Occupancy Ray-shape Sampling (ORS)

• Occupancy Ray-shape Sampling (ORS) is a 3D volumetric strategy that projects occupancy data along structured rays or tubes, encoding scene geometry and semantics.
• It reduces computational complexity from O(N³) to O(N²) by aligning sampling with image pixels or grid cells, enabling efficient 3D reconstruction and neural rendering.
• ORS integrates with models like DualDiff, Ray-ONet, and CLONeR to yield improved reconstruction metrics, better rendering quality, and lower memory usage.

Occupancy Ray-shape Sampling (ORS) refers to a family of 3D volumetric encoding, sampling, and representation strategies that project or query occupancy information along structured rays or tubes, typically aligned with the image plane or grid axes, to yield dense, efficient, and semantically informative descriptors for downstream tasks such as 3D scene reconstruction, conditional generative modeling, and neural rendering. This approach subsumes a variety of formulations, including per-pixel structured ray sampling from camera viewpoints, occupancy-guided sampling for radiance field rendering, and axis-aligned tube-based run-length representations. ORS fundamentally restructures the encoding of 3D volumes, trading dense, cubic sampling and expensive inference for camera- or axis-aligned, per-ray sampling and efficient neural architectures.

1. Conceptual Overview and Definitions

ORS is characterized by mapping a high-dimensional occupancy grid—typically resulting from multi-view fusion, LiDAR accumulation, or shape voxelization—into structured 1D signals along rays or tubes. In dense camera-centric versions, such as in "DualDiff: Dual-branch Diffusion Model for Autonomous Driving with Semantic Fusion" (Li et al., 3 May 2025) and "Ray-ONet: Efficient 3D Reconstruction From A Single RGB Image" (Bian et al., 2021), rays are cast per pixel from the camera image plane through the 3D volume. For each ray, occupancy states are sampled at predetermined depths, producing a compact per-pixel descriptor encoding spatial structure and semantic content.

In axis-aligned or grid-centric settings, e.g., "SeqXY2SeqZ: Structure Learning for 3D Shapes by Sequentially Predicting 1D Occupancy Segments From 2D Coordinates" (Han et al., 2020), the 3D grid is reorganized into a set of 1D tubes (rays) along a chosen axis, with occupancy segments encoded efficiently via run-length representations. In the rendering context, such as CLONeR (Carlson et al., 2022), occupancy grids are leveraged to guide importance sampling of points along rays for downstream radiance integration.

2. Ray-based Mathematical Formulation and Algorithmics

The mathematical core of ORS involves ray definition, point sampling, and occupancy query. In canonical camera-centric ORS (Li et al., 3 May 2025, Bian et al., 2021):

• Each pixel's homogeneous coordinate $s_\text{img} \in \mathbb{R}^3$ is back-projected via camera intrinsics $K \in \mathbb{R}^{3\times3}$ and extrinsics $T \in SE(3)$.
• The corresponding ray is

$$\mathbf{r} = \operatorname{normalize}\left(T^{-1} K^{-1} s_\text{img} - p_{\text{ego}}\right)$$

with $p_{\text{ego}}$ as the camera center.

• A depth schedule $N = \{n_1,\ldots,n_N\}$ defines 3D sample locations

$$\hat s_\text{ego}^{(u,v,i)} = p_{\text{ego}} + n_i \mathbf{r}$$

• Occupancy is retrieved from the grid $O\in\mathbb{R}^{H\times W\times D}$ at each sample via trilinear interpolation:

$$v(u,v,i) = O\left(\hat s_\text{ego}^{(u,v,i)}\right)$$

Organizing all per-pixel rays yields tensor $v \in \mathbb{R}^{U\times V\times N}$.

Variants—such as importance sampling in CLONeR (Carlson et al., 2022)—use a learned probabilistic occupancy grid to adapt sampling density along rays:

• Half the points are uniformly distributed; the other half are sampled proportional to estimated occupancy probability

$p_x = \sigma(l_x) = \frac{1}{1+\exp(-l_x)}$

with $l_x$ the trilinear-interpolated log-odds at location $x$. This concentrates computation on likely surfaces.

Axis-aligned tube approaches (Han et al., 2020) define for each $(x,y)$ coordinate an ordered sequence of occupancy segments along $z$: $\mathbf{t}_i = \{ v_{i,z} | z=1,\ldots,R \}$ where each tube is represented by its segment start and end indices.

3. Architectural Integration and Model Design

In conditional generative frameworks such as DualDiff (Li et al., 3 May 2025), ORS acts as the central semantic representation, conditioning diffusion models with dense, spatially aligned 3D context:

• ORS tensors are split into semantic foreground/background via occupancy masks.
• Each branch undergoes a sequence of (self-attention, spatial fusion, deformable-attention) with other scene descriptors (numerical attributes, text), followed by cross-attention-based conditioning.
• ORS outputs serve as ControlNet residuals injected into a frozen UNet-based diffusion model.

In Ray-ONet (Bian et al., 2021), ORS underlies efficient 3D reconstruction:

• Per-pixel ray features (via 2D convolutions and local mixing) are passed through an MLP, outputting $M$ occupancy values along each ray in a single forward pass.

