Map-dependent energy

Qualitative aspect

This energy conveys two constraints related to :

• the distance between the nodes,
• the angle formed by three successive nodes.

Expression

x_i : node in the network (random variable),
x_i^c: associated node on the map.
$\alpha_{i,j,k}$: the angle formed by three consecutive nodes x_i,x_j and x_k.
$\alpha^c_{i,j,k}$: the angle formed by the same nodes in the map.

$$U_{c} (x) = \sum_{d(x_i,x_j)=1}U^1_{c} (x_i,x_j) + \sum_{\scr... ...{c}d(x_i,x_j)=1\\ d(x_j,x_k)=1\end{array}}U^2_{c} (x_i,x_j,x_k)$$

With :

U¹_c (x_i,x_j) = f_i,j(dist(x_i,x_j))

$$U^2_{c} (x_i,x_j,x_k) = g_{i,j,k}(\alpha_{i,j,k})$$

• $f_{i,j}(\lambda)$, convex function defined on $\mathbb{R} ^{+}$, taking its minimum value at the point $\lambda= dist(x^c_i,x^c_j)$
• $g_{i,j,k}(\alpha_{i,j,k})$ takes its minimum value for $\alpha^c_{i,j,k}$ , and increases with the difference between $\alpha_{i,j,k}$ and $\alpha^c_{i,j,k}$ .

These energy terms :

• are invariant through a global translation or rotation of the network x,
• are based on the hypothesis of reliability of the relative position of the nodes on the map.

Illustration

• Potential U¹_c : the road given by the map is in dashed line, so the road 1 has a potential U¹_c inferior to the one of road 2.
  

• Potential U²_c : the potential associated to the series of three nodes in position related to road 1 is lower to the one associated to road 2.