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April  2018, 12(2): 433-460. doi: 10.3934/ipi.2018019

Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms

François Monard

University of California Santa Cruz, Department of Mathematics, 1156 High St. Santa Cruz, CA 95064, USA

Received  May 2017 Revised  October 2017 Published  February 2018

Fund Project: The author is supported by NSF grant DMS-1712790.

This article extends the author's past work [11] to attenuated X-ray transforms, where the attenuation is complex-valued and only depends on position. We give a positive and constructive answer to the attenuated tensor tomography problem on the Euclidean unit disc in fan-beam coordinates. For a tensor of arbitrary order, we propose an equivalent tensor of the same order which can be uniquely and stably reconstructed from its attenuated transform, as well as an explicit and efficient procedure to do so.

Keywords: Tensor tomography, fan-beam coordinates, attenuated X-ray transform, radon transform, Doppler transform, killing decomposition, invariant distributions.
Mathematics Subject Classification: Primary: 35R30, 44A12; Secondary: 30E20, 92C55, 58J32.
Citation: François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems & Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019
References:
[1]

G. Ainsworth, The attenuated magnetic ray transform on surfaces, Inverse Problems and Imaging, 7 (2013), 27-46.   Google Scholar

[2]

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

[3]

Y. M. Assylbekov, F. Monard and G. Uhlmann, Inversion formulas and range characterizations for the attenuated geodesic ray transform, Journal de Mathématiques Pures et Appliquées (to appear), arXiv: 1609.04361. Google Scholar

[4]

G. Bal, On the attenuated radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418.   Google Scholar

[5]

G. Bal and A. Tamasan, Inverse source problem in transport equations, SIAM J. Math. Anal., 39 (2007), 57-76.   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.   Google Scholar

[7]

D. Finch, The attenuated X-ray transform: Recent developments, Inside Out, Inverse Problems and Applications, 47-66, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003.  Google Scholar

[8]

S. Holman and P. Stefanov, The weighted doppler transform, Inverse Problems and Imaging, 4 (2010), 111-130.   Google Scholar

[9]

S. G. Kazantsev and A. A. Bukhgeim, Inversion of the scalar and vector attenuated x-ray transforms in a unit disc, J. Inv. Ill-Posed Problems, 15 (2007), 735-765.   Google Scholar

[10]

S. G. Kazantsev and A. A. Bukhgeim, Singular value decomposition for the 2d fan-beam radon transform of tensor fields, Journal of Inverse and Ill-posed Problems, 12 (2004), 245-278.   Google Scholar

[11]

F. Monard, Efficient tensor tomography in fan-beam coordinates, Inverse Probl. Imaging, 10 (2016), 433-459.   Google Scholar

[12]

F. Monard, Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, SIAM J. Math. Anal., 48 (2016), 1155-1177.   Google Scholar

[13]

F. Monard and G. P. Paternain, The geodesic X-ray transform with a $GL(n, \mathbb{C})$ -connection, to appear in Journal of Geometric Analysis, arXiv: 1610.09571. Google Scholar

[14]

F. Natterer, Inversion of the attenuated radon transform, Inverse Problems, 17 (2001), 113-119.   Google Scholar

[15]

F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001.  Google Scholar

[16]

R. G. Novikov, An inversion formula for the attenuated x-ray transformation, Ark. Math., 40 (2002), 145-167, (Rapport de recherche 00/05-3 Université de Nantes, Laboratoire de Mathématiques).  Google Scholar

[17]

G. Paternain, Inside Out II, chapter Inverse problems for connections, 2012. Google Scholar

[18]

G. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and higgs fields, Geom. Funct. Anal. (GAFA), 22 (2012), 1460-1489.   Google Scholar

[19]

G. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, Inventiones Math., 193 (2013), 229-247.   Google Scholar

[20]

G. P. Paternain, M. Salo and G. Uhlmann, Spectral rigidity and invariant distributions on anosov surfaces, Journal of Differential Geometry, 98 (2014), 147-181.  doi: 10.4310/jdg/1406137697.  Google Scholar

[21]

G. P. Paternain, M. Salo and G. Uhlmann, Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Mathematische Annalen, 363 (2015), 305-362.  doi: 10.1007/s00208-015-1169-0.  Google Scholar

[22]

G. P. Paternain and H. Zhou, Invariant distributions and the geodesic ray transform, Analysis & PDE, 9 (2016), 1903-1930.  doi: 10.2140/apde.2016.9.1903.  Google Scholar

[23]

