The balloon-borne Compton Spectrometer and Imager (COSI) had a successful 46-day flight in 2016. The instrument is sensitive to photons in the energy range $0.2$-$5$ MeV. Compton telescopes have the advantage of a unique imaging response and provide the possibility of strong background suppression. With its high-purity germanium detectors, COSI can precisely map $gamma$-ray line emission. The strongest persistent and diffuse $gamma$-ray line signal is the 511 keV emission line from the annihilation of electrons with positrons from the direction of the Galactic centre. While many sources have been proposed to explain the amount of positrons, $dot{N}_{mathrm{e^+}} sim 10^{50},mathrm{e^+,yr^{-1}}$, the true contributions remain unsolved. In this study, we aim at imaging the 511 keV sky with COSI and pursue a full-forward modelling approach, using a simulated and binned imaging response. For the strong instrumental background, we describe an empirical approach to take the balloon environment into account. We perform two alternative methods to describe the signal: Richardson-Lucy deconvolution, an iterative method towards the maximum likelihood solution, and model fitting with pre-defined emission templates. Consistently with both methods, we find a 511 keV bulge signal with a flux between $0.9$ and $3.1 times 10^{-3},mathrm{ph,cm^{-2},s^{-1}}$, confirming earlier measurements, and also indications of more extended emission. The upper limit we find for the 511 keV disk, $< 4.3 times 10^{-3},mathrm{ph,cm^{-2},s^{-1}}$, is consistent with previous detections. For large-scale emission with weak gradients, coded aperture mask instruments suffer from their inability to distinguish isotropic emission from instrumental background, while Compton-telescopes provide a clear imaging response, independent of the true emission.
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