Fourier Splatting: Frequency-Domain Rendering

• Fourier Splatting is a technique that parameterizes rendering primitives using Fourier analysis to represent arbitrary, adaptive shapes in the frequency domain.
• It supports scalable real-time performance by adjusting level-of-detail through truncated Fourier coefficients, ensuring high visual fidelity across varying bandwidth constraints.
• Advanced optimization methods like the straight-through estimator and HYDRA densification enhance gradient stability and efficiency, benefiting applications in SLAM, novel view synthesis, and 3D mapping.

Fourier Splatting refers to a family of techniques that leverage Fourier analysis or Fourier-parameterized kernels to achieve scalable, efficient, and high-fidelity rendering or mapping with explicit splatted primitives. Recent developments have established Fourier Splatting as a unifying concept in scalable radiance field rendering and real-time mapping via adaptive densification, enabling bandwidth-aware reconstruction and rendering with explicit geometric or density primitives.

1. Mathematical Foundations of Fourier Splatting

Fourier Splatting generalizes the notion of spatial “splats” by encoding their geometric or radiometric properties in the frequency domain. Two principal approaches have been described:

• Fourier-encoded planar surfels parameterize the boundary of each surfel by a truncated complex Fourier series. For each surfel $i$ with center in its tangent plane, the boundary is defined as

$$r_i(\theta) = R_i \left| \sum_{k=0}^{K-1} c_{i,k} e^{i k \theta} \right|,\quad c_{i,k} = \bar r_{i,k} e^{i\phi_{i,k}}$$

ensuring arbitrary closed-form boundaries controlled by the number of Fourier coefficients $K$. Squared-$\ell_1$ normalization on real amplitudes $\bar r_{i,k}$ ensures $r_i(\theta) \leq R_i$ for all $\theta$.

• Fourier-based adaptive densification identifies high- and low-frequency content in images using the 2D Fourier transform and uses this information to guide spatial initialization of splatted Gaussians, ensuring that higher frequencies receive denser, more finely resolved primitive coverage.

The closed-form Fourier transform of a 3D Gaussian is

$$\mathcal{F}\{G\}(\omega) = (2\pi)^{3/2} |\Sigma|^{1/2} \exp\left(-\frac{1}{2}\omega^T \Sigma \omega + i\omega^T\mu\right)$$

with properties directly exploited for efficient rendering and filter design (Xu et al., 3 Mar 2025, Jurca et al., 20 Mar 2026).

2. Explicit Primitive Parameterization and Rendering

Explicit primitives in Fourier Splatting are parametrized by frequency-domain coefficients, with each primitive controlling not only geometric extent but also level of detail (LoD):

• Surfels with Fourier boundaries: Each planar surfel employs the boundary function $r_i(\theta)$, permitting the representation of noncircular, arbitrarily complex closed shapes. Rendering is achieved by intersecting view rays with the surfel plane, computing the signed distance to the Fourier boundary, and applying a power-window kernel:

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

where $\rho_j, \theta_j$ are polar coordinates of the intersection, $o_i$ is trainable opacity, and $\sigma_i$ tunes profile sharpness.

• Level-of-Detail (LoD) Control: Runtime scalability is achieved by truncating the Fourier sum to $K' < K$ coefficients:

$$r_i^{(K')}(\theta) = R_i \left| \sum_{k=0}^{K'-1} c_{i,k} e^{i k \theta} \right|$$

LoD is smoothly degradable, allowing a single trained model to support multiple bandwidth/performance envelopes without retraining.

• Radiance Accumulation: Per-pixel color or other attributes are composited through front-to-back alpha blending:

$$C(x) = \sum_{i} c_i\,\alpha_{i}(x) \prod_{j<i} (1-\alpha_{j}(x))$$

where $c_i$ encodes radiance (or color) for primitive $i$ (Jurca et al., 20 Mar 2026).

3. Optimization and Densification Techniques

Optimization in Fourier Splatting involves addressing non-differentiabilities arising from sharp primitive boundaries and maintaining stability during densification:

• Straight-Through Estimator (STE): To overcome vanishing gradients outside the hard-window boundary, a surrogate window is used in the backward pass:

$$\tilde{w}(x)=\mathrm{sigmoid}(\beta x)\left(\frac{\mathrm{softplus}(\beta x)}{\beta}\right)^{\sigma_i} + \gamma\,\mathrm{sigmoid}(\beta x)$$

This ensures gradient flow to Fourier coefficients for stable optimization, particularly in low-frequency regimes.

• HYDRA Densification (MCMC-based): Densification is performed by recursively decomposing complex surfels via MCMC proposals, primarily through scale-preserving splits and lobe-based decompositions. Surfels are split along valleys of their Fourier boundary, with child primitive parameters determined by a small neural network predicting lobe offsets and parameter adjustments.
• Regularization: Shape regularity (pancake-like ellipsoids) and boundary constraints are enforced via explicit regularizers, preventing uncontrolled scaling and ensuring geometric fidelity (Jurca et al., 20 Mar 2026, Xu et al., 3 Mar 2025).

