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

Abstract

<p style='text-indent:20px;'>In two dimensions, we consider the problem of inversion of the attenuated <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline-formula>-ray transform of a compactly supported function from data restricted to lines leaning on a given arc. We provide a method to reconstruct the function on the convex hull of this arc. The attenuation is assumed known. The method of proof uses the Hilbert transform associated with <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula>-analytic functions in the sense of Bukhgeim.</p>

Numerical reconstruction of radiative sources from partial boundary measurements

“…This section provides a brief review of the theoretical background used in our numerical reconstruction. For details we refer to [9,10].…”

“…), as shown in [25,26]. Solutions to (10) are said to be L 2 -analytic in the sense of Bukhgeim [3]. An important property of L 2 -analytic sequences is that they obey a Cauchy-like integral formula.…”

“…It is known [20,24] that the spectrum of iH c on L 2 (−c, c) is the interval [−1, 1]. Fortunately, 1 is not in the point spectrum of iH c [10,33], thus allowing to invert (I − iH c ) in its range. Unfortunately, the range of (I − iH c ) is a dense proper subset of L 2 (−c, c) yielding an unstable inversion with the discretization schemes rendered ill-conditioned.…”

A source reconstruction method in two dimensional radiative transport using boundary data measured on an arc

“…Specific to two dimensional domains, our approach is based on the Cauchy problem with partial data for a Beltrami-like equation associated with Aanalytic maps in the sense of Bukhgeim [7], and extends the authors' previous work [16], which used measurements on the entire boundary, to this specific partial data case. The new insight is that, similar to the non-scattering case [18], the trace u| Λ determines u| L provided the scattering kernel has finite Fourier content as in (3). The role of the finite Fourier content has been independently recognized in [23].…”

“…(b) Recover for each n ě M , the trace v ´n L by solving(39). (3) By Bukhgeim-Cauchy formula(18), extend v ´n for n ě M from the boundary Λ Y L to Ω `.…”