Pattern Recognition

$m\prime =m+{\left(a,0,\dots ,0\right)}^{\text{T}}={\left(Z_0+a,Z_1,Z_{-1},\dots ,Z_n,Z_{-n}\right)}^{\text{T}}.\left(2.106\right)$

A translation only affects the first coefficient Z0. Actually, this coefficient is nothing else than the center of mass of the contour. Hence, omitting this coefficient (or implicitly setting it to zero) describes the same contour with its center of mass moved to the origin. A translation invariant feature vector is therefore

$\tilde{m}={\left(Z_1.Z_{-1},\dots ,Z_n,Z_{-n}\right)}^{\text{T}}.\left(2.107\right)$

Scaling invariance is also easy to obtain. If the contour z(l) is scaled by a real, positive value a∈ <0 to z′(