> ## Documentation Index
> Fetch the complete documentation index at: https://aegean.ai/llms.txt
> Use this file to discover all available pages before exploring further.

# Likelihood Field

> The likelihood field model for range sensors.

## The Likelihood Field Model

Instead of casting a ray for each beam during localization, the model:

* Precomputes the distance transform $d(\mathbf p)$ to the nearest occupied cell.
* Computes the likelihood as:

$$
p(z_t^k \mid \mathbf x_t, m) = \eta \exp\!\left(-\frac{(d(\mathbf e_k))^2}{2\sigma_d^2}\right),
$$

where $\mathbf e_k$ is beam endpoint predicted without obstacle truncation.

This method is faster, but it models short and maximum-range measurements in less detail.

## Particle Filter Localization

For particle $i$ with pose $\mathbf x_t^{[i]}$, the weight is:

$$
w_t^{[i]} \propto w_{t-1}^{[i]} \prod_{k \in \mathcal K} p(z_t^k \mid \mathbf x_t^{[i]}, m),
$$

You can subsample the beams in $\mathcal K$ to reduce computation. Compute the sum in log space for numerical stability.

## 9. Correlation and Beam Selection

Adjacent beams are correlated because they observe the same surfaces. The independence assumption is therefore optimistic. You can reduce this effect by:

* Subsample (e.g., every 4th beam).
* Clamp each per-beam likelihood to a lower bound.
* Use adaptive beam selection near discontinuities.

## 10. Noise Sources and Effects

| Source | Effect on Distribution |
| - | - |
| Surface reflectivity (sonar) | Increases short and random components |
| Transparent / specular surfaces | Increases random and maximum-range components |
| Multi-path | Broadens $p_{\text{hit}}$ |
| Timing quantization | Produces slight discretization at small ranges |
| Motion distortion (lidar while moving) | Systematic curvature in scans |

Typical compensations include deskewing with IMU or odometry data and calibrating the mixture weights.

**Key references**: (Lukežič et al., 2016; Song et al., 2016; Kendall et al., 2015)

## References

* Kendall, A., Grimes, M., Cipolla, R. (2015). *PoseNet: A Convolutional Network for Real-Time 6-DOF Camera Relocalization*. arXiv \[cs.CV].
* Lukežič, A., Vojíř, T., Čehovin, L., Matas, J., Kristan, M. (2016). *Discriminative correlation filter with channel and Spatial Reliability*. arXiv \[cs.CV].
* Song, J., Choi, J., Love, D. (2016). *Common Codebook Millimeter Wave Beam Design: Designing Beams for Both Sounding and Communication with Uniform Planar Arrays*. arXiv \[cs.IT].

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