Many images (and signals) admit sparse representations in the sense that they are well approximated by linear combinations of a small number of functions taken from know sets. The topic of sparse and redundant representations, often termed as a sparse regression or sparse coding, has attracted tremendous interest from the research community in the last ten years. This interest stems from the role that the low dimensional models play in many signal and image areas such as compression, restoration, classification, and design of priors and regularizers, just to name a few. In this talk we use the sparse approximations for phase and magnitude of a complex-valued wavefield. While our techniques are quite general here they are illustrated for processing phase-shifting interferometry measurements. It is assumed that the observations are Poissonian (photon counting). In this way we are targeting at optimal sparse reconstruction of both phase and magnitude taking into consideration all details of the observation formation. Contrary to the standard variational approaches we propose a vector optimization with two objective functions leading to decoupling of inverse and denoising operations. This reconstruction is framed as a maximum likelihood constrained nonlinear optimization problem. It is demonstrated by simulation that proposed recursive algorithm is efficient, demonstrates high accuracy and better imaging performance in comparison with the current state-of-the-art.