Statistical interpretation of the importance of phase information in signal and image reconstruction

In the Fourier representation of signals and images, phases have long been realized to be more important than magnitudes in the reconstruction. In this paper, a justification is presented from a statistical viewpoint. The main result shows that under random magnitudes, the DC component of the inverse Fourier transform converges to a positive value, while all the other components converge to zero. For random phases, such a result does not exist.