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

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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 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.

    Mathematics Subject Classification: Primary: 35J56, 30E20; Secondary: 45E05.

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  • Figure 1.  Geometric setup: $ \partial{ \Omega^+} = \Lambda\cup L $

    Figure 2.  Setting of numerical experiments. The gray regions correspond to the support of the source $ f $, whereas the dotted circles delineate regions of high attenuation. Note that $ f $ is to be reconstructed only in the upper semi-disc

    Figure 3.  Partial measurement data $ u(\zeta,\theta)|_{\Lambda\times {{\mathbf S}^ 1}} $ obtained by numerical computation of the attenuated Radon transform (4). The red curves are their polar coordinate representation $ \{ (u(\zeta,\theta),\theta) \::\: \theta \in {{\mathbf S}^ 1}\} $ and that at $ \zeta = (0,1) $ is magnified in the right graph. The pink areas show the regions of (known) higher absorption

    Figure 4.  Numerical solutions on $ L $ to the singular integral equation (13), from $ u_{0} $ to $ u_{-5} $, real parts (left) and imaginary parts (right)

    Figure 5.  Numerically reconstructed source $ f $ in $ \Omega^{+} $ (left), and its section on $ y = -\sqrt{3}x $ (right), which is indicated by the dotted line in the left figure

    Figure 6.  Partial measurement data $ u(\zeta,\theta)|_{\Lambda\times {{\mathbf S}^ 1}} $ with $ 10.9\% $ noise in the relative $ L^2 $ norm, depicted in the same manner as in Figure 3.

    Figure 7.  Numerically reconstructed source $ f $ from partial measurement data with $ 10.9\% $ noise shown in Figure 6.

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