On the de-noising of time-frequency images
I don't have experience in this area but I see that it has been studied: Minimum entropy approach to denoising time-frequency distributions


  In this paper, we introduce an entropy based approach to denoising time-frequency distributions. This new approach uses the spectrogram decomposition of time-frequency kernels proposed by Cunningham and Williams. In order to denoise the time-frequency distribution, we combine those spectrograms with smallest entropy values, thus ensuring that each spectrogram is well concentrated on the time-frequency plane and contains as little noise as possible. Renyi entropy is used as the measure to quantify the complexity of each spectrogram. The threshold for the number of spectrograms to combine is chosen adaptively based on the tradeoff between entropy and variance.


Essentially your problem is one of signal/source separation; the additive unmixing of a bunch of structured signals. In order to proceed you need to model your signals. Obviously the one of interest is periodic and is centered about some frequency, so you need to estimate the period (along the x axis) and the center frequency (on the y axis). Then you can characterize the others (noise). For starters, it seems that they come in nice curves.

With a model in hand I would consult a book like Handbook of Blind Source Separation: Independent Component Analysis and Applications.