Likelihood Field

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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.

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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.

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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.

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10. Noise Sources and Effects

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)

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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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