Accurate PSF-matched photometry and photometric redshifts for the
  Extreme Deep Field with the Chebyshev-Fourier functions
  Photometric redshifts, which have become the cornerstone of several of the
largest astronomical surveys like PanStarrs, DES, J-PAS or the LSST, require
precise measurements of galaxy photometry in different bands using a consistent
physical aperture. This is not trivial, due to the variation in the shape and
width of the Point Spread Function (PSF) introduced by wavelength differences,
instrument positions and atmospheric conditions. Current methods to correct for
this effect rely on a detailed knowledge of the PSF characteristics as a
function of the survey coordinates, which can be difficult due to the relative
paucity of stars tracking the PSF behaviour. Here we show that it is possible
to measure accurate, consistent multicolour photometry without knowing the
shape of PSF. The Chebyshev-Fourier Functions (CHEFs) can fit the observed
profile of each object and produce high signal-to-noise integrated flux
measurements unaffected by the PSF. These total fluxes, which encompass all the
galaxy populations, are much more useful for Galaxy Evolution studies than
aperture photometry. We compare the total magnitudes and colours obtained using
our software to traditional photometry with SExtractor, using real data from
the COSMOS survey and the Hubble Ultra Deep Field. We also apply the CHEFs
technique to the recently published Extreme Deep Field and compare the results
to those from ColorPro on the HUDF. We produce a photometric catalogue with
35732 sources (10823 with S/N>5), reaching a photometric redshift precision of
2% due to the extraordinary depth and wavelength coverage of the XDF images.
