  In two dimensions, we consider the problem of inversion of the attenuated
$X$-ray transform of a compactly supported function from data restricted to
lines leaning on a given arc. We provide a method to reconstruct the function
on the convex hull of this arc. The attenuation is assumed known. The method of
proof uses the Hilbert transform associated with $A$-analytic functions in the
sense of Bukhgeim.
