Euclidean invariance of $\xi _{kl}$:

The form of the energy functional expressed in equation 6 has a well-defined symmetry: it is invariant under the action of $E(2)$; it is invariant under translations $\{{\bf r},\phi\} \to \{{\bf r}+{\bf b}, \phi\}$, rotations $\{{\bf r}, \phi\} \to \{R_{\theta}{\bf r}, \phi + \theta\}$ and reflections $\{{\bf r}, \phi\} \to \{\acute{\kappa}R_{-2\theta}{\bf r}, -(\phi - 2\theta)\}$.

The argument $q$, $s$ and $t$ in equation 6 remain unchanged, because as shown in section 1.1 the action of $E(2)$ generates isometric objects, or it can be verified as following. The invariance of $s = \vert\vert{\bf r}_k -{\bf r}_l\vert\vert$ can be established as:

$$\begin{array}{l} \vert\vert\tau_{\bf {b}}\varrho{\bf r}_k - ... ...:\vert\vert{\bf r}_k -{\bf r}_l\vert\vert^2 = s^2 \end{array}$$ (7)

where $<,>$ denotes the dot product and $\varrho$ is either a rotation or a reflection; since, $<\tau_{\bf {b}}\varrho{\bf r}_k,\tau_{\bf {b}}\varrho{\bf r}_l>$ $=$ $<{\bf r}_k,\varrho^{-1}\tau_{\bf {b}}^{-1}\tau_{\bf {b}}\varrho{\bf r}_l>$ $=$ $<{\bf r}_k,{\bf r}_l>$, and similarly for the first and second terms in the first line of the above equation. Similarly, the invariance of $q$ and $t$ can also be established.

Translation invariance of equation 6 is evident because:

$$\begin{array}{l} \xi_{kl}(\tau_{\bf b} \cdot \omega_b \:\ver... ...= \xi_{kl}(\omega_b \:\vert\: \omega_k, \omega_l) \end{array}$$ (8)

Invariance with respect to a rotation $\theta$ follows from:

$$\begin{array}{l} \xi_{kl}(R_{\theta} \cdot \omega_b \:\vert\... ...= \xi_{kl}(\omega_b \:\vert\: \omega_k, \omega_l) \end{array}$$ (9)

and invariance under a reflection $\kappa$ about the an axis holds since:

$$\begin{array}{l} \xi_{kl}(\kappa \cdot \omega_b \:\vert\: \k... ...= \xi_{kl}(\omega_b \:\vert\: \omega_k, \omega_l) \end{array}$$ (10)

It must be noted that the energy functional given in equation 6 incorporates the Gestalt principles of proximity, collinearity, parallelism, and good continuation. Equation 6 is at the heart of the perceptual grouping process. Its Euclidean invariance, as shown above, means that equation 5 remains invariant, and the perceptual grouping process will produce the same groupings - longer linear lines. All higher-level structures are extracted using these longer linear lines.