Planar Gaussian Coordinates

• Planar Gaussian coordinates are analytical tools that encode, reconstruct, and analyze geometric, probabilistic, and topological features in 2D spaces and manifolds.
• They utilize methods like Hermite polynomial expansions, structured Gaussian splatting, and multifractal measures to enable precise modeling in signal processing and neural rendering.
• Key applications span network information theory, neural reconstruction, stochastic geometry, and diagrammatic expansions on symmetric spaces, driving both theory and practical innovations.

Planar Gaussian coordinates are a set of analytical and computational frameworks that leverage the properties of Gaussian distributions, fields, and function expansions to encode, reconstruct, and analyze geometric, probabilistic, and topological features in two-dimensional Euclidean or manifold settings. These coordinates enable fine-grained analysis of network information flow, planar reconstructions in neural rendering, stochastic processes, and critical phenomena in random fields by either representing input densities as perturbations of reference Gaussians (typically via Hermite polynomial bases) or by parameterizing geometric primitives (Gaussian splats) in scene reconstruction and neural graphics. The following sections detail the mathematical, algorithmic, and practical dimensions of planar Gaussian coordinates as developed in key research contributions.

1. Hermite Polynomial Coordinate Systems for Gaussian Networks

The foundational approach in network information theory is to “linearize” the space of input probability densities around a reference Gaussian via expansion in normalized Hermite polynomials. Given input density $f(x)$ with variance $p$, one writes:

$$f(x) = g_p(x)\cdot (1 + L(x)), \qquad g_p(x) = \frac{1}{\sqrt{2\pi p}} \exp\left(-\frac{x^2}{2p}\right)$$

where $L(x)$ is a “direction” function expanded as $L(x) = \sum_k \alpha_k H_k^{(p)}(x)$, with $H_k^{(p)}(x)$ the orthonormal Hermite polynomials in $L_2(g_p)$.

This coordinate system satisfies the constraints:

$$\int L(x) g_p(x)dx = 0 \qquad \int x^2 L(x)g_p(x)dx = 0$$

ensuring normalization and variance preservation. The Hermite basis is especially powerful because when such a perturbed density passes through an additive white Gaussian noise (AWGN) channel of variance $v$, the convolution operator induces a scaling on each Hermite component:

$$(+) H_k^{(p)} = \left(\frac{p}{p+v}\right)^{k/2} H_k^{(p+v)}$$

Thus, each polynomial “coordinate” is an eigenfunction of the channel operator, and its amplitude is dampened according to the noise level. This machinery allows “spectral” analysis of non-Gaussian code ensembles in settings such as broadcast channels and interference channels, providing exact characterizations of mutual information perturbations and enabling the discovery of regimes where non-Gaussian input distributions outperform the standard Gaussian codebook (Abbe et al., 2010).

2. Planar Gaussian Splatting and Structured Gaussian Coordinates in Neural Rendering

Planar Gaussian Splatting (PGS) and its successors (e.g., GSPlane) adopt a distinct but related notion of Gaussian coordinates for 3D geometric reconstruction from image data. Here, each Gaussian primitive is parameterized by its spatial mean $\mu \in \mathbb{R}^3$, covariance $\Sigma$, and additional attributes (color, learned plane descriptors, surface normal). The role of planar Gaussian coordinates is to enforce that splats align with planar structures when reconstructing man-made environments.

The GSPlane method re-parameterizes planar Gaussian coordinates as normalized linear combinations of three robust basis points $F_1, F_2, F_3$ lying on the same plane:

$$V'_p = \omega_1 F_1 + \omega_2 F_2 + \omega_3 F_3, \qquad \omega_1 + \omega_2 + \omega_3 = 1$$

where $(\omega_1, \omega_2, \omega_3)$ are learnable parameters for each Gaussian in the plane cluster. This parameterization constrains each Gaussian to the identified plane and enables consistent enforcement of geometric priors, which are extracted from sophisticated segmentation and normal prediction models (e.g., SAM, Metric3Dv2) (Gan et al., 20 Oct 2025, Zanjani et al., 2 Dec 2024).

To ensure only truly planar Gaussians are structurally constrained, the Dynamic Gaussian Re-classifier (DGR) adaptively monitors the gradient magnitudes during optimization and switches “false positives” back to unconstrained xyz parameterization if their gradients exceed a threshold determined by non-planar Gaussians.

