Lie group transformations of objects in images

If an image of an object such as an aeroplane is rotated in the plane of the image, and if some feature space representation is chosen which is a projection onto some set of smooth functions, as in the calculation of moments, wavelets or Fourier coefficients, then each angle defines a point in the feature space. Providing the object is not symmetric under rotation, this defines a one-one map (an injection) of the circle group of rotations, SO(2, R) into the feature space. The injection is smooth (infinitely differentiable). Corresponding results hold if the object is rotated in three dimensions, or shifted in space and viewed from a fixed position: we obtain a smooth injection of SO(3, R) or the translation group.