Centralised moments

The definition of a discrete centralised moment as described by Hu[6] is: $\mu_{pq}$Two dimensional centralised moment

$$\mu_{pq} = \sum_{x=1}^{M} \sum_{y=1}^N (x-\overline{x})^{p} (y-\overline{y})^q P_{xy}$$ (19)

This is essentially a translated Cartesian moment, which means that the centralised moments are invariant under translation. To enable invariance to scale, normalised moments $\eta _{pq}$$\eta _{pq}$Two dimensional scale-normalised centralised moment are used [20], given by:

$$\eta_{pq} = \frac{\mu_{pq}}{\mu_{00}^{\gamma}}%%~~~~~~\gamma = \frac{p+q}{2} + 1 ~~~~~~\forall p+q\geq2$$ (20)

where :

$$\gamma = \frac{p+q}{2} + 1 ~~~~~~~~~~~~ \forall (p+q)\geq2$$ (21)