August  2016, 10(3): 585-616. doi: 10.3934/ipi.2016013

On the stability of some imaging functionals

1. 

Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027

2. 

Department of Mathematics, Colorado State University, Fort Collins CO 80523, United States

3. 

Department of Mathematics, Stanford University, Stanford, CA 94305

Received  January 2015 Published  August 2016

This work is devoted to the stability/resolution analysis of several imaging functionals in complex environments. We consider both linear functionals in the wavefield as well as quadratic functionals based on wavefield correlations. Using simplified measurement settings and reduced functionals that retain the main features of functionals used in practice, we obtain optimal asymptotic estimates of the signal-to-noise ratios depending on the main physical parameters of the problem. We consider random media with possibly long-range dependence and with a correlation length that is less than or equal to the central wavelength of the source we aim to reconstruct. This corresponds to the wave propagation regimes of radiative transfer or homogenization.
Citation: Guillaume Bal, Olivier Pinaud, Lenya Ryzhik. On the stability of some imaging functionals. Inverse Problems and Imaging, 2016, 10 (3) : 585-616. doi: 10.3934/ipi.2016013
References:
[1]

H. Ammari, E. Bretin, J. Garnier and V. Jugnon, Coherent interferometry algorithms for photoacoustic imaging, SIAM J. Numer. Anal., 50 (2012), 2259-2280. doi: 10.1137/100814275.

[2]

H. Ammari, J. Garnier, V. Jugnon and H. Kang, Stability and resolution analysis for a topological derivative based imaging functional, SIAM Journal of Control and Optimization, 50 (2012), 48-76. doi: 10.1137/100812501.

[3]

A. Baggeroer, W. Kuperman and P. Mikhalevsky, An overview of matched-field methods in ocean acoustics, IEEE Journal of Ocean Engineering, 18 (1993), 401-424. doi: 10.1109/48.262292.

[4]

G. Bal, On the self-averaging of wave energy in random media, SIAM Mult. Mod. Simul., 2 (2004), 398-420. doi: 10.1137/S1540345903426298.

[5]

G. Bal, I. Langmore and O. Pinaud, Single scattering estimates for the scintillation function of waves in random media, J. Math. Phys., 51 (2010), 022903, 18pp. doi: 10.1063/1.3276437.

[6]

G. Bal and O. Pinaud, Dynamics of wave scintillation in random media, CPDE, 35 (2010), 1176-1235. doi: 10.1080/03605301003801557.

[7]

G. Bal and O. Pinaud, Imaging using transport models for wave-wave correlations, M3AS, 21 (2011), 1071-1093. doi: 10.1142/S0218202511005258.

[8]

G. Bal and O. Pinaud, Analysis of the double scattering scintillation of waves in random media, CPDE, 38 (2013), 945-984. doi: 10.1080/03605302.2013.777451.

[9]

G. Bal and K. Ren, Transport-based imaging in random media, SIAM Applied Math., 68 (2008), 1738-1762. doi: 10.1137/070690122.

[10]

G. Bal, Homogenization in random media and effective medium theory for high frequency waves, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 473-492 (electronic). doi: 10.3934/dcdsb.2007.8.473.

[11]

N. Bleistein, J. K. Cohen and J. W. Stockwell Jr., Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, vol. 13 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2001, Geophysics and Planetary Sciences. doi: 10.1007/978-1-4613-0001-4.

[12]

L. Borcea, J. Garnier, G. Papanicolaou and C. Tsogka, Enhanced statistical stability in coherent interferometric imaging, Inverse Problems, 27 (2011), 085004, 33 pp. doi: 10.1088/0266-5611/27/8/085004.

[13]

L. Borcea, L. Issa and C. Tsogka, Source localization in random acoustic waveguides, Multiscale Model. Simul., 8 (2010), 1981-2022. doi: 10.1137/100782711.

[14]

L. Borcea, G. Papanicolaou and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), 1419-1460. doi: 10.1088/0266-5611/21/4/015.

[15]

L. Borcea, G. Papanicolaou and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Problems, 22 (2006), 1405-1436. doi: 10.1088/0266-5611/22/4/016.

[16]

J. F. claerbout, Fundamentals of Geophysical Data Processing: With Applications to Petroleum Prospecting, Blackwell scientific, Palo Alto, 1985.

[17]

S. Dolan, C. Bean and R. B., The broad-band fractal nature of heterogeneity in the upper crust from petrophysical logs, Geophys. J. Int., 132 (1998), 489-507. doi: 10.1046/j.1365-246X.1998.00410.x.

[18]

M. Fink and C. Prada, Acoustic time-reversal mirrors, Imaging of Complex Media with Acoustic and Seismic Waves, 84 (2002), 17-43. doi: 10.1007/3-540-44680-X_2.

[19]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, vol. 56 of Stochastic Modelling and Applied Probability, Springer, New York, 2007. doi: 10.1007/978-0-387-49808-9_4.

[20]

J. Garnier and K. Sølna, Background velocity estimation with cross correlations of incoherent waves in the parabolic scaling, Inverse Problems, 25 (2009), 045005, 34 pp. doi: 10.1088/0266-5611/25/4/045005.

[21]

J. Garnier and G. Papanicolaou, Pulse propagation and time reversal in random waveguides, SIAM J. Appl. Math., 67 (2007), 1718-1739 (electronic). doi: 10.1137/060659235.

[22]

I. M. Gelfand and G. E. Shilov, Generalized Functions, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977], Properties and operations, Translated from the Russian by Eugene Saletan.

[23]

P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[24]

F. C. Karal Jr. and J. B. Keller, Elastic, electromagnetic, and other waves in a random medium, J. Mathematical Phys., 5 (1964), 537-547. doi: 10.1063/1.1704145.

