A Class of Global Optimization Problems as Models of the Phase Unwrapping Problem

Abstract

Let I={(i,j) ― i=1, 2,..., N ₁, j=1, 2,..., N ₂} and let U=U_i,j, (i,j)∈I be a discrete real function defined on I. Let [ċ]_2π be ċ modulus 2π, we define W:I → π, π) as follows W=[U]_2π. The function U will be called phase function and the function W will be called wrapped phase function. The phase unwrapping problem consists in recovering U from some knowledge of W. This problem is not well defined, that is infinitely many functions U correspond to the same function W, and must be `regularized' to be satisfactorily solvable. We propose several formulations of the phase unwrapping problem as an integer nonlinear minimum cost flow problem on a network. Numerical algorithms to solve the minimum cost flow problems obtained are proposed. The phase unwrapping problem is the key problem in interferometry, we restrict our attention to the SAR (Synthetic Aperture Radar) interferometry problem. We compare the different formulations of the phase unwrapping problem proposed starting from the analysis of the numerical experience obtained with the numerical algorithms proposed on synthetic and real SAR interferometry data. The real data are taken from the ERS missions of the European Space Agency (ESA).