Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author

    * Corresponding author 
Abstract Full Text(HTML) Figure(7) Related Papers Cited by
  • 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.


    \begin{equation} \\ \end{equation}
  • 加载中
  • 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.

  • [1] E. V. Arbuzov and A. L. Bukhgeim, Carleman's formulas for $A$-analytic functions in a half-plane, J. Inv. Ill-Posed Problems, 5, (1997), 491–505. doi: 10.1515/jiip.1997.5.6.491.
    [2] E. V. ArzubovA. L. Bukhgeim and S. G. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Adv. Math., 8 (1998), 1-20. 
    [3] G. Bal, On the attenuated Radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418.  doi: 10.1088/0266-5611/20/2/006.
    [4] J. Boman, On stable inversion of the attenuated Radon transform with half data, Integral Geometry and Tomography, Contemp. Math., 405 (2006), 19-26. 
    [5] J. Boman, On stable region-of-interest reconstruction in tomography: Examples of non-existence of bounded inverse, Inverse Problems, 32 (2016), 125005. doi: 10.1088/0266-5611/32/12/125005.
    [6] J. Boman and J. -O. Strömberg, Novikov's inversion formula for the attenuated Radon transform–A new approach, J. Geom. Anal., 14 (2004), 185-198.  doi: 10.1007/BF02922067.
    [7] J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Math. J., 55 (1987), 943-948.  doi: 10.1215/S0012-7094-87-05547-5.
    [8] A. L. Bukhgeim, Inversion Formulas in Inverse Problems, in Linear Operators and Ill-Posed Problems (eds. M. M. Lavrentev and L. Ya. Savalev), Plenum, New York, (1995).
    [9] R. Clackdoyle and M. Defrise, Tomographic reconstruction in the 21st century, IEEE Signal Processing Magazine, 27 (2010), 60-80. 
    [10] D. V. Finch, The attenuated x-ray transform: Recent developments, Inside Out: Inverse Problems and Applications, Math. Sci. Res. Inst. Publ., 47 (2003), 47-66. 
    [11] H. FujiwaraK. Sadiq and A. Tamasan, Numerical reconstruction of radiative sources in an absorbing and non-diffusing scattering medium in two dimensions, SIAM J. Imaging Sci., 13 (2020), 535-555.  doi: 10.1137/19M1282921.
    [12] H. Fujiwara, K. Sadiq and A. Tamasan, On a local inversion of the X-ray transform from one sided data, Recent developments on inverse problems for PDE and their applications
    [13] S. Helgason, The Radon Transform, 2$^{nd}$ edition, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4757-1463-0.
    [14] S. G. Kazantsev and A. A. Bukhgeim, Inversion of the scalar and vector attenuated X-ray transforms in a unit disc, J. Inverse Ill-Posed Probl., 15 (2007), 735-765.  doi: 10.1515/jiip.2007.040.
    [15] W. Koppelman and J. D. Pincus, Spectral representations for finite Hilbert transformations, Math. Z., 71 (1959), 399-407.  doi: 10.1007/BF01181411.
    [16] P. Kuchment and I. Shneiberg, Some inversion formulae in the single photon emission computed tomography, Appl. Anal., 53 (1994), 221-231.  doi: 10.1080/00036819408840258.
    [17] F. Monard, Efficient tensor tomography in fan-beam coordinates. II: Attenuated transforms, Inverse Probl. Imaging, 12 (2018), 433-460.  doi: 10.3934/ipi.2018019.
    [18] N. I. Muskhelishvili, Singular Integral Equations, 2$^{nd}$ edition, Dover, New York, 1992.
    [19] F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986.
    [20] F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119.  doi: 10.1088/0266-5611/17/1/309.
    [21] F. NooM. DefriseJ. D. Pack and R. Clackdoyle, Image reconstruction from truncated data in single-photon emission computed tomography with uniform attenuation, Inverse Problems, 23 (2007), 645-667.  doi: 10.1088/0266-5611/23/2/011.
    [22] F. Noo and J. M. Wagner, Image reconstruction in 2D SPECT with $180^\circ$ acquisition, Inverse Problems, 17 (2001), 1357-1371.  doi: 10.1088/0266-5611/17/5/308.
    [23] R. G. Novikov, Une formule d'inversion pour la transformation d'un rayonnement X atténué, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), 1059-1063.  doi: 10.1016/S0764-4442(01)01965-6.
    [24] R. G. Novikov, On the range characterization for the two-dimensional attenuated x-ray transformation, Inverse Problems, 18 (2002), 677-700.  doi: 10.1088/0266-5611/18/3/310.
    [25] S. Okada and D. Elliott, The finite Hilbert transform in $L^2$, Math. Nachr., 153 (1991), 43-56.  doi: 10.1002/mana.19911530105.
    [26] X. PanC. Kao and C. Metz, A family of $\pi$-scheme exponential Radon transforms and the uniqueness of their inverses, Inverse Problems, 18 (2002), 825-836.  doi: 10.1088/0266-5611/18/3/319.
    [27] J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sachs. Akad. Wiss. Leipzig, Math-Nat., 69 (1917), 262-267. 
    [28] H. Rullgård, An explicit inversion formula for the exponential Radon transform using data from $180^\circ$, Ark. Mat., 42 (2004), 353-362.  doi: 10.1007/BF02385485.
    [29] H. Rullgård, Stability of the inverse problem for the attenuated Radon transform with $180^\circ$ data, Inverse Problems, 20 (2004), 781-797.  doi: 10.1088/0266-5611/20/3/008.
    [30] K. Sadiq and A. Tamasan, On the range of the attenuated Radon transform in strictly convex sets, Trans. Amer. Math. Soc., 367 (2015), 5375-5398.  doi: 10.1090/S0002-9947-2014-06307-1.
    [31] K. Sadiq and A. Tamasan, On the range characterization of the two dimensional attenuated Doppler transform, SIAM J. Math. Anal., 47 (2015), 2001-2021.  doi: 10.1137/140984282.
    [32] K. SadiqO. Scherzer and A. Tamasan, On the X-ray transform of planar symmetric 2-tensors, J. Math. Anal. Appl., 442 (2016), 31-49.  doi: 10.1016/j.jmaa.2016.04.018.
    [33] F. G. Tricomi, Integral Equations, Interscience, New York, 1957.
    [34] H. Widom, Singular integral equations in $L_p$, Trans. Amer. Math. Soc., 97 (1960), 131-160.  doi: 10.2307/1993367.
  • 加载中



Article Metrics

HTML views(379) PDF downloads(254) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint