Full Field Inversion in Photoacoustic Tomography with Variable Sound Speed
Abstract
:1. Introduction
2. Full Field Detection Photoacoustic Tomography
2.1. Mathematical Model
2.2. Description of the Inverse Problem
2.3. TwoStep Reconstruction
 ■
 Inverse Radon transform: Assume that projection data ${g}_{R}={\mathbf{X}}_{R}{\mathbf{W}}_{T}f$ are given and consider the zero extension $g:[0,\pi ]\times {\mathbb{R}}^{2}\to \mathbb{R}$ by $g(\theta ,x,z)={g}_{R}(\theta ,x,z)$ for $(\theta ,x,z)\in {M}_{R}$ and $g(\theta ,x,z)=0$ otherwise. We then define an approximation to ${\mathbf{W}}_{T}f$ by applying an inversion formula of the Radon transform in planes ${\mathbb{R}}^{2}\times \left\{z\right\}$. Using the wellknown filtered backprojection inversion formula for the Radon transform [42] yields$${\mathbf{W}}_{T}f(x,y,z)\simeq {\mathbf{X}}^{\u266f}g(x,y,z)\frac{1}{2{\pi}^{2}}{\int}_{0}^{\pi}\mathrm{P}.\mathrm{V}.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\int}_{\mathbb{R}}\frac{\left(\right)open="("\; close=")">{\partial}_{s}g}{(\theta ,s,z)}(xcos(\theta )+ysin(\theta \left)\right)s$$
 ■
 Final time wave inversion: For the second step, assume that an approximation $h\simeq {\mathbf{W}}_{T}f$ to the 3D acoustic field at time T is given. Recovering the initial pressure then yields the final time wave inversion problem$$\mathrm{Recover}\phantom{\rule{4.pt}{0ex}}f\phantom{\rule{4.pt}{0ex}}\mathrm{from}\phantom{\rule{4.pt}{0ex}}\mathrm{data}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}h={\mathbf{W}}_{T}f+\mathrm{noise}\phantom{\rule{0.166667em}{0ex}}.$$To the best of our knowledge, the final time wave inversion problem (9) has not been considered so far and its investigation will be the main theoretical focus of this work.
3. Final Time Wave Inversion Problem for Variable Sound Speed
3.1. Uniqueness and Stability Theorem
3.2. Proof of Theorem 1
3.3. Continuous Adjoint Operator
3.4. Application of the Steepest 6hhod
Algorithm 1: Steepest descent method for solving ${\mathbf{W}}_{T}f=g$. 

4. Numerical Simulations
4.1. Discretization
4.2. Data Simulation
4.3. Reconstruction Results
4.4. Quantitative Error Analysis
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. kSpace Method
Algorithm A1: kspace method for numerically solving (1)–(3). 

References
 Beard, P. Biomedical photoacoustic imaging. Interface Focus 2011, 1, 602–631. [Google Scholar] [CrossRef] [PubMed][Green Version]
 Kruger, R.A.; Kopecky, K.K.; Aisen, A.M.; Reinecke, R.D.; Kruger, G.A.; Kiser, W.L. Thermoacoustic CT with Radio waves: A medical imaging paradigm. Radiology 1999, 200, 275–278. [Google Scholar] [CrossRef] [PubMed]
 Wang, L.V. Multiscale photoacoustic microscopy and computed tomography. Nat. Photonics 2009, 3, 503–509. [Google Scholar] [CrossRef][Green Version]
 Wang, K.; Anastasio, M. Photoacoustic and thermoacoustic tomography: Image formation principles. In Handbook of Mathematical Methods in Imaging; Springer: Berlin, Germany, 2011; pp. 781–815. [Google Scholar]
 Xu, M.; Wang, L.V. Photoacoustic imaging in biomedicine. Rev. Sci. Instrum. 2006, 77, 041101. [Google Scholar] [CrossRef][Green Version]
 Agranovsky, M.; Kuchment, P.; Kunyansky, L. On reconstruction formulas and algorithms for the thermoacoustic tomography. In Photoacoustic Imaging and Spectroscopy; Wang, L.V., Ed.; CRC Press: Boca Raton, FL, USA, 2009; pp. 89–101. [Google Scholar]
