Papoulis theorem states that if f(x,y) is piecewise continuous and has non-zero values in only a finite part of the x-y plane, then moments of all order exist and the moment sequence is uniquely determined by f(x,y), and conversely moment sequence uniquely determines f(x,y).
https://www.researchgate.net/publication/216861380_Behavior_of_Normalized_Moments_under_Distortion_and_Optimization?ev=prf_pub
Article Behavior of Normalized Moments under Distortion and Optimization
I've seen MRFs constructed to mimic specified (local) moments. The process starts with a very diffuse probability model (maximum entropy), and constraints are added until the model's behaviour / statistics match the observed phenomenon behaviour / statistics. See for example - Zhu, Song Chun, Yingnian Wu, and David Mumford. "Filters, random fields and maximum entropy (FRAME): Towards a unified theory for texture modeling." Spoiler alert: the authors generate images from a probability model by constraining the statistics (aka moments) of their probability model (MRF) to statistics observed on the training images for various fabrics.
The next two videos don't directly answer your question, but demonstrate a complicated function (video) reconstructed from relatively few input signals. Check out "image reconstruction from brain activity", http://www.youtube.com/watch?v=FR0qJ17Rsvc or http://www.youtube.com/watch?v=daY7uO0eftA .
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