December  2020, 14(6): 967-983. doi: 10.3934/ipi.2020044

Optimal recovery of a radiating source with multiple frequencies along one line

1. 

Department of Mathematical Sciences, Faculty of Information Technology and Electrical Engineering

2. 

NTNU – Norwegian University of Science and Technology, Trondheim, Norway

3. 

Technical University of Denmark, Department of Applied Mathematics and Computer Science, Lyngby, Denmark

4. 

University of Jyväskylä, Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland

5. 

Department of Mathematics and Statistics, P.O 68 (Pietari Kalmin katu 5), 00014 University of Helsinki, Finland

6. 

Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland

* Corresponding author: Tommi Brander

Received  May 2019 Revised  May 2020 Published  December 2020 Early access  August 2020

We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over levelsets of the integrated attenuation. This leads to a generalized Laplace transform. We also discuss some numerical approaches and demonstrate the results with several examples.

Citation: Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems and Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044
References:
[1]

R. E. Alvarez and A. Macovski, Energy-selective reconstructions in X-ray computerised tomography, Physics in Medicine & Biology, 21 (1976), 733.

[2]

T. Brander and J. Siltakoski, Recovering a variable exponent, Preprint, URL https://arXiv.org/abs/2002.06076.

[3]

T. Brander and D. Winterrose, Variable exponent Calderón's problem in one dimension, Annales Academiæ Scientiarum Fennicæ, Mathematica, 44 (2019), 925–943. doi: 10.5186/aasfm.2019.4459.

[4]

R. A. Brooks and G. Di Chiro, Beam hardening in X-ray reconstructive tomography, Physics in Medicine & Biology, 21 (1976), 390. doi: 10.1088/0031-9155/21/3/004.

[5]

H. ChoiV. GintingF. Jafari and R. Mnatsakanov, Modified Radon transform inversion using moments, J. Inverse Ill-Posed Probl, 28 (2020), 1-15.  doi: 10.1515/jiip-2018-0090.

[6]

S. R. Deans, The Radon Transform and Some of Its Applications, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1983. URL https://books.google.com.mt/books?id=xSCc0KGi0u0C.

[7]

C. Dellacherie and P.-A. Meyer, Probabilities and Potential, vol. 29 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam – New York, 1978.

[8]

N. Eldredge, Closure of Polynomials of A Function in $L^2$, MathOverflow, 2018, URL https://mathoverflow.net/a/292978/1445

[9]

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

[10]

D. Gourion and D. Noll, The inverse problem of emission tomography, Inverse Problems, 18 (2002), 1435-1460.  doi: 10.1088/0266-5611/18/5/315.

[11]

P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, Fundamentals of Algorithms, 7. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898718836.

[12]

M. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, 49 (1952), 409-436.  doi: 10.6028/jres.049.044.

[13]

J. Ilmavirta and F. Monard, Integral geometry on manifolds with boundary and applications, in The Radon Transform: The First 100 Years and Beyond, (eds. R. Ramlau and O. Scherzer), de Gruyter, 2019, 1–73. URL http://users.jyu.fi/ jojapeil/pub/integral-geometry-review.pdf. doi: 10.1515/9783110560855-004.

[14]

J. D. Ingle Jr and S. R. Crouch, Spectrochemical Analysis, Prentice Hall College Book Division, Old Tappan, NJ, USA, 1988.

[15]

O. Kallenberg, Foundations of Modern Probability, Probability and its Applications (New York). Springer-Verlag, New York, 1997.

[16]

V. P. Krishnan, R. Manna, S. K. Sahoo and V. A. Sharafutdinov, Momentum ray transforms, Inverse Problems and Imaging, 13 (2019), 679–701, URL http://aimsciences.org//article/id/d88823a5-827c-4c4b-909a-c27daa0b74ec. doi: 10.3934/ipi.2019031.

[17]

L. A. Lehmann and R. E. Alvarez, Energy-selective radiography a review, in Digital Radiography: Selected Topics (eds. J. G. Kereiakes, S. R. Thomas and C. G. Orton), Springer US, Boston, MA, (1986), 145–188. doi: 10.1007/978-1-4684-5068-2_7.

[18]

R. M. Lewitt and S. Matej, Overview of methods for image reconstruction from projections in emission computed tomography, Proceedings of the IEEE, 91 (2003), 1588-1611.  doi: 10.1109/JPROC.2003.817882.

[19]

C. H. McColloughS. LengL. Yu and J. G. Fletcher, Dual- and multi-energy CT: Principles, technical approaches, and clinical applications, Radiology, 276 (2015), 637-653.  doi: 10.1148/radiol.2015142631.

