Ultracold atoms have become a major platform for the simulation of physical phenomena in a controlled setting. Single-shot images are the standard readout of experiments with ultracold atoms. An efficient extraction of observables from them is thus crucial to gather information on these systems. In our work, we demonstrate how artificial neural networks can optimize this extraction to accurately regress atomic densities, correlations, and their local potential landscape from a drastically reduced number of single-shot images compared to standard averaging approaches. Strikingly, neural networks also allow the regression of momentum-space observables from real-space single-shot images and vice versa.