  A non-iterative method for the construction of the Short-Time Fourier
Transform (STFT) phase from the magnitude is presented. The method is based on
the direct relationship between the partial derivatives of the phase and the
logarithm of the magnitude of the un-sampled STFT with respect to the Gaussian
window. Although the theory holds in the continuous setting only, the
experiments show that the algorithm performs well even in the discretized
setting (Discrete Gabor transform) with low redundancy using the sampled
Gaussian window, the truncated Gaussian window and even other compactly
supported windows like the Hann window.
  Due to the non-iterative nature, the algorithm is very fast and it is
suitable for long audio signals. Moreover, solutions of iterative phase
reconstruction algorithms can be improved considerably by initializing them
with the phase estimate provided by the present algorithm.
  We present an extensive comparison with the state-of-the-art algorithms in a
reproducible manner.
