Action of the Euclidean group - Action by translation, rotation, and reflection

It is well-known that the group of all isometries of $\Re^2$ is the Euclidean group. To see this, let $\Gamma$ be an isometry of $\Re^2$, and let ${\bf b} = \Gamma(0)$. Then $\varrho = \tau_{-{\bf b}}\Gamma$ is an isometry of $\Re^2$, satisfying $\varrho(0) = 0$. It can be shown that if $\varrho(0) = 0$, then $\varrho$ is linear [9], and thus, $\Gamma = \tau^{-1}_{-{\bf b}}\varrho = \tau_{\bf b}\varrho$ is a product of a linear isometry and a translation. Further, it can also be shown that the linear isometries are represented by the orthogonal group $O(2, \Re)$ of $2 \times 2$ orthogonal matrices that represent reflections and rotations. Hence, the product of the translation group and the orthogonal group is the group of isometries of $\Re^2$ (called Euclidean group $E(2)$). The normality of the translation group in $E(2)$ can used to deduce that $E(2) \cong O(2, \Re) \bowtie \Re^2$, where $\bowtie$ denotes semi direct product.

The rest of the paper is organized as follows: section 2 explains the perceptual grouping process to extract structure and the color histogram as representations of isotropic mapping, section 3 describes the texture analysis via a channel energy model as a representation of anisotropic mapping, section 4 outlines the integration of isotropic and anisotropic mappings, section 5 describes the results obtained, and finally, section 6 provides the conclusions.