Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year


IMPACT FACTOR 2016: 1.097

CiteScore 2016: 1.09

SCImago Journal Rank (SJR) 2016: 0.872
Source Normalized Impact per Paper (SNIP) 2016: 0.887

Mathematical Citation Quotient (MCQ) 2016: 0.75

Online
ISSN
1609-9389
See all formats and pricing
More options …
Ahead of print

Issues

Operator Learning Approach for the Limited View Problem in Photoacoustic Tomography

Florian Dreier / Sergiy Pereverzyev JrORCID iD: http://orcid.org/0000-0002-8627-3995 / Markus HaltmeierORCID iD: http://orcid.org/0000-0001-5715-0331
Published Online: 2018-04-10 | DOI: https://doi.org/10.1515/cmam-2018-0008

Abstract

In photoacoustic tomography, one is interested to recover the initial pressure distribution inside a tissue from the corresponding measurements of the induced acoustic wave on the boundary of a region enclosing the tissue. In the limited view problem, the wave boundary measurements are given on the part of the boundary, whereas in the full view problem, the measurements are known on the whole boundary. For the full view problem, there exist various fast and robust reconstruction methods. These methods give severe reconstruction artifacts when they are applied directly to the limited view data. One approach for reducing such artefacts is trying to extend the limited view data to the whole region boundary, and then use existing reconstruction methods for the full view data. In this paper, we propose an operator learning approach for constructing an operator that gives an approximate extension of the limited view data. We consider the behavior of a reconstruction formula on the extended limited view data that is given by our proposed approach. Approximation errors of our approach are analyzed. We also present numerical results with the proposed extension approach supporting our theoretical analysis.

Keywords: Photoacoustic Tomography; Wave Equation; Limited View Problem; Inversion Formula, Universal Back-Projection; Data Extension; Operator Learning

MSC 2010: 65R32; 35L05; 92C55

References

  • [1]

    M. Agranovsky, D. Finch and P. Kuchment, Range conditions for a spherical mean transform, Inverse Probl. Imaging 3 (2009), no. 3, 373–382. CrossrefGoogle Scholar

  • [2]

    M. A. Alvarez, L. Rosasco and N. D. Lawrence, Kernels for vector-valued functions: A review, Found. Trends Mach. Learn. 4 (2012), no. 3, 195–266. CrossrefGoogle Scholar

  • [3]

    G. Ambartsoumian and P. Kuchment, A range description for the planar circular Radon transform, SIAM J. Math. Anal. 38 (2006), no. 2, 681–692. CrossrefGoogle Scholar

  • [4]

    L. L. Barannyk, J. Frikel and L. V. Nguyen, On artifacts in limited data spherical Radon transform: Curved observation surface, Inverse Problems 32 (2016), no. 1, Article ID 015012. Google Scholar

  • [5]

    P. Beard, Biomedical photoacoustic imaging, Interf. Focus 1 (2011), no. 4, 602–631. CrossrefGoogle Scholar

  • [6]

    P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier and G. Paltauf, Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors, Inverse Problems 23 (2007), no. 6, S65–S80. CrossrefGoogle Scholar

  • [7]

    P. Burgholzer, G. J. Matt, M. Haltmeier and G. Paltauf, Exact and approximate imaging methods for photoacoustic tomography using an arbitrary detection surface, Phys. Rev. E 75 (2007), no. 4, Article ID 046706. Google Scholar

  • [8]

    R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II, Interscience, New York 1962. Google Scholar

  • [9]

    D. Finch, M. Haltmeier and R. Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM J. Appl. Math. 68 (2007), no. 2, 392–412. CrossrefGoogle Scholar

  • [10]

    D. Finch, S. K. Patch and R. Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal. 35 (2004), no. 5, 1213–1240. CrossrefGoogle Scholar

  • [11]

    D. Finch and R. Rakesh, The range of the spherical mean value operator for functions supported in a ball, Inverse Problems 22 (2006), no. 3, 923–938. CrossrefGoogle Scholar

  • [12]

    D. Finch and R. Rakesh, Recovering a function from its spherical mean values in two and three dimensions, Photoacoustic Imaging and Spectroscopy, CRC Press, Boca Raton (2009), 77–88. Google Scholar

  • [13]

    J. Frikel and E. T. Quinto, Characterization and reduction of artifacts in limited angle tomography, Inverse Problems 29 (2013), no. 12, Article ID 125007. Google Scholar

  • [14]

