# American Institute of Mathematical Sciences

2008, 2(2): 225-250. doi: 10.3934/ipi.2008.2.225

## Enhanced imaging from multiply scattered waves

 1 Department of Mathematics and Statistics, University of Limerick, Castletroy, Limerick, Ireland 2 Department of Mathematics and Statistics, University of Limerick, Castletroy, Limeric, Ireland

Received  September 2007 Revised  December 2007 Published  April 2008

Many imaging methods involve probing a material with a wave and observing the back-scattered wave. The back-scattered wave measurements are used to compute an image of the internal structure of the material. Many of the conventional methods make the assumption that the wave has scattered just once from the region to be imaged before returning to the sensor to be recorded. The purpose of this paper is to show how this restriction can be partially removed and also how its removal leads to an enhanced image, free of the artifacts often associated with the conventionally reconstructed image.
Citation: Romina Gaburro, Clifford J Nolan. Enhanced imaging from multiply scattered waves. Inverse Problems & Imaging, 2008, 2 (2) : 225-250. doi: 10.3934/ipi.2008.2.225
##### References:
 [1] G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform,, J. Math. Phys., 26 (1985), 99. doi: 10.1063/1.526755. [2] G. Beylkin and R. Burridge, Linearized inverse scattering problems in acoustic and elasticity,, Wave Motion, 12 (1990), 15. doi: 10.1016/0165-2125(90)90017-X. [3] N. Bleistein, J. K. Cohen and J. W. Stockwell, "The Matematics of Multidimensional Seismic Inversion,", Springer-Verlag, (2000). [4] M. Cheney, A mathematical tutorial on synthetic aperture radar,, SIAM Review, 43 (2001), 301. doi: 10.1137/S0036144500368859. [5] M. Cheney and R. J. Bonneau, Imaging that exploits multipath scattering from point scatterers,, Inverse Problems, 20 (2004), 1691. doi: 10.1088/0266-5611/20/5/023. [6] J. J. Duistermaat, "Fourier Integral Operators. Progress in Mathematics, 130, ", Birkhauser, (1996). [7] R. Gaburro, C. J. Nolan, T. Dowling and M. Cheney, Imaging from multiply scattered waves,, Proc. SPIE 6513, (2007). doi: 10.1117/12.712569. [8] A. Grigis and J. Sjöstrand, "Microlocal Analysis for Differential Operators: an Introduction,", London Mathematical Sciety Lecture Note Series, 196 (1994). [9] G. T. Herman, H. K. Tuy, K. J. Langenberg and P. C. Sabatier, "Basic Methods of Tomography and Inverse Problems,", Adam Hilger, (1988). [10] P. Morse and H. Feshbach, "Methods of Theoretical Physics," Vol. 1,, McGraw-Hill, (1953). [11] C. J. Nolan, Scattering near a fold caustic,, SIAM J. of Appl. Math, 61 (2000), 659. doi: 10.1137/S0036139999356107. [12] C. J. Nolan and M. Cheney, Synthetic aperture inversion for arbitrary flight paths and non-flat topography,, IEEE Trans. on Image Processing, 12 (2003), 1035. doi: 10.1109/TIP.2003.814243. [13] C. J. Nolan and M. Cheney, Synthetic aperture inversion,, Inverse Problems, 18 (2002), 221. doi: 10.1088/0266-5611/18/1/315. [14] C. J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging,, J. Fourier Analysis and its Applications, 10 (2004), 133. [15] C. J. Nolan, M. Cheney, T. Dowling and R. Gaburro, Enhanced angular resolution from multiply scattered waves,, Inverse Problems, 22 (2006), 1817. doi: 10.1088/0266-5611/22/5/017. [16] C. J. Nolan and W. W. Symes, Global solution of a linearized inverse problem for the acoustic wave equation,, Comm. in PDE, 22 (1997), 919. doi: 10.1080/03605309708821289. [17] M. Soumekh, Bistatic synthetic aperture radar inversion with application in dynamic object imaging,, IEEE Trans. on Signal Processing, 39 (1991), 2044. doi: 10.1109/78.134436. [18] X. Saint Raymond, "Elementary Introduction to the Theory of Pseudodifferential Operators. Studies in Advanced Mathematics,", CRC Press, (1991). [19] F. Treves, "Introduction to Pseudodifferential and Fourier Integral Operators," Vol. Iand II,, Plenum Press, (1980). [20] L. M. H. Ulander and P. O. Frölund, Ultra-wideband SAR interferometry,, IEEE Trans. Geosci. Remote Sensing, 36 (1998), 1540. doi: 10.1109/36.718858. [21] L. M. H. Ulander and H. Hellsten, Low-frequency ultra-wideband array-antenna SAR for stationary and moving target imaging,, in Proce. Conf. SPIE 13th Annu. Int. Symp. Aerosense, (1999). [22] C. E. Yarman, B. Yazici and M. Cheney, Bistatic synthetic aperture radar imaging for arbitrary flight trajectories,, submitted to IEEE-TIP, (2007).

