Discrete Fourier Transform. Symmetry question

I use this example to frame the question that follows, with the sublists
representing matrix rows ...
img={{4, 7, 1, 4}, {6, 0, 7, 5}, {7, 1, 2, 2}, {4, 0, 1, 7}}
fftImg={{14.5, 2.5-2.5I, 1.5, 2.5+2.5I}, {1+1.5I, -1, -3+2.5I, -2I}, {-0.5,
1.5+3.5I, -1.5, 1.5-3.5I}, {1-1.5I, 2I, -3-2.5I, -1}}

Could someone please tell me what the symmetry constraints are on a 2-D
Fourier transform given a real-valued input. Put another way, if I
*construct* the 2-D transform, what symmetry relationships must I enforce to
guarantee that the inverse transform is real . It's clear enough to me in
the 1-D case (Conjugate[x_k] == x_{n-k}). From looking at the example, that
symmetry condition seems to hold on each row of the DFT as well. It does
not, however, appear to be a sufficient condition to produce a real inverse
of the fftImg (or have I messed something up)? Is there some sort of similar
constraint on the *columns* that is not obvious to me? And lastly, while its
clear what the value 14.5 represents, it isn't clear to me what the first
value in each row indicates?

While it would be nice if you're willing to answer the questions, a pointer
to a suitable text might substitute!

Many thanks.



Mark Diamond





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