  The X-ray spectra of Anomalous X-ray Pulsars have long been fit by smooth,
empirical models such as the sum of a black-body plus a power law. These
reproduce the ~0.5 to 10 keV range well, but fail at lower and higher energies,
grossly over-predicting the optical and under-predicting the hard X-ray
emission. A poorly constrained source of uncertainty in determining the true,
intrinsic spectra, in particular at lower energies, is the amount of
interstellar extinction. In previous studies, extinction column densities with
small statistical errors were derived as part of the fits of the spectra to
simple continuum models. Different choices of model, however, each produced
statistically acceptable fits, but a wide range of columns. Here, we attempt to
measure the interstellar extinction in a model-independent way, using
individual absorption edges of the elements O, Fe, Ne, Mg and Si in X-ray
grating spectra taken with XMM-Newton. We find that our inferred equivalent
hydrogen column density NH for 4U 0142+61 is a factor of 1.4 lower than the
typically quoted value from black-body plus power-law fits, and is now
consistent with estimates based on the dust scattering halo and visual
extinction. For three other sources, we find column densities consistent with
earlier estimates. We use our measurements to recover the intrinsic spectra of
the AXPs empirically, without making assumptions on what the intrinsic spectral
shapes ought to be. We find that the power-law components that dominate at
higher energies do not extend below the thermal peak.
