Higher-level structures:

The fundamental perceptual grouping proposed in [8] for higher-level structures can be modeled as the following. The proximity of two edge segments $\omega_k$ and $\omega_l$ can be modeled by the relation $\Lambda(s)\:\delta({\bf {r}}_k-{\bf {{r}}}_l - s{\bf {e}}_{{kl}})$, whereas the variation in the orientations of $\omega_k$ and $\omega_l$ can be controlled by the relation $\tilde{\Lambda}(p)\: \delta(\phi_k - \phi_l - p)$, where the variable $p = \phi_k - \phi_l$, $p \in [0,2\pi]$ and $\tilde{\Lambda}$ is a constant function (not equal to zero) with compact support (similar to $\Lambda$). The length of overlap of lines is determined by orthogonal projection, and remains invariant because, as shown in section 1.1, the action of $E(2)$ generates isometric objects. Using an argument similar to the one shown above, it can be verified these relations are invariant under the action of $E(2)$. Hence, equation 1 also remains invariant, i.e., ${\bf X}_{\cal S}$ obtained by the mapping $\psi$ is invariant after the action of $E(2)$ - invariant to orientation and position.