Invariant–Covariant

A function is invariant under a certain family of transformations if its value does not change when a transformation from this family is applied to its argument. A function is covariant when it commutes with the transformation, i.e., applying the transformation to the argument of the function has the same effect as applying the transformation to the output of the function. A few examples may help to explain the difference. The area of a 2D surface is invariant under 2D rotations, since rotating a 2D surface does not make it any smaller or bigger. But the orientation of the major axis of inertia of the surface is covariant under the same family of transformations, since rotating a 2D surface will affect the orientation of its major axis in exactly the same way. Based on these definitions, it is clear that the so-called local scale and/or affine invariant features are in fact only covariant. The descriptors derived from them, on the other hand, are usually invariant due to a...