Show Summary Details
More options …

# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

6 Issues per year

IMPACT FACTOR 2017: 0.941
5-year IMPACT FACTOR: 0.953

CiteScore 2017: 0.91

SCImago Journal Rank (SJR) 2017: 0.461
Source Normalized Impact per Paper (SNIP) 2017: 1.022

Mathematical Citation Quotient (MCQ) 2016: 0.38

Online
ISSN
1569-3945
See all formats and pricing
More options …

# Photoacoustic tomography with spatially varying compressibility and density

Zakaria Belhachmi
• Laboratoire de Mathématiques LMIA, Université de Haute Alsace, 4, rue des Frères Lumière, 68200 Mulhouse, France
• Email
• Other articles by this author:
/ Thomas Glatz
• Corresponding author
• Computational Science Center, University of Vienna, Oskar-Morgenstern Platz 1, A-1090 Vienna, Austria
• Email
• Other articles by this author:
/ Otmar Scherzer
• Computational Science Center, University of Vienna, Oskar-Morgenstern Platz 1, A-1090 Vienna, Austria; and Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria
• Email
• Other articles by this author:
Published Online: 2016-09-29 | DOI: https://doi.org/10.1515/jiip-2015-0113

## Abstract

This paper investigates photoacoustic tomography with two spatially varying acoustic parameters, the compressibility and the density. We consider the reconstruction of the absorption density parameter (imaging parameter of Photoacoustics) with complete and partial measurement data. We investigate and analyze three different numerical methods for solving the imaging problem and compare the results.

MSC 2010: 65J22; 35R30

## References

• [1]

• [2]

Agranovsky M. and Kuchment P., Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed, Inverse Problems 23 (2007), no. 5, 2089–2102. Google Scholar

• [3]

Aubin T., Nonlinear Analysis on Manifolds, Monge–Ampère Equations, Grundlehren Math. Wiss. 252, Springer, Berlin, 1982. Google Scholar

• [4]

Bao G. and Symes W. W., A trace theorem for solutions of partial differential equations, Math. Methods Appl. Sci. 14 (1991), no. 8, 553–562. Google Scholar

• [5]

Bao G. and Symes W. W., Trace regularity for a second order hyperbolic equation with non-smooth coefficients, J. Math. Anal. Appl. 174 (1993), 370–389. Google Scholar

• [6]

Belhachmi Z., Glatz T. and Scherzer O., A direct method for photoacoustic tomography with inhomogeneous sound speed, Inverse Problems 32 (2016), no. 4, Article ID 045005. Google Scholar

• [7]

Colton D. and Kress R., Inverse Acoustic and Electromagnetic Scattering Theory, Appl. Math. Sci. 93, Springer, Berlin, 1992. Google Scholar

• [8]

Engl H. W., Hanke M. and Neubauer A., Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic Publishers, Dordrecht, 1996. Google Scholar

• [9]

Evans L. C., Partial Differential Equations, 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, 2010. Google Scholar

• [10]

Groetsch C. W., The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984. Google Scholar

• [11]

Hanke M., Conjugate Gradient Type Methods for Ill-Posed Problems, Pitman Res. Notes in Math. 327, Longman Scientific & Technical, Harlow, 1995. Google Scholar

• [12]

Hörmander L., A uniqueness theorem for second order hyperbolic differential equations, Comm. Partial Differential Equations 17 (1992), 699–714. Google Scholar

• [13]

Hristova Y., Time reversal in thermoacoustic tomography—An error estimate, Inverse Problems 25 (2009), no. 5, Article ID 055008. Google Scholar

• [14]

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

• [15]

Huang C., Nie L., Schoonover R. W., Wang L. and Anastasio M. A., Photoacoustic computed tomography correcting for heterogeneity and attenuation, J. Biomed. Opt. 17 (2012), Article ID 061211. Google Scholar

• [16]

Kuchment P., The Radon Transform and Medical Imaging, Society for Industrial and Applied Mathematics, Philadelphia, 2014. Google Scholar

• [17]

Kuchment P. and Kunyansky L., Mathematics of thermoacoustic tomography, European J. Appl. Math. 19 (2008), 191–224. Google Scholar

• [18]

Nashed M., Generalized Inverses and Applications, Academic Press, New York, 1976. Google Scholar

• [19]

Qia J., Stefanov P., Uhlmann G. and Zhao H., An efficient Neumann series-based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci. 4 (2011), no. 3, 850–883. Google Scholar

• [20]

Robbiano L., Théorème d’unicité adapté au contrôle des solutions des problèmes hyperboliques, Comm. Partial Differential Equations 16 (1991), no. 4–5, 789–800. Google Scholar

• [21]

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

• [22]

Tataru D., Unique continuation for solutions to PDEs; between Hörmander’s theorem and Holmgren’s theorem, Comm. Partial Differential Equations 20 (1995), no. 5–6, 855–884. Google Scholar

• [23]

Tataru D., On the regularity of boundary traces for the wave equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 26 (1998), no. 1, 185–206. Google Scholar

• [24]

Tittelfitz J., Thermoacoustic tomography in elastic media, Inverse Problems 28 (2012), no. 5, Article ID 055004. Google Scholar

• [25]

Treeby B. E. and Cox B. T., K-Wave: MATLAB toolbox for the simulation and reconstruction of photoacoustic wace fields, J. Biomed. Opt. 15 (2010), Article ID 021314.

• [26]

Treeby B. E., Varslot T. K., Zhang E. Z., Laufer J. G. and Beard P. C., Automatic sound speed selection in photoacoustic image reconstruction using an autofocus approach, J. Biomed. Opt. 16 (2011), no. 9, 090501–3. Google Scholar

• [27]

Vainberg B. R., On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as $t\to \mathrm{\infty }$ of solutions of non-stationary problems, Russian Math. Surveys 30 (1975), no. 2, 1–58. Google Scholar

• [28]

Wang L. V., Photoacoustic Imaging and Spectroscopy, Opt. Sci. Eng., CRC Press, Boca Raton, 2009. Google Scholar

• [29]

Xu M. and Wang L. V., Exact frequency-domain reconstruction for thermoacoustic tomography–I: Planar geometry, IEEE Trans. Med. Imaging 21 (2002), no. 7, 823–828. Google Scholar

• [30]

Xu M. and Wang L. V., Time-domain reconstruction for thermoacoustic tomography in a spherical geometry, IEEE Trans. Med. Imaging 21 (2002), no. 7, 814–822. Google Scholar

• [31]

Xu M. and Wang L. V., Universal back-projection algorithm for photoacoustic computed tomography, Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71 (2005), no. 1, Article ID 016706.

• [32]

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

• [33]

Xu M., Xu Y. and Wang L. V., Time-domain reconstruction algorithms and numerical simulations for thermoacoustic tomography in various geometries, IEEE Trans. Biomed. Eng. 50 (2003), no. 9, 1086–1099. Google Scholar

• [34]

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

Revised: 2016-07-08

Accepted: 2016-09-08

Published Online: 2016-09-29

Published in Print: 2017-02-01

Funding Source: Austrian Science Fund

Award identifier / Grant number: P26687-N25

The work of Thomas Glatz and Otmar Scherzer is supported by the Austrian Science Fund (FWF), Project P26687-N25 Interdisciplinary Coupled Physics Imaging.

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 1, Pages 119–133, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

Export Citation