View Cone Sampling (VCS)

• VCS is a probabilistic ray sampling approach that replaces single-ray sampling with a Gaussian-distributed bundle of rays within a small angular cone matching the human fovea.
• It uses Gaussian angular sampling and weighted intersections to create smoother, more robust saliency maps that tackle high-frequency textures and complex mesh geometries.
• Quantitative studies show that VCS offers improved consistency and coverage—up to 31× enhancement—over traditional methods in VR eye-tracking pipelines.

View Cone Sampling (VCS) is a probabilistic ray sampling methodology designed to improve 3D mesh saliency ground-truth (GT) acquisition in virtual reality (VR), particularly for eye-tracking pipelines. Unlike classical single-ray gaze sampling, VCS simulates the finite spatial extent of the human foveal receptive field by casting a Gaussian-distributed bundle of rays within a small angular cone centered on the gaze direction. This approach enhances robustness to high-frequency textures, geometric sparsity, and topologically complex mesh regions, mitigating aliasing artifacts and signal discontinuities prevalent in single-ray methods (Zheng et al., 6 Jan 2026).

1. Conceptual Motivation and Definition

VCS arises from the inadequacy of conventional single-ray sampling to account for the spatial spread of human visual attention. Traditional VR eye-tracking maps a user's gaze to a solitary zero-area ray intersecting the 3D mesh, providing limited surface coverage and being highly sensitive to missed or noisy intersections, especially on textured or punctured surfaces. VCS instead forms a circular sampling cone $\mathcal{C}$ of apex $O$ (eye position), axis $d_0$ (gaze vector), and full angle $R_f$ (typically $R_f \approx 5^\circ$), mimicking the approximate angular extent of the human fovea. Sampling $M$ rays per cone, distributed according to a zero-mean angular Gaussian in cone-centric coordinates, each intersection is weighted to reflect its angular proximity to the central axis, producing saliency maps that more faithfully represent perceptual foveation and local attention (Zheng et al., 6 Jan 2026).

2. Mathematical Formalism of Gaussian Angular Sampling

Let $\theta$ be the angular deviation from the central axis $d_0$, $\phi$ the azimuthal angle, and $\sigma_1 = R_f / 6$ to ensure $\pm 3\sigma_1$ covers nearly the entire cone, following the Gaussian $3\sigma$ rule. The angular deviation $\theta$ is sampled from a truncated normal $\mathcal{N}(0,\sigma_1^2)$ with $|\theta|\leq R_f/2$, and $\phi \sim U(0,2\pi)$. Using the Box–Muller transform, for uniform $u_1, u_2\sim U(0,1)$:

$z = \sqrt{-2\ln u_1}\;\sin(2\pi u_2)$

$R_s = \sigma_1\, z$

$R_s \in \left[-\frac{R_f}{2}, \frac{R_f}{2}\right]$

$R_r \sim U(0, 2\pi)$

The final world-space direction for the $n$th sample ray, $D_n$, is

$D_n = M_C\, R_z(R_r)\, R_x(R_s)\, d_0$

where $M_C$ is the alignment matrix mapping the local $z$ to $d_0$, and $R_x$, $R_z$ are standard rotation matrices. The angular joint probability density is

$$p(\theta, \phi) = \frac{1}{\sigma_1 \sqrt{2\pi}} \exp\left( -\frac{\theta^2}{2\sigma_1^2} \right) \frac{1}{2\pi}$$

valid for $0 \leq |\theta|\leq R_f/2$.

3. Algorithmic Procedure and Implementation Details

Within eye-tracking acquisition, the core pseudocode workflow for VCS comprises:

Input: eye‐pose (O, head‐orientation), corneal-reflect gaze vector g₀, cone angle R_f, per-cone sample count M, σ₁=R_f/6, backface threshold τ=0.1 Output: list of valid intersections InfList 1. Transform gaze to world space: d₀ ← normalize( M_head * g₀ ) 2. Compute cone alignment: M_C ← ComputeAlignment( [0,0,1], d₀ ) 3. For n = 1...M: u₁, u₂ ∼ U(0,1) z ← sqrt(-2 log u₁) * sin(2π u₂) R_s ← clamp(σ₁ * z, -R_f/2, +R_f/2) R_r ← U(0,2π) d_local ← R_z(R_r) * R_x(R_s) * [0,0,1]^T Dₙ ← M_C * d_local hit ← Physics.Raycast(O, Dₙ) if hit and dot(hit.normal, -normalize(Dₙ)) > τ: InfList.append({faceID=hit.face, point=hit.pos, normal=hit.normal}) return InfList

Within real-time VR pipelines, GPU/CPU-resident BVHs or engine-level colliders enable efficient ray intersection. Per-face saliency counts $S_\text{raw}(f_i)$ are incremented for each valid hit, and adjacency lists for mesh faces/vertices are maintained for subsequent geodesic smoothing operations.

