  Context: The Planck data releases have provided the community with
sub-millimetre and radio observations of the full-sky at unprecedented
resolutions. We make use of the Planck 353, 545 and 857 GHz maps alongside the
IRAS 3000 GHz map. These maps contain information on the cosmic microwave
background (CMB), cosmic infrared background (CIB), extragalactic point sources
and diffuse thermal dust emission. Aims: We aim to determine the modified black
body (MBB) model parameters of thermal dust emission in total intensity and
produce all sky maps of pure thermal dust, having separated this Galactic
component from the CMB and CIB. Methods: This separation is completed using a
new, sparsity-based, parametric method which we refer to as premise. The method
comprises of three main stages: 1) filtering of the raw data to reduce the
effect of the CIB on the MBB fit. 2) fitting an MBB model to the filtered data
across super-pixels of various sizes determined by the algorithm itself and 3)
refining these super-pixel estimates into full resolution maps of the MBB
parameters. Results: We present our maps of MBB temperature, spectral index and
optical depth at 5 arcmin resolution and compare our estimates to those of
GNILC as well as the two-step MBB fit presented by the Planck collaboration in
2013. Conclusions: By exploiting sparsity we avoid the need for smoothing,
enabling us to produce the first full resolution MBB parameter maps from
intensity measurements of thermal dust emission.We consider the premise
parameter estimates to be competitive with the existing state-of-the-art
solutions, outperforming these methods within low signal-to-noise regions as we
account for the CIB without removing thermal dust emission through
over-smoothing.
