Figure 1.
Geometric setup: $ \partial{ \Omega^+} = \Lambda\cup L $
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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
| 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 |
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 and Imaging,
2022, 16
(1)
: 215-228.
doi: 10.3934/ipi.2021047
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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. |
| [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.
|
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. |
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. |
| [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.
|
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. |
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
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