"Codes" on images and iterative phase unwrapping

Many imaging techniques, including magnetic resonance imaging and interferometric synthetic aperture radar, produce "phase-wrapped" images. In a phase-wrapped image, the original image values are measured modulus a known wavelength, A. The goal of phase unwrapping is to produce an estimate of the original image using an a priori preference for smooth images. We formulate phase unwrapping as the problem of computing a vector field that is an estimate of the gradient field of the original image. A preference for smooth images is obtained using a Gaussian prior on the vector field. For a vector field to be a gradient field, it must satisfy the constraint that the sum of the vectors around every closed loop is zero. We enforce this constraint using "zero-curl checks" in a factor graph on the vector field. The sum-product algorithm in this factor graph is used to approximately compute the posterior probabilities of the vectors. Hard decisions are used to produce a vector field, which is integrated to obtain the unwrapped image. Experimental results show that this method can work significantly better than existing techniques for phase unwrapping. Although phase unwrapping for general image priors is NP-hard, we conjecture that the sum-product algorithm in an appropriate factor graph will lead to a near-optimal unwrapping algorithm for Gaussian process sources.