# Likelihood Field Model

- tags
- Map Matching

## Key Idea

Project an individual sensor measurement \(z_t^k\) into the global coordinate frame of map \(m\). Discards max-range readings.

Assumes three types of noise, similar to Range Finder Model:

- Measurement noise: Gaussian
- Failures: point-mass distribution at \(z_{\text{max}}\)
- Random measurements: Uniform distribution \(p_{\text{rand}}\)

The model is a mixture of these 3 densities:

\begin{equation} z_{\mathrm{hit}} \cdot p_{\mathrm{hit}}+z_{\mathrm{rand}} \cdot p_{\mathrm{rand}}+z_{\mathrm{max}} \cdot p_{\mathrm{max}} \end{equation}

## Issues

- Does not explicitly model dynamic objects that cause short readings
- Treats sensors as being able to see through walls: ray casting replaced by nearest neighbour function: incapable of determining whether a path to a point is intercepted by an obstacle in the map
- Does not account for map uncertainty

These issues can be addressed via extensions to the algorithm.