August  2020, 14(4): 665-682. doi: 10.3934/ipi.2020030

Thermoacoustic Tomography with circular integrating detectors and variable wave speed

Department of Mathematics, Purdue University, 150 N University St., West Lafayette, IN 47907, USA

Received  July 2019 Revised  February 2020 Published  May 2020

Fund Project: Author partly supported by NSF Grant DMS-1600327

We explore Thermoacoustic Tomography with circular integrating detectors assuming variable, smooth wave speed. We show that the measurement operator in this case is a Fourier Integral Operator and examine how the singularities in initial data and measured data are related through the canonical relation of this operator. We prove which of those singularities in the initial data are visible from a fixed open subset of the set on which measurements are taken. In addition, numerical results are shown for both full and partial data.

Citation: Chase Mathison. Thermoacoustic Tomography with circular integrating detectors and variable wave speed. Inverse Problems and Imaging, 2020, 14 (4) : 665-682. doi: 10.3934/ipi.2020030
References:
[1]

M. J. Grote and I. Sim, Efficient PML for the wave equation, preprint, arXiv: 1001.0319.

[2]

M. HaltmeierO. ScherzerP. Burgholzer and G. Paltauf, Thermoacoustic computed tomography with large planar receivers, Inverse Problems, 20 (2004), 1663-1673.  doi: 10.1088/0266-5611/20/5/021.

[3]

L. Hörmander, Fourier integral operators. Ⅰ., Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052.

[4]

L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-00136-9.

[5]

F. John, Partial Differential Equations, Applied Mathematical Sciences, 1, 1$^st$ edition, Springer-Verlag, New York-Berlin, 1971.

[6]

R. A. KrugerW. L. Kiser Jr.D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography using a conventional linear transducer array, Med. Phys., 30 (2003), 856-860.  doi: 10.1118/1.1565340.

[7]

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224.  doi: 10.1137/1.9781611973297.

[8]

A. A. Oraevsky, S. L. Jacques, R. O. Esenaliev and F. K. Tittel, Laser-based optoacoustic imaging in biological tissues, in Laser-Tissue Interaction V; and Ultraviolet Radiation Hazards, 2134, Proc. SPIE, 1994,122–129. doi: 10.1117/12.182927.

[9]

E. Quinto, Radon transforms on curves in the plane, Lect. Appl. Math., 30 (1994), 231-244. 

[10]

P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16 pp. doi: 10.1088/0266-5611/25/7/075011.

[11]

P. Stefanov and Y. Yang, Thermo- and photoacoustic tomography with variable speed and planar detectors, SIAM J. Math. Anal., 49 (2017), 297-310.  doi: 10.1137/16M1073716.

[12]

M. E. Taylor, Pseudodifferential Operators, in Princeton Mathematical Series, 34, Princeton University Press, Princeton, NJ, 1981,146–191.

[13]

M. E. Taylor, Partial differential equations. I. Basic theory, in Applied Mathematical Sciences, 115, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4419-7055-8.

[14]

F. Trèves, Introduction to pseudodifferential and Fourier integral operators, in The University Series in Mathematics, 2, Plenum Press, New York-London, 1980.

[15]

M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine, Review of Scientific Instruments, 77 (2006), 041101. doi: 10.1063/1.2195024.

[16]

G. ZangerlO. Scherzer and M. Haltmeier, Circular integrating detectors in photo and thermoacoustic tomography, Inverse Probl. Sci. Eng., 17 (2009), 133-142.  doi: 10.1080/17415970802166782.

[17]

G. ZangerlO. Scherzer and M. Haltmeier, Exact series reconstruction in photoacoustic tomography with circular integrating detectors, Commun. Math. Sci., 7 (2009), 665-678.  doi: 10.4310/CMS.2009.v7.n3.a8.

show all references

References:
[1]

M. J. Grote and I. Sim, Efficient PML for the wave equation, preprint, arXiv: 1001.0319.

[2]

M. HaltmeierO. ScherzerP. Burgholzer and G. Paltauf, Thermoacoustic computed tomography with large planar receivers, Inverse Problems, 20 (2004), 1663-1673.  doi: 10.1088/0266-5611/20/5/021.

[3]

L. Hörmander, Fourier integral operators. Ⅰ., Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052.

[4]

L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-00136-9.

[5]

F. John, Partial Differential Equations, Applied Mathematical Sciences, 1, 1$^st$ edition, Springer-Verlag, New York-Berlin, 1971.

[6]

R. A. KrugerW. L. Kiser Jr.D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography using a conventional linear transducer array, Med. Phys., 30 (2003), 856-860.  doi: 10.1118/1.1565340.

[7]

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224.  doi: 10.1137/1.9781611973297.

[8]

A. A. Oraevsky, S. L. Jacques, R. O. Esenaliev and F. K. Tittel, Laser-based optoacoustic imaging in biological tissues, in Laser-Tissue Interaction V; and Ultraviolet Radiation Hazards, 2134, Proc. SPIE, 1994,122–129. doi: 10.1117/12.182927.

