What are the known conditions for the log of the Fourier transform of a 2D real discrete signal to have no branch cuts? Suppose we sample a real 2D signal, f(x,y), at N by N evenly spaced points in x and y. Then we compute the Fourier transform of the sampled signal, F(u,v), and then take the log of F. There will be a branch cut in log for each point where the magnitude of F(u,v) goes to zero. Are there conditions we can impose on f such that the magnitude of F is never 0 and hence the log(F) has no branch cuts? 
Experimentation shows that there are almost always branch cuts for most real signals. However if the signal f(x,y) is pre-multiplied by a Gaussian function centered at (N/2,N/2) and tapering to near zero at the edges the input array these branch cuts are moved to the higher frequency portion of F(u,v) and eventually disappear. So the main question is: is there an analytic condition on the inputs that prevents branch cuts on Log(F(u,v))?
 A: This turns out to be too long for a comment.
I am not sure if one can say much in the discrete two-dimensional case, but something is known in one-dimension for continuous signals:
There is a theorem of Logan that a signal with Fourier-transform supported in one octave (i.e. in some $[-2B,-B]\cup[B,2B]$) is characterized (up to a multiplicative constant) by its zero-crossings if the signal has no zeros in common with its Hilbert transform. The paper is

B.F. Logan, Information in the Zero Crossings of Bandpass Signals, The Bell System Technical Journal, 56(4), 1977

People worked on generalizations to higher dimensions, but up to my knowledge no usable results have been found. There is this work on the higher dimensional case

Susan R. Curtis and Alan V. Oppenheim, Reconstruction of multidimensional signals from zero crossings, Journal of the Optical Society of America A Vol. 4, Issue 1, pp. 221-231 (1987) here

and also the PhD thesis of S.R. Curtis on the same topic is available (pdf link).
