  We present a new method for joint likelihood deconvolution (Jolideco) of a
set of astronomical observations of the same sky region in the presence of
Poisson noise. The observations may be obtained from different instruments with
different resolution, and different point spread functions. Jolideco
reconstructs a single flux image by optimizing the posterior distribution based
on the joint Poisson likelihood of all observations under a patch-based image
prior. The patch prior is parameterised via a Gaussian Mixture model which we
train on high-signal-to-noise astronomical images, including data from the
James Webb Telescope and the GLEAM radio survey. This prior favors correlation
structures among the reconstructed pixel intensities that are characteristic of
those observed in the training images. It is, however, not informative for the
mean or scale of the reconstruction. By applying the method to simulated data
we show that the combination of multiple observations and the patch-based prior
leads to much improved reconstruction quality in many different source
scenarios and signal to noise regimes. We demonstrate that with the patch prior
Jolideco yields superior reconstruction quality relative to alternative
standard methods such as the Richardson-Lucy method. We illustrate the results
of Jolideco applied to example data from the Chandra X-ray Observatory and the
Fermi-LAT Gamma-ray Space Telescope. By comparing the measured width of a
counts based and the corresponding Jolideco flux profile of an X-ray filament
in SNR 1E 0102.2-721} we find the deconvolved width of 0.58+- 0.02 arcsec to be
consistent with the theoretical expectation derived from the known width of the
PSF.