[20]

P. Milanfar, Geometric Estimation and Reconstruction from Tomographic Data, PhD thesis, Massachusetts Institute of Technology, 1993.

[21]

P. MilanfarW. C. Karl and A. S. Willsky, A moment-based variational approach to tomographic reconstruction, IEEE Transactions on Image Processing, 5 (1996), 459-470.  doi: 10.1109/83.491319.

[22]

J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, vol. 10 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972344.

[23]

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

[24]

R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Arkiv För Matematik, 40 (2002), 145–167. doi: 10.1007/BF02384507.

[25]

G. P. PaternainM. Salo and G. Uhlmann, Tensor tomography: Progress and challenges, Chinese Annals of Mathematics. Series B, 35 (2014), 399-428.  doi: 10.1007/s11401-014-0834-z.

[26]

G. P. PaternainM. 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.

[27]

W. Rudin, Real and Complex Analysis, McGraw–Hill Series in Higher Mathematics. McGraw–Hill Book Co., New York-Düsseldorf–Johannesburgn, 1974.

[28]

K. Schmüdgen, The Moment Problem, vol. 277 of Graduate Texts in Mathematics, Springer International Publishing, 2017.

[29]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.

[30]

K. Taguchi and J. S. Iwanczyk, Vision 20/20: Single photon counting x-ray detectors in medical imaging, Medical Physics, 40 (2013), 100901. doi: 10.1118/1.4820371.

[31]

A. WelchG. T. GullbergP. E. ChristianJ. Li and B. M. Tsui, An investigation of dual energy transmission measurements in simultaneous transmission emission imaging, IEEE Transactions on Nuclear Science, 42 (1995), 2331-2338.  doi: 10.1109/23.489437.

[32]

D. V. Widder, The Laplace Transform, Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941.

show all references

References:
[1]

R. E. Alvarez and A. Macovski, Energy-selective reconstructions in X-ray computerised tomography, Physics in Medicine & Biology, 21 (1976), 733.

[2]

T. Brander and J. Siltakoski, Recovering a variable exponent, Preprint, URL https://arXiv.org/abs/2002.06076.

[3]

T. Brander and D. Winterrose, Variable exponent Calderón's problem in one dimension, Annales Academiæ Scientiarum Fennicæ, Mathematica, 44 (2019), 925–943. doi: 10.5186/aasfm.2019.4459.

[4]

R. A. Brooks and G. Di Chiro, Beam hardening in X-ray reconstructive tomography, Physics in Medicine & Biology, 21 (1976), 390. doi: 10.1088/0031-9155/21/3/004.

[5]

H. ChoiV. GintingF. Jafari and R. Mnatsakanov, Modified Radon transform inversion using moments, J. Inverse Ill-Posed Probl, 28 (2020), 1-15.  doi: 10.1515/jiip-2018-0090.

[6]

S. R. Deans, The Radon Transform and Some of Its Applications, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1983. URL https://books.google.com.mt/books?id=xSCc0KGi0u0C.

[7]

C. Dellacherie and P.-A. Meyer, Probabilities and Potential, vol. 29 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam – New York, 1978.

[8]

N. Eldredge, Closure of Polynomials of A Function in $L^2$, MathOverflow, 2018, URL https://mathoverflow.net/a/292978/1445

[9]

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

[10]

D. Gourion and D. Noll, The inverse problem of emission tomography, Inverse Problems, 18 (2002), 1435-1460.  doi: 10.1088/0266-5611/18/5/315.

[11]

P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, Fundamentals of Algorithms, 7. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898718836.

[12]

M. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, 49 (1952), 409-436.  doi: 10.6028/jres.049.044.

[13]

J. Ilmavirta and F. Monard, Integral geometry on manifolds with boundary and applications, in The Radon Transform: The First 100 Years and Beyond, (eds. R. Ramlau and O. Scherzer), de Gruyter, 2019, 1–73. URL http://users.jyu.fi/ jojapeil/pub/integral-geometry-review.pdf. doi: 10.1515/9783110560855-004.

[14]

J. D. Ingle Jr and S. R. Crouch, Spectrochemical Analysis, Prentice Hall College Book Division, Old Tappan, NJ, USA, 1988.

[15]

O. Kallenberg, Foundations of Modern Probability, Probability and its Applications (New York). Springer-Verlag, New York, 1997.

[16]

V. P. Krishnan, R. Manna, S. K. Sahoo and V. A. Sharafutdinov, Momentum ray transforms, Inverse Problems and Imaging, 13 (2019), 679–701, URL http://aimsciences.org//article/id/d88823a5-827c-4c4b-909a-c27daa0b74ec. doi: 10.3934/ipi.2019031.

