From Fienup's phase retrieval techniques to regularized inversion for in-line holography: tutorial

International audience; This paper includes a tutorial on how to reconstruct in-line holograms using an inverse problems approach, starting with modeling the observations, selecting regularizations and constraints, and ending with the design of a reconstruction algorithm. A special focus is made on the connections between the numerous alternating projection strategies derived from Fienup's phase retrieval technique and the inverse problems framework. In particular, an interpretation of Fienup's algorithm as iterates of a proximal gradient descent for a particular cost function is given. Reconstructions from simulated and experimental holograms of micrometric beads illustrate the theoretical developments. The results show that the transition from alternating projection techniques to the inverse problems formulation is straightforward and advantageous.