On PDE-based spectrogram image restoration. Application to wolf chorus noise reduction and comparison with other algorithms

We investigate the use of image processing techniques based on partial differential equations applied to the image produced by time-frequency representations of one-dimensional signals, such as the spectrogram. Specifically, we use the PDE model introduced by ´ Alvarez, Lions and Morel for noise smoothing and edge enhancement, which we show to be stable under signal and window perturbations in the spectrogram image. We demonstrate by numerical examples that the corresponding numerical algorithm applied to the spectrogram of a noisy signal reduces the noise and produce an enhancement of the instantaneous frequency lines, allowing to track these lines more accurately than with the original spectrogram. We apply this technique both to synthetic signals and to wolves chorus field recorded signals, which was the original motivation of this work. Finally, we compare our results with some classical signal denoising algorithms and with wavelet based image denoising methods and give some objective measures of the performance of our method. We emphasize that the 1D signal restoration is not the purpose of our work but the spectrogram noise reduction for later instantaneous frequency estimation.