Ray-Centered Local Triangle Parameterization

• Ray-centered local triangle parameterization is a geometric technique that anchors triangle frames to emitter rays, ensuring robust gradient propagation and consistent coverage.
• It employs a local coordinate system using an anchor ray and orthonormal basis, reducing the complexity of global quaternion regression and mitigating degeneracies.
• This approach enhances differentiable rendering and LiDAR simulation, with experiments showing over 11 dB PSNR improvement compared to traditional methods.

Ray-centered local triangle parameterization is a class of geometric representation in which planar triangle elements used for rendering or simulation are oriented, positioned, and parameterized locally with respect to an associated emitter ray or viewing direction. This formalism plays a central role in modern high-fidelity 3D scene reconstruction (notably, differentiable triangle splatting as in FLAT) and efficient emitter-centric filtering in ray casting, such as real-time LiDAR simulation. By anchoring triangle coordinate frames to the local ray, this parameterization enables stability, efficient coverage computations, and improved gradient propagation, particularly when predicting explicit surface primitives directly from compact generative codes or when culling triangles emitter-centrically.

1. Definition and Motivation

Ray-centered parameterization associates with each primitive triangle a local frame constructed around an "anchor ray"—defined by a point of origin and a direction in $\mathbb{R}^3$ (for rendering, often camera-space; for LiDAR, emitter-space). The triangle’s location, orientation, and sometimes shape are parameterized in this local frame, rather than in a fixed global/world frame. This construction allows the triangle to be directly relevant to the incoming ray, addressing geometric degeneracies (such as edge-on configurations) and ensuring efficient spatial indexing and coverage modeling.

The paradigm addresses two core challenges:

• For differentiable triangle rendering, especially from latent codes, robust triangle orientation is essential for non-vanishing photometric gradients and stable learning.
• In high-throughput ray simulation (e.g., LiDAR), emitter-centric parameterization enables triangle-to-ray filtering and culling without repeated global acceleration structure rebuilding.

2. Local Ray-Centered Coordinate System Construction

The reference frame for a triangle is constructed as follows:

• Anchor Ray: Defined by origin $r_0 \in \mathbb{R}^3$ and direction $r_d \in \mathbb{S}^2$.
• Orthonormal Basis: Set $w = r_d$ (local z-axis). Select vector $a$ not collinear with $w$ (e.g., $a = (0,0,1)$), and orthogonalize using the Gram–Schmidt process:
  • $u = \mathrm{normalize}(a - (a \cdot w) w)$
  • $v = w \times u$
• Transformation Matrices: Columns of $[u\;|\;v\;|\;w]$ constitute the 3×3 matrix mapping local coordinates to world frame. The rotation $r_0 \in \mathbb{R}^3$0 (whose rows are $r_0 \in \mathbb{R}^3$1) projects world points to the local frame.

The triangle centroid's world position is set as $r_0 \in \mathbb{R}^3$2, where $r_0 \in \mathbb{R}^3$3 is a predicted depth along the ray, and $r_0 \in \mathbb{R}^3$4 serves as the local origin.

3. Triangle Vertex and Normal Parameterization

In the canonical ray-centered frame, an equilateral 2D triangle with zero centroid is defined in the $r_0 \in \mathbb{R}^3$5-plane ($r_0 \in \mathbb{R}^3$6). Vertex locations are transformed in several steps:

• Shape: A Cholesky-style $r_0 \in \mathbb{R}^3$7 lower-triangular matrix $r_0 \in \mathbb{R}^3$8 parameterizes in-plane shape and anisotropy via three regressed scalars—$r_0 \in \mathbb{R}^3$9, $r_d \in \mathbb{S}^2$0, $r_d \in \mathbb{S}^2$1—yielding $r_d \in \mathbb{S}^2$2 for local canonical vertices $r_d \in \mathbb{S}^2$3.
• Residual Rotation: Instead of a global unconstrained quaternion, orientation is predicted as three residual Euler angles $r_d \in \mathbb{S}^2$4: tilts about $r_d \in \mathbb{S}^2$5 and $r_d \in \mathbb{S}^2$6, plus in-plane spin about $r_d \in \mathbb{S}^2$7. The composite rotation $r_d \in \mathbb{S}^2$8 (with standard axis-angle exponentiation), acts in the local frame.
• Vertex Lifting: Each 2D point $r_d \in \mathbb{S}^2$9 is lifted to 3D $w = r_d$0, rotated by $w = r_d$1, mapped to world space by $w = r_d$2, and re-centered at $w = r_d$3:

