Phases of complex functions from the amplitudes of the functions and the amplitudes of the Fourier and Mellin transforms

For the most part, the phase-retrieval problem has been dominated by the use of the Hilbert transform on the logarithm of the absolute value of the wave function. In our approach, by contrast, we use the intensity associated with the function and the intensities in Fourier- or Mellin-transform space corresponding to a set of apertures to obtain the phases within a constant by means of simple formulas. The set of apertures required is those arbitrarily close to a fixed aperture and its complement. The higher-dimensional cases are also treated for the case of the Fourier transform.