Microlocal aspects of common offset synthetic aperture radar imaging

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August 2011, 5(3): 659-674. doi: 10.3934/ipi.2011.5.659

Microlocal aspects of common offset synthetic aperture radar imaging

Venkateswaran P. Krishnan ^1, and Eric Todd Quinto ^1,

1.	Department of Mathematics, Tufts University, 503, Boston Avenue, Medford, MA 02155, United States, United States

Received August 2010 Revised February 2011 Published August 2011

Abstract
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In this article, we analyze the microlocal properties of the linearized forward scattering operator $F$ and the reconstruction operator $F^{*}F$ appearing in bistatic synthetic aperture radar imaging. In our model, the radar source and detector travel along a line a fixed distance apart. We show that $F$ is a Fourier integral operator, and we give the mapping properties of the projections from the canonical relation of $F$, showing that the right projection is a blow-down and the left projection is a fold. We then show that $F^{*}F$ is a singular FIO belonging to the class $I^{3,0}$.

Keywords: Fourier Integral Operators, Microlocal Analysis, SAR Imaging, Scattering., Radar.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Venkateswaran P. Krishnan, Eric Todd Quinto. Microlocal aspects of common offset synthetic aperture radar imaging. Inverse Problems & Imaging, 2011, 5 (3) : 659-674. doi: 10.3934/ipi.2011.5.659

References:

[1]	L.-E. Andersson, On the determination of a function from spherical averages,, SIAM J. Math. Anal., 19 (1988), 214. doi: 10.1137/0519016.
[2]	M. Cheney, A mathematical tutorial on synthetic aperture radar,, SIAM Rev., 43 (2001), 301. doi: 10.1137/S0036144500368859.
[3]	M. Cheney and B. Borden, "Fundamentals of Radar Imaging,", CBMS-NSF Regional Conference Series in Applied Mathematics, 79 (2009).
[4]	J. Duistermaat, "Fourier Integral Operators,", Birkhäuser Boston, (1996).
[5]	J. Cohen and H. Bleistein, Velocity inversion procedure for acoustic waves,, Geophysics, 44 (1979), 1077. doi: 10.1190/1.1440996.
[6]	T. Dowling, "Radar Imaging Using Multiply Scattered Waves,", Ph.D. Thesis, (2009).
[7]	R. Felea, Composition of Fourier integral operators with fold and blowdown singularities,, Comm. Partial Differential Equations, 30 (2005), 1717.
[8]	R. Felea, Displacement of artefacts in inverse scattering,, Inverse Problems, 23 (2007), 1519. doi: 10.1088/0266-5611/23/4/009.
[9]	R. Felea and A. Greenleaf, An FIO calculus for marine seismic imaging: Folds and cross caps,, Comm. P.D.E., 33 (2008), 45. doi: 10.1080/03605300701318716.
[10]	A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform,, Duke Math. J., 58 (1989), 205. doi: 10.1215/S0012-7094-89-05811-0.
[11]	A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols,, J. Functional Anal., 89 (1990), 202. doi: 10.1016/0022-1236(90)90011-9.
[12]	A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms,, Ann. Inst. Fourier (Grenoble), 40 (1990), 443.
[13]	A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry,, in, 113 (1989), 121.
[14]	V. Guillemin, "Cosmology in $(2 + 1)$-Dimensions, Cyclic Models, and Deformations of $M_{2,1}$,", Annals of Mathematics Studies, 121 (1989).
[15]	V. Guillemin and S. Sternberg, "Geometric Asymptotics,", Mathematical Surveys, (1977).
[16]	V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols,, Duke Math. J., 48 (1981), 251. doi: 10.1215/S0012-7094-81-04814-6.
[17]	L. Hörmander, Fourier integral operators, I,, Acta Mathematica, 127 (1971), 79. doi: 10.1007/BF02392052.
[18]	A. M. Horne and G. Yates, "Bistatic Synthetic Aperture Radar,", Proc. IEEE Radar Conf., (2002), 6.
[19]	L. Hörmander, "The Analysis of Linear Partial Differential Operators, I-IV,", Springer-Verlag, (1983).
[20]	A. K. Louis and E. T. Quinto, Local tomographic methods in SONAR,, in, (2000), 147.
[21]	R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem,, Comm. Pure Appl. Math., 32 (1979), 483.
[22]	C. J. Nolan and M. Cheney, Synthetic aperture inversion,, Inverse Problems, 18 (2002), 221. doi: 10.1088/0266-5611/18/1/315.
[23]	C. J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging,, J. Fourier Anal. Appl., 10 (2004), 133. doi: 10.1007/s00041-004-8008-0.
[24]	E. T. Quinto, The dependence of the generalized Radon transform on defining measures,, Trans. Amer. Math. Soc., 257 (1980), 331. doi: 10.1090/S0002-9947-1980-0552261-8.
[25]	F. Tréves, "Introduction to Pseudodifferential and Fourier Integral Operators," Vol. 1 and 2,, Plenum Press, (1980).
[26]	C. E. Yarman, B. Yazıcı and M. Cheney, Bistatic synthetic aperture radar imaging with arbitrary trajectories,, IEEE Transactions on Image Processing, 17 (2008), 84. doi: 10.1109/TIP.2007.911812.

