Adriana Gonzalez

Robust phase unwrapping by convex optimization

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.

Refractive index map reconstruction in optical deflectometry using total-variation regularization

We study the resolution of an inverse problem arising in Optical Deflectometry: the reconstruction of refractive index map of transparent materials from light deflection measurements under multiple orientations. This problem is solved from a standard convex optimization procedure aiming at minimizing the Total Variation of the map (as a prior information) while matching the observed data (fidelity constraint). In this process, the forward measurement operator, mapping any index map to its deflectometric observation, is simplified in the Fourier domain according to a modified Projection-Slice theorem.