Multiwave tomography with reflectors: Landweber's iteration

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April 2017, 11(2): 373-401. doi: 10.3934/ipi.2017018

Multiwave tomography with reflectors: Landweber's iteration

Plamen Stefanov ^*, and Yang Yang

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

^*First author partly supported by NSF Grant DMS-1301646

Received April 2016 Revised November 2016 Published March 2017

We use the Landweber method for numerical simulations for the multiwave tomography problem with a reflecting boundary and compare it with the averaged time reversal method. We also analyze the rate of convergence and the dependence on the step size for the Landweber iterations on a Hilbert space.

Keywords: Thermoacoustic tomography, microlocal analysis, Landweber iterations, medical imaging.

Mathematics Subject Classification: Primary: 35R25.

Citation: Plamen Stefanov, Yang Yang. Multiwave tomography with reflectors: Landweber's iteration. Inverse Problems and Imaging, 2017, 11 (2) : 373-401. doi: 10.3934/ipi.2017018

References:

[1]	S. Acosta and C. Montalto, Multiwave imaging in an enclosure with variable wave speed Inverse Problems, 31 (2015), 065009, 12pp. doi: 10.1088/0266-5611/31/6/065009.
[2]	S. Acosta and C. Montalto, Photoacoustic imaging taking into account thermodynamic attenuation Inverse Problems, 32 (2016), arXiv: 1602.01872. doi: 10.1088/0266-5611/32/11/115001.
[3]	N. Albin, O. P. Bruno, T. Y. Cheung and R. O. Cleveland, Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams, The Journal of the Acoustical Society of America, 132 (2012), 2371-2387. doi: 10.1121/1.4742722.
[4]	R. M. Alford, K. R. Kelly and D. M. Boore, Accuracy of finite-difference modeling of the acoustic wave equation, Geophysics, 39 (1974), 834-842. doi: 10.1190/1.1440470.
[5]	S. Arridge, B. Marta, B. Cox, F. Lucka and B. Treeby, On the adjoint operator in photoacoustic tomography Inverse Problems, 32 (2016), 110512, 19pp, arXiv: 1602.02027. doi: 10.1088/0266-5611/32/11/115012.
[6]	D. Auroux and J. Blum, Back and forth nudging algorithm for data assimilation problems, Comptes Rendus Mathematique, 340 (2005), 873-878. doi: 10.1016/j.crma.2005.05.006.
[7]	C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.
[8]	Z. Belhachmi, T. Glatz and O. Scherzer, Photoacoustic tomography with spatially varying compressibility and density, J. Inverse Ill-Posed Probl., 25 (2017), 119-133, arXiv:1512.07411 (2015). doi: 10.1515/jiip-2015-0113.
[9]	______, A direct method for photoacoustic tomography with inhomogeneous sound speed Inverse Problems, 32 (2016), 045005, 25pp. doi: 10.1088/0266-5611/32/4/045005.
[10]	C. Byrne, Iterative Algorithms in Inverse Problems, http://faculty.uml.edu/cbyrne/ITER2.pdf, 2006.
[11]	O. Chervova and L. Oksanen, Time reversal method with stabilizing boundary conditions for Photoacoustic tomography Inverse Problems, 32 (2016), arXiv: 1605.07817 (2016). doi: 10.1088/0266-5611/32/12/125004.
[12]	B. T. Cox, S. R. Arridge and P. C. Beard, Photoacoustic tomography with a limited-aperture planar sensor and a reverberant cavity, Inverse Problems, 23 (2007), S95-S112. doi: 10.1088/0266-5611/23/6/S08.
[13]	M. A. Dablain, The application of high-order differencing to the scalar wave equation, Geophysics, 51 (1986), 54-66. doi: 10.1190/1.1442040.
[14]	T. Fei and K. Larner, Geophysics, 60 (1995), 1830-1842.
[15]	Finch, Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240 (electronic). doi: 10.1137/S0036141002417814.
[16]	J. A. Goldstein and M. Wacker, The energy space and norm growth for abstract wave equations, Applied Mathematics Letters, 16 (2003), 767-772. doi: 10.1016/S0893-9659(03)00080-6.
[17]	M. Hanke, Accelerated Landweber iterations for the solution of ill-posed equations, Numer. Math., 60 (1991), 341-373. doi: 10.1007/BF01385727.
