Having sub-sampled the WHT random projection of an image or similar natural data you can use an iterative restoration method based on binomial filtering. You apply the inverse random projection to the data, smooth it with the filter, project back into the random domain and then correct only the sub-sample, leaving the other data unchanged. Repeat the process multiple times.
Or put another way:
A natural image for example has some randomness but also has a lot of embedded order. For example a lot of sine and cosine embedded waves. And those can be extracted to do things like jpeg compression.
You can go the other way though and randomly flip the signs of the data in the natural image. If you go looking for embedded sine and cosine waves you won’t find any. Instead you will find Gaussian noise from the Normal random distribution.
Are those random numbers useless? No, on the contrary each single one of them contains its fair share of information about all the natural image. And you only need pick a few of them, it doesn’t matter which, do some inversion and smoothing a few times and you get some reasonable representation of the original image back.