Fourier Splatting: Generalized Fourier encoded primitives for scalable radiance fields

Abstract: Novel view synthesis has recently been revolutionized by 3D Gaussian Splatting (3DGS), which enables real-time rendering through explicit primitive rasterization. However, existing methods tie visual fidelity strictly to the number of primitives: quality downscaling is achieved only through pruning primitives. We propose the first inherently scalable primitive for radiance field rendering. Fourier Splatting employs scalable primitives with arbitrary closed shapes obtained by parameterizing planar surfels with Fourier encoded descriptors. This formulation allows a single trained model to be rendered at varying levels of detail simply by truncating Fourier coefficients at runtime. To facilitate stable optimization, we employ a straight-through estimator for gradient extension beyond the primitive boundary, and introduce HYDRA, a densification strategy that decomposes complex primitives into simpler constituents within the MCMC framework. Our method achieves state-of-the-art rendering quality among planar-primitive frameworks and comparable perceptual metrics compared to leading volumetric representations on standard benchmarks, providing a versatile solution for bandwidth-constrained high-fidelity rendering.

• The paper introduces a Fourier-encoded primitive that decouples reconstruction fidelity from primitive count by dynamically adjusting Fourier coefficients.
• It employs a differentiable rendering framework with a straight-through estimator and hybrid lobe decomposition to optimize both boundary complexity and spatial coverage.
• Experiments demonstrate smooth level-of-detail adaptation and enhanced geometric fidelity, outperforming traditional splatting approaches.

Fourier Splatting: Generalized Fourier Encoded Primitives for Scalable Radiance Fields

Introduction and Motivation

Radiance field representations have established themselves as the backbone of photorealistic novel view synthesis. Implicit neural approaches, such as NeRFs [mildenhall_nerf_2020], have demonstrated high-fidelity reconstructions but are limited by rendering inefficiencies due to dense ray marching. The introduction of explicit primitive-based methods, notably 3D Gaussian Splatting (3DGS) [kerbl_3d_2023], has proven highly successful for real-time rendering by treating scenes as collections of explicit anisotropic Gaussian primitives. However, prior splatting frameworks fundamentally tie reconstruction fidelity to the primitive count—scalability is achieved by pruning or increasing the number of primitives, not by refining the complexity of individual primitives.

The primary contribution of Fourier Splatting is the introduction of the first inherently scalable primitive for radiance field rendering, where the expressiveness of each primitive is parameterized by a truncatable Fourier boundary in the tangent plane. This representation enables dynamic runtime adaptation of reconstruction quality through modification of the number of retained Fourier coefficients, thus decoupling the rate-distortion tradeoff from primitive density.

Figure 1: Overview of Fourier Splatting using generalized Fourier encoded primitives. Increasing the retained frequency count $K$ yields progressively more complex primitive shapes, enhancing scalability and reconstruction fidelity.

Technical Architecture

Primitive Parameterization

Each surfel in Fourier Splatting is encoded by a center, a tangent frame, opacity, spherical harmonics for view-dependent color, circumradius, Fourier amplitudes and phases, and a sharpness parameter. The critical innovation is the Fourier-encoded boundary, defined as:

$$r(\theta) = R \left| \sum_{k=0}^{K-1} \bar{r}_k e^{i\phi_k} e^{ik\theta} \right|$$

where the $K$ complex coefficients control the primitive's shape within its tangent plane. Amplitude normalization enforces spatial bounds to ensure the boundary does not exceed the circumradius.

Figure 2: Primitive parameterization—the boundary $r_i(\theta)$ is defined in the tangent plane within the circumradius $R_i$, enabling arbitrary closed shapes.

Differentiable Rendering and Optimization

Rendering utilizes a polar window function, with per-pixel opacity:

$$\alpha_{i,j} = o_i \cdot \max\left(0, \frac{r_i(\theta_j)-\rho_j}{r_i(\theta_j)} \right)^{\sigma_i}$$

to ensure spatial locality and hard boundaries. However, the zero-gradient region outside the window hampers expressive optimization. This is mitigated by a straight-through estimator (STE), which injects surrogate gradients outside the primitive boundary, ensuring robust learning of both boundary complexity and spatial coverage.

Figure 3: The straight-through estimator (STE) provides gradients outside the primitive's support, alleviating optimization stalling and enabling expressive boundary deformation.

Densification and Decomposition

Fourier Splatting employs an MCMC-compatible densification strategy, supporting deterministic primitive relocation and addition within a fixed budget, while preserving the target distribution. HYDRA, the hybrid decomposition mechanism, operates in two modes:

• Scale-preserving split: Small primitives are split symmetrically via relocation formulas.
• Learned lobe decomposition: For complex boundaries, an MLP partitions multi-lobed shapes along deep valleys in $r(\theta)$, splitting the primitive into simpler descriptive lobes for improved geometric fidelity.

