Phase-shifting VU factorization for interferometry

The problem of phase recovery in phase-shifting interferometry (PSI), when the introduced phase-shifts are unknown, has been analyzed by existing methods through the modification or combination of the least-squares and the principal component analysis (PCA) approaches. In this paper, we show that the PSI sequence of images can be written in a matrix form as the product of two matrices V and U, where the matrix V contains the components of the modulating phase and the matrix U contains the components of the phase-shifts. The VU matrix describes the PSI images without restrictions on the background illumination or the phase shifts, thus, the phase-shifts can be given arbitrarily and they do not need to cover the interval [0, 2π]. In particular, we introduce a numerical method that can separate these matrices V and U from a given set of PSI images, which enables us to recover both the modulating phase and the phase-shifts, albeit, we are only interested in the modulating phase. We refer to this process as phase-shifting VU factorization. This factorization method is fast, stable, and converges well to the desired solution. An important feature of the method is that the convergence of the VU factorization is robust to the initial guess and the only parameter it requires is the convergence accuracy. The method can reach a convergence accuracy of the order of 10^-3 in a few iterations. Results from our method using both simulated and experimental interferograms are shown and compared with recovered modulating phases obtained by three other state-of-the-art approaches: the PCA, least squares and a combination of both.