×

A new optimization approach to sparse reconstruction of log-conductivity in acousto-electric tomography. (English) Zbl 1401.35351

Summary: A new optimization approach to sparse reconstruction of log-conductivity in acousto-electric tomography is presented. This approach considers the minimization of a Tikhonov least-squares functional of measured power density together with an \(L^1\)-norm penalization of the log-conductivity to promote sparsity and a Perona–Malik functional to enhance edges. This minimization is performed subject to the differential constraint given by two instances of the EIT equation. The resulting infinite-dimensional optimization problem is approximated by finite differences and solved efficiently by a proximal scheme. Numerous pointers to existing literature are given to motivate the present framework and results of numerical experiments are presented that demonstrate the many advantages of the new approach.

MSC:

35R30 Inverse problems for PDEs
49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics

Software:

iPiano
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. S. Alberti and H. Ammari, Disjoint sparsity for signal separation and applications to hybrid inverse problems in medical imaging, Appl. Comput. Harmon. Anal., 42 (2015), pp. 319–349. · Zbl 1354.94002
[2] G. S. Alberti, H. Ammari, and K. Ruan, Multi-frequency acousto-electromagnetic tomography, in A Panorama of Mathematics: Pure and Applied, Contemp. Math. 658, 2016, pp. 67–79. · Zbl 1348.35309
[3] G. Alessandrini and V. Nesi, Univalent \(σ\)-harmonic mappings, Arch. Ration. Mech. Anal., 158 (2001), pp. 155–171. · Zbl 0977.31006
[4] H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter, and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), pp. 1557–1573. · Zbl 1156.35101
[5] U. M. Ascher and E. Haber, A multigrid method for distributed parameter estimation problems, Electron. Trans. Numer. Anal., 18 (2003), pp. 1–18. · Zbl 1031.65108
[6] K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173 (1994), pp. 37–60. · Zbl 0815.30015
[7] G. Bal, Hybrid inverse problems and redundant systems of partial differential equations, in Inverse Problems and Applications, Contemp. Math. 615, 2014, pp. 15–48. · Zbl 1330.35521
[8] G. Bal, E. Bonnetier, F. Monard, and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Probl. Imaging, 7 (2013), pp. 353–375. · Zbl 1267.35249
[9] G. Bal, K. Hoffmann, and K. Knudsen, Propagation of singularities for linearised hybrid data impedance tomography, Inverse Problems, 34 (2017), 024001. · Zbl 1433.65262
[10] G. Bal, W. Naetar, O. Scherzer, and J. Schotland, The Levenberg–Marquardt iteration for numerical inversion of the power density operator, J. Inverse Ill-Posed Probl., 21 (2013), pp. 265–280. · Zbl 1273.35307
[11] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), pp. 183–202. · Zbl 1175.94009
[12] L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), pp. R99–136. · Zbl 1031.35147
[13] A. Borzì and V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations, Comput. Sci. Eng., SIAM, Philadelphia, 2011. · Zbl 1240.90001
[14] K. Bryan and T. Leise, Making do with less: An introduction to compressed sensing, SIAM Rev., 55 (2013), pp. 547–566. · Zbl 1306.94009
[15] A. P. Calderòn, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rìo de Janeiro, 1980), Soceira de Brasil. Matemática, Rio de Janeiro, 1980, pp. 65–73.
[16] E. J. Candes, J. K. Romberg, and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., 59 (2006), pp. 1207–1223. · Zbl 1098.94009
[17] Y. Capdeboscq, On a counter-example to quantitative Jacobian bounds, J. Éc. polytech. Math., 2 (2015), pp. 171–178. · Zbl 1327.35432
[18] Y. Capdeboscq, J. Fehrenbach, F. de Gournay, and O. Kavian, Imaging by modification: Numerical reconstruction of local conductivities from corresponding power density measurements, SIAM J. Imaging Sci., 2 (2009), pp. 1003–1030. · Zbl 1180.35549
[19] F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), pp. 182–193. · Zbl 0746.65091
[20] T. F. Chan, S. Esedoglu, and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math., 66 (2006), pp. 1632–1648. · Zbl 1117.94002
[21] M. Ceheney, D. Isaacson, and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), pp. 85–101. · Zbl 0927.35130
[22] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), pp. 1168–1200. · Zbl 1179.94031
[23] D. L. Donoho and M. Elad, Maximal sparsity representation via \(l_1\) minimization, Proc. Natl. Acad. Sci. USA, 100 (2003), pp. 2197–2202. · Zbl 1064.94011
