Fast solution of phase unwrapping partial differential equation using wavelets

Abstract

Phase unwrapping is the most critical step in the processing of synthetic aperture radar interferometry. The phase obtained by SAR interferometry is wrapped over a range from $-\pi$ to $\pi$. Phase unwrapping must be performed to obtain the true phase. The least square approach attains the unwrapped phase by minimizing the difference between the discrete partial derivatives of the wrapped phase and the discrete partial derivatives of the unwrapped solution. The least square solution will result in discrete version of the Poisson's partial differential equation. Solving the discretized Poisson's equation with the classical method of Gauss-Seidel relaxation has extremely slow convergence. In this paper we have used Wavelet techniques which overcome this limitation by transforming low-frequency components of error into high frequency components which consequently can be removed quickly by using the Gauss-Seidel relaxation method. In Discrete Wavelet Transform (DWT) two operators, decomposition (analysis) and reconstruction (synthesis), are used. In the decomposition stage an image is separated into one low-frequency component (approximation) and three high-frequency components (details). In the reconstruction stage, the image is reconstructed by synthesizing the approximated and detail components. We tested our algorithm on both simulated and real data and on both unweighted and weighted forms of discretized Poisson's equation. The experimental results show the effectiveness of the proposed method.

Keywords

• Wavelets