Two-dimensional phase unwrapping based on Fourier transforms and the Yukawa potential spectrum

The two-dimensional phase unwrapping problem (PHUP) has been solved with discrete Fourier transforms (FTs) and many other techniques traditionally. Nevertheless, a formal way of solving the continuous Poisson equation for the PHUP, with the use of continuous FT and based on distribution theory, has not been reported yet, to our knowledge. The well-known specific solution of this equation is given in general by a convolution of a continuous Laplacian estimate with a particular Green function, whose FT does not exist mathematically. However, an alternative Green function called the Yukawa potential, with a guaranteed Fourier spectrum, can be considered for solving an approximated Poisson equation, inducing a standard procedure of a FT-based unwrapping algorithm. Thus, the general steps for this approach are described in this work by considering some reconstructions with synthetic and real data.