Robust phase unwrapping by convex optimization

Abstract

The 2-D phase unwrapping problem aims at retrieving a "phase" image from its modulo 2π observations. Many applications, such as interferometry or synthetic aperture radar imaging, are concerned by this problem since they proceed by recording complex or modulated data from which a "wrapped" phase is extracted. Although 1-D phase unwrapping is trivial, a challenge remains in higher dimensions to overcome two common problems: noise and discontinuities in the true phase image. In contrast to state-of-the-art techniques, this work aims at simultaneously unwrap and denoise the phase image. We propose a robust convex optimization approach that enforces data fidelity constraints expressed in the corrupted phase derivative domain while promoting a sparse phase prior. The resulting optimization problem is solved by the Chambolle-Pock primal-dual scheme. We show that under different observation noise levels, our approach compares favorably to those that perform the unwrapping and denoising in two separate steps.

Isotropic inverse-problem approach for two-dimensional phase unwrapping

“…Also as above, the Research Article reconstructed phase images were made consistent by using the operator Eq. (15). The results in Fig.…”

“…Moreover, our data fidelity term D penalizes both horizontal ϵ x and vertical ϵ y components of ϵ in a joint fashion. This is significantly different from traditional formulations in the literature [13,15], where the l 1 -norm is penalized in a separable fashion as ‖ϵ‖ l 1 ‖ϵ x ‖ l 1 ‖ϵ y ‖ l 1 . In fact, there is a clear analogy between our formulation Eq.…”

“…Upon convergence, the solutions of all the variational algorithms were made consistent with a single application of the operator Eq. (15).…”

“…Our formulation thus allows the phase image to contain edges that are of arbitrary orientation, which is distinct from traditional approaches in literature [13][14][15][16][17].…”

“…Similarly, Rivera and Marroquin [14] have investigated nonconvex optimization strategies relying on half-quadratic regularization. More recently, González and Jacques [15] have proposed an iterative unwrapping method based on l 1 minimization that additionally promotes sparsity of the solution in the wavelet domain. Ying et al [16] have proposed an iterative method based on dynamic programming that models the phase as a Markov random field.…”