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

Partial inversion of the 2D attenuated $X$-ray transform with data on an arc