Depth-Ray Framework Overview

• Depth-Ray Framework is a method that parameterizes depth along rays to achieve pixel-wise and atomic-layer resolution in imaging and simulation.
• It integrates atomic-layer slicing, implicit field learning, and discrete optimization to enhance applications from X-ray reflectometry to multi-view stereo.
• The framework unifies ray-wise decomposition for robust statistical depth estimation, sensor simulation, and multi-agent fusion, improving computational efficiency and accuracy.

A Depth-Ray Framework refers to any analytical or computational scheme that parameterizes depth structure, measurement, or inference along optical or geometric rays, typically with the goal of achieving pixel-wise or atomic-layer resolution in applications spanning X-ray reflectometry, multi-view computer vision, sensor simulation, and statistical depth in data analysis. The defining characteristic is explicit ray-wise decomposition—where depth or related quantities (e.g., refractive index, SDF value, occupancy) are modeled or optimized along rays, often leveraging multi-layer slicing, implicit field learning, event-based ray accumulation, or discrete optimization via ray potentials.

1. Ray-wise Parameterization of Depth

Depth-ray parameterization treats geometric or physical data as functions or distributions defined along individual rays, allowing for highly localized inference and reconstruction. In resonant soft X-ray reflectometry, this means slicing a heterostructure into atomically thin layers and modeling their optical properties (refractive indices $n_j(\omega)$) as ray-dependent. In multi-view stereo, depth at each pixel is inferred by minimizing a cost or regressing a zero-crossing of a 1D implicit field along the camera ray. Dual-pixel sensor simulation uses ray tracing to construct pixel-wise point spread functions (PSFs) according to real lens and sensor geometry. In statistical data depth, the halfspace depth of a point is defined with respect to the family of rays (halfspaces) passing through it, linking geometric coverage to location inference.

Application Domain	Raywise Parameterization	Depth Representation
X-ray reflectivity	Slicing into atomic layers along $z$	Refractive index profile $n_j(\omega)$
Stereo/MVS/Neural rendering	1D SDF/implicit field $f(r,t)$	Zero-crossing depth $t^*$ per ray
DP sensor simulation	Ray-traced PSF per pixel	Left/right DP images
Event camera depth	Ray densities in disparity space	DSI accumulation per ray
Statistical depth	Halfspaces/rays through $x$	Depth as infimum over covering rays

2. Atomic-layer Slicing and Optical Heterogeneity

In resonant soft X-ray reflectivity, conventional slab models fail on resonance because the optical response varies strongly between atomic planes. The Depth-Ray Framework replaces slab models with atomic slicing, where each crystallographic layer ($\approx 2$–$5$ Å) is treated as an independent slice with its own refractive index, parameterized by local density $\rho_j$ and atomic scattering factors $f_{1,j}$, $f_{2,j}$: $$n_j(\omega) = 1 - \delta_j(\omega) + i \beta_j(\omega)$$

$$\delta_j = \frac{r_0\,\lambda^2}{2\pi} N_j f_{1,j},\;\; \beta_j = \frac{r_0\,\lambda^2}{2\pi} N_j f_{2,j}$$

Recursive Parratt or transfer-matrix methods compute the reflectivity $R(Q)$, maintaining distinct optical properties for each layer. Atomic layer slicing achieves depth sensitivity $\leq 2$ Å, enabling the extraction of interface terminations, stacking order, and local spectroscopic variations—even electronic reconstructions at the surface (e.g., Mn valence shift in LaSrMnO$_4$) (Zwiebler et al., 2015).

3. Zero-crossing Implicit Fields and Neural Ray-based Depth

Many recent multi-view stereo methods replace computationally expensive full cost volumes with ray-based 1D implicit fields. For each camera ray with origin $o$ and direction $d$, the surface is modeled as the zero-crossing $t^*$ of a signed distance function $f(o,d,t)$ with respect to metric depth along the ray: $x^* = o + t^* d; \quad f(o,d,t^*) = 0$ Sequential models (LSTM/transformer) process features sampled at discrete points along $t \in [t_c-\Delta, t_c+\Delta]$, regressing signed distances and the crossing location. Monotonicity of $f$ along narrow bands is exploited for regularization, improving accuracy and convergence. Epipolar transformers aggregate multi-view features at each point on the ray. RayMVSNet and RayMVSNet++ achieved state-of-the-art on DTU and Tanks & Temples (overall reconstruction error $\sim 0.33$ mm, F-score $59.48\%$) (Xi et al., 2022, Shi et al., 2023). Incorporation of local-frustum attentional gating further improves contextual aggregation.

