From Visibility to Modulation Profile and Back Again

Given the visibilities (4), it is trivial to find the modulation profile (2): just take the real part of (4) and add the D.C. (flux) term a₀. But reversing the process is more difficult, because at any given angle $\phi$, the imaginary part might be of arbitrary size, and the real part of gives no immediate information about it. This problem is directly analogous to the problem of determining the length of a one-dimensional, rotating object from images of its shadow. At any given instant, the shadow does not give enough information to determine the length. But since the length is constant, shadows at different times can provide the length (amplitude) and orientation angle (phase) of the object. This provides the solution to our ``phase problem''.