E-RayZer: Geometry-Driven Inference Methods

• E-RayZer is a collection of frameworks that use explicit geometry and physical constraints, applied in 3D vision, cosmic ray detection, and anisotropic ray-tracing.
• It enforces structured, interpretable representations through methods such as per-pixel 3D Gaussians, analytic energy fluence fitting, and variational finite-element modeling.
• E-RayZer methods deliver state-of-the-art performance and robust transferability, yielding physically grounded outputs for complex real-world applications.

E-RayZer denotes a set of methodologies and frameworks, independently developed in multiple scientific domains, that leverage explicit, geometry-based or ray-based formulations for either physical inference or neural representation learning. The term is used across contexts including the inverse problem in seismics and elastic media, robust energy estimation from cosmic-ray radio signals, and, most recently, self-supervised pre-training for 3D vision via explicit scene reconstruction. Common to all instantiations is the focus on physically or spatially grounded inference, often bypassing or improving upon tradition by directly enforcing geometric or energetic constraints.

1. E-RayZer in Self-supervised 3D Vision

The E-RayZer framework in 3D vision refers to a self-supervised model for learning 3D-aware representations directly from multi-view image sets without the need for ground-truth 3D data (Zhao et al., 11 Dec 2025). The key contribution is its shift from latent or implicit representations, as in NeRFs or RayZer, to explicit geometry, specifically through the use of parametric 3D Gaussians for scene modeling.

Given a batch of $V$ images $$\mathcal{I} = \{I_i \in \mathbb{R}^{H \times W \times 3}\}_{i=1}^V$$, the model regresses shared camera intrinsics $\mathbf{K}$ and per-view extrinsics $\{\mathbf{T}_i \in SE(3)\}$ through a multi-view transformer $f_\theta^{\text{cam}}$. These are mapped into Plücker-ray space, enabling subsequent explicit 3D reconstruction via a scene encoder $f_\psi^{\text{scene}}$. Each pixel in the reference images emits a parametric Gaussian:

$$g = (d, q, C, s, \alpha), \quad \text{with} \; d \in \mathbb{R}, q \in \mathbb{R}^4, C \in \mathbb{R}^{M \times 3}, s \in \mathbb{R}^3, \alpha \in \mathbb{R}$$

These per-pixel Gaussians collectively reconstruct the target views using a differentiable Gaussian-splat renderer $\pi$. The supervisory signal is a photometric loss:

$$\mathcal{L}_{\text{photo}} = \sum_{\ell \in \mathcal{I}_{\text{tgt}}} \| I_\ell - \hat{I}_\ell \|_2^2 + \lambda\, \mathrm{Percep}(I_\ell,\hat{I}_\ell)$$

All parameters, including geometry and pose, are optimized end-to-end from scratch.

A progressive learning curriculum is used: training begins with high visual-overlap image pairs, gradually introducing harder, low-overlap pairs as measured by semantic and geometric overlap metrics (e.g., cosine similarity of DINOv2 features or covisibility via a pretrained UFM network). This curriculum ensures stable convergence even for wide-baseline or diverse sequences.

E-RayZer demonstrates superior results over latent-space approaches such as RayZer: it attains RPA@5° metrics of 84.5–90.8% (RayZer: 0–0.2%) while matching or exceeding fully supervised baselines like VGGT on both in-domain and out-of-domain datasets. Its features align with physical 3D structures and transfer robustly to downstream tasks (multi-view depth, pose, and flow estimation), outperforming state-of-the-art visual pre-training models such as DINOv3, CroCo v2, and VideoMAE V2.

2. E-RayZer for Energy Estimation in Cosmic Ray Radio Detection

In astroparticle physics, “E-RayZer” designates the energy estimator deployed by the Auger Engineering Radio Array (AERA) for determining the energy of ultra-high-energy cosmic rays from radio emission measurements in the 30–80 MHz band (Collaboration et al., 2015).

