Reconstruction ( inverse mapping ) of the spatial image via Fourier transform require knowledge of phase in addition to amplitude of the warped image.
7.
Since it is one-one, the inverse mapping of " f " is holomorphic from the image onto the Riemann surface.
8.
The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of structure, is often the inverse mapping of the geometric Frobenius.
9.
The methods solves for a smooth time indexed vector field such that flows along the field which start at the data points will end at a lower-dimensional linear subspace, thereby attempting to preserve pairwise differences under both the forward and inverse mapping.