Algebraic Phase Unwrapping Based on Two-Dimensional Spline Smoothing Over Triangles

Two-dimensional (2D) phase unwrapping is an estimation problem of a continuous phase function, over a 2D domain, from its wrapped samples. In this paper, we propose a novel approach for high-resolution 2D phase unwrapping. In the first step - SPline Smoothing (SPS), we construct a pair of the smoothest spline functions which minimize the energies of their local changes while interpolating, respectively, the cosine and the sine of given wrapped samples. If these functions have no common zero over the domain, the proposed estimate of the continuous phase function can be obtained by algebraic phase unwrapping in the second step - Algebraic Phase Unwrapping (APU). To avoid the occurrence of common zeros in SPS due to phase noise in the observed wrapped samples, we also propose a denoising step - Denoising by Selective Smoothing (DSS) - as preprocessing, which selectively smooths unreliable wrapped samples by using convex optimization. The smoothness of the proposed unwrapped phase function is guaranteed globally over the domain without losing any desired consistency with all reliable wrapped samples. Numerical experiments for terrain height estimation demonstrate the effectiveness of the proposed 2D phase unwrapping scheme.