Bayesian Occupancy Filter

The fundamental mechanism is the recursive application of Bayes' rule to update a posterior probability of occupancy for each cell. Given a prior belief P(O|z_1:t-1) and a new sensor measurement z_t, the filter computes a new posterior P(O|z_1:t) using an inverse sensor model. This Markov assumption—that the current state is sufficient for prediction—enables computationally efficient, real-time updates without storing the entire measurement history.

BOFs rely on an inverse sensor model P(O|z) to interpret raw data. This model maps a sensor measurement directly to a probability of occupancy, inherently accounting for sensor noise, false positives, and false negatives. For example, a LiDAR beam's return might be modeled as a high probability of occupancy at the detected range, a low probability along the beam's path, and a uniform prior in occluded space. This explicit noise modeling is what distinguishes BOFs from naive binary mapping.

Static occupancy grids fail in dynamic scenes. BOFs extend the framework by incorporating a motion model that predicts how occupancy probabilities evolve between sensor updates. This is often implemented using a Bayes filter with a transition function that propagates cell probabilities based on estimated object velocities. The result is a 4D grid (3D space + time) or a grid where moving objects are tracked as probability distributions, preventing the 'ghosting' artifacts that plague static maps.

A key strength is the principled handling of clutter and false alarms. Unlike deterministic methods that must make a hard decision on each measurement's validity, BOFs maintain uncertainty. A single spurious sensor return only slightly nudges a cell's probability, requiring multiple consistent observations to converge to a definitive occupied or free state. This inherent robustness is formalized through the log-odds representation, which prevents numerical underflow and simplifies the Bayesian update to a simple addition.

The Bayesian framework provides a mathematically elegant method for heterogeneous sensor fusion. Because each sensor's inverse model P(O|z_i) is independent, fusing data from a LiDAR, radar, and camera is achieved by sequentially applying their respective probabilistic updates to the same grid. The order of updates does not affect the final posterior probability, a property known as commutativity. This allows a BOF to seamlessly combine the long-range velocity precision of radar with the dense geometric accuracy of LiDAR.

To avoid numerical instability near probabilities of 0 or 1, BOFs operate in log-odds space. The probability p is transformed to l = log(p/(1-p)). The Bayesian update then becomes a simple, computationally cheap addition: l_new = l_prior + l_measurement. This linearizes the fusion process and allows the filter to represent absolute certainty (infinity) gracefully, making it ideal for resource-constrained embedded systems on autonomous platforms.

The foundational spatial representation that discretizes the environment into a grid of independent cells, each storing a probability of being occupied or free. The Bayesian Occupancy Filter extends this static model by adding a temporal dimension, recursively updating each cell's belief using a Binary Bayes Filter with the log-odds representation to avoid numerical instability near probabilities of 0 or 1.

• Static grids: Used in SLAM for map building
• Dynamic grids: BOF tracks moving objects via 4D cells (x, y, vx, vy)
• Inverse sensor model: Maps raw sensor returns to occupancy probabilities

A two-step probabilistic cycle that forms the mathematical core of the BOF. The prediction step propagates the occupancy distribution of each cell forward in time using a motion model, while the update step fuses new sensor observations via Bayes' rule. This recursive formulation means the filter only needs the previous state and current measurement, not the entire history.

• Markov assumption: Future state depends only on current state
• Chapman-Kolmogorov equation: Governs the prediction propagation
• Conjugate priors: Beta distributions simplify occupancy updates

A probabilistic function P(O | Z) that maps a sensor measurement Z to a likelihood of occupancy O, characterizing the sensor's detection profile. Unlike a forward model that predicts measurements from states, the inverse model directly provides the evidence needed for Bayesian updates. Key parameters include the probability of detection (Pd) and probability of false alarm (Pfa).

• LiDAR models: Gaussian beams with range-dependent uncertainty
• Radar models: Account for specularity and multi-path reflections
• Learned models: Neural networks trained on labeled point clouds

The BOF extends the 2D spatial grid into a 4D spatio-velocity space, where each cell encodes both position (x, y) and velocity (vx, vy). This allows the filter to represent and track moving objects without explicit data association. The velocity dimension is typically discretized into bins, and the full 4D distribution is updated at each time step.

• Curse of dimensionality: 4D grids demand significant memory
• Sparse representations: Only occupied velocity cells are stored
• Marginalization: Summing over velocity yields a standard occupancy grid

An alternative implementation that replaces the discrete 4D grid with a weighted particle set to combat the curse of dimensionality. Each particle represents a hypothesis of an object's position and velocity. The particle filter naturally handles non-linear motion models and non-Gaussian distributions, making it suitable for tracking highly maneuverable objects like pedestrians.

• Sequential Importance Resampling (SIR): Prevents particle degeneracy
• Adaptive particle count: Adjusts based on scene complexity
• Hybrid approaches: Combine grid cells for static, particles for dynamic

Post-processing step that groups adjacent occupied cells in the 4D BOF output into discrete object hypotheses. Connected component labeling on the velocity-averaged grid identifies individual obstacles, while the velocity distribution within each cluster provides an estimate of the object's speed and heading. This bridges the gap between the grid-level filter and higher-level tracking.

• DBSCAN: Density-based clustering robust to noise
• Cluster velocity: Weighted average of cell velocity distributions
• Track initiation: New clusters spawn object tracks for prediction