Uniqueness of phase retrieval from three measurements

In this paper we consider the question of finding an as small as possible family of operators \((T_j)_{j\in J}\) on \(L^2(R)\) that does phase retrieval: every \(\varphi\) is uniquely determined (up to a constant phase factor) by the phaseless data \((|T_j\varphi|)_{j\in J}\). This problem arises in various fields of applied sciences where usually the operators obey further restrictions. Of particular interest here are so-called {\em coded diffraction paterns} where the operators are of the form \(T_j\varphi=\mathcal{F}m_j\varphi\), \(\mathcal{F}\) the Fourier transform and \(m_j\in L^\infty(R)\) are "masks". Here we explicitely construct three real-valued masks \(m_1,m_2,m_3\in L^\infty(R)\) so that the associated coded diffraction patterns do phase retrieval. This implies that the three self-adjoint operators \(T_j\varphi=\mathcal{F}[m_j\mathcal{F}^{-1}\varphi]\) also do phase retrieval. The proof uses complex analysis.We then show that some natural analogues of these operators in the finite dimensional setting do not always lead to the same uniqueness result due to an undersampling effect.