Euclidean isotropy of ${\bf X}_{\cal S}$

Let $\omega = \{\omega_i\}$ represent the collection of objects of interest present in an image, where each object $\omega _i$, is a collection of $\omega_{i_k} = \{{\bf r}, \phi\} \in \Re^2 \times {\cal S}^1$, where ${\bf r} = \{x,y\} \in \Re^2$ is a coordinate pair, ${\cal S}^1$ is the unit circle, $\phi \in {\cal S}^1$ represents the orientation of $\omega_{i}$. At the lowest level of vision $\omega_{i_k}$'s, are represented by points on an edge segment $\omega _i$ (where $\omega _i$ is obtained by using Burns' edge detector [15]). At the next level of perceptual grouping, certain $\omega_{i}$ will be combined to generate a higher-level structure. Such a structure obtained from the grouping of $\omega_{i}$'s may be called $\omega_j$ for consistency of notation, although it should be understood that $\omega_j$ now represents a structure at a higher-level than $\omega _i$. (Refer to figure 2.)

We have defined a mapping $\psi: \omega \to \Re^d$, (where $d$ is the dimensionality of the feature space), to be isotropic if it is invariant to the action of the Euclidean group:

$$\psi(E \cdot \omega) = \psi(\omega)$$ (2)

where $E$ is the Euclidean group $E(2)$ - the semi-direct product of the group of linear isometries and the translation group - such that:

$$E \cdot\omega = \{E_j \cdot \omega_i \:\vert\: E_j \in E, \omega_i \in \omega\}$$ (3)

The extraction of the feature vector, ${\bf X}_{\cal S}$, is represented by $\psi$. The action of the Euclidean group on $\omega _i$ transforms each $\omega_{i_k} \in \omega_i$ and is given as (refer to figure 3):

$$\begin{array}{lll} \tau_{\bf b} \cdot ({\bf r}, \phi) & = & ... ...e{\kappa}R_{-2\theta}{\bf r}, -(\phi - 2\theta))\\ \end{array}$$ (4)

where $\tau_{b} \in T(2)$, ($b \in \Re^2$), represents a member of the translation group of $\Re^2$, $T(2)$, such that $\tau_{\bf b}({\bf r}) = {\bf r} + {\bf b}, {\bf r} \in \Re^2$, $R_{\theta} \in O(2,\Re)$ is a rotation by an angle $\theta$, $\kappa$ is a reflection along an axis in $\Re^2$, and $\acute{\kappa}$ is the reflection along the $x$-axis, $(x,y) \to (x,-y)$. The action $\kappa \cdot ({\bf r}, \phi) = R_{\theta}\acute{\kappa}R_{-\theta}\cdot({\bf r}, \phi) = \acute{\kappa}R_{-2\theta}\cdot({\bf r}, \phi)$ (by using the identity $R_{\theta}\acute{\kappa} = \acute{\kappa}R_{-\theta}$), because reflection along an arbitrary axis is equivalent to rotation of $\Re^2$ by an angle $-\theta$ to align the axis of reflection with the original $x$-axis, followed by a reflection in the $x$-axis, and then rotation by an angle $\theta$.

Figure 2: $\omega _3$, $\omega _4$, $\omega _5$ and $\omega _6$ combined to form $\omega _7$.

Figure 3: Action of $E(2)$ on an edge segment, $\omega _i$. ${\bf r}_k$ and ${\bf r}_l$ represent the end-points of $\omega _i$.