  Because the galaxies of the Local Group have such large angular sizes, much
of their diffuse, large-angular-scale emission is filtered out by the Herschel
data reduction process. In this work, we restore this previously missed dust in
Herschel observations of the Large Magellanic Cloud, Small Magellanic Cloud,
M31, and M33. We do this by combining Herschel data (including new reductions
for the Magellanic Clouds), in Fourier space, with lower-resolution data from
all-sky surveys (Planck, IRAS, and COBE) that did not miss the extended
emission. With these new maps, we find that a significant amount of emission
was missing from uncorrected Herschel data of these galaxies; over 20% in some
bands. Our new photometry also resolves the disagreement between fluxes
reported from older HERITAGE Magellanic Cloud Herschel reductions, and fluxes
reported from other telescopes. More emission is restored in shorter wavelength
bands, especially in the galaxies' peripheries, making these regions 20-40%
bluer than before. We also find that the Herschel-PACS instrument response
conflicts with the all-sky data, over the 20-90' angular scales to which they
are both sensitive, by up to 31%. By binning our new data based on hydrogen
column density, we are able to detect emission from dust at low ISM densities
(at $\Sigma_{\rm H} < 1\,{\rm M_{\odot} pc^{-2}}$ in some cases), and are able
to detect emission at much lower densities (a factor of 2.2 lower on average,
and more than a factor of 7 lower in several cases) than was possible with
uncorrected data.
