Least-squares phase estimation from the phase difference

The purpose of this paper is to clarify the relationship between the isolated zero points in modulus distribution and the least-squares phase estimation from the phase difference. The concepts of phase and phase difference are reaffirmed. In addition, a necessary condition that makes the least-squares phase estimation feasible is derived by applying the concept of complete observability in estimation theory to the measurement of the phase difference. The occurrence of isolated zero points causes the conventional least-squares phase estimation to fail because the phase difference defined by this concept does not satisfy the necessary condition when isolated zero points occur. This condition also generates a new type of least-squares phase estimation that is feasible for phase retrieval even if zero points exist. One algorithm for realizing this new type of least-squares phase estimation is proposed, and its effectiveness is verified by using computer simulations. Two types of phase unwrapping are also presented: one is the exponential function type; the other results from the proposed least-squares phase-estimation algorithm.