Discrete Green’s methods and their application to two-dimensional phase unwrapping

Abstract

A fully self-contained discrete framework with discrete equivalents of Stokes's, Gauss's, and Green's theorems is presented. The formulation is analogous to that of continuous operators, but totally discrete in nature, and the exact relationships derived are shown to hold provided that a set of predefined rules is followed in building discrete contours and domains. The method allows for an analytical rigor that is not guaranteed if one translates the classical continuous formulations onto a discretized approximated framework. We clarify several issues related to the use of discrete operators, which may play a crucial role in specific applications such as the two-dimensional phase-unwrapping problem, chosen as our main application example, and we show that reconstruction on irregular domains and/or in the presence of undersampling and noise is better formulated in the discrete framework than in the continuous domain.