[18]	M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37. doi: 10.1007/s002110050158.
[19]	B. Holman and L. Kunyansky, Gradual time reversal in thermo-and photo-acoustic tomography within a resonant cavity Inverse Problems, 31 (2015), 035008, 25pp. doi: 10.1088/0266-5611/31/3/035008.
[20]	A. Homan, Multi-wave imaging in attenuating media, Inverse Probl. Imaging, 7 (2013), 1235-1250. doi: 10.3934/ipi.2013.7.1235.
[21]	K. R. Kelly, R. W. Ward, S. Treitel and R. M. Alford, Synthetic seismograms: A finite-difference approach, Geophysics, 41 (1976), 2-27. doi: 10.1190/1.1440605.
[22]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, second ed., Applied Mathematical Sciences, vol. 120, Springer, New York, 2011. doi: 10.1007/978-1-4419-8474-6.
[23]	R.A. Kruger, W. L. Kiser, 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.
[24]	R. A. Kruger, D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography--technical considerations, Med Phys, 26 (1999), 1832-1837. doi: 10.1118/1.598688.
[25]	P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography, Handbook of Mathematical Methods in Imaging (Otmar Scherzer, ed.), Springer New York, 2011, pp. 817-865.
[26]	L. Kunyansky, B. Holman and B. T. Cox, Photoacoustic tomography in a rectangular reflecting cavity Inverse Problems, 29 (2013), 125010, 20pp. doi: 10.1088/0266-5611/29/12/125010.
[27]	L. V. Nguyen and L. A. Kunyansky, A dissipative time reversal technique for photoacoustic tomography in a cavity, SIAM J. Imaging Sci., 9 (2016), 748-769. doi: 10.1137/15M1049683.
[28]	J. Qian, P. Stefanov, G. Uhlmann and H. Zhao, An efficient Neumann series-based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci., 4 (2011), 850-883. doi: 10.1137/100817280.
[29]	M. Reed and B. Simon, Methods of Modern Mathematical Physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979, Scattering theory.
[30]	______, Methods of Modern Mathematical Physics. I, second ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980, Functional analysis.
[31]	P. Stefanov and G. Uhlmann, Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal., 256 (2009), 2842-2866. doi: 10.1016/j.jfa.2008.10.017.
[32]	______, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16pp.
[33]	______, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004, 26pp.
[34]	______, Multi-wave methods via ultrasound, Inside Out, MSRI Publications, 60 (2012), 271-324. .
[35]	P. Stefanov and Y. Yang, Multiwave tomography in a closed domain: Averaged sharp time reversal Inverse Problems, 31 (2015), 065007, 23pp. doi: 10.1088/0266-5611/31/6/065007.
[36]	D. Tataru, Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884. doi: 10.1080/03605309508821117.
[37]	______, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 26 (1998), 185-206
[38]	G. M. Vainikko, Error estimates of the successive approximation method for ill-posed problems, Automat. Remote Control, (1980), 84-92.
[39]	G. M. Vainikko and A. Yu. Veretennikov, Iteration Procedures in Ill-Posed Problems (in russian) "Nauka", Moscow, 1986.
[40]	M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine Review of Scientific Instruments, 77 (2006), 041101. doi: 10.1063/1.2195024.
[41]	Y. Xu and L. V. Wang, Rhesus monkey brain imaging through intact skull with thermoacoustic tomography, IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 53 (2006), 542-548.
[42]	X. Yang and L. V. Wang, Monkey brain cortex imaging by photoacoustic tomography Proc. SPIE, 6856 (2008), 762396. doi: 10.1117/12.762396.
[43]	E. Zuazua, Numerics for the control of partial differential equations, Reference Work Entry: Encyclopedia of Applied and Computational Mathematics, (2015), 1076-1080. doi: 10.1007/978-3-540-70529-1_362.

