A Factorized Variational Technique for Phase Unwrapping in Markov Random Field

Abstract

Some types of medical and topographic imaging device produce images in which the pixel values are "phase-wrapped", i.e., measured modulus a known scalar. Phase unwrapping can be viewed as the problem of inferring the integer number of relative shifts between each and every pair of neighboring pixels, subject tO an a priori preference for smooth surfaces, and a zero curl constraint, which requires that the shifts must sum to 0 around every loop. We formulate phase unwrapping as a probabilistic inference problem in a Markov random field where the prior favors the zero curl constraint. We derive a relaxed, factorized variational method that infers approximations to the marginal probabilities of the integer shifts between pairs of neighboring pixels. The original, unwrapped image can then be obtained by integrating the integer shifts. We compare our mean field technique with the least squares method on a synthetic 100 × 100 image, and give results on a larger 512 × 512 image measured using synthetic aperature radar from Sandia National Laboratories.