4. Fourier Guidance in Adaptive Map Construction

In SLAM and dense 3D mapping, Fourier Splatting supports high-fidelity, adaptive map construction:

• Frequency-aware Densification: Incoming RGB (or RGB-D) frames are Fourier-transformed, and high-frequency content is isolated by applying a high-pass Gaussian filter in the frequency domain. Spatial masks for high- and low-frequency regions are obtained via magnitude thresholding on the inverse FFT image.
• Spawn Masking and Primitive Insertion: High-frequency mask pixels spawn small, dense Gaussians; low-frequency regions spawn larger, sparser Gaussians. New Gaussians are only created where rendered alpha, depth, or color fail to explain observed content (via opacity, depth, or color error masking).
• Sparse/Dense Map Fusion: Parallel maps are maintained—a dense Gaussian map for image-based optimization (RGB, depth, SSIM), and a sparse subset for fast camera tracking using Generalized Iterative Closest Point (GICP). Joint optimization alternates between dense reconstruction and efficient pose estimation, both benefiting from Fourier-based initialization.
• Mapping and Tracking Workflow: Mapping threads perform FFT-based analysis and primitive spawning, while tracking threads use downsampled Gaussians and GICP for real-time frame-to-frame alignment (Xu et al., 3 Mar 2025).

5. Quantitative Performance and Benchmarking

Empirical evaluation in both radiance field rendering and SLAM demonstrates the scalability and fidelity enabled by Fourier Splatting:

Method	Mip360 PSNR↑	SSIM↑	LPIPS↓	TNT PSNR↑	SSIM↑	LPIPS↓
3DGS	26.98	0.813	0.214	23.14	0.841	0.183
Tri–Splat	26.98	0.812	0.191	23.14	0.857	0.143
Fourier Splat	27.65	0.824	0.193	24.15	0.868	0.137

• Rendering: Fourier Splatting achieves real-time (30 fps @ 1080p) performance with $10^5$ primitives on contemporary GPUs; per-primitive computational cost is $57 + 8K$ operations per pixel, enabling graceful scalability as $K$ is varied.
• SLAM: Fourier-based Gaussian Splatting yields ∼36 FPS on Replica and TUM RGB-D with ATE RMSE $\sim$0.15–2.0 cm, dense reconstructions at PSNR $\sim$38.7 dB, SSIM $\sim$0.975, and LPIPS $\sim$0.041 (Xu et al., 3 Mar 2025, Jurca et al., 20 Mar 2026).

Graceful degradation in LoD is reflected by monotonic PSNR decrease under coefficient truncation, with artifact-minimizing behavior compared to pruning-based schemes. Ablations show that dropping the STE, lobe-split, or random offset modules result in measurable PSNR decreases, underscoring these components' functional necessity.

6. Practical Considerations and Application Scenarios

Fourier Splatting has been applied in contexts requiring high spatial adaptivity, bandwidth-aware scalability, and real-time interactive performance:

• Novel View Synthesis: Fourier-parameterized surfels model arbitrary closed boundaries, capturing sharp edges and thin structures with fewer primitives compared to blob- or triangle-based splats.
• SLAM and Mapping: Frequency-guided densification outperforms agnostic initialization by minimizing redundancy and focusing parameter optimization where it is spectrally necessary, lowering convergence time and improving allocation efficiency.
• Bandwidth-Constrained Environments: Dynamic LoD adjustment via runtime Fourier coefficient truncation permits bandwidth-utility tradeoff with a single model, facilitating applications in streaming or edge-computing scenarios.

All computationally intensive steps are off-loaded to hardware-accelerated pipelines (e.g., cuFFT for FFT operations, CUDA for splatting and mask calculation).

7. Relationship to Related Primitive Splatting Methods

Fourier Splatting generalizes earlier approaches such as 3D Gaussian Splatting (3DGS), Tri–Splat, and Octree-GS. Unlike methods where granularity and fidelity are controlled solely by the explicit number and scale of primitives, Fourier Splatting decouples primitive count from rendered detail via frequency-domain boundary encoding. This enables quality downscaling without aggressive pruning and achieves SOTA among planar-primitive renderers, narrowing the quality gap with volumetric NeRF-based techniques while maintaining explicit, real-time rasterization (Jurca et al., 20 Mar 2026).

A plausible implication is that Fourier Splatting's spectral adaptivity will serve as an architectural motif for future unified representations, jointly optimizing explicit geometry and radiance for scalable, bandwidth-adaptive rendering and mapping.

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References:

• "Fourier Splatting: Generalized Fourier encoded primitives for scalable radiance fields" (Jurca et al., 20 Mar 2026)
• "FGS-SLAM: Fourier-based Gaussian Splatting for Real-time SLAM with Sparse and Dense Map Fusion" (Xu et al., 3 Mar 2025)