Phase Unwrapping of Coarsely Sampled Maps Using Higher-Order Methods

Abstract

We consider the problem of 2-D phase unwrapping that of reconstructing the absolute phase (up to a constant) given a noisy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\pi $ </tex-math></inline-formula> -wrapped phase map as input. In particular, we present an algorithm that excels at unwrapping low-resolution phase maps, that is, maps obtained by coarsely sampling a field of view. Our key observation is that the magnitudes of higher-order differences on phase maps are typically much lower than corresponding first-order differences. We develop this insight into formulating “higher-order Itoh conditions.” Using this, we build an optimization-based framework that leverages higher-order information to estimate the first-order derivatives of the unwrapped phase with the aid of appropriate total variation and irrotationality-based regularizers. The first-order derivatives are then integrated using a minimum spanning tree (MST) approach to produce the unwrapped phase map. We compare the performance on synthetic terrain maps and real-world data obtained from interferometric synthetic aperture radar (InSAR) with other contemporary algorithms to demonstrate superior performance on key metrics such as absolute error, Feature SIMilarity (FSIM), and image sharpness for low-resolution phase maps.