[AidanChurch]

The Discrete-Time Fourier Series is invertible. The problem is that for each pixel the Fourier series is a complex number and what is plotted is the magnitude (so the phase is not shown). Reconstructing a signal from only the magnitudes of the Fourier coefficients is ill-posed, i.e. there are multiple signals with the same Fourier coefficient magnitudes.

Still there is active research on reconstructing signals from only the Fourier magnitudes (under several assumptions). This is called the 'phase-retrieval problem'. http://en.wikipedia.org/wiki/Phase_retrieval