show all references

References:

[1]	S. Acosta and C. Montalto, Multiwave imaging in an enclosure with variable wave speed Inverse Problems, 31 (2015), 065009, 12pp. doi: 10.1088/0266-5611/31/6/065009.
[2]	S. Acosta and C. Montalto, Photoacoustic imaging taking into account thermodynamic attenuation Inverse Problems, 32 (2016), arXiv: 1602.01872. doi: 10.1088/0266-5611/32/11/115001.
[3]	N. Albin, O. P. Bruno, T. Y. Cheung and R. O. Cleveland, Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams, The Journal of the Acoustical Society of America, 132 (2012), 2371-2387. doi: 10.1121/1.4742722.
[4]	R. M. Alford, K. R. Kelly and D. M. Boore, Accuracy of finite-difference modeling of the acoustic wave equation, Geophysics, 39 (1974), 834-842. doi: 10.1190/1.1440470.
[5]	S. Arridge, B. Marta, B. Cox, F. Lucka and B. Treeby, On the adjoint operator in photoacoustic tomography Inverse Problems, 32 (2016), 110512, 19pp, arXiv: 1602.02027. doi: 10.1088/0266-5611/32/11/115012.
[6]	D. Auroux and J. Blum, Back and forth nudging algorithm for data assimilation problems, Comptes Rendus Mathematique, 340 (2005), 873-878. doi: 10.1016/j.crma.2005.05.006.
[7]	C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.
[8]	Z. Belhachmi, T. Glatz and O. Scherzer, Photoacoustic tomography with spatially varying compressibility and density, J. Inverse Ill-Posed Probl., 25 (2017), 119-133, arXiv:1512.07411 (2015). doi: 10.1515/jiip-2015-0113.
[9]	______, A direct method for photoacoustic tomography with inhomogeneous sound speed Inverse Problems, 32 (2016), 045005, 25pp. doi: 10.1088/0266-5611/32/4/045005.
[10]	C. Byrne, Iterative Algorithms in Inverse Problems, http://faculty.uml.edu/cbyrne/ITER2.pdf, 2006.
[11]	O. Chervova and L. Oksanen, Time reversal method with stabilizing boundary conditions for Photoacoustic tomography Inverse Problems, 32 (2016), arXiv: 1605.07817 (2016). doi: 10.1088/0266-5611/32/12/125004.
[12]	B. T. Cox, S. R. Arridge and P. C. Beard, Photoacoustic tomography with a limited-aperture planar sensor and a reverberant cavity, Inverse Problems, 23 (2007), S95-S112. doi: 10.1088/0266-5611/23/6/S08.
[13]	M. A. Dablain, The application of high-order differencing to the scalar wave equation, Geophysics, 51 (1986), 54-66. doi: 10.1190/1.1442040.
[14]	T. Fei and K. Larner, Geophysics, 60 (1995), 1830-1842.
[15]	Finch, Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240 (electronic). doi: 10.1137/S0036141002417814.
[16]	J. A. Goldstein and M. Wacker, The energy space and norm growth for abstract wave equations, Applied Mathematics Letters, 16 (2003), 767-772. doi: 10.1016/S0893-9659(03)00080-6.
[17]	M. Hanke, Accelerated Landweber iterations for the solution of ill-posed equations, Numer. Math., 60 (1991), 341-373. doi: 10.1007/BF01385727.
[18]	M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37. doi: 10.1007/s002110050158.
[19]	B. Holman and L. Kunyansky, Gradual time reversal in thermo-and photo-acoustic tomography within a resonant cavity Inverse Problems, 31 (2015), 035008, 25pp. doi: 10.1088/0266-5611/31/3/035008.
[20]	A. Homan, Multi-wave imaging in attenuating media, Inverse Probl. Imaging, 7 (2013), 1235-1250. doi: 10.3934/ipi.2013.7.1235.
