I have a phase retrieval problem, wherein the phase that needs to be determined lies in the range $[-2\pi, 2\pi]$.
So, if we represent a phase distribution by $\phi$, such that $\phi$ belongs in the range [-pi,pi], we have: $$|FT{\exp(i2\phi)}| = X$$
When I run the HIO (hybrid input-output) algorithm, it never gives the right results (of course), because of the ambiguity of arctangent function (am using MATLAB) in this range.
However, is it possible to recover the phase $\phi$ belonging to $[-\pi,\pi]$ from this known quantity $X$? I tried to solve this problem by trying to evaluate the quantity $$|FT{\exp(i\phi)}|$$ using the known value $X$. But could not put any mathematical relation in place that would allow this.