 Filbir, F.; Kunis, S.; Seyfried, R. Effective discretization of direct reconstruction schemes for photoacoustic imaging in spherical geometries. SIAM J. Numer. Anal. 2014, 52, 2722–2742. [Google Scholar] [CrossRef]
 Finch, D. The spherical mean value operator with centers on a sphere. Inverse Probl. 2007, 23, 37–49. [Google Scholar] [CrossRef]
 Finch, D.; Haltmeier, M. Rakesh Inversion of spherical means and the wave equation in even dimensions. SIAM J. Appl. Math. 2007, 68, 392–412. [Google Scholar] [CrossRef]
 Finch, D.; Patch, S.K. Determining a function from its mean values over a family of spheres. SIAM J. Math. Anal. 2004, 35, 1213–1240. [Google Scholar] [CrossRef]
 Haltmeier, M. Universal inversion formulas for recovering a function from spherical means. SIAM J. Math. Anal. 2014, 46, 214–232. [Google Scholar] [CrossRef]
 Haltmeier, M. Exact Reconstruction Formula for the Spherical Mean Radon Transform on Ellipsoids. Inverse Probl. 2014, 30, 035001. [Google Scholar] [CrossRef]
 Haltmeier, M.; Pereverzyev, S., Jr. The universal backprojection formula for spherical means and the wave equation on certain quadric hypersurfaces. J. Math. Anal. Appl. 2015, 429, 366–382. [Google Scholar] [CrossRef]
 Haltmeier, M.; Schuster, T.; Scherzer, O. Filtered backprojection for thermoacoustic computed tomography in spherical geometry. Math. Methods Appl. Sci. 2005, 28, 1919–1937. [Google Scholar] [CrossRef][Green Version]
 Kuchment, P.; Kunyansky, L.A. Mathematics of thermoacoustic and photoacoustic tomography. Eur. J. Appl. Math. 2008, 19, 191–224. [Google Scholar] [CrossRef]
 Kunyansky, L.A. Explicit inversion formulae for the spherical mean Radon transform. Inverse Probl. 2007, 23, 373–383. [Google Scholar] [CrossRef][Green Version]
 Kunyansky, L.A. A series solution and a fast algorithm for the inversion of the spherical mean Radon transform. Inverse Probl. 2007, 23, S11–S20. [Google Scholar] [CrossRef][Green Version]
 Natterer, F. Photoacoustic inversion in convex domains. Inverse Probl. Imaging 2012, 6, 315–320. [Google Scholar] [CrossRef][Green Version]
 Nguyen, L.V. A family of inversion formulas for thermoacoustic tomography. Inverse Probl. Imaging 2009, 3, 649–675. [Google Scholar] [CrossRef]
 Xu, M.; Wang, L.V. Universal backprojection algorithm for photoacoustic computed tomography. Phys. Rev. E 2005, 71, 016706. [Google Scholar] [CrossRef] [PubMed]
 Haltmeier, M.; Zangerl, G. Spatial resolution in photoacoustic tomography: Effects of detector size and detector bandwidth. Inverse Probl. 2010, 26, 125002. [Google Scholar] [CrossRef]
 Roitner, H.; Haltmeier, M.; Nuster, R.; O’Leary, D.P.; Berer, T.; Paltauf, G.; Grün, H.; Burgholzer, P. Deblurring algorithms accounting for the finite detector size in photoacoustic tomography. J. Biomed. Opt. 2014, 19, 056011. [Google Scholar] [CrossRef] [PubMed]
 Rosenthal, A.; Ntziachristos, V.; Razansky, D. Modelbased optoacoustic inversion with arbitraryshape detectors. Med. Phys. 2011, 38, 4285–4295. [Google Scholar] [CrossRef]
 Wang, L.V. An imaging model incorporating ultrasonic transducer properties for threedimensional optoacoustic tomography. IEEE Trans. Med. Imaging 2011, 30, 203–214. [Google Scholar] [CrossRef] [PubMed]
