Algebraic phase unwrapping along the real axis: extensions and stabilizations

The unwrapped phase of a complex function is defined with a line integral of the gradient of the arctangent of the ratio of the real and imaginary parts of the function. The phase unwrapping, which is a problem to reconstruct the unwrapped phase of an unknown complex function from its finite observed samples, has been a key for estimating useful physical quantity in many signal and image processing applications. In the light of the functional data analysis, it is natural to estimate first the unknown complex function by a certain piecewise complex polynomial and then to compute the exact unwrapped phase of the piecewise complex polynomial with the algebraic phase unwrapping algorithms (Yamada et al. in IEEE Trans Signal Process 46(6), 1639–1664, 1998; Yamada and Bose in IEEE Trans Circuits Syst I Fundam Theory Appl 49(3), 298–304, 2002; Yamada and Oguchi in Multidimens Syst Signal Process 22(1–3), 191–211, 2011). In this paper, we propose several useful extensions and numerical stabilizations of the algebraic phase unwrapping along the real axis which was established originally in Yamada and Oguchi (Multidimens Syst Signal Process 22(1–3), 191–211, 2011). The proposed extensions include (i) removal of a certain critical assumption premised in the original algebraic phase unwrapping, and (ii) algebraic phase unwrapping for a pair of bivariate polynomials. Moreover, in order to resolve certain numerical instabilities caused by the coefficient growth in an inductive step in the original algorithm, we propose to compute directly a certain subresultant sequence without passing through the inductive step. The extensive numerical experiments exemplify the notable improvement, in the performance of the algebraic phase unwrapping, made by the proposed numerical stabilization.