  Figure 4: Learned lobe decomposition identifies and segments boundary lobes, providing expressive densification in regions with complex geometry.

Experimental Evaluation

Rendering Quality and State-of-the-Art Comparison

Quantitative and qualitative evaluations on Mip-NeRF 360 and Tanks and Temples benchmarks demonstrate that Fourier Splatting sets a new standard among planar-primitive methods, outperforming both 2DGS [huang_2d_2024] and Triangle Splatting [held_triangle_2025] with substantial PSNR and SSIM gains. On planar methods, Fourier Splatting is the only approach capable of dynamic level-of-detail adaptation without post-training optimization.

Scalability and Level-of-Detail

A key strength is inherent scalability: during inference, restricting the number of transmitted Fourier coefficients at runtime leads to monotonic and graceful degradation of rendering quality, unlike prune-based methods that can induce catastrophic drop in quality by removing entire spatial regions.

Figure 5: Qualitative scalability analysis shows that reducing the coefficients leads to gradual loss of detail, outperforming octree-based pruning approaches such as Octree-GS.

Figure 6: Visual comparisons among ground truth, prior methods, and Fourier Splatting highlight superior recovery of fine details and accurate surfaces.

Figure 7: The ablation validates the impact of lobe decomposition, the STE, and cloning noise, demonstrating strong isolation of each component’s contribution to PSNR, SSIM, and LPIPS.

Analysis of Scalability

(Figure 7, as right half of the composite)

Figure 7: Quantitative LoD curve. Performance drops smoothly as Fourier coefficients are truncated, confirming continuous, controllable rate-distortion scalability per primitive.

Surface Reconstruction

Figure 8: Mesh reconstruction results display high geometric accuracy and faithful color preservation from the extracted surface.

Figure 9: Surface alignment visualizations demonstrate precise adherence to ground-truth geometry.

Limitations

Current design focuses on maximizing fidelity for a given primitive budget, not minimizing bitrate. There is significant potential in combining scale regularization and higher frequency boundaries to further reduce model size, especially for transmission scenarios. The per-primitive arithmetic cost is marginally higher than 2DGS: $57 + 8K$ operations per pixel per primitive, but this is justified by the increased per-primitive expressiveness.

Implications and Future Directions

Fourier Splatting introduces a fully continuous, scalable axis for 3D radiance field expressiveness, supporting direct runtime rate-distortion tradeoffs—crucial for streaming and bandwidth-constrained high-fidelity 3D applications. By decoupling quality from primitive count, it fundamentally alters the design space for scalable neural rendering and offers an extensible foundation that can integrate with emerging vector quantization or distributed compression frameworks (see [navaneet_compgs_2025], [chen_hac_2025], [papantonakis_reducing_2024]).

Combining per-primitive scalability with hierarchical LoD structures (as in Octree-GS [ren_octree-gs_2025-1]) or neural codebook compression points toward highly adaptive 3D representations. Further exploration of nonplanar, volumetric generalizations of the Fourier parameterization could unlock new performance bounds in large-scale scene synthesis and real-time XR applications.

Conclusion

Fourier Splatting brings principled, progressive scalability to radiance field rendering by introducing a primitive whose spatial expressiveness is parameterized via a Fourier-encoded boundary. The tight integration of differentiable rendering, stable optimization through STEs, and hybrid lobe decomposition enables both state-of-the-art visual quality and robust, continuous level-of-detail adaptation. This architecture sets a new direction for efficient, high-fidelity scene representations, with broad implications for neural compression, adaptive streaming, and scalable 3D content delivery.

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• [kerbl_3d_2023]: 3D Gaussian Splatting for Real-Time Radiance Field Rendering
• [huang_2d_2024]: 2D Gaussian Splatting for Geometrically Accurate Radiance Fields
• [held_triangle_2025]: Triangle Splatting for Real-Time Radiance Field Rendering
• [navaneet_compgs_2025]: CompGS: Smaller and Faster Gaussian Splatting with Vector Quantization
• [chen_hac_2025]: HAC: Hash-Grid Assisted Context for 3D Gaussian Splatting Compression
• [papantonakis_reducing_2024]: Reducing the Memory Footprint of 3D Gaussian Splatting
• [ren_octree-gs_2025-1]: Octree-GS: Towards Consistent Real-time Rendering with LOD-Structured 3D Gaussians