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In this paper, we propose to present the general ingredients involved in an inverse problems methodology dedicated to the reconstruction of in-line holograms, and compare it with the classical Gercherg-Saxton or Fienup alternating projections strategies for phase retrieval [1,2,3]. An inverse approach [4,5] consists in retrieving an optimal solution to a reconstruction/estimation problem from a dataset, knowing an approximate model of its formation process. The problem is generally formulated as an optimization problem that aims at fitting the model to the data, while favoring a priori knowledge on the targeted information using regularizations and constraints. An appropriate resolution method has to be designed, based on a convex optimization framework. We develop the end-to-end inverse problems methodology on a case-study : the reconstruction of an in-line hologram of a collection of weakly dephasing objects. This simple problem allows us to explain current physical considerations (type of objects, diffraction physics) to derive the appropriate model, and to present classical constraints and regularizations that can be used in image reconstruction. Starting from these ingredients, we introduce a simple yet efficient method to solve this inverse problem, belonging to the class of proximal gradient algorithms [6,7]. A special focus is made on the connections between the numerous alternating projections 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. We discuss the advantages provided by the inverse problems methodology. We illustrate both strategies on reconstructions from simulated and experimental holograms of micrometric beads. The results show that the transition from alternating projection techniques to the inverse problems formulation is straightforward and advantageous.
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