  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.
