent abnormal scale inflation, an upper bound constraint is introduced in the implementation:

$$ \sigma_{\text{new}} \le k \cdot \frac{\sum_i w_i \sigma_i}{\sum_i w_i} $$

3.4.5 Rotation Merge

The rotation of a Gaussian ellipsoid is represented using a unit quaternion. Let the rotation of the $i$-th Gaussian be:

$$ \mathbf{q}_i = (x_i, y_i, z_i, w_i), \qquad \lVert \mathbf{q}_i \rVert = 1 $$

• (1) Hemisphere Alignment

Since quaternions have sign ambiguity (i.e., $\mathbf{q}$ and $-\mathbf{q}$ represent the same rotation), all quaternions are first aligned to the same hemisphere before merging.
Using a reference quaternion $\mathbf{q}_{\mathrm{ref}}$, each quaternion is adjusted as:

$$ \mathbf{q}_i' = \begin{cases} \mathbf{q}_i, & \mathbf{q}_{\mathrm{ref}} \cdot \mathbf{q}_i \ge 0 \\ -\mathbf{q}_i, & \text{otherwise} \end{cases} $$

where $\cdot$ denotes the quaternion inner product.

• (2) Weighted Quaternion Accumulation

After hemisphere alignment, quaternions are accumulated using weighted summation:

$$ \tilde{\mathbf{q}} = \sum_i w_i , \mathbf{q}_i' $$

where $w_i$ is the weight associated with the $i$-th Gaussian.

• (3) Normalization

The accumulated quaternion is normalized to obtain the merged rotation:

$$\mathbf{q}_{\mathrm{new}}=\frac{\tilde{\mathbf{q}}}{\lVert \tilde{\mathbf{q}} \rVert}$$

This quaternion represents the principal axis orientation of the merged Gaussian ellipsoid.

3.7 Hash Grid and 27-Cell Neighborhood

When the grid cell edge length is set to $(r)$, any two points within distance $(r)$ must differ by at most $([-1, 1])$ grid units along each axis. Therefore, only:

$$ 3 \times 3 \times 3 = 27 $$

neighboring grid cells need to be examined to ensure no valid neighbors are missed.

4. Computational Complexity

Let (N) be the number of input Gaussians and (M) the number of representatives after thinning（typically $( M \ll N ) $）。

• Sorting:

$$O(N \log N) $$

• Radius-based thinning and merging (with hash grid acceleration):
  Each Gaussian checks only a constant number of candidates on average:

$$O(N)$$

• Overall time complexity:

$$O(N \log N)$$

• Space complexity:

$$O(N)$$

5. Advantages and Effects

5.1 Visual Quality

• Eliminates grid artifacts introduced by voxel-based thinning
• Suppresses Gaussian stacking and ghosting in high-density regions
• Preserves structural continuity and boundary stability

5.2 Engineering and Performance

• Near-linear complexity, suitable for million- to ten-million-scale datasets
• Simple hash grid structure, easy to implement on CPU, WASM, or GPU
• No dependence on covariance decomposition or KL-divergence computation, ensuring numerical stability

5.3 General Applicability

• No multi-level LOD hierarchy required; suitable for single-layer model optimization
• Can serve as a preprocessing, compression, or loading optimization module for 3DGS data
• Easily decoupled and integrated with rendering schedulers and streaming systems

6. Conclusion

The radius-based spatial thinning and merge scheme suppresses redundant Gaussian primitives through spatial distance constraints. Without relying on regular voxel partitioning or information-theoretic metrics, it achieves a high-quality and scalable thinning approach for 3D Gaussian data, balancing visual fidelity with engineering performance and making it well suited for deployment in large-scale 3DGS systems.

Moreover, this scheme naturally serves as a solid foundation for Level of Detail (LOD) construction. By progressively increasing the radius threshold ( r ), a hierarchy of Gaussian representations from fine to coarse can be generated while preserving consistent merging rules and stable spatial distributions. Each radius corresponds to a distinct spatial resolution, ensuring a minimum inter-Gaussian distance at every level and avoiding the abrupt transitions and structural discontinuities commonly observed in voxel-based LOD methods.

Because all levels follow the same radius constraint and merging strategy, strong geometric, density, and visual coherence is maintained across LODs. This allows seamless integration with runtime scheduling mechanisms based on view distance, screen-space error (SSE), or other adaptive criteria, enabling smooth, stable, and engineering-friendly multi-level rendering and streaming of 3D Gaussian Splatting data.