L. Pestov and G. Uhlmann, On the characterization of the range and inversion formulas for the geodesic X-ray transform, International Math. Research Notices, 80 (2004), 4331-4347.   Google Scholar

[24]

L. Pestov and G. Uhlmann, Two-dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2005), 1093-1110.  doi: 10.4007/annals.2005.161.1093.  Google Scholar

[25]

K. Sadiq, O. Scherzer and A. Tamasan, On the x-ray transform of planar symmetric 2-tensors, Journal of Mathematical Analysis and Applications, 442 (2016), 31-49.  doi: 10.1016/j.jmaa.2016.04.018.  Google Scholar

[26]

K. Sadiq and A. Tamasan, On the range characterization of the two-dimensional attenuated doppler transform, SIAM Journal on Mathematical Analysis, 47 (2015), 2001-2021.  doi: 10.1137/140984282.  Google Scholar

[27]

K. Sadiq and A. Tamasan, On the range of the attenuated radon transform in strictly convex sets, Transactions of the American Mathematical Society, 367 (2015), 5375-5398.   Google Scholar

[28]

M. Salo and G. Uhlmann, The Attenuated Ray Transform on Simple Surfaces, J. Diff. Geom., 88 (2011), 161-187.  doi: 10.4310/jdg/1317758872.  Google Scholar

[29]

V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994.  Google Scholar

[30]

G. Sparr, K. Strahlen, K. Lindstrom and H. Persson, Doppler tomography for vector fields, Inverse Problems, 11 (1995), 1051-1061.  doi: 10.1088/0266-5611/11/5/009.  Google Scholar

[31]

P. Stefanov and G. Uhlmann, An inverse source problem in optical molecular imaging, Analysis & PDE, 1 (2008), 115-126.  doi: 10.2140/apde.2008.1.115.  Google Scholar

[32]

E. M. Stein and R. Shakarchi, Real Analysis. Measure Theory, Integration and Hilbert Spaces, Princeton University Press, 2005.  Google Scholar

[33]

W. A. Strauss, Partial Differential Equations, Second edition. John Wiley & Sons, Ltd., Chichester, 2008.  Google Scholar

[34]

A. Tamasan, Tomographic reconstruction of vector fields in variable background media, Inverse Problems, 23 (2007), 2197-2205.  doi: 10.1088/0266-5611/23/5/022.  Google Scholar

[35]

H. Zhou, Generic injectivity and stability of inverse problems for connections, Communications in Partial Differential Equations, 42 (2017), 780-801.  doi: 10.1080/03605302.2017.1295061.  Google Scholar

show all references

References:
[1]

G. Ainsworth, The attenuated magnetic ray transform on surfaces, Inverse Problems and Imaging, 7 (2013), 27-46.   Google Scholar

[2]

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

[3]

Y. M. Assylbekov, F. Monard and G. Uhlmann, Inversion formulas and range characterizations for the attenuated geodesic ray transform, Journal de Mathématiques Pures et Appliquées (to appear), arXiv: 1609.04361. Google Scholar

[4]

G. Bal, On the attenuated radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418.   Google Scholar

[5]

G. Bal and A. Tamasan, Inverse source problem in transport equations, SIAM J. Math. Anal., 39 (2007), 57-76.   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.   Google Scholar

[7]

D. Finch, The attenuated X-ray transform: Recent developments, Inside Out, Inverse Problems and Applications, 47-66, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003.  Google Scholar

[8]

S. Holman and P. Stefanov, The weighted doppler transform, Inverse Problems and Imaging, 4 (2010), 111-130.   Google Scholar

[9]

S. G. Kazantsev and A. A. Bukhgeim, Inversion of the scalar and vector attenuated x-ray transforms in a unit disc, J. Inv. Ill-Posed Problems, 15 (2007), 735-765.   Google Scholar

[10]

S. G. Kazantsev and A. A. Bukhgeim, Singular value decomposition for the 2d fan-beam radon transform of tensor fields, Journal of Inverse and Ill-posed Problems, 12 (2004), 245-278.   Google Scholar

[11]

F. Monard, Efficient tensor tomography in fan-beam coordinates, Inverse Probl. Imaging, 10 (2016), 433-459.   Google Scholar

[12]

F. Monard, Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, SIAM J. Math. Anal., 48 (2016), 1155-1177.   Google Scholar

[13]

F. Monard and G. P. Paternain, The geodesic X-ray transform with a $GL(n, \mathbb{C})$ -connection, to appear in Journal of Geometric Analysis, arXiv: 1610.09571. Google Scholar