 Burgholzer, P.; Hofer, C.; Paltauf, G.; Haltmeier, M.; Scherzer, O. Thermoacoustic tomography with integrating area and line detectors. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 2005, 52, 1577–1583. [Google Scholar] [CrossRef] [PubMed]
 Burgholzer, P.; BauerMarschallinger, J.; Grün, H.; Haltmeier, M.; Paltauf, G. Temporal backprojection algorithms for photoacoustic tomography with integrating line detectors. Inverse Probl. 2007, 23, S65–S80. [Google Scholar] [CrossRef]
 Haltmeier, M.; Scherzer, O.; Burgholzer, P.; Paltauf, G. Thermoacoustic computed tomography with large planar receivers. Inverse Probl. 2004, 20, 1663. [Google Scholar] [CrossRef]
 Paltauf, G.; Nuster, R.; Haltmeier, M. Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors. Inverse Probl. 2007, 23, S81–S94. [Google Scholar] [CrossRef][Green Version]
 Zangerl, G.; Scherzer, O.; Haltmeier, M. Exact series reconstruction in photoacoustic tomography with circular integrating detectors. Commun. Math. Sci. 2009, 7, 665–678. [Google Scholar][Green Version]
 Nuster, R.; Zangerl, G.; Haltmeier, M.; Paltauf, G. Full field detection in photoacoustic tomography. Opt. Express 2010, 18, 6288–6299. [Google Scholar] [CrossRef]
 Nuster, R.; Slezak, P.; Paltauf, G. High resolution threedimensional photoacoustic tomography with CCDcamera based ultrasound detection. Biomed. Opt. Express 2014, 5, 2635–2647. [Google Scholar] [CrossRef]
 Niederhauser, J.J.; Weber, D.F.H.P.; Frenz, M. Realtime optoacoustic imaging using a Schlieren transducer. Appl. Phys. Lett. 2002, 81, 571–573. [Google Scholar] [CrossRef]
 Niederhauser, J.J.; Jäger, M.; Frenz, M. Realtime threedimensional optoacoustic imaging using an acoustic lens system. Appl. Phys. Lett. 2004, 85, 846–848. [Google Scholar] [CrossRef]
 Jin, X.; Wang, L.V. Thermoacoustic tomography with correction for acoustic speed variations. Phys. Med. Biol. 2006, 51, 6437. [Google Scholar] [CrossRef] [PubMed]
 Ku, G.; Fornage, B.D.; Xing, J.; Xu, M.; Hunt, K.K.; Wang, L.V. Thermoacoustic and photoacoustic tomography of thick biological tissues toward breast imaging. Med. Phys. 1995, 22, 1605–1609. [Google Scholar] [CrossRef] [PubMed]
 Belhachmi, Z.; Glatz, T.; Scherzer, O. A direct method for photoacoustic tomography with inhomogeneous sound speed. Inverse Probl. 2016, 32, 045005. [Google Scholar] [CrossRef][Green Version]
 Arridge, S.R.; Betcke, M.M.; Cox, B.T.; Lucka, F.; Treeby, B.E. On the adjoint operator in photoacoustic tomography. Inverse Probl. 2016, 32, 115012. [Google Scholar] [CrossRef][Green Version]
 Haltmeier, M.; Nguyen, L.V. Analysis of Iterative Methods in Photoacoustic Tomography with variable Sound Speed. SIAM J. Imaging Sci. 2017, 19, 751–781. [Google Scholar] [CrossRef]
 Huang, C.; Wang, K.; Nie, L.; Wang, L.V.; Anastasio, M.A. Fullwave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media. IEEE Trans. Med. Imaging 2013, 32, 1097–1110. [Google Scholar] [CrossRef] [PubMed]
 Nguyen, L.V.; Haltmeier, M. Reconstruction algorithms for photoacoustic tomography in heterogenous damping media. arXiv, 2018; arXiv:1808.06176v1. [Google Scholar]
 Stefanov, P.; Uhlmann, G. Thermoacoustic tomography with variable sound speed. Inverse Probl. 2009, 25, 075011. [Google Scholar] [CrossRef][Green Version]
 Natterer, F. The Mathematics of Computerized Tomography; SIAM: Philadelphia, PA, USA, 1986. [Google Scholar]
 Kuchment, P. The Radon Transform and Medical Imaging; SIAM: Philadelphia, PA, USA, 2014; Volume 85. [Google Scholar]
