shift invariance

[Shlomo Geva]

Regarding shift invariance: 

One might learn something about the problem by looking at the Fourier Transform
in the context of shift invariance.

One may perform a Discrete Fourier Transform (DFT), and take advantage
of the shift invariance properties of the magnitudes components,
discarding the phase and representing objects by a feature vector
consisting of the the magnitudes in frequency domain alone.
(This is not new and also extends to higher dimensionalities)

This approach will solve many practical problems, but has an in-principle 
difficulty in that this procedure does not produce a unique mapping from
objects to invariant features.
For example, start from any object and obtain its invariant representation
as above. By choosing arbitrary phase components and performing an inverse 
DFT we can get arbitrarily many object representations.
Note that these objects are extremely unlikely to look like an original
shifted object!

If by chance - and it may be very remote - two of
the objects you wish to recognize with shift invariance, have identical
magnitudes in the frequency domain then this method will obviously fail.

Now I'd like to make a conjecture.
It appears to make sense to assume that this difficulty is inherent
to the shift invariance requirement itself. If this is so 
then unless you have an additional constraint imposed on objects -
they cannot be allowed to be identical under the invariant feature
extraction transformation you wish to employ -
then you cannot solve the problem. In other words, one needs a guarantee that
all permissible objects are uniquely transformed by the procedure.
It seems to follow that
no general procedure, that does not take into account the nature of the objects
for which the procedure is intended, can exist.

I am wondering if anyone could clarify whether this is a valid argument.

Shlomo Geva

s.geva at qut.edu.au