In-line digital holography (DH) and lensless microscopy are 3D imaging techniques used to reconstruct the volume of micro-objects in many fields. However, their performances are limited by the pixel size of the sensor. Recently, various pixel super-resolution algorithms for digital holography have been proposed. A hologram with improved resolution was produced from a stack of laterally shifted holograms, resulting in better resolved reconstruction than a single low-resolution hologram. Algorithms for super-resolved reconstructions based on inverse problems approaches have already been shown to improve the 3D reconstruction of opaque spheres. Maximum a posteriori approaches have also been shown capable of reconstructing the object field more accurately and more efficiently and to extend the usual field-of-view. Here we propose an inverse problem formulation for DH pixel super-resolution and an algorithm that alternates registration and reconstruction steps. The method is described in detail and used to reconstruct synthetic and experimental holograms of sparse 2D objects. We show that our approach improves both the shift estimation and reconstruction quality. Moreover, the reconstructed field-of-view can be expanded by up to a factor 3, thus making it possible to multiply the analyzed area ninefold. © 2016 Optical Society of America.