Symmetry properties

A measure of asymmetry in an image is given by its skewness, where here the description is a statistical measure of a distribution's degree of deviation from symmetry about the mean [9]. The third order moments (skewness and bi-correlations) will be zero if the distribution is symmetric eg. Gaussian. The degree of skewness can be determined using the two third order moments, $\mu_{30}$ and $\mu_{03}$. Prokop [12] used these moments as a basis to define the coefficients of skewness. The direction of skewness can be determined by analysing the signs of these results.

More generally, Li [8] described the basis function $x^py^q$ (in Equation 1.16), as a weighting function which extracts features of the image $f(x,y)$ concerning the symmetry in the irradiance distribution. Li used this property to show how low order $(p+q)^{th}$ normalised centralised moments (Equation 1.20) produce descriptions which are directly comparable to the existence of symmetry within the image. Here symmetry is being detected about the COM of the image, hence the use of the centralised moments. The first seven scale-normalised centralised moments ( $\eta_{11},\eta_{20},\eta_{02},\eta_{21},\eta_{12},\eta_{30},\eta_{03}$) were analysed using typed characters as binary input images. It was shown that by looking at the sign and the magnitude of the centralised moments, character recognition based on symmetry properties is possible. Here follows a summary of this work. Shapes that are either symmetric about the $x$ or $y$ axes will produce $\eta_{11}=0$. For shapes symmetrical about the $y$ axis $\eta_{12}=0$ and $\eta_{30}=0$, Figure 1.5a and Table 1.1. However for shapes symmetric about the $x$ axis, $\eta_{03}=0$ and $\eta_{12}$ is positive, Figure 1.5b and Table 1.1. Further to this the following generalities are true:

$$\eta_{pq}=0~~~~~~~~~\forall p=0,2,4..~~;~~q=1,3,5..$$ (34)

for shapes symmetric about the $x$ axis. However shapes which are asymmetrical about the $x$ axis produce:

$$\eta_{pq}<0~~\forall p=0,2,4..~~;~~q=1,3,5..$$ (35)

and:

$$\eta_{p0}>0~,~\eta_{0p}>0~~~~~~~\forall p=0,2,4..~~~~f(x,y)>0$$ (36)

In this way it can seen that the sign of the normalised centralised moments can be arranged to give qualitative information about the shape being described (i.e. the existence of symmetry), while the magnitude of the centralised moments gives a quantitative description (i.e. their size and density).

Figure 1.5: Axes of symmetry for typed characters.

$\parbox{7cm}{ \center{ \rotatebox{-90}{\scalebox{0.3}{\includegraphics{images/theory/letter_M.ps}}} } }$$\parbox{7cm}{ \center{ \rotatebox{-90}{\scalebox{0.3}{\includegraphics{images/theory/letter_C.ps}}} } }$
$\parbox{7cm}{\center{(a)}}$$\parbox{7cm}{\center{(b)}}$

Table 1.1: Typed characters $\eta _{pq}$ values indicating symmetry.

$\parbox{9.5cm}{ \begin{tabular}{\vert l\vert\vert c\vert c\vert c\ve... ...+ & - & 0 & 0 & - \\ C & 0 & + & + & 0 & + & + & 0 \\ \hline \end{tabular} }$