Statistical Mechanical Analysis of Phase Unwrapping -One-Dimensional Model

We study a phase unwrapping model in the one-dimensional space on the basis of Bayesian inference using the maximizer of posterior marginals (MPM) estimate by the statistical mechanical methods. We propose a model in which the recursion relations to obtain statistical quantities such as MPM estimates are derived. We introduce the three state Potts model to handle the discontinuities in observed data, and propose two methods, the step and direct methods. We derive the recursion relations for MPM estimates of hyperparameters and phase differences in both methods, and investigate the random and regular phase differences, and previously studied other type of random phase differences. We find that the phase differences are inferred fairly well in rather wide ranges of noise amplitudes. The ranges depend on samples and the system sizes. Furthermore, we find that the step method has performance in phase unwrapping comparable to the direct method, and that it is much faster in numerical computation and applicable to much larger system sizes than the direct method.