[14]

F. Natterer, Inversion of the attenuated radon transform, Inverse Problems, 17 (2001), 113-119.   Google Scholar

[15]

F. Natterer, The Mathematics of Computerized Tomography, SIAM, 2001.  Google Scholar

[16]

R. G. Novikov, An inversion formula for the attenuated x-ray transformation, Ark. Math., 40 (2002), 145-167, (Rapport de recherche 00/05-3 Université de Nantes, Laboratoire de Mathématiques).  Google Scholar

[17]

G. Paternain, Inside Out II, chapter Inverse problems for connections, 2012. Google Scholar

[18]

G. Paternain, M. Salo and G. Uhlmann, The attenuated ray transform for connections and higgs fields, Geom. Funct. Anal. (GAFA), 22 (2012), 1460-1489.   Google Scholar

[19]

G. Paternain, M. Salo and G. Uhlmann, Tensor tomography on surfaces, Inventiones Math., 193 (2013), 229-247.   Google Scholar

[20]

G. P. Paternain, M. Salo and G. Uhlmann, Spectral rigidity and invariant distributions on anosov surfaces, Journal of Differential Geometry, 98 (2014), 147-181.  doi: 10.4310/jdg/1406137697.  Google Scholar

[21]

G. P. Paternain, M. Salo and G. Uhlmann, Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Mathematische Annalen, 363 (2015), 305-362.  doi: 10.1007/s00208-015-1169-0.  Google Scholar

[22]

G. P. Paternain and H. Zhou, Invariant distributions and the geodesic ray transform, Analysis & PDE, 9 (2016), 1903-1930.  doi: 10.2140/apde.2016.9.1903.  Google Scholar

[23]

L. Pestov and G. Uhlmann, On the characterization of the range and inversion formulas for the geodesic X-ray transform, International Math. Research Notices, 80 (2004), 4331-4347.   Google Scholar

[24]

L. Pestov and G. Uhlmann, Two-dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2005), 1093-1110.  doi: 10.4007/annals.2005.161.1093.  Google Scholar

[25]

K. Sadiq, O. Scherzer and A. Tamasan, On the x-ray transform of planar symmetric 2-tensors, Journal of Mathematical Analysis and Applications, 442 (2016), 31-49.  doi: 10.1016/j.jmaa.2016.04.018.  Google Scholar

[26]

K. Sadiq and A. Tamasan, On the range characterization of the two-dimensional attenuated doppler transform, SIAM Journal on Mathematical Analysis, 47 (2015), 2001-2021.  doi: 10.1137/140984282.  Google Scholar

[27]

K. Sadiq and A. Tamasan, On the range of the attenuated radon transform in strictly convex sets, Transactions of the American Mathematical Society, 367 (2015), 5375-5398.   Google Scholar

[28]

M. Salo and G. Uhlmann, The Attenuated Ray Transform on Simple Surfaces, J. Diff. Geom., 88 (2011), 161-187.  doi: 10.4310/jdg/1317758872.  Google Scholar

[29]

V. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994.  Google Scholar

[30]

G. Sparr, K. Strahlen, K. Lindstrom and H. Persson, Doppler tomography for vector fields, Inverse Problems, 11 (1995), 1051-1061.  doi: 10.1088/0266-5611/11/5/009.  Google Scholar

[31]

P. Stefanov and G. Uhlmann, An inverse source problem in optical molecular imaging, Analysis & PDE, 1 (2008), 115-126.  doi: 10.2140/apde.2008.1.115.  Google Scholar

[32]

E. M. Stein and R. Shakarchi, Real Analysis. Measure Theory, Integration and Hilbert Spaces, Princeton University Press, 2005.  Google Scholar

[33]

W. A. Strauss, Partial Differential Equations, Second edition. John Wiley & Sons, Ltd., Chichester, 2008.  Google Scholar

[34]

A. Tamasan, Tomographic reconstruction of vector fields in variable background media, Inverse Problems, 23 (2007), 2197-2205.  doi: 10.1088/0266-5611/23/5/022.  Google Scholar

[35]

H. Zhou, Generic injectivity and stability of inverse problems for connections, Communications in Partial Differential Equations, 42 (2017), 780-801.  doi: 10.1080/03605302.2017.1295061.  Google Scholar

Figure 1.  Fan-beam coordinates, scattering relation $\mathcal{S}$ and antipodal scattering relation ${{\mathcal{S}}_{A}}$
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