Reconstruction of Audio Signal from its Absolute Spectrogram

I have the absolute Spectrogram of an audio signals.

I lost the phase data of the Spectogram because of various processing applied on the original spectrogram of the signal.

I'm trying to reconstruct the audio signal in a meaningful (Audibly) manner from teh absolute value only of the Spectrogram.
The obvious inverse won't work (The DFT inverse of the absolute, Since the Phase is significant).

The Spectrogram is a result of fusion of few audio signals as I'm trying to create a smooth transition between audio signals.

Anyone has experience with the problem? Anyone has experience with this procedure? Could anyone refer me to a code, article, etc...

Thanks.

Answer

One thing commonly done (for example in the source separation community) is to use the phase data of the original signal (before transformation where applied to it) - the result is much better than null or random phase, and not so far from algorithms aiming at reconstructing the phase information from scratch.

A classic reconstruction algorithm is Griffin&Lim's, described in the paper "Signal estimation from modified short-time Fourier transform". This is an iterative algorithm, each iteration requires a full STFT / inverse STFT, which makes it quite costly.

This problem is indeed an active area of research, a search for STFT + reconstruction + magnitude will yield plenty of papers aiming at improving on Griffin&Lim in terms of signal quality and/or computational efficiency.