A Robust Algebraic Phase Unwrapping Based on Spline Approximation

摘要

The phase unwrapping is a problem to find, at any specified point, the value of the continuous phase function which contains valuable information in many applications. If a given data sequence of 2-D real vectors is modeled as samples of a complex polynomial, the exact unwrapped phase can be computed through algorithms named the algebraic phase unwrapping. In this paper, to promote the understanding and extend the practicability of the algebraic phase unwrapping, we propose a widely applicable phase unwrapping technique, by combining the spline smoothing and the algebraic phase unwrapping, for a given data sequence of 2-D noisy vectors. The spline smoothing works as an optimal preprocessing in the sense that it is the unique solution to a variational problem for minimizing the sum of "fidelity" to the data and "roughness" of the curve. Fortunately, since the standard spline smoothing and its various generalizations produce always low-order piecewise real polynomials, we can compute the exact unwrapped phase for the pair of piecewise polynomials without suffering from a certain numerical instability observed typically in applications of the algebraic phase unwrapping to a complex polynomial of large degree.