    J. Frikel and E. T. Quinto, Artifacts in incomplete data tomography with applications to photoacoustic tomography and sonar, SIAM J. Appl. Math. 75 (2015), no. 2, 703–725. CrossrefGoogle Scholar

  • [15]

    H. Grün, T. Berer, P. Burgholzer, R. Nuster and G. Paltauf, Three-dimensional photoacoustic imaging using fiber-based line detectors, J. Biomed. Optics 15 (2010), no. 2, Article ID 021306. Google Scholar

  • [16]

    M. Haltmeier, Frequency domain reconstruction for photo- and thermoacoustic tomography with line detectors, Math. Models Methods Appl. Sci. 19 (2009), no. 2, 283–306. CrossrefGoogle Scholar

  • [17]

    M. Haltmeier, Inversion of circular means and the wave equation on convex planar domains, Comput. Math. Appl. 65 (2013), no. 7, 1025–1036. CrossrefGoogle Scholar

  • [18]

    M. Haltmeier, Universal inversion formulas for recovering a function from spherical means, SIAM J. Math. Anal. 46 (2014), no. 1, 214–232. CrossrefGoogle Scholar

  • [19]

    M. Haltmeier and L. V. Nguyen, Analysis of iterative methods in photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci. 10 (2017), no. 2, 751–781. CrossrefGoogle Scholar

  • [20]

    M. Haltmeier and S. Pereverzyev Jr., Recovering a function from circular means or wave data on the boundary of parabolic domains, SIAM J. Imaging Sci. 8 (2015), no. 1, 592–610. CrossrefGoogle Scholar

  • [21]

    M. Haltmeier and S. Pereverzyev Jr., The universal back-projection formula for spherical means and the wave equation on certain quadric hypersurfaces, J. Math. Anal. Appl. 429 (2015), no. 1, 366–382. CrossrefGoogle Scholar

  • [22]

    T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, 2nd ed., Springer Ser. Statist., Springer, New York, 2009. Google Scholar

  • [23]

    G. T. Herman, Fundamentals of Computerized Tomography. Image Reconstruction from Projections, 2nd ed., Adv. Pattern Recognit., Springer, Dordrecht, 2009. Google Scholar

  • [24]

    Y. Hristova, P. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems 24 (2008), no. 5, Article ID 055006. Google Scholar

  • [25]

    C. Huang, K. Wang, L. Nie, L. V. Wang and M. A. Anastasio, Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media, IEEE Trans. Med. Imag. 32 (2013), no. 6, 1097–1110. CrossrefGoogle Scholar

  • [26]

    P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math. 19 (2008), no. 2, 191–224. Google Scholar

  • [27]

    P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography, Handbook of Mathematical Methods in Imaging. Vol. 1, 2, 3, Springer, New York (2015), 1117–1167. Google Scholar

  • [28]

    L. Kunyansky, Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra, Inverse Problems 27 (2011), no. 2, Article ID 025012. Google Scholar

  • [29]

    L. A. Kunyansky, A series solution and a fast algorithm for the inversion of the spherical mean Radon transform, Inverse Problems 23 (2007), no. 6, S11–S20. CrossrefGoogle Scholar

  • [30]

    L. A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems 23 (2007), no. 1, 373–383. CrossrefGoogle Scholar

  • [31]

    C. Li and L. V. Wang, Photoacoustic tomography and sensing in biomedicine, Phys. Med. Biol. 54 (2009), no. 19, R59–R97. CrossrefGoogle Scholar

  • [32]

    S. Matej and R. M. Lewitt, Practical considerations for 3-D image reconstruction using spherically symmetric volume elements, IEEE Trans. Med. Imag. 15 (1996), no. 1, 68–78. CrossrefGoogle Scholar

  • [33]

    C. A. Micchelli and M. Pontil, On learning vector-valued functions, Neural Comput. 17 (2005), no. 1, 177–204. CrossrefGoogle Scholar

  • [34]

    F. Natterer, Photo-acoustic inversion in convex domains, Inverse Probl. Imaging 6 (2012), no. 2, 315–320. CrossrefGoogle Scholar

  • [35]

    L. V. Nguyen, On a reconstruction formula for spherical Radon transform: A microlocal analytic point of view, Anal. Math. Phys. 4 (2014), no. 3, 199–220. CrossrefGoogle Scholar

  • [36]

    L. V. Nguyen, On artifacts in limited data spherical Radon transform: Flat observation surfaces, SIAM J. Math. Anal. 47 (2015), no. 4, 2984–3004. CrossrefGoogle Scholar

