Image features & supervised learning - removing meteoradar noise

I found interesting article about meteodata processing with neural network :

http://blog.forecast.io/cleaning-radar-images-using-neural-nets-computer-vision/

How those guys can filter noise ? I know many neural network samples where computer recognizes object e.g. OCR, shape recognition, face recognition... but i don't know how to implement noise removal - neural network have to mark noisy area somehow and that's tricky - sometimes you do have cloud and noise mixed together.

How to implement such thing, do you know any helpful articles/link on this topic ?

Sample image (noisy / without noise) : http://blog.forecast.io/wp-content/uploads/2013/05/conus.gif