 Quinto, E. Singular value decompositions and inversion methods for the exterior Radon transform and a spherical transform. J. Math. Anal. Appl. 1983, 95, 437–448. [Google Scholar] [CrossRef][Green Version]
 Quinto, E. Tomographic reconstructions from incomplete datanumerical inversion of the exterior Radon transform. Inverse Probl. 1988, 4, 867. [Google Scholar] [CrossRef]
 Vainberg, B. On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as t→∞ of solutions of nonstationary problems. Rus. Math. Surv. 1975, 30, 1–58. [Google Scholar] [CrossRef]
 Tréves, F. Introduction to Pseudodifferential and Fourier Integral Operators Volume 2: Fourier Integral Operators; Springer Science & Business Media: Berlin, Germany, 1980. [Google Scholar]
 John, F. Partial Differential Equations. In Applied Mathematical Sciences, 4th ed.; Springer: New York, NY, USA, 1982; Volume 1. [Google Scholar]
 Nguyen, L.V. On singularities and instability of reconstruction in thermoacoustic tomography. Tomogr. Inverse Transp. Theory Contemp. Math. 2011, 559, 163–170. [Google Scholar]
 Hörmander, L. Fourier integral operators. I. Acta Math. 1971, 127, 79–183. [Google Scholar] [CrossRef]
 Taylor, M.E. Pseudodifferential Operators, Volume 34 of Princeton Mathematical Series; Princeton University Press: Princeton, NJ, USA; Springer Science & Business Media: Berlin, Germany, 1981. [Google Scholar]
 Cox, B.T.; Kara, S.; Arridge, S.R.; Beard, P.C. kspace propagation models for acoustically heterogeneous media: Application to biomedical photoacoustics. J. Acoust. Soc. Am. 2007, 121, 3453–3464. [Google Scholar] [CrossRef]
 Mast, T.D.; Souriau, L.P.; Liu, D.L.; Tabei, M.; Nachman, A.I.; Waag, R.C. A kspace method for largescale models of wave propagation in tissue. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 2002, 48, 341–354. [Google Scholar] [CrossRef]
$\frac{\parallel \mathbf{g}{\mathbf{g}}_{\u2605}\parallel}{\parallel {\mathbf{g}}_{\u2605}\parallel}$  $\frac{\parallel {\mathcal{X}}^{\u266f}\mathbf{g}{\mathbf{h}}_{\u2605}\parallel}{\parallel {\mathbf{h}}_{\u2605}\parallel}$  $\frac{\parallel \mathcal{W}{\mathbf{f}}_{\mathit{rec}}{\mathbf{h}}_{\u2605}\parallel}{\parallel {\mathbf{h}}_{\u2605}\parallel}$  $\frac{\parallel {\mathbf{f}}_{\mathit{rec}}{\mathbf{f}}_{\u2605}\parallel}{\parallel {\mathbf{f}}_{\u2605}\parallel}$  

Noise  Error (after Step 1)  Residual (for Step 2)  Error (after Step 2)  
Correct SS (simulated data)  0  $\overline{)0.208}$  $\overline{)0.052}$  $\overline{)0.170}$ 
Correct SS (noisy data)  0.853  1.760  0.874  0.227 
Wrong SS (simulated data)  0  0.208  0.510  0.560 
Fourier method [31]  0      0.564 
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Zangerl, G.; Haltmeier, M.; Nguyen, L.V.; Nuster, R. Full Field Inversion in Photoacoustic Tomography with Variable Sound Speed. Appl. Sci. 2019, 9, 1563. https://doi.org/10.3390/app9081563
Zangerl G, Haltmeier M, Nguyen LV, Nuster R. Full Field Inversion in Photoacoustic Tomography with Variable Sound Speed. Applied Sciences. 2019; 9(8):1563. https://doi.org/10.3390/app9081563
Chicago/Turabian StyleZangerl, Gerhard, Markus Haltmeier, Linh V. Nguyen, and Robert Nuster. 2019. "Full Field Inversion in Photoacoustic Tomography with Variable Sound Speed" Applied Sciences 9, no. 8: 1563. https://doi.org/10.3390/app9081563