CLONeR's ORS (Carlson et al., 2022) enables occupancy-guided volumetric rendering by decoupling scene occupancy estimation from color, and using occupancy-only MLPs to guide all subsequent ray sampling.

Axis-aligned tube architectures (e.g., SeqXY2SeqZ (Han et al., 2020)) employ 2D-to-1D sequence-to-sequence models, predicting run-length segmentations per tube given global context.

4. Computational Efficiency and Sampling Complexity

ORS yields major reductions in sampling and inference complexity relative to classical volumetric approaches:

• Traditional ONet and NeRF variants evaluate on every 3D grid point, incurring $O(N^3)$ complexity at resolution $N$.
• ORS reduces this to $O(N^2)$ by matching the number of rays to image pixels (or grid cells on a 2D face), and predicting all samples per ray in parallel (Bian et al., 2021).
• Tube-based run-length representations further reduce memory and compute, as only non-empty segments are stored and predicted (Han et al., 2020).
• In volumetric rendering (e.g., CLONeR), ORS-guided sampling places higher sample density near surface regions, increasing rendering “hit-rate” while cutting redundant evaluations in empty space (Carlson et al., 2022).

The table summarizes sampling/inference complexity:

Method	Sample Placement	Per-shape Network Calls	Reference
Volumetric ONet	3D grid (dense)	$O(N^3)$	(Bian et al., 2021)
Ray-ONet ORS	2D image rays, $M$ steps	$O(N^2)$	(Bian et al., 2021)
Tube (Seq2Seq)	2D grid, run-length	$O(R^2 J_{\max})$	(Han et al., 2020)
CLONeR ORS	Guided ray sample (LiDAR)	$O(K)$, $K\ll N^3$	(Carlson et al., 2022)

5. Quantitative Impact and Benchmark Results

Experiments across multiple papers demonstrate substantial improvements from ORS adoption:

• In DualDiff (Li et al., 3 May 2025) on nuScenes and Waymo, replacing a BEV map with ORS improved Road mIoU from ~61% to ~62.2% and FID from 16.20 to 13.26; the full model reached FID 10.99, Vehicle mIoU 30.22, and 3D-mAP 13.99.
• Ray-ONet (Bian et al., 2021) on ShapeNet achieved mean volumetric IoU 0.633 (vs. 0.593 for ONet), Chamfer-L1 0.153 (vs. 0.194), and 24× speedup; unseen-category IoU improved from 0.278 to 0.375.
• CLONeR (Carlson et al., 2022) demonstrated PSNR 20.04 dB vs. 17.66 dB (w/o occupancy grid), and a training time of <13 minutes on an A100, versus >2 hours for vanilla NeRF; depth error (absErrRel) reduced from 0.314 to 0.073.
• SeqXY2SeqZ (Han et al., 2020) yielded mean IoU 90.35% for auto-encoding $64^3$ grids (vs. OccNet 89.00%), and peak memory of 0.28 GB (vs. OccNet 1.15 GB).

Ablations show diminished returns beyond moderate numbers of ray samples per pixel or tube, and critical dependence on the cross-modal context (e.g., attention in SeqXY2SeqZ or multi-branch fusion in DualDiff).

6. Extension, Scope, and Limitations

ORS is broadly applicable for 3D representation learning, generative modeling, and implicit function learning. Its structured sampling is particularly advantageous in driving scenes (DualDiff), single-view shape inference (Ray-ONet), LiDAR/raster fusion (CLONeR), and memory-efficient voxel representations (SeqXY2SeqZ).

Key strengths include:

• Exact alignment with camera rays, yielding viewpoint-consistent features and compact per-pixel 3D context (Li et al., 3 May 2025, Bian et al., 2021).
• Efficiency: conversion of $H\times W\times D$ volumes to $U\times V\times N$ tensors, and major reductions in compute/memory (Bian et al., 2021, Carlson et al., 2022, Han et al., 2020).
• Retention of fine geometry and semantics, beyond coarse mask or bounding-box descriptors.

Potential limitations include:

• Requirement for dense occupancy volumes or reliable multi-modal fusion methods to construct $O$.
• In axis-aligned schemes, possible bias if shape structure does not align well with grid axes; this suggests applications with irregular geometry may benefit more from camera-centric or learned ray orientation.
• The choice of the number and distribution of sampling steps $N$ along rays is critical; oversampling impacts computational efficiency, while undersampling may miss fine detail.

Further investigation into adaptive ray allocation, more expressive cross-ray context modeling, and integration with dynamic or time-varying scenes is ongoing.

7. References to Key Works

• "DualDiff: Dual-branch Diffusion Model for Autonomous Driving with Semantic Fusion" (Li et al., 3 May 2025)
• "CLONeR: Camera-Lidar Fusion for Occupancy Grid-aided Neural Representations" (Carlson et al., 2022)
• "Ray-ONet: Efficient 3D Reconstruction From A Single RGB Image" (Bian et al., 2021)
• "SeqXY2SeqZ: Structure Learning for 3D Shapes by Sequentially Predicting 1D Occupancy Segments From 2D Coordinates" (Han et al., 2020)