The process unfolds as follows:

1. Electric-Field Reconstruction: Dual-polarization voltages recorded at each AERA station are cleaned of narrow-band RFI, calibrated for channel response, and converted into vector electric-field traces $\vec{E}(t)$, incorporating the known shower geometry from surface detectors.
2. Energy Fluence Calculation: At each antenna location $\vec{r}$, the instantaneous Poynting flux $S(t) = \epsilon_0 c |\vec{E}(t)|^2$ is integrated over a $\pm$100 ns window, subtracting a background window to yield the energy fluence $f(\vec{r})$ in eV/m$^2$.
3. Lateral Distribution and Functional Fit: The 2D spatial profile of fluence is fit by an empirical function accounting for geomagnetic and charge-excess interference and the Cherenkov ring:

$$f_{\text{fit}}(\vec{r}) = A \Big[ e^{-|\vec{r}+C_1\hat{e}_{v \times B} - \vec{r}_{\text{core}}|^2/\sigma^2} - C_0 e^{-|\vec{r}+C_2\hat{e}_{v \times B} - \vec{r}_{\text{core}}|^2/(C_3 e^{C_4\sigma})^2} \Big]$$

1. Radiation Energy and Geomagnetic Correction: The integrated radiation energy is given by

$$E_{\text{rad}} = A \pi \left[ \sigma^2 - C_0 C_3^2 e^{2C_4\sigma} \right]$$

and is corrected for geomagnetic angle dependence as

$S_{\text{radio}} = E_{\text{rad}} / \sin^2\alpha$

This $S_{\text{radio}}$ serves as the "E-RayZer" estimator.

2. Absolute Calibration: Calibration against the surface detector yields the scaling

$$S_{\text{radio}} = A_1 \times 10^7 \text{eV} \left( \frac{E_{\text{CR}}}{10^{18} \text{eV}} \right)^B$$

with measured $A_1 = 1.58 \pm 0.07$, $B = 1.98 \pm 0.04$, confirming quadratic scaling as expected for coherent emission. Inverse mapping gives

$$E_{\text{CR}} = 10^{18} \text{eV} \cdot \sqrt{ S_{\text{radio}} / (1.58 \times 10^7 \text{eV}) }$$

The method achieves radio-only energy resolution of $22\%$ (full data) and $17\%$ (high-quality subset), with systematic uncertainties dominated by antenna calibration and absolute scale settings. The approach is site-portable, contingent only on local magnetic field strength and the measured radio-band radiation energy.

3. E-RayZer in Ray-based Inverse Problems in Anisotropic Media

The term “E-RayZer” is also used as a synonym for the eigenray variational approach to ray-tracing and two-point boundary-value problems in 3D heterogeneous anisotropic elastic media (Koren et al., 2020).

The framework is constructed on variational principles. The ray path $x(s)$, parameterized by arclength $s$, connects source $S$ and receiver $R$. Ray direction is $r(s) = dx/ds$ with $r \cdot r = 1$. The medium's group velocity $V_\text{ray}(x, r)$ depends on both position and direction due to anisotropy.

The stationary path solves the Euler-Lagrange equation for Fermat's principle of stationary traveltime:

$$T[x] = \int_{s_0}^{s_1} L(x(s), r(s)) ds, \quad L(x, r) = \frac{V_{r \cdot r}}{V_\text{ray}(x, r)}$$

The stationary path $x^*(s)$ solves:

$$\frac{d}{ds} \left(\frac{\partial L}{\partial r}\right) - \frac{\partial L}{\partial x} = 0$$

Defining $p = \partial L/\partial r$ (slowness vector) and $$L_x = \partial L/\partial x = (1/V_\text{ray}^2) \nabla_x V_\text{ray}$$, the core ODE system is:

$$\frac{dx}{ds} = r, \qquad \frac{dp}{ds} = \frac{1}{V_\text{ray}^2} \nabla_x V_\text{ray}(x, r)$$

Stationary rays are further classified based on the second variation:

• strict minimum (well-behaved amplitudes, positive-definite second variation)
• saddle point (caustics, requiring Maslov/uniform asymptotics)
• maximum (rare in lossless media).

Finite-element implementation employs Hermite interpolation for $x(s)$ and $r(s)$, ensuring $C^1$ continuity and accurate evaluation of gradients and Hessians along the discretized path. This choice is justified by the requirements of anisotropic velocity field and the variational formulation.

