  Herschel observations of interstellar clouds support a paradigm for star
formation in which filaments play a central role. One of the foundations of
this paradigm is the finding, based on studies of the radial column density
profiles, that nearby filaments share a common inner width of ~0.1 pc. The
existence of a characteristic filament width has been questioned, however, as
it seems inconsistent with the scale-free nature of the power spectrum of cloud
images.
  In an effort to clarify the origin of this apparent discrepancy, we examined
the power spectra of the Herschel 250 micron images of the Polaris, Aquila, and
Taurus clouds and performed a number of simple numerical experiments by
injecting synthetic filaments in both the Herschel images and synthetic
background images.
  We constructed several populations of synthetic filaments of 0.1 pc width
with realistic area filling factors ($A_{\rm fil}$) and distributions of column
density contrasts ($\delta_c$). After adding synthetic filaments to the
Herschel images, we re-computed the image power spectra and compared the
results with the original, essentially scale-free power spectra. We used the
$\chi^2$ variance of the residuals between the best power-law fit and the
output power spectrum as a diagnostic of the presence of a significant
departure from a scale-free power spectrum.
  We found that the $\chi^2$ variance depends primarily on the combined
parameter $\delta_c^2\, A_{\rm fil}$. Analysis of the real Herschel data shows
that the observed filamentary structure contributes only 1/5 of the power in
the power spectra at angular frequencies where an effect of the characteristic
filament width is expected.
  We conclude that the scale-free power spectra of Herschel images remain
consistent with the existence of a characteristic filament width ~0.1 pc and do
not invalidate the conclusions drawn from studies of the filament profiles.
