Ray-Based Geometric Representation

• Ray-based geometric representation is a technique that models 3D geometry by tracking ray-surface interactions rather than relying on point clouds or voxel grids.
• It employs layered, feature-augmented, and implicit methods to capture key surface intersection events, aiding tasks like differentiable rendering and tomographic reconstruction.
• This approach enhances precision in 3D reconstruction and physical simulation while addressing challenges such as occlusion, discretization, and computational complexity.

Ray-based geometric representation defines and manipulates 3D geometry by describing the interaction of rays—parameterized lines in space—with geometric or physical structures. Rather than encoding geometry by point clouds, meshes, or voxel grids, these representations center on the set of surface intersection events, paths, or invariants induced by rays traversing a scene. This approach supports a wide array of modern tasks, from differentiable rendering and neural shape inference to high-precision tomography, scene reconstruction, and the geometric analysis of wave propagation.

1. Formal Models and Mathematical Definitions

Ray-based geometric methods universally begin by parameterizing rays. For a camera or source at position $o \in \mathbb{R}^3$, with directional vector $d \in \mathbb{S}^2$ (the unit sphere), a ray is $r(t) = o + t d$, typically for $t \geq 0$. Core ray-based representations diverge according to the geometric quantities they associate to $r$:

• Surface Intersections: Layered or sequential representations collect all surface hits along each ray, as in LaRI (Layered Ray Intersections), where for each image pixel, a stack $\{p_k = o + u_k d\}_{k=1}^L$ records intersection points, forming a $V \in \mathbb{R}^{H \times W \times L \times 3}$ tensor and an associated validity mask $M$ (Li et al., 25 Apr 2025).
• Feature-augmented Intersections: X-Ray packs not only depths $t_k$ along each pixel’s ray but also normal and color attributes at each intersection, producing a layered sequence $F_k = [H_k; D_k; N_k; C_k]$ suitable for diffusion models (Hu et al., 2024).
• Implicit Intersection Points: PRIF models the ray-surface interaction directly as a learned mapping $f_\theta: (f_r, d_r) \mapsto (a_r, s_r)$, predicting both hit/miss and surface depth per input ray, bypassing sphere or grid tracing (Feng et al., 2022).
• Medial Atoms: MARF parameterizes ray-space via neural prediction of $K$ "medial atoms"—sphere centers and radii $(c_i, r_i)$ for candidate maximal inscribed balls—then computes ray-sphere intersection analytically to yield surface hits and derivatives (Sundt et al., 2023).
• Volume and Particle Models: Gaussian ray tracing and RaySplats assign to each primitive (e.g., a Gaussian ellipsoid) closed-form or numerically robust hit, density, or opacity along a ray, summing (or compositing) their contributions analytically or through traversal (Chen et al., 1 Feb 2026, Moenne-Loccoz et al., 2024, Byrski et al., 31 Jan 2025).
• Ray Bundles and Fields: For localization or scene alignment, methods such as GRLoc regress sets of rays or points in world space, enabling differentiable optimization of camera pose via Procrustes alignment of predicted and canonical ray bundles (Li et al., 17 Nov 2025).
• Ray Mapping and Mass Transport: In optics, continuous ray-mapping approaches compute diffeomorphic maps between source and target distributions, leading to PDEs whose solutions (e.g., via optimal mass transport) define the surface through which rays may traverse (Bösel et al., 2015).

2. Algorithmic and Computational Aspects

Ray-based representations require accurate and efficient computation of both ray-geometry interactions and associated attributes.