  • [37]

    G. Paltauf, R. Nuster, M. Haltmeier and P. Burgholzer, Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors, Inverse Problems 23 (2007), no. 6, S81–S94. CrossrefGoogle Scholar

  • [38]

    G. Paltauf, R. Nuster, M. Haltmeier and P. Burgholzer, Photoacoustic tomography using a Mach–Zehnder interferometer as an acoustic line detector, App. Opt. 46 (2007), no. 16, 3352–3358. CrossrefGoogle Scholar

  • [39]

    G. Paltauf, J. A. Viator, S. A. Prahl and S. L. Jacques, Iterative reconstruction algorithm for optoacoustic imaging, J. Acoust. Soc. Am. 112 (2002), no. 4, 1536–1544. CrossrefGoogle Scholar

  • [40]

    S. K. Patch, Thermoacoustic tomography – Consistency conditions and the partial scan problem, Phys. Med. Biol. 49 (2004), 2305–2315. CrossrefGoogle Scholar

  • [41]

    S. K. Patch, Photoacoustic and thermoacoustic tomography: Consistency conditions and the partial scan problem, Photoacoustic Imaging and Spectroscopy, CRC Press, Boca Raton (2009), 103–116. Google Scholar

  • [42]

    A. Rosenthal, V. Ntziachristos and D. Razansky, Acoustic inversion in optoacoustic tomography: A review, Curr. Med. Imag. Rev. 9 (2013), no. 4, 318–336. Google Scholar

  • [43]

    J. Schwab, S. Pereverzyev, Jr. and M. Haltmeier, A Galerkin least squares approach for photoacoustic tomography, SIAM J. Numer. Anal. 56 (2018), no. 1, 160–184. CrossrefGoogle Scholar

  • [44]

    P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems 25 (2009), no. 7, Article ID 075011. Google Scholar

  • [45]

    P. Stefanov and G. Uhlmann, Is a curved flight path in SAR better than a straight one?, SIAM J. Appl. Math. 73 (2013), no. 4, 1596–1612. CrossrefGoogle Scholar

  • [46]

    K. Wang, R. W. Schoonover, R. Su, A. Oraevsky and M. A. Anastasio, Discrete imaging models for three-dimensional optoacoustic tomography using radially symmetric expansion functions, IEEE Trans. Med. Imag. 33 (2014), no. 5, 1180–1193. CrossrefGoogle Scholar

  • [47]

    J. Xia, J. Yao and L. V. Wang, Photoacoustic tomography: Principles and advances, Prog. Electromagn. Res. 147 (2014), 1–22. CrossrefGoogle Scholar

  • [48]

    M. Xu and L. V. Wang, Universal back-projection algorithm for photoacoustic computed tomography, Phys. Rev. E 71 (2005), no. 1, Article ID 0167067. Google Scholar

  • [49]

    M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine, Rev. Sci. Instruments 77 (2006), no. 4, Article ID 041101. Google Scholar

  • [50]

    Y. Xu, L. V. Wang, G. Ambartsoumian and P. Kuchment, Reconstructions in limited-view thermoacoustic tomography, Med. Phys. 31 (2004), no. 4, 724–733. CrossrefGoogle Scholar

  • [51]

    Y. Xu, M. Xu and L. V. Wang, Exact frequency-domain reconstruction for thermoacoustic tomography–II: Cylindrical geometry, IEEE Trans. Med. Imag. 21 (2002), 829–833. CrossrefGoogle Scholar

  • [52]

    L. Yao and H. Jiang, Photoacoustic image reconstruction from few-detector and limited-angle data, Biomed. Opt. Express 2 (2011), no. 9, 2649–2654. CrossrefGoogle Scholar

  • [53]

    G. Zangerl, O. Scherzer and M. Haltmeier, Exact series reconstruction in photoacoustic tomography with circular integrating detectors, Commun. Math. Sci. 7 (2009), no. 3, 665–678. CrossrefGoogle Scholar

About the article

Received: 2017-09-04

Revised: 2018-03-09

Accepted: 2018-03-13

Published Online: 2018-04-10


Funding Source: Austrian Science Fund

Award identifier / Grant number: P 29514-N32

The authors gratefully acknowledge the support of the Tyrolean Science Fund (TWF). The second author gratefully acknowledges the support of the Austrian Science Fund (FWF), project P 29514-N32.


Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0008.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in