The connection between the crystallographic phase problem and the feasible set approach is explored. It is argued that solving the crystallographic phase problem is formally equivalent to a feasible set problem using a statistical operator interpretable via a log-likelihood functional, projection onto the non-convex set of experimental structure factors coupled with a phase-extension constraint and mapping onto atomic positions. In no way does this disagree with or dispute any of the existing statistical relationships available in the literature; instead it expands understanding of how the algorithms work. Making this connection opens the door to the application of a number of well developed mathematical tools in functional analysis. Furthermore, a number of known results in image recovery can be exploited both to optimize existing algorithms and to develop new and improved algorithms.