L1-Norm Redundant Delaunay Phase Unwrapping and Gradient Correction

This article deals with arrays of real numbers which have been reduced modulo 2h into theinterval [−h, h] where h > 0 is a positive real number. Such an array is said to be wrappedmodulo 2h. Often, the elements of these arrays correspond to values observed at points in animage-like 2D space which are connected by a graph structure. The process of retrieving theoriginal array from which the wrapped image originates is called unwrapping. Of course, thewrapping process is not one-to-one, and the quality of the recovered unwrapped version dependson the smoothness of the original array. The goal of unwrapping is to define a most plausibleleft inverse (as will be defined in a precise way) to the non-injective modulation operator mod2h using heuristic arguments and regularity assumptions on the original signal. Followingthe guidelines described in [M. Constantini, A Novel Phase Unwrapping Method Based onNetwork Programming, IEEE Transactions on Geoscience and Remote Sensing, 1998] and [M.Constantini et al., A general formulation for redundant integration of finite differences andphase unwrapping on a sparse multidimensional domain, IEEE Transactions on Geoscience andRemote Sensing, 2012], this is made possible by correcting an approximate gradient into a globalgradient using either linear programming or, in some cases, minimum-cost flow techniques to solve an L1-norm optimization problem. Such a gradient-correcting technique can also be usedin general for finding a most plausible gradient and reconstructing a signal. The online demo associated with this paper implements the aforementioned methods.