FEDReD I: 3D extinction and stellar maps by Bayesian deconvolution
  Context. While Gaia enables to probe in great detail the extended local
neighbourhood, the thin disk structure at larger distances remains sparsely
explored.
  Aims. We aim here to build a non-parametric 3D model of the thin disc
structures handling both the extinction and the stellar density simultaneously.
  Methods. We developed a Bayesian deconvolution method in two dimensions:
extinction and distance. It uses a reference catalogue which completeness
information defines the selection function. It is designed so that any
complementary information from other catalogues can be added. It has also been
designed to be robust to outliers, frequent in crowded fields, and differential
extinction. The prior information is designed to be minimal: only a reference
H-R diagram. We derived for this an empirical H-R diagram of the thin disk
using Gaia DR2 data and synthetic isochrone-based H-R diagrams can also be
used.
  Results. We validated the method on simulations and real fields using 2MASS
and UKIDSS data complemented by Gaia DR2 photometry and parallaxes. We detail
the results of two test fields: a 2MASS field centred around the NGC 4815 open
cluster which shows an over-density of both extinction and stellar density at
the cluster distance, and a UKIDSS field at $l=10^{\circ}$ where we recover the
position of the Galactic bar.
