Affine phase retrieval of quaternion signals

Quaternion algebra $ \mathbb {H} $ H is an extension of the complex number field, which is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has interested some mathematicians due to its applications in signal analysis and image processing. This paper addresses quaternionic affine phase retrieval (QAPR) in quaternion Euclidean spaces $ \mathbb {H}<^>{M} $ HM, which aims to exactly recover a signal in $ \mathbb {H}<^>{M} $ HM from the magnitudes of its affine measurements. We introduce the concepts of QAPR and phaselift operator in $ \mathbb {H}<^>{M} $ HM. Then, we characterize QAPR in terms of the real Jacobian matrix, prove that 5M is the minimal measurement number for QAPR in $ \mathbb {H}<^>{M} $ HM, study the stability of QAPR-sequences, and use the phaselift techniques to give some sufficient conditions on QAPR for $ \mathbb {H}<^>{M} $ HM which provide us with a method to construct QAPR-sequences.