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## Invariant Measures

Several of the statistical measures we have met are invariant measures, which is to say that the value of the measure does not vary with, for example, the position of the region in the image, or perhaps its orientation or scale.

Thus, while the centre of gravity of a region obviously varies with its position, its area does not.

While area does vary with scale (and thus closeness of the camera to the object, for example), compactness  as defined above is invariant with respect to scale as well as position and orientation. Invariant measures can be quite useful in recognising objects.

Other invariant measures we have met before are:

• Moments of inertia (Eqn. 55) are invariant with respect to scaling, rotation and translation.
• Eccentricity is invariant with respect to scaling, rotation and translation.
• Fourier Transforms are rotation invariant. We can use the Fourier transform to compute Fourier descriptors of an object which are invariant with respect to position and orientation.

David Marshall 1994-1997