  Astronomical imaging based on photon count data is a non-trivial task. In
this context we show how to denoise, deconvolve, and decompose multi-domain
photon observations. The primary objective is to incorporate accurate and well
motivated likelihood and prior models in order to give reliable estimates about
morphologically different but superimposed photon flux components present in
the data set. Thereby we denoise and deconvolve photon counts, while
simultaneously decomposing them into diffuse, point-like and uninteresting
background radiation fluxes. The decomposition is based on a probabilistic
hierarchical Bayesian parameter model within the framework of information field
theory (IFT). In contrast to its predecessor D$^3$PO, D$^4$PO reconstructs
multi-domain components. Thereby each component is defined over its own direct
product of multiple independent domains, for example location and energy.
D$^4$PO has the capability to reconstruct correlation structures over each of
the sub-domains of a component separately. Thereby the inferred correlations
implicitly define the morphologically different source components, except for
the spatial correlations of the point-like flux. Point-like source fluxes are
spatially uncorrelated by definition. The capabilities of the algorithm are
demonstrated by means of a synthetic, but realistic, mock data set, providing
spectral and spatial information about each detected photon. D$^4$PO
successfully denoised, deconvolved, and decomposed a photon count image into
diffuse, point-like and background flux, each being functions of location as
well as energy. Moreover, uncertainty estimates of the reconstructed fields as
well as of their correlation structure are provided employing their posterior
density function and accounting for the manifolds the domains reside on.