• Intersecting Primitives: For ellipsoidal Gaussians, intersection is solved by inverting $(r(t)-m)^\top \Sigma^{-1}(r(t)-m) = Q$, a quadratic in $t$. Closed-form solutions exist for calculating hit locations, densities, and blending weights for both rendering and tomography (Chen et al., 1 Feb 2026, Byrski et al., 31 Jan 2025).
• Sequential and Layered Prediction: Modern networks predict entire sequences of intersection points per ray (multi-layer depth). LaRI and X-Ray architectures leverage per-patch or per-pixel feature extraction using transformer/CNN encoders, followed by decoders outputting $\mathbb{R}^{H \times W \times L \times 3}$ intersection or frame tensors. Stopping indices (for layer validity) and masks are predicted via segmentation-style heads (Li et al., 25 Apr 2025, Hu et al., 2024).
• Fusion and Aggregation: PointEMRay constructs "geometric frame buffers" containing depth, normals, and masks from coarse ray-tube sampling plus neural refinement, enabling aggregation over multiple views for multi-bounce simulation (Yang et al., 2024).
• Differentiable Ray Integration: For end-to-end optimization, closed-form or autograd-friendly ray evaluations are critical, e.g., explicit line integrals through basis functions or analytic backward passes in neural rendering (Haouchat et al., 26 Mar 2025, Byrski et al., 31 Jan 2025).
• Higher-Order Basis Ray Tracing: Tomographic and image-reconstruction models use ray-tracing over grids of overlapping shifted basis functions (splines, box-splines), supporting more accurate physical forward and adjoint operators than standard voxel-based models (Haouchat et al., 26 Mar 2025).

3. Applications and Comparative Performance

Ray-based geometric representations are central in multiple 3D inference, rendering, and physical simulation domains:

• Single-view 3D Reconstruction: Representations like LaRI reconstruct visible and occluded surfaces from a single image in one network forward pass. LaRI shows competitive F-scores and Chamfer distance in object and scene benchmarks while being more data- and parameter-efficient than generative baselines (Li et al., 25 Apr 2025).
• 3D Generation and Editing: X-Ray’s video-like ray-interaction sequences underpin high-resolution 3D shape synthesis from an image, outperforming mesh- and NeRF-style methods on Chamfer and surface completion (Hu et al., 2024).
• Tomographic Inverse Problems: Analytic ray-Gaussian integration yields superior projection accuracy, physically consistent correction for nonlinearities (e.g., PET arc correction), and improved reconstruction metrics (PSNR, SSIM) over rasterized or splatting methods (Chen et al., 1 Feb 2026).
• Electromagnetic Field Simulation on Point Clouds: PointEMRay’s screen-based ray-tube and GFB approach achieves low error (<3 dB RMSE) and high speed (sub-second, multi-bounce fields on 50k points) for SBR on raw point clouds (Yang et al., 2024).
• Structured Wavefields: Ray theory and Poincaré-sphere parametrization of ray orbits facilitate analysis of wavefront caustics, amplitude and phase propagation, and caustic corrections for complex geometries, maintaining accuracy with orders-of-magnitude less computation than full-wave solvers (Ren et al., 2024, Alonso et al., 2016).
• Geometric Visual Localization: Predicting ray bundles and pointmaps explicitly enables decoupled estimation of camera rotation and translation, yielding large gains in pose estimation error compared to direct APR regressors (Li et al., 17 Nov 2025).

4. Limitations and Theoretical Considerations

Despite their advantages, ray-based geometric representations introduce several challenges:

• Occlusions and Tangency: Layered methods (e.g., LaRI, X-Ray) may miss thin or tangent surfaces, as the density of intersection layers is inadequate in such regions (Li et al., 25 Apr 2025, Hu et al., 2024).
• Data Sparsity and Discretization: For deep layers or high geometric complexity, fixed per-pixel layer counts may be insufficient, and diffusion models may produce noisy or sparse outputs in those regimes (Hu et al., 2024).
• Algorithmic Complexity: Ray-tracing approaches are typically more computationally intensive than rasterization, especially when secondary effects, multiple bounces, or generalized primitives are modeled. BVH and other acceleration strategies mitigate but do not eliminate this cost (Byrski et al., 31 Jan 2025, Moenne-Loccoz et al., 2024).
• View-Dependence and Multi-view Consistency: Neural ray fields not anchored in geometric invariants can suffer from “wobble” across views. Techniques like MARF’s medial atoms or explicit consistency losses partly address this (Sundt et al., 2023).
• Limited Real-world Generalization: Training data for layered models is often restricted to synthetic or indoor scenes; extension to outdoor or large-scale environments remains ongoing (Li et al., 25 Apr 2025).
• Physical and Theoretical Constraints: Many frameworks assume far-field, non-diffractive, or perfect-conductor conditions, limiting direct applicability to some domains. Some only approximate or neglect wave effects, which are critical in certain physical modeling tasks (Yang et al., 2024, Ren et al., 2024).

