**1 Answer**

written 20 months ago by |

**Solution:**

Lower-order moments, central moments, and normalized central moments characterize the region R and are invariant to translations and scaling of R but are variant to rotations of the foreground region R.

Invariance to the rotation of R is obtained by finding the principal angle ɸ. It is a measure of the orientation of the region R. It can be expressed in terms of second-order central moments.

$ \text { Principal angle of } R \text { is defined as } \phi=\frac{1}{2} \operatorname{atan} 2\left(2 \mu_{11}, \mu_{20}-\mu_{02}\right) $

To interpret the principal angle, consider a line L(β) drawn through the centroid of R at an the angle of β with respect to the x-axis.

If we have a coordinate plane and the initial arm of the angle is at zero degrees and then we have a terminal arm somewhere.

The moment of inertia will depend on the angle β. The angle at which the moment of inertia is minimized is the principal angle β = ɸ. The principal angle is defined for the elongated objects.