Maximum smoothness consistent unwrapping of n-dimensional phase fields

The accurate recovery of an n-Dimensional field from observations that are wrapped into a particular interval (e.g., [-pi, pi]) is a problematic step in many applications, such as holography, interferometry, optical metrology, magnetic resonance imaging and analog-to-digital conversion in digital photography. While methods designed for this purpose abound (mainly in the 2-Dimensional case), most fail if the original unwrapped field contains abrupt changes that become aliased. In this paper we present Maximum Smoothness Consistent Unwrapping (MSCU), a novel and general method that overcomes said limitation. The method operates in two stages: in the first, a region without aliased changes which is as large as possible (which we call the consistent region) is found, and in the second, the unwrapped phase in the consistent region is propagated to the inconsistent areas to get the final result. MSCU has the following advantages: it is well founded theoretically, which allows the assessment of the reliability of its results; it is directly applicable to fields in any number of dimensions and when the signal is wrapped module any real number P; it has no free parameters to adjust; and it is computationally efficient and easy to implement. We present a formal derivation of the method and illustrations of its performance, both in synthetic fields -where we compare it with that of other state-of-the-art methods- and in real data (2-D data from optical speckle interferometry and 3-D data from magnetic susceptibility images obtained by magnetic resonance acquisitions). In this paper we focus on phase unwrapping applications, but the presented method may be directly applied to the case of other wrapping intervals as well, as for instance in High Dynamic Range image processing.