5. Advances, Hybridizations, and Future Directions

Recent and ongoing research pursues the synthesis of ray-based approaches with other geometric and physical frameworks:

• Hybrid Representations: Fusing layered ray maps with implicit SDFs or learning adaptive layer counts per ray aims to increase surface completeness and fill gaps between discrete intersections (Li et al., 25 Apr 2025).
• Generalized Primitives: Extending beyond Gaussians to higher-order, anisotropic, or modulated kernels (e.g., generalized Gaussians, cosine kernels) reduces particle hit counts, increases efficiency, and better models semi-transparent and textured materials (Moenne-Loccoz et al., 2024).
• Differentiable and Inverse Design: Ray mapping methods driven by variational optimal mass transport provide physically-plausible surfaces in optical design, potentially informing neural or learning-based surface construction (Bösel et al., 2015).
• Domain Extensions: Application to reflective, refractive, or multi-frequency materials, real-world LiDAR and RGB-D scans, SLAM-style online updates, or scene-level geometry with complex wave effects remains active (Li et al., 25 Apr 2025, Yang et al., 2024).
• Efficient Ray Sampling for Learning: Structured ray distributions, hierarchical and adaptive grouping, and feature-augmented sketches (e.g., RaySense) allow statistically robust analysis and fast neural inference in high-dimensional shape spaces (Liu et al., 2019).

6. Theoretical Unification and Physical Modeling

Ray-based geometric representations extend from purely geometric approaches to those incorporating full physical modeling:

• Ray-based Wave Theory: RTW’s integration of intrinsic wavefront curvature into the ray framework enables amplitude and phase tracking, hybrid diffraction correction at caustics, and recovers classical and quantum properties in a unified differential-geometry formalism (Ren et al., 2024).
• Non-Euclidean Spaces and Geometries: Frameworks such as ray-marching Thurston geometries generalize signed distance, ray tracing, and light transport to arbitrary Riemannian model spaces and their quotients, supporting non-Euclidean visualization, geometric computation, and real-time shading (Coulon et al., 2020).
• Geometric Optics and Surface Reconstruction: Ray-mapping approaches grounded in conservation of energy and Snell’s law yield globally integrable surfaces via PDEs and analytic boundary integration, further linking ray representations to the underlying physical optics (Bösel et al., 2015).

7. Summary Table of Representative Ray-based Geometric Approaches

Approach	Ray Parameterization	Key Output/Primitive
LaRI, X-Ray	Pixels $\to$ rays $r(t)$	Layered points, attributes per ray
PRIF, MARF	Ray direction and auxiliary points	Intersection depth, medial atoms
RaySplats, 3D Gaussian Ray Tracing	Ray–primitive (Gaussian)	Analytic/sampled densities/opacity
RaySense	Random rays over point sets	Nearest neighbor features/sketches
GitNet, GRLoc	Image columns/bundles of rays	Bundled features, pose fields
RTW, Poincaré Sphere	Rays with wavefront curvature/labels	Field amplitude, Poincaré orbits

These ray-based geometric frameworks are increasingly central in the unification of geometric, photometric, and physical modeling for 3D computer vision, graphics, computational wave physics, and inverse problems. As computational resources grow and hybrid physical-learning systems mature, ray-based representations are anticipated to expand further into large-scale, real-world, and physically-rich domains.