Image Moment - Rotation Invariant Moments

Rotation Invariant Moments

It is possible to calculate moments which are invariant under translation, changes in scale, and also rotation. Most frequently used are the Hu set of invariant moments :

\begin{align} I_1 =\ & \eta_{20} + \eta_{02} \\ I_2 =\ & (\eta_{20} - \eta_{02})^2 + (4\eta_{11})^2 \\ I_3 =\ & (\eta_{30} - 3\eta_{12})^2 + (3\eta_{21} - \eta_{03})^2 \\ I_4 =\ & (\eta_{30} + \eta_{12})^2 + (\eta_{21} + \eta_{03})^2 \\ I_5 =\ & (\eta_{30} - 3\eta_{12}) (\eta_{30} + \eta_{12}) + \\ \ & (3\eta_{21} - \eta_{03}) (\eta_{21} + \eta_{03}) \\ I_6 =\ & (\eta_{20} - \eta_{02}) + 4\eta_{11}(\eta_{30} + \eta_{12})(\eta_{21} + \eta_{03}) \\ I_7 =\ & (3\eta_{21} - \eta_{03})(\eta_{30} + \eta_{12}) - \\ \ & (\eta_{30} - 3\eta_{12})(\eta_{21} + \eta_{03}). \end{align}

The first one, I1, is analogous to the moment of inertia around the image's centroid, where the pixels' intensities are analogous to physical density. The last one, I7, is skew invariant, which enables it to distinguish mirror images of otherwise identical images.

A general theory on deriving complete and independent sets of rotation invariant moments was proposed by J. Flusser and T. Suk. They showed that the traditional Hu's invariant set is not independent nor complete. I3 is not very useful as it is dependent on the others. In the original Hu's set there is a missing third order independent moment invariant:

\begin{align} I_8 =\ & \eta_{11} - (\eta_{20}-\eta_{02}) (\eta_{30}+\eta_{12}) (\eta_{03}+\eta_{21}) \end{align}

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