4. Ray-based Optimization Schemes

Discrete and continuous optimization along rays appears in several domains. For semantic 3D reconstruction, ray potentials are defined by the cost of the first occupied voxel along each observation ray, coupling semantic likelihoods $P_{\text{semantic}}$ with stereo depth likelihoods $P_{\text{depth}}$: $\psi_r(\mathbf{x}^r) = c_r(\ell, d)$ Binary and multi-label formulations are made graph-representable (submodular) via QPBO relaxation, solvable via graph cuts and $\alpha$-expansion (Savinov et al., 2019). In multi-view implicit volumetric reconstruction, Signed Ray Distance Functions (SRDF) encode depth consistency by evaluating

$SRDF_j(X) = D_j(X) - Z_j(X)$

for all points $X$ along camera rays, with the global energy maximizing joint zero-level consistency and photometric agreement across views (Zins et al., 2022).

5. Ray-traced Simulation and Event-ray Density Accumulation

Simulation frameworks for dual-pixel sensors and event cameras model pixel formation based on full ray tracing. In DP simulation, multi-surface lens propagation and microlens models result in pixelwise PSF estimation, capturing the real sensor geometry and closing the domain gap to physical DP images (He et al., 14 Mar 2025).

Event cameras produce asynchronous streams of spatial-temporal events. Depth-Ray approaches map events into disparity-space images (DSIs) by back-projecting each event along its 3D ray with known pose/calibration, then accumulating ray densities per pixel/depth. This representation enables high-completeness and accuracy (e.g., $>42\%$ MAE reduction in stereo settings) with constant memory and parallelizable local processing via small Sub-DSI volumes. Neural architectures (3D conv + GRU) yield precise depth from event-based representations (Hitzges et al., 22 Apr 2025).

6. Multi-view Consistency and Advanced Ray Fusion

Modern neural implicit representations (RayDF, Depth Anything 3) introduce ray-surface distance fields, learning $f_\Theta(r)$ for each ray code $r$ (angular coordinates) and enforcing multi-view geometry consistency via dual-ray visibility classifiers and explicit loss terms: $$\ell_{mv}(r) = \frac{|h - d| + \sum_{m} v^m |h^m - s^m|}{1 + \sum_m v^m}$$ Rendering is highly efficient—RayDF can reconstruct $800 \times 800$ depth maps $\sim$ 1000$\times$ faster than coordinate-based SDFs (Liu et al., 2023).

RayFusion-enhanced collaborative perception systems deploy ray-occupancy vectors ($\rho^k$) extracted from collaborators’ camera depth-head outputs and perform multi-camera attention fusion to reduce redundancy, suppress false positives, and improve 3D localization under bandwidth/latency constraints (Wang et al., 9 Oct 2025, Kim et al., 23 Sep 2024).

7. Statistical Depth, Robust Estimation, and The Ray-Basis Theorem

In nonparametric statistics, the notion of data depth—particularly halfspace depth—is inherently ray-based. The halfspace depth $D(x;\mu)$ of $x$ with respect to measure $\mu$ is the infimum measure over all closed halfspaces through $x$. The Ray-Basis theorem characterizes deepest regions (median sets) via covering families of rays (halfspaces). Trimmed regions, floating bodies, and monotonicity of the depth function are tightly linked to the covering structure formed by rays through data points. Algorithmic implications include efficient computation of robust covering medians and depth regions (Laketa et al., 2021).

Conclusion

The Depth-Ray Framework generalizes across fields as a ray-decomposed approach to depth modeling, inference, and optimization. Its implementations range from atomically-resolved reflectometry and robust statistical estimators to highly memory-efficient, accurate neural stereo and event-camera depth reconstruction, multi-agent sensor fusion, and physical sensor simulation. The explicit use of rays—either as the locus of optimization, simulation, aggregation, or measurement—yields superior resolution, flexibility, and often computational tractability, unifying disparate approaches to depth recovery under a common ray-based paradigm.