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In article <1992Sep7.110102.25536@bnr.co.uk> pgh@bnr.co.uk (Peter Hamer) writes:

   In the referenced article spz@specklec.mpifr-bonn.mpg.de (S.P.Zeidler) writes:
   >In <9209011949.AA19843@hp-and.an.hp.com> susanm@hp-and.an.hp.com (Susan Margulies) writes:
   >One thing that you might want to take into account is that the Fourier
   >transform of an ordinary image is hermitic, ie if you do a transform
   >AxB real image -> AxB complex Fourier transform, you got quite some
   >redundancies.

To be precise, half of your transform is redundant. If you pack things
carefully, you can even transform an N x M real array into an N/2 x M
complex array. If you are less aggressive in packing and waste a
little space, you can trivially get an (N+2)/2 x (M+2) complex output
array for your N x M real input.

   Serious question: so why not use the Hartley transform which does not
   introduce these redundancies in the first place?

I have tried both. It's relatively easy not to generate the half of
the Fourier transform that you don't need, and the result is the
familar and easy to use complex representation. It's messy and error
prone trying to rewrite the simple formulas (e.g., for filtering) in
the Fourier domain for the Hartley transform, and, as far as I can
tell, the result isn't any faster (if anything, it's likely to be
slower because you need more array indexing).

   >Doing instead AxB image -> -(A/2+1)xB phase and (A/2+1)xB modulus
   >will give you a nicely viewable (and displayable) Fourier transform.

   ... or use your favourite Fourier package and display the Hartley transform?

If you look at the Hartley transform of an image, you are looking at
both phase and magnitude information, interwoven in some visually
complicated way. The nice thing about the Fourier transform is the
separation into phase and magnitude.

   Regards, Peter

   PS The Hartley transform can be regarded as a `tidier' Fourier transform.

      a) Uses cos(x)+sin(x) rather than cos(x)+i*sin(x).
      b) The inverse is the same operation as the transform.
      c) Takes real => real.
      d) Does not introduce redundancies into the representation.
      e) Even clearer relationship to the sine and cosine transforms.

It looks neat on paper, but I don't think it's tidier in practice.

      d) Easy to convert between Hartley and Fourier transforms.

If you are going to compute the FHT and then reconstitute Fourier
coefficients (or phase/magnitude information) from the result, you are
probably better off using the Fourier transform in the first place.

					Thomas.