[21]	K. R. Kelly, R. W. Ward, S. Treitel and R. M. Alford, Synthetic seismograms: A finite-difference approach, Geophysics, 41 (1976), 2-27. doi: 10.1190/1.1440605.
[22]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, second ed., Applied Mathematical Sciences, vol. 120, Springer, New York, 2011. doi: 10.1007/978-1-4419-8474-6.
[23]	R.A. Kruger, W. L. Kiser, 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.
[24]	R. A. Kruger, D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography--technical considerations, Med Phys, 26 (1999), 1832-1837. doi: 10.1118/1.598688.
[25]	P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography, Handbook of Mathematical Methods in Imaging (Otmar Scherzer, ed.), Springer New York, 2011, pp. 817-865.
[26]	L. Kunyansky, B. Holman and B. T. Cox, Photoacoustic tomography in a rectangular reflecting cavity Inverse Problems, 29 (2013), 125010, 20pp. doi: 10.1088/0266-5611/29/12/125010.
[27]	L. V. Nguyen and L. A. Kunyansky, A dissipative time reversal technique for photoacoustic tomography in a cavity, SIAM J. Imaging Sci., 9 (2016), 748-769. doi: 10.1137/15M1049683.
[28]	J. Qian, P. Stefanov, G. Uhlmann and H. Zhao, An efficient Neumann series-based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci., 4 (2011), 850-883. doi: 10.1137/100817280.
[29]	M. Reed and B. Simon, Methods of Modern Mathematical Physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979, Scattering theory.
[30]	______, Methods of Modern Mathematical Physics. I, second ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980, Functional analysis.
[31]	P. Stefanov and G. Uhlmann, Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal., 256 (2009), 2842-2866. doi: 10.1016/j.jfa.2008.10.017.
[32]	______, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16pp.
[33]	______, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004, 26pp.
[34]	______, Multi-wave methods via ultrasound, Inside Out, MSRI Publications, 60 (2012), 271-324. .
[35]	P. Stefanov and Y. Yang, Multiwave tomography in a closed domain: Averaged sharp time reversal Inverse Problems, 31 (2015), 065007, 23pp. doi: 10.1088/0266-5611/31/6/065007.
[36]	D. Tataru, Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884. doi: 10.1080/03605309508821117.
[37]	______, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 26 (1998), 185-206
[38]	G. M. Vainikko, Error estimates of the successive approximation method for ill-posed problems, Automat. Remote Control, (1980), 84-92.
[39]	G. M. Vainikko and A. Yu. Veretennikov, Iteration Procedures in Ill-Posed Problems (in russian) "Nauka", Moscow, 1986.
[40]	M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine Review of Scientific Instruments, 77 (2006), 041101. doi: 10.1063/1.2195024.
[41]	Y. Xu and L. V. Wang, Rhesus monkey brain imaging through intact skull with thermoacoustic tomography, IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 53 (2006), 542-548.
[42]	X. Yang and L. V. Wang, Monkey brain cortex imaging by photoacoustic tomography Proc. SPIE, 6856 (2008), 762396. doi: 10.1117/12.762396.
[43]	E. Zuazua, Numerics for the control of partial differential equations, Reference Work Entry: Encyclopedia of Applied and Computational Mathematics, (2015), 1076-1080. doi: 10.1007/978-3-540-70529-1_362.