[25]

P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.

[26]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, 2nd edition, Academic Press, Inc., New York, 1980.

[27]

L. Ryzhik, G. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370. doi: 10.1016/S0165-2125(96)00021-2.

[28]

P. J. Shull, Nondestructive Evaluation. Theory, Techniques and Applications, Marcel Dekker, New York, 2002.

[29]

C. Sidi and F. Dalaudier, Turbulence in the stratified atmosphere: Recent theoretical devel- opments and experimental results, Adv. in Space Res., 10 (1990), 25-36.

show all references

References:
[1]

H. Ammari, E. Bretin, J. Garnier and V. Jugnon, Coherent interferometry algorithms for photoacoustic imaging, SIAM J. Numer. Anal., 50 (2012), 2259-2280. doi: 10.1137/100814275.

[2]

H. Ammari, J. Garnier, V. Jugnon and H. Kang, Stability and resolution analysis for a topological derivative based imaging functional, SIAM Journal of Control and Optimization, 50 (2012), 48-76. doi: 10.1137/100812501.

[3]

A. Baggeroer, W. Kuperman and P. Mikhalevsky, An overview of matched-field methods in ocean acoustics, IEEE Journal of Ocean Engineering, 18 (1993), 401-424. doi: 10.1109/48.262292.

[4]

G. Bal, On the self-averaging of wave energy in random media, SIAM Mult. Mod. Simul., 2 (2004), 398-420. doi: 10.1137/S1540345903426298.

[5]

G. Bal, I. Langmore and O. Pinaud, Single scattering estimates for the scintillation function of waves in random media, J. Math. Phys., 51 (2010), 022903, 18pp. doi: 10.1063/1.3276437.

[6]

G. Bal and O. Pinaud, Dynamics of wave scintillation in random media, CPDE, 35 (2010), 1176-1235. doi: 10.1080/03605301003801557.

[7]

G. Bal and O. Pinaud, Imaging using transport models for wave-wave correlations, M3AS, 21 (2011), 1071-1093. doi: 10.1142/S0218202511005258.

[8]

G. Bal and O. Pinaud, Analysis of the double scattering scintillation of waves in random media, CPDE, 38 (2013), 945-984. doi: 10.1080/03605302.2013.777451.

[9]

G. Bal and K. Ren, Transport-based imaging in random media, SIAM Applied Math., 68 (2008), 1738-1762. doi: 10.1137/070690122.

[10]

G. Bal, Homogenization in random media and effective medium theory for high frequency waves, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 473-492 (electronic). doi: 10.3934/dcdsb.2007.8.473.

[11]

N. Bleistein, J. K. Cohen and J. W. Stockwell Jr., Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, vol. 13 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2001, Geophysics and Planetary Sciences. doi: 10.1007/978-1-4613-0001-4.

[12]

L. Borcea, J. Garnier, G. Papanicolaou and C. Tsogka, Enhanced statistical stability in coherent interferometric imaging, Inverse Problems, 27 (2011), 085004, 33 pp. doi: 10.1088/0266-5611/27/8/085004.

[13]

L. Borcea, L. Issa and C. Tsogka, Source localization in random acoustic waveguides, Multiscale Model. Simul., 8 (2010), 1981-2022. doi: 10.1137/100782711.

[14]

L. Borcea, G. Papanicolaou and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), 1419-1460. doi: 10.1088/0266-5611/21/4/015.

[15]

L. Borcea, G. Papanicolaou and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Problems, 22 (2006), 1405-1436. doi: 10.1088/0266-5611/22/4/016.

[16]

J. F. claerbout, Fundamentals of Geophysical Data Processing: With Applications to Petroleum Prospecting, Blackwell scientific, Palo Alto, 1985.

[17]

S. Dolan, C. Bean and R. B., The broad-band fractal nature of heterogeneity in the upper crust from petrophysical logs, Geophys. J. Int., 132 (1998), 489-507. doi: 10.1046/j.1365-246X.1998.00410.x.

[18]

M. Fink and C. Prada, Acoustic time-reversal mirrors, Imaging of Complex Media with Acoustic and Seismic Waves, 84 (2002), 17-43. doi: 10.1007/3-540-44680-X_2.

[19]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, vol. 56 of Stochastic Modelling and Applied Probability, Springer, New York, 2007. doi: 10.1007/978-0-387-49808-9_4.

[20]

J. Garnier and K. Sølna, Background velocity estimation with cross correlations of incoherent waves in the parabolic scaling, Inverse Problems, 25 (2009), 045005, 34 pp. doi: 10.1088/0266-5611/25/4/045005.

[21]

J. Garnier and G. Papanicolaou, Pulse propagation and time reversal in random waveguides, SIAM J. Appl. Math., 67 (2007), 1718-1739 (electronic). doi: 10.1137/060659235.

[22]

I. M. Gelfand and G. E. Shilov, Generalized Functions, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977], Properties and operations, Translated from the Russian by Eugene Saletan.

[23]

P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[24]

F. C. Karal Jr. and J. B. Keller, Elastic, electromagnetic, and other waves in a random medium, J. Mathematical Phys., 5 (1964), 537-547. doi: 10.1063/1.1704145.

[25]

P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.

[26]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, 2nd edition, Academic Press, Inc., New York, 1980.

[27]

L. Ryzhik, G. Papanicolaou and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), 327-370. doi: 10.1016/S0165-2125(96)00021-2.

[28]

P. J. Shull, Nondestructive Evaluation. Theory, Techniques and Applications, Marcel Dekker, New York, 2002.

[29]

C. Sidi and F. Dalaudier, Turbulence in the stratified atmosphere: Recent theoretical devel- opments and experimental results, Adv. in Space Res., 10 (1990), 25-36.

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