Flattened Gaussian Primitives

• Flattened Gaussian primitives are Gaussian functions with support concentrated along lower-dimensional subspaces achieved by reducing one or more eigenvalues of the covariance matrix.
• They are applied in computer vision, wave optics, and kernel methods, enhancing neural rendering, scene reconstruction, and flat-top beam modeling.
• These methods yield improvements in geometric accuracy and parameter efficiency, but require careful regularization to prevent overfitting in sparse-view scenarios.

A flattened Gaussian primitive is a Gaussian function whose support is concentrated along one or more lower-dimensional subspaces (e.g., a thin surface in 3D, a flat region in 2D or an elongated ellipsoid). Such primitives are foundational in modern computer vision (scene reconstruction, neural rendering, cross-view localization), wave optics (flattened Gaussian beams), and kernel methods (radial basis functions in the flat limit). Flattening is typically achieved by setting one or more eigenvalues of the covariance matrix to zero or a small positive value, resulting in high anisotropy and enforcing near-planarity or other geometric constraints. The mathematical and computational treatment of flattened Gaussians depends on the application context: splatting and blending in differentiable rendering, analytic continuation in beam propagation, or variational analysis in functional approximation.

1. Mathematical Definitions of Flattened Gaussian Primitives

A generic 3D Gaussian primitive is defined by its mean $\mu \in \mathbb{R}^3$ and covariance $\Sigma \in \mathbb{R}^{3 \times 3}_{\succ 0}$:

$$G(x) = \exp\left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right).$$

Flattened Gaussians are constructed by constraining $\Sigma$ such that one or more eigenvalues are much smaller than the rest, effectively making the Gaussian infinitely thin along certain directions. For example:

• In surface splatting (Qu et al., 15 Jul 2025, Gu et al., 18 Nov 2025, Taktasheva et al., 19 Sep 2025), flattening is achieved by factorizing $\Sigma = R S^2 R^\top$ with $S = \operatorname{diag}(s_1, s_2, s_3)$ and setting $s_\mathrm{min} \approx 0$.
• In hybrid 2D/3D representations (Taktasheva et al., 19 Sep 2025), a planar Gaussian primitive is defined on a plane $P_p$ parameterized by $(o_p, n_p)$ with in-plane mean $\mu^{2D}_k$ and covariance $\Sigma^{2D}_k$; the out-of-plane variance is zero.

For analytical beam profiles (Borghi, 2023):

• The flattened Gaussian beam (FG) of order $\nu$ is defined as:

$$U_\nu(r) = U_0 \exp(-r^2/w_0^2) \cdot {}_1F_1(-\nu; 1; r^2/w_0^2)$$

where ${}_1F_1$ is Kummer’s confluent hypergeometric function. Here, flattening corresponds to larger $\nu$ (high flatness).

2. Role in Surface Reconstruction and Neural Rendering

In 3D Gaussian splatting (GS), flattened primitives are used to provide an accurate and continuous interface to local surface geometry (Qu et al., 15 Jul 2025, Gu et al., 18 Nov 2025, Taktasheva et al., 19 Sep 2025):

• Point-centered ellipses: For every 3D point, a thin Gaussian is aligned to the estimated local tangent, with the smallest covariance in the normal direction.
• Mixed-primitive strategy: MP-GS (Qu et al., 15 Jul 2025) clusters points into primitives of 1 (ellipse), 2 (line), or 3 (triangle) vertices and constructs flat Gaussians accordingly. This improves geometric coverage and blending, especially for sharp features and planar regions.
• Compositional splatting and blending: Flattened primitives are rendered by projecting their support onto the image plane, computing region-aware blending weights, and applying $\alpha$-compositing front-to-back over depth-ordered splats.

In hybrid 2D/3D photometric reconstruction (Taktasheva et al., 19 Sep 2025), planar Gaussians are optimized in tandem with freeform 3D Gaussians. This eliminates "semi-transparent" artifacts on flat surfaces and improves depth accuracy without penalizing appearance modeling for the rest of the scene. The process dynamically fits and refines planes, assigns flattened Gaussians, and alternates between photometric and geometric optimization.

3. Functional Approximation and RKHS Flat Limit

In kernel approximation, "flattening" the Gaussian kernel (i.e., taking large length-scale $t \to \infty$) produces a continuum from adaptive, nonlinear interpolation to classical polynomial interpolation and Gaussian quadrature (Karvonen et al., 2019). Explicitly, the RKHS of the Gaussian kernel $K_t(x, y) = \exp(-\|x-y\|^2 / (2 t^2))$ contains functions $f(x) = e^{-\|x\|^2/(2t^2)} \sum f_\alpha x^\alpha$. As $t \to \infty$, the damping vanishes and functions approach low-degree polynomials.

