Analytic Phase Retrieval Based on Intensity Measurements

Abstract

This paper concerns the reconstruction of a function f in the Hardy space of the unit disc $\mathbb{D}$ by using a sample value f (a) and certain n-intensity measurements $\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|$, where a₁, ⋯, ${a_n} \in \mathbb{D}$, and ${E_{{a_1} \cdots {a_n}}}$ is the n-th term of the Gram-Schmidt orthogonalization of the Szegö kernels ${k_{{a_1}}}, \cdots ,{k_{{a_n}}}$, or their multiple forms. Three schemes are presented. The first two schemes each directly obtain all the function values f (z). In the first one we use Nevanlinna’s inner and outer function factorization which merely requires the 1-intensity measurements equivalent to know the modulus |f (z)|. In the second scheme we do not use deep complex analysis, but require some 2- and 3-intensity measurements. The third scheme, as an application of AFD, gives sparse representation of f (z) converging quickly in the energy sense, depending on consecutively selected maximal n-intensity measurements $\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|$.