Predicted-Occupancy Grid (POG) Overview

Updated 22 December 2025

Predicted-Occupancy Grid (POG) is a probabilistic space-time representation that discretizes an environment into cells with future occupancy likelihoods.
It integrates model-based simulations and machine learning pipelines, such as Random Forests with autoencoders, to predict multi-agent trajectories and manage uncertainty.
POGs support autonomous vehicle risk assessment and safe trajectory planning by translating sensor data and agent dynamics into actionable occupancy maps.

A Predicted-Occupancy Grid (POG) is a probabilistic, space-time representation of the future state of a spatial environment discretized into cells, where each cell’s value corresponds to the likelihood of being occupied by relevant agents (vehicles, pedestrians, etc.) at some specified future time horizon. POGs provide a unified framework for capturing forecasted dynamics in complex, multi-agent traffic scenarios and form a foundational component for decision-making in active safety systems, trajectory planning, and risk assessment in autonomous vehicles (Nadarajan et al., 15 Dec 2025, Nadarajan et al., 15 Dec 2025, Nadarajan et al., 15 Dec 2025). This article overviews the formal definition, data representations, estimation methodologies, learning pipelines, evaluation results, and application contexts for POGs.

1. Mathematical Definition and Data Structures

A POG is formally defined on a two-dimensional grid $G_{t_{\rm pred}} = \{g_{t_{\rm pred}}^{i,j}\}_{i,j}$ , covering the relevant spatial region at a future prediction time $t_{\rm pred}$ . For each cell, the occupancy probability is either explicitly modeled or statistically estimated:

$POG^{t_{\rm pred}} = \{\,p^{i,j}_{t_{\rm pred}}\,|\,i=1,\ldots,I;\,j=1,\ldots,J\,\}$

$p^{i,j}_{t_{\rm pred}} = P(o_{t_{\rm pred}}^{i,j} = 1\,|\,current\,scene)$

In multi-agent settings, $p^{i,j}_{t_{\rm pred}}$ aggregates over multiple dynamic hypotheses for each relevant agent:

$p^{i,j}_{t_{\rm pred}} = \min\left\{1,\, \sum_{e=1}^{L}\sum_{s=1}^S P(h^s_{e,t_{\rm pred}})\, \mathbf{1}_{\{g_{t_{\rm pred}}^{i,j} \in h^s_{e}\}}\,\right\}$

where $P(h^s_{e,t_{\rm pred}})$ is the probability of trajectory hypothesis $s$ for agent $e$ , and $\mathbf{1}$ is the cell inclusion indicator (Nadarajan et al., 15 Dec 2025, Nadarajan et al., 15 Dec 2025).

2. Feature Engineering: Augmented Occupancy Grids

The input representation to POG estimation leverages an Augmented Occupancy Grid (AOG) constructed from instantaneous sensor data and inferred agent states. Each cell is encoded as:

$\mathbf{x}_0^{i,j} = \begin{cases} [1,\,v_e,\,D_e,\,a_{x,e},\,a_{y,e}]^\top & \text{if a moving object %%%%7%%%% occupies %%%%8%%%%} \ [1,\,0,\,0,\,0,\,0]^\top & \text{empty lanes or static infrastructure} \ \end{cases}$

resulting in $\mathrm{AOG} \in \mathbb{R}^{I \times J \times 5}$ (Nadarajan et al., 15 Dec 2025, Nadarajan et al., 15 Dec 2025).

This encoding facilitates capturing both agent kinematics (velocity, acceleration, orientation) and the static scene context, serving as the basis for downstream probabilistic mapping.

3. Model-Based and Machine Learning Approaches for POG Estimation

3.1. Model-Based POG Aggregation

A classical approach involves explicit generation of all plausible future trajectories for each traffic participant based on behavioral hypotheses and physical models (multi-hypothesis dynamics, lane-following, turning, etc.), then rasterizing each into the grid at the prediction horizon (Nadarajan et al., 15 Dec 2025):

For $S$ hypotheses per agent, run dynamic simulation models (two-track for cars, single-track for bicycles) to produce future positions.
For each cell, collect all passing hypotheses, aggregate probabilities, and cap at unity for collision detection.
Computational complexity is $\mathcal{O}(L\,S\,C_{\rm dyn})$ for $L$ agents.

This approach accurately models uncertainty and interaction but becomes computationally prohibitive in real time for large $S$ and $L$ .

