Generalized phase retrieval in quaternion Euclidean spaces

Recently, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses quaternionic generalized phase retrieval (QGPR) problem in quaternion Euclidean spaces ℍM$$ {\mathrm{\mathbb{H}}}^M $$. We introduce the concept of QGPR which aims to reconstruct a signal f$$ f $$ in ℍM$$ {\mathrm{\mathbb{H}}}^M $$ from the quadratic measurements {fFlf∗}l=0N−1$$ {\left\{f{F}_l{f}^{\ast}\right\}}_{l=0}^{N-1} $$, where each Fl$$ {F}_l $$ is an M×M$$ M\times M $$ self‐adjoint quaternion matrix. We characterize QGPR sequences in terms of their real Jacobian matrices, prove that the set of QGPR sequences is an open set in some sense, and present some phaselift‐based sufficient conditions on QGPR which gives a method to construct QGPR sequences.