We present a new framework 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 (PSFs). Jolideco reconstructs a single flux image by optimizing the posterior distribution based on the joint Poisson likelihood of all observations under one of several prior distributions. Most notably, we employ a patch-based image prior that is parameterized via a Gaussian mixture model, which we train on high-signal-to-noise astronomical images, including data from the JWST 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 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 0102.2–7219, we find the deconvolved width of 0farcs58 ± 0farcs02 to be consistent with the theoretical expectation derived from the known width of the PSF.