4. Comparative Robustness and Quantitative Effectiveness

VCS exhibits substantial robustness and accuracy improvements over single-ray methods. Qualitatively, VCS fills spatial and topological gaps unaddressed by single-ray sampling, yielding smooth, blob-like attention maps even before subsequent diffusion. In saliency alignment and statistical stability, the following representative metrics were obtained (Zheng et al., 6 Jan 2026):

Metric	Single Ray (SR)	VCS (with HCD)
Internal Consistency (IC)	0.0557	0.8137
Correlation Coefficient (CC)	0.1970	0.4829
KL-Divergence	3.2092	1.1278
sAUC	0.7865	0.8288

Additionally, sampling coverage is improved by factors ranging from $4\times$ to $31\times$ for mesh sizes up to 1M faces. Ablation studies indicate VCS-derived saliency peaks are more tightly aligned to ground-truth eye-tracking densities.

5. Hyperparameterization and Tuning Strategies

Salient hyperparameters in VCS are:

• Cone apex angle $R_f$: $5^\circ$ (aligned to foveal receptive field estimates).
• Angular standard deviation $\sigma_1$: $R_f/6$, ensuring 99.7% inclusion within the cone.
• Rays per cone $M$: $200$–$500$, trading sampling density with real-time performance ($>60$ Hz on GTX-1080 hardware).
• Backface threshold $\tau$: $0.1$ (rejects rays at incidence angles $>84.3^\circ$).
• Post-diffusion geodesic $\sigma_2$: $0.02$ (applied in subsequent HCD geodesic smoothing; not directly part of VCS itself).

If $M$ is too small, under-sampled regions yield noisy or unstable saliency with degraded IC. Excessive $M$ yields diminishing quality improvements but increases computational cost. The $\sigma_1/R_f$ ratio regulates the central weighting of rays; $\sigma_1=R_f/6$ emerges as a strong empirical choice.

6. Integration in VR Eye-Tracking Pipelines

VCS is deployed as follows:

• Raw eye-tracker outputs (pupil center, corneal glints) are mapped to a 3D gaze vector $g_0$ in eye-camera coordinates, then transformed to world-space $d_0$ using the head's 6-DoF pose.
• The alignment matrix $M_C$ is computed per frame to map canonical $z$ to $d_0$.
• Efficient mesh intersection is achieved through precomputed BVHs or physics engine infrastructure (e.g., Unity3D colliders).
• Each cone sampling event records valid hits, with optional per-hit Gaussian weighting $w_n = \exp(-R_s^2 / 2\sigma_1^2)$.
• Almost all of the pipeline operates at real-time frame rates with 200–500 rays per cone, with mesh adjacency information reserved for smoothing and diffusion.

Handling non-manifold mesh configurations and excluding back-facing or grazing intersections (by dot product threshold) are key for precise and topologically valid hit registration.

7. Visual Interpretation and Figure Annotations

Figure 1(b) (cross-sectional diagram) illustrates the eye at the cone apex with the central gaze (solid line) and a radially fanning bundle of rays. Rays are densest near the axis and sparse towards the cone perimeter. Figure 1(c) (projection diagram) depicts the ray distribution in top view: a Gaussian radial density with central clustering fading towards the boundary, confirming the probabilistic spread of sampling directions.

In synthesis, VCS operationalizes a “foveal-field” model that supersedes single-pixel gaze sampling. The approach's Gaussian angular ray distribution and precise hit filtering establish a robust and perception-aligned foundation for mesh saliency acquisition. When combined downstream with Hybrid Manifold–Euclidean Diffusion, VCS enables perceptual fidelity and statistical robustness, even for large-scale, high-resolution, and topologically intricate meshes (Zheng et al., 6 Jan 2026).