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##### References:
 [1] G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform,, J. Math. Phys., 26 (1985), 99. doi: 10.1063/1.526755. [2] G. Beylkin and R. Burridge, Linearized inverse scattering problems in acoustic and elasticity,, Wave Motion, 12 (1990), 15. doi: 10.1016/0165-2125(90)90017-X. [3] N. Bleistein, J. K. Cohen and J. W. Stockwell, "The Matematics of Multidimensional Seismic Inversion,", Springer-Verlag, (2000). [4] M. Cheney, A mathematical tutorial on synthetic aperture radar,, SIAM Review, 43 (2001), 301. doi: 10.1137/S0036144500368859. [5] M. Cheney and R. J. Bonneau, Imaging that exploits multipath scattering from point scatterers,, Inverse Problems, 20 (2004), 1691. doi: 10.1088/0266-5611/20/5/023. [6] J. J. Duistermaat, "Fourier Integral Operators. Progress in Mathematics, 130, ", Birkhauser, (1996). [7] R. Gaburro, C. J. Nolan, T. Dowling and M. Cheney, Imaging from multiply scattered waves,, Proc. SPIE 6513, (2007). doi: 10.1117/12.712569. [8] A. Grigis and J. Sjöstrand, "Microlocal Analysis for Differential Operators: an Introduction,", London Mathematical Sciety Lecture Note Series, 196 (1994). [9] G. T. Herman, H. K. Tuy, K. J. Langenberg and P. C. Sabatier, "Basic Methods of Tomography and Inverse Problems,", Adam Hilger, (1988). [10] P. Morse and H. Feshbach, "Methods of Theoretical Physics," Vol. 1,, McGraw-Hill, (1953). [11] C. J. Nolan, Scattering near a fold caustic,, SIAM J. of Appl. Math, 61 (2000), 659. doi: 10.1137/S0036139999356107. [12] C. J. Nolan and M. Cheney, Synthetic aperture inversion for arbitrary flight paths and non-flat topography,, IEEE Trans. on Image Processing, 12 (2003), 1035. doi: 10.1109/TIP.2003.814243. [13] C. J. Nolan and M. Cheney, Synthetic aperture inversion,, Inverse Problems, 18 (2002), 221. doi: 10.1088/0266-5611/18/1/315. [14] C. J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging,, J. Fourier Analysis and its Applications, 10 (2004), 133. [15] C. J. Nolan, M. Cheney, T. Dowling and R. Gaburro, Enhanced angular resolution from multiply scattered waves,, Inverse Problems, 22 (2006), 1817. doi: 10.1088/0266-5611/22/5/017. [16] C. J. Nolan and W. W. Symes, Global solution of a linearized inverse problem for the acoustic wave equation,, Comm. in PDE, 22 (1997), 919. doi: 10.1080/03605309708821289. [17] M. Soumekh, Bistatic synthetic aperture radar inversion with application in dynamic object imaging,, IEEE Trans. on Signal Processing, 39 (1991), 2044. doi: 10.1109/78.134436. [18] X. Saint Raymond, "Elementary Introduction to the Theory of Pseudodifferential Operators. Studies in Advanced Mathematics,", CRC Press, (1991). [19] F. Treves, "Introduction to Pseudodifferential and Fourier Integral Operators," Vol. Iand II,, Plenum Press, (1980). [20] L. M. H. Ulander and P. O. Frölund, Ultra-wideband SAR interferometry,, IEEE Trans. Geosci. Remote Sensing, 36 (1998), 1540. doi: 10.1109/36.718858. [21] L. M. H. Ulander and H. Hellsten, Low-frequency ultra-wideband array-antenna SAR for stationary and moving target imaging,, in Proce. Conf. SPIE 13th Annu. Int. Symp. Aerosense, (1999). [22] C. E. Yarman, B. Yazici and M. Cheney, Bistatic synthetic aperture radar imaging for arbitrary flight trajectories,, submitted to IEEE-TIP, (2007).
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