Worst-case optimal cubature/interpolation with flat Gaussian kernels is shown to converge to polynomial methods:

• For fixed interpolation nodes $X$, kernel interpolants converge to polynomial interpolants of degree $m$.
• When both nodes and weights are optimized (1D), the limit yields the classical Gaussian quadrature rule, maximally exact for polynomials of degree $2N-1$.

This analysis clarifies that kernel-based and polynomial methods are parameterically linked via the Gaussian's flatness (Karvonen et al., 2019).

4. Applications in Optics and Wave Propagation

The "flattened Gaussian beam" formalism (Borghi, 2023) is central in paraxial wave optics for modeling flat-top, high-uniformity beams:

• For integer order $\nu$, FG$_N$ beams correspond to finite sums of radial Laguerre-Gaussian modes, with the flatness of the beam increasing with $N$.
• Analytic continuation (non-integer $\nu$) is provided via Kummer's function or the incomplete gamma function representation, enabling fine control over beam profiles.
• Propagation through generic (ABCD) optical systems is given in closed form:

$$U_\nu(r; z) = U_0/[A + i B \lambda/(\pi w_0^2)] \exp\left[ -\frac{k r^2}{2(A + i B \lambda / \pi w_0^2)} \right] {}_1F_1\left(-\nu; 1; \frac{k r^2}{2(A + i B \lambda / \pi w_0^2)}\right)$$

The FG basis is complete for modeling cylindrically symmetric flat-top beams with explicit normalization and partial orthogonality properties.

5. Computational Techniques and Challenges

Flattened Gaussians demand numerical stability in both parameterization and optimization:

• Covariance matrices are parameterized via eigen-decomposition ($\Sigma = R S^2 R^\top$) to ensure positive semidefiniteness and controlled flattening.
• Mixed-primitive methods (Qu et al., 15 Jul 2025) require dynamic initialization (hierarchical clustering from sparse point clouds), as well as vertex pruning for optimal storage and accuracy.
• In sparsely supervised scenarios, extreme flattening introduces overfitting risk (thin surfels fit training views at the cost of poor generalization). Remedies include depth/normal regularization, feature-space alignment, and stereo-consistency losses (Gu et al., 18 Nov 2025).

In computer-generated holography (Zhan et al., 19 Nov 2025), complex-valued 2D Gaussian primitives with strong flattening reduce parameter counts by 10:1, enabling fast, memory-efficient (up to 2.5× VRAM saving) and high-fidelity hologram synthesis. Differentiable rasterization, optimized light-propagation kernels (band-limited angular spectrum), and structure-guided phase-only conversions are integral for tractable optimization and physical realization.

6. Impact and Ablation Results Across Domains

Key empirical findings across domains:

Method/Setting	Metric	Flattened/Hybrid vs. Baseline	Reference
MP-GS (mixed/flat) vs. point ellipses	Chamfer (DTU)	0.46 mm vs. 0.61–0.64 mm	(Qu et al., 15 Jul 2025)
Hybrid 2D/3D GS (ScanNet++)	Depth RMSE	0.27 m vs. 0.44 m	(Taktasheva et al., 19 Sep 2025)
SparseSurf (3-view DTU)	Chamfer	1.05 vs. 1.37 (FatesGS)	(Gu et al., 18 Nov 2025)
Flat Gaussians for BEV localization	Mean Error	2.86 m vs. 6.81 m (IPM)	(Wang et al., 13 Feb 2025)
Complex 2D Gaussians (holography)	PSNR	29–31 dB with 10×-fewer parameters	(Zhan et al., 19 Nov 2025)

These results demonstrate that flattened Gaussian primitives offer statistically significant gains for geometric accuracy, parameter efficiency, and inference speed, provided that appropriate regularization and hybrid or compositional frameworks are utilized.

7. Limitations, Overfitting, and Regularization

Flattened primitives are susceptible to overfitting under limited or sparse viewpoints (Gu et al., 18 Nov 2025):

• Excessive anisotropy relinquishes the smoothing bias introduced by isotropic kernels, allowing Gaussians to "slide" freely over surfaces or memorize training views.
• Proposed mitigations include depth/normal supervision (monocular priors, multi-view consistency, stereo alignment) and multi-view feature distillation (rendered CNN feature matching, pseudo-view consistency).

A plausible implication is that hybrid methods (combining flattened and full-covariance Gaussians, or integrating geometric regularizers with photometric losses) should be preferred in settings with incomplete or ambiguous supervision.

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Flattened Gaussian primitives constitute a mathematically robust, computationally efficient, and empirically validated toolset for high-fidelity scene representation, functional approximation, wave propagation, and structured light modeling. Their impact extends from foundational analysis (RKHS flat limits) to practical innovations in neural rendering, cross-view localization, and holography. Further research is expected to refine regularization strategies, enable fully dynamic primitive type adaptation, and expand their adoption in large-scale 3D scene understanding and physics-based modeling.