L1-Norm Redundant Delaunay Phase Unwrapping and Gradient Correction
Alexandre Achard-de Lustrac, Roland Akiki, Axel Davy, Jean-Michel Morel
published
2025-12-27
reference
Alexandre Achard-de Lustrac, Roland Akiki, Axel Davy, and Jean-Michel Morel, L1-Norm Redundant Delaunay Phase Unwrapping and Gradient Correction, Image Processing On Line, 15 (2025), pp. 108–162. https://doi.org/10.5201/ipol.2025.583

Communicated by B. Puyssegur and J. Anger
Demo edited by R. Akiki and A. Achard de Lustrac

Abstract

This article deals with arrays of real numbers which have been reduced modulo 2h into the interval [-h,h] where h>0 is a positive real number. Such an array is said to be wrapped modulo 2h. Often, the elements of these arrays correspond to values observed at points in an image-like 2D space which are connected by a graph structure. The process of retrieving the original array from which the wrapped image originates is called unwrapping. Of course, the wrapping process is not one-to-one, and the quality of the recovered unwrapped version depends on the smoothness of the original array. The goal of unwrapping is to define a most plausible left inverse (as will be defined in a precise way) to the non-injective modulation operator mod 2h using heuristic arguments and regularity assumptions on the original signal. Following the guidelines described in [M. Constantini, A Novel Phase Unwrapping Method Based on Network Programming, IEEE Transactions on Geoscience and Remote Sensing, 1998] and [M. Constantini et al., A general formulation for redundant integration of finite differences and phase unwrapping on a sparse multidimensional domain, IEEE Transactions on Geoscience and Remote Sensing, 2012], this is made possible by correcting an approximate gradient into a global gradient 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 used in general for finding a most plausible gradient and reconstructing a signal. The online demo associated with this paper implements the aforementioned methods.

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