[17]

L. A. Lehmann and R. E. Alvarez, Energy-selective radiography a review, in Digital Radiography: Selected Topics (eds. J. G. Kereiakes, S. R. Thomas and C. G. Orton), Springer US, Boston, MA, (1986), 145–188. doi: 10.1007/978-1-4684-5068-2_7.

[18]

R. M. Lewitt and S. Matej, Overview of methods for image reconstruction from projections in emission computed tomography, Proceedings of the IEEE, 91 (2003), 1588-1611.  doi: 10.1109/JPROC.2003.817882.

[19]

C. H. McColloughS. LengL. Yu and J. G. Fletcher, Dual- and multi-energy CT: Principles, technical approaches, and clinical applications, Radiology, 276 (2015), 637-653.  doi: 10.1148/radiol.2015142631.

[20]

P. Milanfar, Geometric Estimation and Reconstruction from Tomographic Data, PhD thesis, Massachusetts Institute of Technology, 1993.

[21]

P. MilanfarW. C. Karl and A. S. Willsky, A moment-based variational approach to tomographic reconstruction, IEEE Transactions on Image Processing, 5 (1996), 459-470.  doi: 10.1109/83.491319.

[22]

J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, vol. 10 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972344.

[23]

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

[24]

R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Arkiv För Matematik, 40 (2002), 145–167. doi: 10.1007/BF02384507.

[25]

G. P. PaternainM. Salo and G. Uhlmann, Tensor tomography: Progress and challenges, Chinese Annals of Mathematics. Series B, 35 (2014), 399-428.  doi: 10.1007/s11401-014-0834-z.

[26]

G. P. PaternainM. 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.

[27]

W. Rudin, Real and Complex Analysis, McGraw–Hill Series in Higher Mathematics. McGraw–Hill Book Co., New York-Düsseldorf–Johannesburgn, 1974.

[28]

K. Schmüdgen, The Moment Problem, vol. 277 of Graduate Texts in Mathematics, Springer International Publishing, 2017.

[29]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. doi: 10.1515/9783110900095.

[30]

K. Taguchi and J. S. Iwanczyk, Vision 20/20: Single photon counting x-ray detectors in medical imaging, Medical Physics, 40 (2013), 100901. doi: 10.1118/1.4820371.

[31]

A. WelchG. T. GullbergP. E. ChristianJ. Li and B. M. Tsui, An investigation of dual energy transmission measurements in simultaneous transmission emission imaging, IEEE Transactions on Nuclear Science, 42 (1995), 2331-2338.  doi: 10.1109/23.489437.

[32]

D. V. Widder, The Laplace Transform, Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941.

Figure 1.  Unknown $ \rho_0 $ (dashed red line) and the numerical solution $ \rho $ (solid blue line) with 0.5% noise level. Example 1 (above) and example 2 (below) with Tikhonov-solution (left), TV-solution (middle) and CGLS-solution (right)
Figure 2.  Unknown $ \rho_0 $ (dashed red line), $ \rho_0 $ averaged over regions where $ p $ is constant (black dot-dash line) and the numerical solution $ \rho $ (solid blue line) with 0.5% noise level. Tikhonov-solution (left), TV-solution (middle) and CGLS-solution (right)
Figure 3.  Unknown $ \rho_0 $ (dashed red line) and the Tikhonov-solution $ \rho $ (solid blue line) with 0.5% noise level and smaller measurement intervals
Table 1.  Averaged relative errors and variances of one hundred solutions on smaller intervals with noise level 0.5%
Intervals $ (0,1) $ $ (0.2,0.8) $ $ (0.3,0.6) $ $ (0.4,0.5) $
$ \overline{\epsilon}_\mathrm{rel} $ 0.117 0.170 0.186 0.320
$ \mathrm{var} $ $ 1.88\cdot 10^{-3} $ $ 1.27\cdot 10^{-2} $ $ 1.27\cdot 10^{-2} $ $ 6.65\cdot 10^{-3} $
Intervals $ (0,1) $ $ (0.2,0.8) $ $ (0.3,0.6) $ $ (0.4,0.5) $
$ \overline{\epsilon}_\mathrm{rel} $ 0.117 0.170 0.186 0.320
$ \mathrm{var} $ $ 1.88\cdot 10^{-3} $ $ 1.27\cdot 10^{-2} $ $ 1.27\cdot 10^{-2} $ $ 6.65\cdot 10^{-3} $
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