4. Methodological Features and Innovations

All E-RayZer instantiations emphasize explicit parametrizations rooted in geometry or physics. In 3D vision, 3D Gaussians result in representations that directly encode spatial cues, preventing the degenerate solutions found in latent approaches (Zhao et al., 11 Dec 2025). In cosmic-ray detection, the methodology is data-driven but grounded in physical radio emission models and analytic lateral energy distribution fitting (Collaboration et al., 2015). In ray-tracing, the variational finite-element approach ensures stable convergence and accurate treatment of path constraints, extending to anisotropic and heterogeneous elastic media where previous initial-value approaches fail (Koren et al., 2020).

A notable theme is the mitigation of ambiguity: E-RayZer methodologies avoid shortcut learning by tightly coupling the representation to the underlying geometry or physics, as seen in the 3D vision curriculum which gradually reduces overlap to ensure convergence to truly 3D representations, or in radio detection where polarization and signal modeling filter spurious contributions.

5. Quantitative Results and Impact Across Domains

E-RayZer achieves state-of-the-art or reference-level results in each domain:

• 3D Vision: Achieves RPA@5° of 84.5–90.8% in unsupervised pose estimation; matches or sometimes exceeds supervised VGGT benchmarks on 3D reconstruction and transfers robustly to multi-view depth and pose tasks. Ablations confirm the necessity of the explicit geometry and curriculum components.
• Cosmic-Ray Energy Estimation: Enables radio-only energy determination with $\approx 20\%$ precision, quadratic scaling with primary energy, and systematic uncertainty dominated by instrumental calibration, establishing a robust alternative to calorimetric estimation with scalable cross-site portability.
• Ray-based Seismics: Provides stable, high-precision computation of stationary ray paths and their dynamic characteristics, particularly useful in anisotropic, heterogeneous environments where classical ray-shooting fails to converge or misclassifies stationary solutions.

The explicit, physically interpretable outputs (e.g., per-pixel Gaussians, analytic energy estimators, stationary ray paths) are significant for transparency, interpretability, and transferability to broader tasks or domains.

6. Limitations and Future Directions

E-RayZer in 3D vision currently lacks explicit geometry regularization such as TV penalties or signed distance smoothness, potentially limiting fine-detail recovery (Zhao et al., 11 Dec 2025). Existing approaches are limited to static scenes, with dynamic scene extension still open. High-resolution scenes challenge memory efficiency due to the dense per-pixel Gaussian storage. In radio applications, leading systematic uncertainties arise from antenna calibration and absolute gain scales; further advances are constrained by these technological bottlenecks.

Across domains, future research directions include integrating learned or physics-informed regularizers, extending to explicit mesh-based or SDF representations, and scaling to dynamic or internet-scale datasets. In seismics, extension of the eigenray variational framework to solve dynamic (Jacobi) equations for geometric spreading, and further linking the Lagrangian and Hamiltonian approaches, are highlighted as ongoing pursuits.

7. Comparative Table of E-RayZer Methodologies

Domain	Core Output	Key Principle	Performance/Precision
3D Vision (Zhao et al., 11 Dec 2025)	3D Gaussians, camera poses	Explicit, self-supervised geom.	84–91% RPA@5° pose est.
Astro/Radio (Collaboration et al., 2015)	Calibrated $S_\text{radio}$	Analytic, data-driven energy fit	$\sim$20% energy res.
Anisotropic Ray (Koren et al., 2020)	Stationary ray path, slowness	Variational FEM, Hermite interp.	Accurate for complex media

Editor's note: RPA = Relative Pose Accuracy @ 5 degrees; res. = resolution; FEM = finite-element method.

• (Zhao et al., 11 Dec 2025) “E-RayZer: Self-supervised 3D Reconstruction as Spatial Visual Pre-training”
• (Collaboration et al., 2015) “Energy Estimation of Cosmic Rays with the Engineering Radio Array of the Pierre Auger Observatory”
• (Koren et al., 2020) “Eigenrays in 3D heterogeneous anisotropic media: Part I -- Kinematics, Variational formulation”