3.2. Machine Learning Pipeline: Random Forests and Autoencoders

To ensure scalability, POG estimation is cast as a supervised mapping from the current AOG to future occupancy probabilities using an ensemble of Random Forests, optionally preceded by dimensionality reduction via stacked denoising autoencoders (SDA) (Nadarajan et al., 15 Dec 2025, Nadarajan et al., 15 Dec 2025, Nadarajan et al., 15 Dec 2025):

Dimensionality reduction: SDAs train layer-wise to produce low-dimensional codes $q^{(l)} = f(W^{(l)}\tilde{p}^{(l)} + b^{(l)})$ , minimizing layer-wise reconstruction loss and regularization.
Random Forest regression: For each cell and prediction horizon, a separate RF is trained to map AOG features (or SDA codes) to quantized occupancy probability levels.

Architectural comparisons (Nadarajan et al., 15 Dec 2025) demonstrate that an SDA + RF pipeline yields the lowest approximation error, albeit at increased ensemble size; deeper architectures can trade accuracy for computational speed.

3.3. Learning Formulation and Losses

The loss functions for POG estimation are standard cross-entropy or mean-squared error:

$\ell_{\rm CE} = -\sum_{n,i,j} \left[ y^{i,j}_n \log \hat{p}^{i,j}_n + (1 - y^{i,j}_n)\log(1 - \hat{p}^{i,j}_n) \right]$

$\ell_{\rm MSE} = \frac{1}{N I J} \sum_{n=1}^{N} \sum_{i,j} (\hat{p}^{i,j}_n - p^{i,j}_n)^2$

Quantized outputs and cross-validation on simulation datasets ensure reliable mapping (Nadarajan et al., 15 Dec 2025, Nadarajan et al., 15 Dec 2025).

4. Quantitative Evaluation and Scalability

Extensive simulation studies have quantified the error of various POG estimation pipelines in realistic urban traffic scenarios (Nadarajan et al., 15 Dec 2025, Nadarajan et al., 15 Dec 2025):

Architecture	Error (Low)	Error (Mid)	Error (High)	Complexity (RFs per horizon)
SDA₁ + RF	0.052	0.034	0.028	6,400
SDA₁→RF→SDA₂	0.074	0.074	0.050	2,000
Conv/Deconv Softmax	0.15	0.145	0.078	Minimal

Random Forest-based approaches achieve $\sim$ 5× speed-up over model-based numerical integration, enabling practical real-time performance (sub-50 ms per update for $80 \times 80 \times 3$ horizons) (Nadarajan et al., 15 Dec 2025).

5. Integration into Risk Assessment and Trajectory Planning

POGs provide direct input to several core components in active safety systems (Nadarajan et al., 15 Dec 2025, Nadarajan et al., 15 Dec 2025):

Criticality estimation: The risk map $R^{i,j} = p_{t_{\rm pred}}^{i,j}\, C(x_{i,j}, \tau)$ highlights high-probability collision regions, supporting early emergency actions.
Trajectory planning: Collision probability of a planned trajectory $\xi$ is computed as $P_{\rm coll}(\xi) = 1 - \prod_{s} (1 - p_{t(s)}^{i(s),j(s)})$ , which is minimized jointly with path length in safe trajectory search.
Scenario clustering: High-dimensional occupancy and predicted occupation patterns enable clustering for scenario-based regulatory and validation testing.

6. Extensions and Real-World Applications

The POG framework generalizes to active safety components in autonomous vehicles, supporting both simulation-based validation and on-road deployment (Nadarajan et al., 15 Dec 2025, Nadarajan et al., 15 Dec 2025):

Real vehicle validation: POG architectures have demonstrated robust prediction in car–bicycle crash clips and four-way intersection scenarios, capturing multi-agent uncertainty and trajectory diversity.
Downstream system integration: POGs are directly consumable by motion planners, collision risk estimators, and emergency action flaggers.

The computational tractability (via Random Forest ensembles and autoencoding) and probabilistic richness make POGs an enabling layer for robust, proactive decision-making in dynamic traffic environments.

References

(Nadarajan et al., 15 Dec 2025): Probability Estimation for Predicted-Occupancy Grids in Vehicle Safety Applications Based on Machine Learning (Nadarajan et al., 15 Dec 2025): Machine Learning Architectures for the Estimation of Predicted Occupancy Grids in Road Traffic (Nadarajan et al., 15 Dec 2025): Predicted-occupancy grids for vehicle safety applications based on autoencoders and the Random Forest algorithm