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February  2022, 16(1): 215-228. doi: 10.3934/ipi.2021047

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

Hiroshi Fujiwara 1, , Kamran Sadiq 2,, and  Alexandru Tamasan 3,

1. 

Graduate School of Informatics, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto 606-8501, Japan
2. 

Johann Radon Institute of Computational and Applied Mathematics (RICAM), Altenbergerstrasse 69, 4040 Linz, Austria
3. 

Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

* Corresponding author

Received  January 2021 Revised  May 2021 Published  February 2022 Early access  July 2021

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.

Keywords: Attenuated X-ray transform, attenuated Radon transform, partial data, region-of-interest, local tomography, A-analytic maps, Hilbert transform.
Mathematics Subject Classification: Primary: 35J56, 30E20; Secondary: 45E05.
Citation: Hiroshi Fujiwara, Kamran Sadiq, Alexandru Tamasan. Partial inversion of the 2D attenuated $ X $-ray transform with data on an arc. Inverse Problems & Imaging, 2022, 16 (1) : 215-228. doi: 10.3934/ipi.2021047
References:
[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.  Google Scholar

[2]

E. V. Arzubov, A. L. Bukhgeim and S. G. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Adv. Math., 8 (1998), 1-20.   Google Scholar

[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.  Google Scholar

[4]

J. Boman, On stable inversion of the attenuated Radon transform with half data, Integral Geometry and Tomography, Contemp. Math., 405 (2006), 19-26.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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). Google Scholar

[9]

R. Clackdoyle and M. Defrise, Tomographic reconstruction in the 21st century, IEEE Signal Processing Magazine, 27 (2010), 60-80.   Google Scholar

[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.   Google Scholar

[11]

H. Fujiwara, K. 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.  Google Scholar

[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 Google Scholar

[13]

S. Helgason, The Radon Transform, 2$^{nd}$ edition, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4757-1463-0.  Google Scholar

[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.  Google Scholar

[15]

W. Koppelman and J. D. Pincus, Spectral representations for finite Hilbert transformations, Math. Z., 71 (1959), 399-407.  doi: 10.1007/BF01181411.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

N. I. Muskhelishvili, Singular Integral Equations, 2$^{nd}$ edition, Dover, New York, 1992.  Google Scholar

[19]

F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986.  Google Scholar

[20]

F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119.  doi: 10.1088/0266-5611/17/1/309.  Google Scholar

[21]

F. Noo, M. Defrise, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

S. Okada and D. Elliott, The finite Hilbert transform in $L^2$, Math. Nachr., 153 (1991), 43-56.  doi: 10.1002/mana.19911530105.  Google Scholar

[26]

X. Pan, C. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

K. Sadiq, O. 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.  Google Scholar

[33]

F. G. Tricomi, Integral Equations, Interscience, New York, 1957.  Google Scholar

[34]

H. Widom, Singular integral equations in $L_p$, Trans. Amer. Math. Soc., 97 (1960), 131-160.  doi: 10.2307/1993367.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

E. V. Arzubov, A. L. Bukhgeim and S. G. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Adv. Math., 8 (1998), 1-20.   Google Scholar

[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.  Google Scholar

[4]

J. Boman, On stable inversion of the attenuated Radon transform with half data, Integral Geometry and Tomography, Contemp. Math., 405 (2006), 19-26.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[9]

R. Clackdoyle and M. Defrise, Tomographic reconstruction in the 21st century, IEEE Signal Processing Magazine, 27 (2010), 60-80.   Google Scholar

[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.   Google Scholar

[11]

H. Fujiwara, K. 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.  Google Scholar

[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 Google Scholar

[13]

S. Helgason, The Radon Transform, 2$^{nd}$ edition, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4757-1463-0.  Google Scholar

[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.  Google Scholar

[15]

W. Koppelman and J. D. Pincus, Spectral representations for finite Hilbert transformations, Math. Z., 71 (1959), 399-407.  doi: 10.1007/BF01181411.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

N. I. Muskhelishvili, Singular Integral Equations, 2$^{nd}$ edition, Dover, New York, 1992.  Google Scholar

[19]

F. Natterer, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986.  Google Scholar

[20]

F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119.  doi: 10.1088/0266-5611/17/1/309.  Google Scholar

[21]

F. Noo, M. Defrise, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

S. Okada and D. Elliott, The finite Hilbert transform in $L^2$, Math. Nachr., 153 (1991), 43-56.  doi: 10.1002/mana.19911530105.  Google Scholar

[26]

X. Pan, C. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

K. Sadiq, O. 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.  Google Scholar

[33]

F. G. Tricomi, Integral Equations, Interscience, New York, 1957.  Google Scholar

[34]

H. Widom, Singular integral equations in $L_p$, Trans. Amer. Math. Soc., 97 (1960), 131-160.  doi: 10.2307/1993367.  Google Scholar

Figure 1.  Geometric setup: $ \partial{ \Omega^+} = \Lambda\cup L $
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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
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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
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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)
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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
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Figure 3.">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.
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Figure 6.">Figure 7.  Numerically reconstructed source $ f $ from partial measurement data with $ 10.9\% $ noise shown in Figure 6.
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