Predicted Occupancy Grid (POG) Overview

• Predicted Occupancy Grid (POG) is a grid-based probabilistic model that forecasts spatial occupancy by encoding dynamic and static agent probabilities.
• It integrates model-based dynamics and machine learning techniques (e.g., SDA, Random Forest, DeconvNet) to predict occupancy with uncertainty under various traffic scenarios.
• Quantitative evaluation using metrics like MAE and MSE validates its effectiveness for real-time risk-aware planning in autonomous systems.

A Predicted Occupancy Grid (POG) is a grid-based probabilistic representation of the spatial environment at a future time horizon, widely deployed in autonomous driving, mobile robotics, and active safety systems. Each grid cell encodes the probability of being occupied by dynamic or static agents, constructed via machine-learning techniques or physics-based models, and supports risk-aware planning under motion and perception uncertainty.

1. Mathematical Formulation and Conceptual Basis

A POG $G_{t_{\text{pred}}}$ is defined on an $I \times J$ regular grid; for each cell $(i,j)$ at prediction instant $t_{\text{pred}}$, the occupancy probability $p_{i,j}^{t_{\text{pred}}}$ quantifies the likelihood of occupation, marginalizing over all hypotheses about moving agents and environment states (Nadarajan et al., 15 Dec 2025Nadarajan et al., 15 Dec 2025). Formally: $$p_{i,j}^{t_{\text{pred}}} = \min\left(1, \sum_{\ell=1}^L \sum_{s=1}^S Z_{v_\ell,s,t_{\text{pred}}}^{i,j} \cdot p(h_{v_\ell,s,t_{\text{pred}}}) \right)$$ where $L$ is the number of traffic participants, $S$ the number of trajectory hypotheses per participant, $Z_{v_\ell,s,t_{\text{pred}}}^{i,j}$ indicates if the $s$-th hypothesis for participant $\ell$ occupies cell $(i,j)$ at $t_{\text{pred}}$, and $p(h_{v_\ell,s,t_{\text{pred}}})$ is the likelihood of each hypothesis (Nadarajan et al., 15 Dec 2025).

For 3D variants (as in autonomous driving and scene understanding), $p_{\text{occ}}(x, y, z)$ denotes the probability that voxel $(x, y, z)$ is occupied given multi-modal sensor data (Chen et al., 24 Aug 2024Lu et al., 2023Wang, 2 Sep 2024).

2. Model-Based and Learning-Based Construction

Model-Based Approaches

Classical POG computation propagates agent dynamics (e.g., two-track/bicycle models) under multiple maneuver-deviation hypotheses, integrating trajectories into grid cells and aggregating occupancy probabilities (Nadarajan et al., 15 Dec 2025). This achieves thorough uncertainty modeling but incurs prohibitive computation ($O(L S T_{\text{integration}})$ for $L$ agents and $S$ hypotheses).

Machine Learning Pipelines

To address scalability, recent approaches encode the current environment as an Augmented Occupancy Grid (AOG)—a multi-channel tensor with occupancy, semantic, and kinematic features. Mapping $f: \text{AOG} \mapsto \text{POG}$ is realized by:

• Stacked Denoising Autoencoders (SDA) for feature dimensionality reduction
• Random Forest regressors or deep convolutional/deconvolutional networks for probabilistic cell prediction (Nadarajan et al., 15 Dec 2025Nadarajan et al., 15 Dec 2025Nadarajan et al., 15 Dec 2025)

$$f(\text{AOG})_{i,j} \approx p_{i,j}^{t_{\text{pred}}}$$

High-dimensional encoded features enable robust prediction across complex traffic scenarios, with uncertainty naturally captured by multi-hypothesis aggregation and softmax/forest outputs.

3. Architecture: Feature Construction, Dimensionality Reduction, and Prediction

Input Feature Engineering

Cells contain not only occupancy ($o_{i,j}$), but attributes such as velocity, orientation, acceleration: $f^{i,j} = [o, v, \psi, a_x, a_y]^T$. Environments are rasterized into $I\times J$ grids with cell size typically 0.5–1 m (Nadarajan et al., 15 Dec 2025Nadarajan et al., 15 Dec 2025).

Dimensionality Reduction

SDA layers (e.g., input feature vector of size $32\,000$ reduced to code size $2\,000$ via 3–4 linear layers with Gaussian noise corruption) yield a compressed latent code optimized for robust inference (Nadarajan et al., 15 Dec 2025).

Prediction Algorithms

• Random Forest ensembles: one RF per cell (or per latent feature), operates on SDA code to output $p_{i,j}^{t_{\text{pred}}}$ (Nadarajan et al., 15 Dec 2025)
• DeconvNet: convolutional encoder and transposed-conv decoder with softmax head for categorical occupancy bins (Nadarajan et al., 15 Dec 2025)
• Hybrid pipelines combining SDA, RF, and neural decoders, balancing accuracy with inference time (Nadarajan et al., 15 Dec 2025Nadarajan et al., 15 Dec 2025)

Probabilistic outputs support multi-modal scenario modeling; discrete quantization stabilizes training ($p \in \{0, 0.25, 0.5, 0.75, 1.0\}$).

4. Evaluation Metrics, Performance, and Computational Complexity

Prediction quality is measured per-cell, typically as mean squared error (MSE) or mean absolute error (MAE) between predicted and quantized ground truth occupancy: $$\varepsilon_{t_{\text{pred}}} = \sqrt{\frac{1}{K} \sum_{(i,j) \in B \cup D} \left( \hat{p}(o_{t_{\text{pred}}}^{ij}) - p_q(o_{t_{\text{pred}}}^{ij}) \right)^2 }$$ Here, $B$ and $D$ define the set of active (predicted/ground-truth) cells, and $K$ their total count (Nadarajan et al., 15 Dec 2025). Errors for low, medium, high-probability bins are reported.

Table: Example Evaluation—1.0 s Prediction Horizon (Nadarajan et al., 15 Dec 2025)

Arch.	MAE (low)	MAE (mid)	MAE (high)
SDA-1+RF	0.0518	0.0337	0.0277
SDA-1+RF+SDA-2	0.0742	0.0739	0.0501
DeconvNet	0.1501	0.1447	0.0777

Random Forest approaches yield lower MAE at medium/high occupancy with moderate computational overhead. Inference time for full grids is sub-20 ms on GPU-parallelized evaluation (Nadarajan et al., 15 Dec 2025).

5. Applications: Planning, Criticality Estimation, and Active Safety

POGs are directly embedded into cyber-physical system planning pipelines:

• Trajectory Planning: Sampling-based planners (RRT, CL-RRT) reject candidate paths traversing cells with high predicted occupancy (Nadarajan et al., 15 Dec 2025Nadarajan et al., 15 Dec 2025)
• Criticality Estimation: The product $$c_{t_{\text{pred}}}^{ij} = p_{\text{other}}(o_{t_{\text{pred}}}^{ij}) \cdot p_{\text{ego}}(o_{t_{\text{pred}}}^{ij})$$ quantifies collision risk; scenario-level criticality triggers pre-crash actions (Nadarajan et al., 15 Dec 2025)
• Scenario Coverage: Clustering diverse POGs systematically generates representative future “worlds” for testing and verification (Nadarajan et al., 15 Dec 2025)

POGs enable risk-aware navigation and assessment of decision logic under motion and perception uncertainties.

6. Extensions and Implementation Guidelines

Key guidelines for adopting POGs in real-time systems:

• Cell size: 0.5 m resolution balances spatial fidelity and computational feasibility
• Quantization: Discrete probability bins ($\geq$5) stabilize learning, finer bins require larger datasets
• Ensemble size: RFs with $\geq$100 trees and full-growth; feature subsampling ($m_{\text{try}} \approx \sqrt{\text{features}}$)
• Data diversity: Training must include varied road geometry, traffic density, and static infrastructure (Nadarajan et al., 15 Dec 2025)
• Parallelization: GPU/multicore evaluation for batch grid inference
• Continuous retraining: Out-of-bag error and environment change detection cue model update schedules

Adhering to these principles supports practical deployment in active safety, prediction-based control, and real-world planning.

7. Comparative Architectures and Outlook

While Random Forest + SDA architectures offer robust probabilistic spatial-temporal prediction, deep neural networks (DeconvNet, ConvLSTM, transformer-based models) and stochastic generative frameworks (VAE, GAN) present competitive alternatives, especially for high-dimensional or multi-modal sensor fusion (Nadarajan et al., 15 Dec 2025Chen et al., 24 Aug 2024Lange et al., 2022). Each framework trades off computational cost, accuracy, and ease of uncertainty quantification.

POGs are a core representation bridging low-level perception, high-level inference, and control in complex dynamic environments, pivotal for the next generation of safety-critical autonomous systems (Nadarajan et al., 15 Dec 2025Nadarajan et al., 15 Dec 2025Nadarajan et al., 15 Dec 2025).