We have invented a new suite of algorithms for **2-D phase unwrapping**, based on iterative probability propagation (the sum-product algorithm). Phase unwrapping in 2-dimensional topologies is a signal processing problem that has been extensively studied over the past 20 years and has many important applications, including *medical imaging*, *radar imaging*, and *satellite imaging*.

The phase unwrapping problem is simply stated: From a 2-dimensional image of scalar values, we measure each value modulus 1. A value of 1.3 is measured as 0.3, a value of 2.3 is measured as 0.3, *etc.* (More generally, the values are measured modulus some known wavelength, but we assume the data is normalized to this wavelength.) Given these wrapped measurements, reconstruct the original image, taking into account prior information such as smoothness in the unwrapped values.

The following videos show a surface being wrapped and then unwrapped using our algorithm and the iterative correction of the wrapping errors. Notice that the wrapped surface can be viewed as a grayscale image, where a bright pixel corresponds to a wrapped value near 1 and a dark pixel corresponds toa wrapped value near 0.

*Click to see video: phase unwrapping in 35 iterations of sum-product algorithm*

*Click to see video: wrapped phase violations during 200 iterations of sum-product algorithm*

Practical applications include unwrapping MRI images, such as the MRI image of the human head shown below, and unwrapping synthetic aperture radar (SAR) topographic maps, such as the map from Sandia National Laboratories, New Mexico, shown below.

Although phase unwrapping in 1 dimension is tractable, phase unwrapping in 2 dimensions is NP-hard integer programming problem. Our conjecture is that there exist a near-optimal phase unwrapping algorithm for Gaussian process priors. We propose the graphical model and approximate inference as the sub-optimal solution.

*Unwrapped Sandia data from above*

For detailed description see our NIPS'01 paper on belief propagation for 2D phase unwrapping.