A time reversal algorithm in acoustic media with Dirac measure approximations

This article is devoted to the study of a photoacoustic tomography model, where one is led to consider the solution of the acoustic wave equation with a source term writing as a separated variables function in time and space, whose temporal component is in some sense close to the derivative of the Dirac distribution at t = 0. This models a continuous wave laser illumination performed during a short interval of time. We introduce an algorithm for reconstructing the space component of the source term from the measure of the solution recorded by sensors during a time T all along the boundary of a connected bounded domain. It is based at the same time on the introduction of an auxiliary equivalent Cauchy problem allowing to derive explicit reconstruction formula and then to use of a deconvolution procedure. Numerical simulations illustrate our approach. Finally, this algorithm is also extended to elasticity wave systems.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  Lihong V. Wang,et al.  Photoacoustic imaging in biomedicine , 2006 .

[3]  Habib Ammari,et al.  Mathematical Modeling in Photoacoustic Imaging of Small Absorbers , 2010, SIAM Rev..

[4]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[5]  Linh V. Nguyen,et al.  Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media , 2008 .

[6]  R. Kowar.  Integral equation models for thermoacoustic imaging of acoustic dissipative tissue , 2010, 1002.4731.

[7]  K. Richard,et al.  Causality Analysis of Frequency Dependent Wave Attenuation , 2009, 0906.4678.

[8]  Otmar Scherzer,et al.  Filtered backprojection for thermoacoustic computed tomography in spherical geometry , 2005, Mathematical Methods in the Applied Sciences.

[9]  Habib Ammari,et al.  Transient elasticity imaging and time reversal , 2011, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[10]  L. Maccone,et al.  Tomography , 1940, British medical journal.

[11]  Otmar Scherzer,et al.  A direct method for photoacoustic tomography with inhomogeneous sound speed , 2015, 1507.01741.

[12]  Tianheng Wang,et al.  Application of laser pulse stretching scheme for efficiently delivering laser energy in photoacoustic imaging. , 2012, Journal of biomedical optics.

[13]  Habib Ammari,et al.  Photoacoustic Imaging for Attenuating Acoustic Media , 2012 .

[14]  Josselin Garnier,et al.  Time-reversal algorithms in viscoelastic media† , 2013, European Journal of Applied Mathematics.

[15]  B. T. Cox,et al.  The challenges for quantitative photoacoustic imaging , 2009, BiOS.

[16]  Yulia Hristova,et al.  Time reversal in thermoacoustic tomography—an error estimate , 2008, 0812.0606.

[17]  Mickael Tanter,et al.  The role of viscosity in the impulse diffraction field of elastic waves induced by the acoustic radiation force , 2004, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[18]  M. Nikolova.  An Algorithm for Total Variation Minimization and Applications , 2004 .

[19]  M. Bergounioux,et al.  An optimal control problem in photoacoustic tomography , 2014 .

[20]  Patrick L. Combettes,et al.  Signal Recovery by Proximal Forward-Backward Splitting , 2005, Multiscale Model. Simul..

[21]  S Catheline,et al.  Time reversal of elastic waves in soft solids. , 2008, Physical review letters.

[22]  S. Emelianov,et al.  Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics. , 1998, Ultrasound in medicine & biology.

[23]  Tan Liu,et al.  Photoacoustic generation by multiple picosecond pulse excitation. , 2010, Medical physics.

[24]  Markus Haltmeier,et al.  Inversion of Spherical Means and the Wave Equation in Even Dimensions , 2007, SIAM J. Appl. Math..

[25]  Karl Kunisch,et al.  Optimal Control of the Undamped Linear Wave Equation with Measure Valued Controls , 2016, SIAM J. Control. Optim..

[26]  Habib Ammari,et al.  A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements , 2007 .

[27]  M. Bergounioux,et al.  Quantitative thermoacoustic tomography with microwaves sources , 2017 .

[28]  J. Greenleaf,et al.  Selected methods for imaging elastic properties of biological tissues. , 2003, Annual review of biomedical engineering.

[29]  M. Fink,et al.  Mathematical foundations of the time reversal mirror , 2002 .

[30]  Josselin Garnier,et al.  Time reversal in attenuating acoustic media , 2010 .

[31]  Guillaume Bal,et al.  Multi-source quantitative photoacoustic tomography in a diffusive regime , 2011 .

[32]  Otmar Scherzer,et al.  Variational Methods in Imaging , 2008, Applied mathematical sciences.

[33]  M. Fink,et al.  Time reversal acoustics , 1996, IEEE Ultrasonics Symposium, 2004.

[34]  Habib Ammari,et al.  An Introduction to Mathematics of Emerging Biomedical Imaging , 2008 .

[35]  Guillaume Bal,et al.  Inverse diffusion theory of photoacoustics , 2009, 0910.2503.

[36]  Time Reversing Waves For Biomedical Applications , 2009 .

[37]  Josselin Garnier,et al.  Mathematical Methods in Elasticity Imaging , 2015 .

[38]  Markus Haltmeier,et al.  Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors , 2007 .

[39]  L. Evans,et al.  Partial Differential Equations , 1941 .

[40]  L. Kunyansky,et al.  Explicit inversion formulae for the spherical mean Radon transform , 2006, math/0609341.

[41]  Jaime H. Ortega,et al.  A source time reversal method for seismicity induced by mining , 2017 .

[42]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[43]  Habib Ammari,et al.  Transient anomaly imaging by the acoustic radiation force , 2010 .

[44]  Linh V. Nguyen.  A family of inversion formulas in thermoacoustic tomography , 2009, 0902.2579.