[9]

E. Quinto, Radon transforms on curves in the plane, Lect. Appl. Math., 30 (1994), 231-244. 

[10]

P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16 pp. doi: 10.1088/0266-5611/25/7/075011.

[11]

P. Stefanov and Y. Yang, Thermo- and photoacoustic tomography with variable speed and planar detectors, SIAM J. Math. Anal., 49 (2017), 297-310.  doi: 10.1137/16M1073716.

[12]

M. E. Taylor, Pseudodifferential Operators, in Princeton Mathematical Series, 34, Princeton University Press, Princeton, NJ, 1981,146–191.

[13]

M. E. Taylor, Partial differential equations. I. Basic theory, in Applied Mathematical Sciences, 115, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4419-7055-8.

[14]

F. Trèves, Introduction to pseudodifferential and Fourier integral operators, in The University Series in Mathematics, 2, Plenum Press, New York-London, 1980.

[15]

M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine, Review of Scientific Instruments, 77 (2006), 041101. doi: 10.1063/1.2195024.

[16]

G. ZangerlO. Scherzer and M. Haltmeier, Circular integrating detectors in photo and thermoacoustic tomography, Inverse Probl. Sci. Eng., 17 (2009), 133-142.  doi: 10.1080/17415970802166782.

[17]

G. ZangerlO. Scherzer and M. Haltmeier, Exact series reconstruction in photoacoustic tomography with circular integrating detectors, Commun. Math. Sci., 7 (2009), 665-678.  doi: 10.4310/CMS.2009.v7.n3.a8.

Figure 1.  Two different experimental setups shown depending on the radius of the integrating detector. On the left is the small radius case, and on the right is the large radius case
Figure 2.  Singularities that may be visible from $ \theta_0 \in \Gamma $ in both the cases (left) $ R-r > 1 $ and (right) $ R = 1, r>2 $ will lie on the geodesics issued from the integrating detectors
Figure 3.  Variable wave speed of $ 1+0.3\sin(8x)\cos(5y)\eta(x,y) $, where $ \eta(x,y)\in C_0^\infty(B_1(0)) $
Figure 4.  Results of reconstruction using $ R = 1 $ and $ r = 2 $ model (Large radius detector model). This reconstruction was made using full data
Figure 5.  Result of reconstruction with partial data using $ R = 2 $, and $ r = 0.8 $ (Small radius detector model). This reconstruction was for $ \theta \in (-\pi/2, 0) $. Shown in the figure are the set on which data is collected as well as some representative circular integrating detectors
Figure 6.  Result of reconstruction with partial data using $ R = 1 $, and $ r = 2 $ (Large radius detector model). This reconstruction was for $ \theta \in (-\pi/2,0) $. Shown in the figure are the set on which data is collected as well as some representative circular integrating detectors
[1]

James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems and Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013

[2]

Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021

[3]

Victor Palamodov. Remarks on the general Funk transform and thermoacoustic tomography. Inverse Problems and Imaging, 2010, 4 (4) : 693-702. doi: 10.3934/ipi.2010.4.693

[4]

Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure and Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1

[5]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems and Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023

[6]

Gaik Ambartsoumian, Leonid Kunyansky. Exterior/interior problem for the circular means transform with applications to intravascular imaging. Inverse Problems and Imaging, 2014, 8 (2) : 339-359. doi: 10.3934/ipi.2014.8.339

[7]

Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems and Imaging, 2007, 1 (1) : 1-11. doi: 10.3934/ipi.2007.1.1

[8]

Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7

[9]

Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems and Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341

[10]

Linh V. Nguyen. A family of inversion formulas in thermoacoustic tomography. Inverse Problems and Imaging, 2009, 3 (4) : 649-675. doi: 10.3934/ipi.2009.3.649

[11]

Shanshan Wang, Yanxia Chen, Taohui Xiao, Lei Zhang, Xin Liu, Hairong Zheng. LANTERN: Learn analysis transform network for dynamic magnetic resonance imaging. Inverse Problems and Imaging, 2021, 15 (6) : 1363-1379. doi: 10.3934/ipi.2020051

[12]

Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems and Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649

[13]

C E Yarman, B Yazıcı. A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group. Inverse Problems and Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457

[14]

Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of borehole seismic data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022026

[15]

Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems and Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879

[16]

Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems and Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111

[17]

Martin Bauer, Thomas Fidler, Markus Grasmair. Local uniqueness of the circular integral invariant. Inverse Problems and Imaging, 2013, 7 (1) : 107-122. doi: 10.3934/ipi.2013.7.107

[18]

Venkateswaran P. Krishnan, Eric Todd Quinto. Microlocal aspects of common offset synthetic aperture radar imaging. Inverse Problems and Imaging, 2011, 5 (3) : 659-674. doi: 10.3934/ipi.2011.5.659

[19]

Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029

[20]

Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of Doppler synthetic aperture radar. Inverse Problems and Imaging, 2019, 13 (6) : 1283-1307. doi: 10.3934/ipi.2019056

2021 Impact Factor: 1.483

Metrics

  • PDF downloads (216)
  • HTML views (146)
  • Cited by (0)

Other articles
by authors

[Back to Top]