$w = r_d$4

• Normal Transformation: The canonical normal $w = r_d$5 is rotated analogously:

$w = r_d$6

4. Emitter-Centric Frames and Ray-Triangle Indexing

For applications such as LiDAR simulation, the emitter-centric triangle parameterization underpins the Gajmer Ray-Casting Algorithm (GRCA) (Gajmer et al., 11 May 2026), in which triangles are filtered and binned relative to an emitter’s frame:

• Emitter Frame Construction: For emitter at $w = r_d$7, axes $w = r_d$8 (forward, right, up) define the rotation $w = r_d$9; local point coordinates are $a$0.
• Apparent-Area Filtering: The apparent area from the emitter is computed as

$a$1

where $a$2 is triangle area, $a$3 is triangle normal, and $a$4 is triangle centroid.

• Channel and Ray Indexing: Spinning LiDAR rays define vertical "channels" ($a$5) and azimuthal steps. Given a channel, triangles are associated with ray-derived cones or planes (GACP surfaces), converting triangle–cone intersection into a quadratic in edge parameters for bounding which rays could strike each triangle. This emitter-centric indexing is computationally tractable and obviates the need for global BVH rebuilding.

5. Advantages in Differentiable Triangle Splatting

Ray-centered parameterization, as introduced in FLAT (Kupyn et al., 23 Jun 2026), addresses training instabilities typical with world-frame quaternion orientation:

• Stable Initial Gradients: The unrotated triangle always faces the anchor ray; thus, from the first training iteration, there is non-zero coverage under soft pixel window functions, allowing immediate photometric gradient backpropagation.
• Avoidance of Vanishing Gradients: World quaternion regression often results in edge-on triangles with degenerate (zero) pixel coverage and stalls in learning. Ray-centered residuals guarantee relevant coverage for every predicted primitive.
• Reduced Parameter Complexity: Only three meaningful local angles must be regressed per triangle, not four unconstrained quaternion values, improving conditioning and preventing normalization pathologies.

Ablation studies in FLAT demonstrate that replacing global quaternion orientation with ray-centric residuals increases PSNR by over 11 dB, enabling the model to escape degenerate output regimes and achieve high-fidelity surface generation.

Experiment	Rotation Type	PSNR (RE10K)	SSIM	LPIPS
Global (world)	Single quaternion	<10	<0.40	>0.40
Ray-centered residual	(θ,ϕ,ψ) about anchor	21.45	0.710	0.245

6. Applications and Impact

Ray-centered triangle parameterization has become essential in several domains:

• Feedforward Scene Generation: Direct decoding from compact video diffusion or other latent representations to explicit triangle splats fundamentally relies on robust local orientation models for geometric accuracy and image-space coverage (Kupyn et al., 23 Jun 2026).
• Real-Time Ray Simulation: High-throughput, dynamic LiDAR simulation leverages emitter-centric parameterization for efficient per-triangle, per-ray culling—enabling massive performance gains over traditional acceleration-structure-based approaches (Gajmer et al., 11 May 2026).
• Differentiable Rendering Pipelines: Explicitly representing surface-adjacent primitives using ray-anchored frames improves learning stability, especially relevant when associating geometry to learned latent codes.

7. Limitations and Future Directions

While ray-centered parameterization is intrinsically suited to camera- and emitter-centric processing, it is inherently local and may not capture global scene coherence without additional constraints or regularization. In simulation contexts, scene-wide topological consistency depends on post-hoc refinement steps or merging strategies. A plausible implication is that hybrid approaches blending global and local parameterizations may further improve overall performance or enable richer downstream simulation capabilities, especially where real-time and offline requirements coexist.

Future research may investigate extensions to higher-order primitives, more expressive local frames, or improved fusion with hierarchical scene representations. The systematic methodology developed in FLAT and GRCA provides a foundation for further advances in efficient surface-centric generative modeling and real-time emitter-based simulation (Kupyn et al., 23 Jun 2026, Gajmer et al., 11 May 2026).