show all references

References:

[1]	L.-E. Andersson, On the determination of a function from spherical averages,, SIAM J. Math. Anal., 19 (1988), 214. doi: 10.1137/0519016.
[2]	M. Cheney, A mathematical tutorial on synthetic aperture radar,, SIAM Rev., 43 (2001), 301. doi: 10.1137/S0036144500368859.
[3]	M. Cheney and B. Borden, "Fundamentals of Radar Imaging,", CBMS-NSF Regional Conference Series in Applied Mathematics, 79 (2009).
[4]	J. Duistermaat, "Fourier Integral Operators,", Birkhäuser Boston, (1996).
[5]	J. Cohen and H. Bleistein, Velocity inversion procedure for acoustic waves,, Geophysics, 44 (1979), 1077. doi: 10.1190/1.1440996.
[6]	T. Dowling, "Radar Imaging Using Multiply Scattered Waves,", Ph.D. Thesis, (2009).
[7]	R. Felea, Composition of Fourier integral operators with fold and blowdown singularities,, Comm. Partial Differential Equations, 30 (2005), 1717.
[8]	R. Felea, Displacement of artefacts in inverse scattering,, Inverse Problems, 23 (2007), 1519. doi: 10.1088/0266-5611/23/4/009.
[9]	R. Felea and A. Greenleaf, An FIO calculus for marine seismic imaging: Folds and cross caps,, Comm. P.D.E., 33 (2008), 45. doi: 10.1080/03605300701318716.
[10]	A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform,, Duke Math. J., 58 (1989), 205. doi: 10.1215/S0012-7094-89-05811-0.
[11]	A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols,, J. Functional Anal., 89 (1990), 202. doi: 10.1016/0022-1236(90)90011-9.
[12]	A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms,, Ann. Inst. Fourier (Grenoble), 40 (1990), 443.
[13]	A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry,, in, 113 (1989), 121.
[14]	V. Guillemin, "Cosmology in $(2 + 1)$-Dimensions, Cyclic Models, and Deformations of $M_{2,1}$,", Annals of Mathematics Studies, 121 (1989).
[15]	V. Guillemin and S. Sternberg, "Geometric Asymptotics,", Mathematical Surveys, (1977).
[16]	V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols,, Duke Math. J., 48 (1981), 251. doi: 10.1215/S0012-7094-81-04814-6.
[17]	L. Hörmander, Fourier integral operators, I,, Acta Mathematica, 127 (1971), 79. doi: 10.1007/BF02392052.
[18]	A. M. Horne and G. Yates, "Bistatic Synthetic Aperture Radar,", Proc. IEEE Radar Conf., (2002), 6.
[19]	L. Hörmander, "The Analysis of Linear Partial Differential Operators, I-IV,", Springer-Verlag, (1983).
[20]	A. K. Louis and E. T. Quinto, Local tomographic methods in SONAR,, in, (2000), 147.
[21]	R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem,, Comm. Pure Appl. Math., 32 (1979), 483.
[22]	C. J. Nolan and M. Cheney, Synthetic aperture inversion,, Inverse Problems, 18 (2002), 221. doi: 10.1088/0266-5611/18/1/315.
[23]	C. J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging,, J. Fourier Anal. Appl., 10 (2004), 133. doi: 10.1007/s00041-004-8008-0.
[24]	E. T. Quinto, The dependence of the generalized Radon transform on defining measures,, Trans. Amer. Math. Soc., 257 (1980), 331. doi: 10.1090/S0002-9947-1980-0552261-8.
[25]	F. Tréves, "Introduction to Pseudodifferential and Fourier Integral Operators," Vol. 1 and 2,, Plenum Press, (1980).
[26]	C. E. Yarman, B. Yazıcı and M. Cheney, Bistatic synthetic aperture radar imaging with arbitrary trajectories,, IEEE Transactions on Image Processing, 17 (2008), 84. doi: 10.1109/TIP.2007.911812.

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