3. Gaussian Multiplicative Chaos and Stochastic Geometric Coordinates

The concept of planar Gaussian coordinates extends into stochastic geometry via the analysis of multifractal measures derived from planar Brownian motion. The construction proceeds by exponentiating the square root of the local time $L_{x,\epsilon}(\tau)$ accumulated by planar Brownian motion in small circles centered at $x$, yielding GMC-type measures:

$$\mu^\gamma(A) = \lim_{\epsilon \to 0} [\log(1/\epsilon)]^{\gamma^2/2} \int_A e^{\gamma \sqrt{L_{x,\epsilon}(\tau)}} dx$$

These measures exist for all $\gamma \in (0,2)$ and are related to a “flat” measure supported on thick points, characterizing the multifractal geometry of the path (Jego, 2018). The planar Gaussian coordinate system, in this stochastic setting, is encoded in the (square root of) local times, which serve as spatial coordinates for capturing scaling limits and fractal structure.

4. Critical Points and Excursion Set Geometry in Planar Gaussian Fields

Planar Gaussian coordinates also arise in the analysis of stationary Gaussian fields $f:\mathbb{R}^2 \to \mathbb{R}$—either as random surfaces or via excursion sets:

$$E_\ell(f) = \{x \in \mathbb{R}^2 : f(x) \geq -\ell \}$$

Key geometric and probabilistic quantities such as the number of critical points in shrinking height windows (Muirhead, 2019), percolation thresholds (Rivera, 2019), and chemical distances (Vernotte, 2023) are directly influenced by the covariance structure, smoothness, and coordinate representation of $f$. The Kac–Rice formula provides mean and variance estimates for critical points:

$$\mathbb{E}[N_R[a,b]] = O(R^2(b-a)),\qquad \mathbb{E}[N_R[a,b]^2] \leq c\,\min\{R^4(b-a)^2 + R^2(b-a), R^4\}$$

revealing transitions dependent on the geometry of coordinates and covariance decay.

Excursion set connectivity, nodal domain boundedness, and scaling relations between chemical and Euclidean distances (e.g., $$d_{\mathrm{chem}}(x,y) \lesssim |x-y|\log^{3/2+\epsilon}|x-y|$$ in supercritical regimes) are all consequences of the interplay between Gaussian field coordinates, regularity properties, and percolation thresholds (Muirhead, 2021, Vernotte, 2023).

5. Planar Feynman Diagrams and Gaussian Coordinates on Symmetric Spaces

Gaussian measures on Riemannian symmetric spaces further exemplify the notion of planar Gaussian coordinates via the reduction of normalization integrals to finite-dimensional vector spaces and diagrammatic expansions. On spaces of positive-definite matrices, the partition function in eigenvalue coordinates involves Vandermonde determinants and admits an interpretation in terms of planar (genus-zero) Feynman diagrams:

$$Z_\beta(\sigma) = \text{const} \int_{\mathbb{R}_+^N} \prod_{i=1}^N \exp\left(-\frac{\log^2 u_i}{2\sigma^2}\right) |\Delta(u)|^\beta du_1\ldots du_N$$

Where $\Delta(u)$ is the Vandermonde determinant and $\beta$ determines the field (real, complex, quaternionic) (Heuveline et al., 2021). In the large-$N$ (planar) limit, saddle-point methods yield a master eigenvalue density $\rho_t$ whose profile captures universal spectral and geometric features—“planar Gaussian coordinates” in this context are the coordinates of eigenvalues in the dominant diagrammatic regime.

6. Applications and Future Directions

Planar Gaussian coordinate systems yield critical advances in:

Application Area	Core Coordinate System	Impact/Insight
Network Information Theory	Hermite polynomial expansion	Spectral analysis, non-Gaussian codes
Planar Reconstruction & Rendering	Structured xyz/basis weights	Mesh quality, editing, physical realism
Stochastic Geometric Analysis	Local time (Brownian motion)	Fractal scaling, chaos measures
Critical Phenomena in Fields	Covariance-based field coordinates	Connectivity, critical thresholds
High-Dimensional Statistics	Eigenvalue coordinates	Universality, saddle-point structure

In neural rendering, planar Gaussian coordinates support mesh refinement and object decoupling (e.g., supportive plane correction), while, in probabilistic geometry, they elucidate the structure of thick points and fractal domains. Network information theory benefits from precise thresholds for code optimality and mutual information analysis.

Further developments will likely focus on adaptive reclassification strategies for Gaussian primitives, optimized mesh layout refinement, and computational efficiency improvements in large-scale inference and reconstruction tasks. The robust analytical machinery developed for planar Gaussian coordinates continues to inform both theoretical advances and state-of-the-art practical systems in geometry, signal processing, communications, and stochastic analysis.