[24] I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics Appl. Math. 28, SIAM, Philadelphia, 1999. · Zbl 0939.49002
[25] P. Guidotti, Anisotropic diffusions of image processing from Perona-Malik on, Adv. Stud. Pure Math., to appear. · Zbl 1367.49007
[26] K. Hoffman and K. Knudsen, Iterative reconstruction methods for hybrid inverse problems in impedance tomography, Sensing Imaging, 15 (2014), 96.
[27] J. Jossinet, The impedivity of freshly excised human breast tissue, Physiol. Meas., 19 (1998), pp. 61–75.
[28] T. Kerner, D. Williams, K. Osterman, F. Reiss, A. Hartov, and K. Paulsen, Electrical impedance imaging at multiple frequencies in phantoms, Physiol. Meas., 21 (2000), pp. 67–77.
[29] S. Kichenassamy, The Perona–Malik Paradox, SIAM J. Appl. Math., 57 (1997), pp. 1328–1342. · Zbl 0887.35071
[30] P. Kuchment and L. Kunyansky, Synthetic focusing in ultrasound modulated tomography, Inverse Probl. Imaging, 46 (2010), pp. 65–67. · Zbl 1286.44002
[31] P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013. · Zbl 1217.35211
[32] P. Kuchment and D. Steinhauer, Stabilizing inverse problems by internal data, Inverse Problems, 28 (2012), 084007. · Zbl 1252.92037
[33] Y. Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal., 153 (2000), pp. 91–151. · Zbl 0958.35060
[34] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.
[35] M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, Compressed sensing MRI, IEEE Signal Process. Mag., 25 (March 2008), pp. 72–82.
[36] N. G. Meyers, An \(L^p\)-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 17 (1963), pp. 189–206. · Zbl 0127.31904
[37] F. Monard and G. Bal, Inverse anisotropic diffusion from power density measurements in two dimensions, Inverse Problems, 28 (2012), 084001. · Zbl 1250.35187
[38] J. Mueller and S. Siltanen, Linear and nonlinear inverse problems with practical applications, Comput. Sci. Eng., SIAM, Philadelphia, 2012. · Zbl 1262.65124
[39] Y. E. Nesterov, A method for solving the convex programming problem with covergence rate \(O(1/k^2)\), Dokl. Akad. Nauk SSSR, 269 (1983), pp. 543–547.
[40] P. Ochs, Y. Chen, T. Brox, and T. Pock, iPiano: Inertial proximal algorithm for nonconvex optimization, SIAM J. Imaging Sci., 7 (2014), pp. 1388–1419. · Zbl 1296.90094
[41] P. Perona and J. Malik, Scale-Space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), pp. 629–639.
[42] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877–898. · Zbl 0358.90053
[43] L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259–268. · Zbl 0780.49028
[44] O. Scherzer and J. Weickert, Relations between regularization and diffusion filtering, J. Math. Imaging Vision, 12 (2000), pp. 43–63. · Zbl 0945.68183
[45] A. Schindele and A. Borzí, Proximal methods for elliptic optimal control problems with sparsity cost functional, Appl. Math., 7 (2016), pp. 967–992.
[46] A. Schindele and A. Borzí, Proximal schemes for parabolic optimal control problems with sparsity promoting cost functionals, Internat. J. Control, 2016, pp. 1–19. ,
[47] A. D. Seagar, D. C. Barber, and B. H. Brown, Theoretical limits to sensitivity and resolution in impedance imaging, Clinical Phys. Physiol. Meas., 8 (1987), pp. 13–31.
[48] L. A. Shepp and B. F. Logan, The Fourier reconstruction of a head section, IEEE Trans. Nucl. Sci., NS-21 (1974), pp. 21–43.
[49] X. Song, Y. Xu, and F. Dong, Sensitivity matrix for ultrasound modulated electrical impedance tomography, in 2016 IEEE International Instrumentation and Measurement Technology Conference Proceedings, IEEE, Piscataway, NJ, 2016, pp. 1–6.
[50] G. Stadler, Elliptic optimal control problems with \(L^1\)-control costs and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), pp. 159–181. · Zbl 1185.49031
[51] F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, AMS, Providence, RI, 2010.
[52] M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, MOS-SIAM Ser. Optim., SIAM, Philadelphia, 2011. · Zbl 1235.49001
[53] J. Weickert, Anisotropic Diffusion in Image Processing, Treubner, Stuttgart, 1998. · Zbl 0886.68131
[54] H. Zhang and L. Wang, Acousto-electric tomography, Photons Plus Ultrasound Imaging and Sensing, Proc. SPIE 5320, 2004, pp. 145–149.
[55] M. S. Zhdanov and G. V. Keller, The Geoelectrical Methods in Geophysical Exploration, Methods Geochem. Geophys. 31, Elsevier, Amsterdam, 1994.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.