Joint reconstruction of an in-focus image and of the background signal in in-line holographic microscopy

•Reconstruction of objects of interest shifted relative to a fixed background in digital holography microscopy.•Unmixing of the coherent background noise and the signal of interest, using only two shifted in-line holograms.•Optimal inverse problems based approach inverting a forward model that accounts for the interferences between the two signals.•Demonstration on shifted holograms of bacteria.