Figure 12. The unstable case, reconstructions with the Landweber method with noise. Left: unfiltered data. Right: filtered data.

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Figure 1. $\log_{10}$ error at each step vs. $\gamma$ with exact data. This is the graph of $\log_{10}\| K\|$ as a function of $\gamma$ for $\mu^2=1$, $\|\mathcal{L}\|^2=20$. The optimal $\gamma^*$ is $\gamma^*=2/21\approx0.0952$. The error increases fast on the right of $\gamma^*$.

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Figure 3. Error vs. $\gamma$ for the Landweber iterations in the stable example. Plotted are errors after $10$ iterations (boxes), after $30$ iterations (diamonds) and after $50$ ones (dots). Right: Error vs. the number of the iterations. The bottom curve with the circles are the ATR errors. The other curves correspond to $\gamma$ ranging from $0.1$ to $0.06$, as on the left, starting form the top to the bottom in the top left corner.

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Figure 2. The functions $g_N(\lambda)$ with $\gamma=1$ and $N=5, 20, 40, 80$. As the number of iterations $N$ increases, the maximum increases as $C_1\sqrt N$ and its location shifts to the left to $\lambda_N=C_2/\sqrt N$.

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Figure 4. A stable case. The smooth increasing curve represents the eigenvalues of $\mathcal{L}^*\mathcal{L}$ in the discrete realization on an $101\!\times\!101$ grid restricted to a $94\!\times\!94$ grid. There are small imaginary parts along the horizontal line in the middle. $\mathcal{L}^*$ is computed as the adjoint to the matrix representation of $\mathcal{L}$. The rough curve represents the squares of the Fourier coefficients of the SL phantom.

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Figure 5. The eigenfunctions of $L^*L$ corresponding to $\lambda_1$, $\lambda_{1,000}$ and $\lambda_{7,500}$

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Figure 6. Error plots in the stable case with data not in the range. Approximate convergence.

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Figure 7. Reconstruction and error plots in the stable case with noise with $200$ iterations; $\gamma=0.01,0.02,0.03,0.04$. A slow divergence. For $\gamma=0.05$ we get a fast divergence.

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Figure 8. Speed $1$ with $T=1.8$ with invisible singularities on the top. From left to right: (a) ATR with 50 steps, a cut near the border, 34\% error. (b) Landweber with a cut near the borders, error $31\%$; (c) Landweber without a cut near the borders, error $36\%$.

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Figure 9. Error vs. $\gamma$ in the unstable case. Left: errors after $10$ iterations (boxes), after $30$ iterations (diamonds) and after $50$ ones (dots). The vertical axis is on a $\log_{10}$ scale and the horizontal axis represents $\gamma$. The lower curve, corresponding to 50 iterations, is flatter than in the stable case. On the right, the curve with the diamond marks is the error with the ATR method.

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Figure 10. The unstable case. The smooth increasing curve represents the eigenvalues of $\mathcal{L}^*\mathcal{L}$ in the discrete realization on an $101\!\times\!101$ grid restricted to a $94\!\times\!94$ grid. $\mathcal{L}^*$ is computed as the adjoint matrix rather than with the wave equation solver. The rough curve represents the squares of the Fourier coefficients of the SL phantom. The arrows indicate the effective lower bound of the power spectrum of the SL phantom, and the first positive eigenvalue modulo small errors, respectively.

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Figure 11. A unstable case with Gaussian noise. Left: error curves with $\gamma$ ranging from $0.03$ to $0.17$ (the diverging curve). The critical value of $\gamma$ looks close to that in the zero noise case in Figure 9. The iterations for $\gamma\le 0.16$ diverge slowly in contrast with the noise free case in Figure 9. Right: Filtered data, $\gamma\le 0.15$. The errors look like in Figure 9.

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Figure 13. Reconstructions with a discontinuous speed, plotted on the left. The original is black and white disks on a gray background. Data on the marked part of the boundary, $T=4$. Center: ATR reconstruction, error $1\%$. Right: Landweber reconstruction, error $1.5\%$.

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Figure 14. Left: the original SL phantom. Right: The reconstructed SL phantom with $c=1$ and $T=2$. The phantom was originally on a $201\times 201$ grid with $\Delta t=\Delta x /\sqrt{2}$. The data was computed on a grid rescaled by a factor of $5.7$ in the spatial variables and $7.41$ times in the time variable; and then rescaled to the the original one on the boundary before inversion. The ripple artifacts are due to high frequency waves in the time reversal moving slower than the speed $c=1$.

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Figure 15. Same situation as in Figure 14 but the phantom contains Gaussians with predominately low frequency content. Left: original. Right: reconstruction with 50 iterations. The relative $L^2$ error is $1.8\%$, the $L^\infty$ error is $3.5\%$.

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