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

Microlocal aspects of common offset synthetic aperture radar imaging

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

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 $I^{3,0}$.
Citation: Venkateswaran P. Krishnan, Eric Todd Quinto. Microlocal aspects of common offset synthetic aperture radar imaging. Inverse Problems and 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), 214-232. doi: 10.1137/0519016.

[2]

M. Cheney, A mathematical tutorial on synthetic aperture radar, SIAM Rev., 43 (2001), 301-312. doi: 10.1137/S0036144500368859.

[3]

M. Cheney and B. Borden, "Fundamentals of Radar Imaging," CBMS-NSF Regional Conference Series in Applied Mathematics, 79, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania, 2009.

[4]

J. Duistermaat, "Fourier Integral Operators," Birkhäuser Boston, Inc., Boston, Massachusetts, 1996.

[5]

J. Cohen and H. Bleistein, Velocity inversion procedure for acoustic waves, Geophysics, 44 (1979), 1077-1085. doi: 10.1190/1.1440996.

[6]

T. Dowling, "Radar Imaging Using Multiply Scattered Waves," Ph.D. Thesis, University of Limerick, Ireland, 2009.

[7]

R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Comm. Partial Differential Equations, 30 (2005), 1717-1740.

[8]

R. Felea, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519-1531. 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), 45-77. doi: 10.1080/03605300701318716.

[10]

A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240. 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), 202-232. 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), 443-466.

[13]

A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry, in "Integral Geometry and Tomography" (Arcata, CA, 1989), 121-136 Contemporary Math., 113 Amer. Math. Soc., Providence, RI, 1990.

[14]

V. Guillemin, "Cosmology in $(2 + 1)$-Dimensions, Cyclic Models, and Deformations of $M_{2,1}$," Annals of Mathematics Studies, 121, Princeton University Press, Princeton, NJ, 1989.

[15]

V. Guillemin and S. Sternberg, "Geometric Asymptotics," Mathematical Surveys, No. 14, American Mathematical Society, Providence, RI, 1977.

[16]

V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267. doi: 10.1215/S0012-7094-81-04814-6.

[17]

L. Hörmander, Fourier integral operators, I, Acta Mathematica, 127 (1971), 79-183. doi: 10.1007/BF02392052.

[18]

A. M. Horne and G. Yates, "Bistatic Synthetic Aperture Radar," Proc. IEEE Radar Conf., 2002, 6-10.

[19]

L. Hörmander, "The Analysis of Linear Partial Differential Operators, I-IV," Springer-Verlag, New York, 1983.

[20]

A. K. Louis and E. T. Quinto, Local tomographic methods in SONAR, in "Surveys on Solution Methods for Inverse Problems" (Vienna/New York) (eds. D. Colton, H. Engl, A. Louis, J. McLaughlin and W. Rundell), Springer, Vienna, 2000, 147-154.

[21]

R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.

[22]

C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221-235. 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), 133-148. 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), 331-346. doi: 10.1090/S0002-9947-1980-0552261-8.

[25]

F. Tréves, "Introduction to Pseudodifferential and Fourier Integral Operators," Vol. 1 and 2, Plenum Press, New York, 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), 84-93. 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), 214-232. doi: 10.1137/0519016.

[2]

M. Cheney, A mathematical tutorial on synthetic aperture radar, SIAM Rev., 43 (2001), 301-312. doi: 10.1137/S0036144500368859.

[3]

M. Cheney and B. Borden, "Fundamentals of Radar Imaging," CBMS-NSF Regional Conference Series in Applied Mathematics, 79, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania, 2009.

[4]

J. Duistermaat, "Fourier Integral Operators," Birkhäuser Boston, Inc., Boston, Massachusetts, 1996.

[5]

J. Cohen and H. Bleistein, Velocity inversion procedure for acoustic waves, Geophysics, 44 (1979), 1077-1085. doi: 10.1190/1.1440996.

[6]

T. Dowling, "Radar Imaging Using Multiply Scattered Waves," Ph.D. Thesis, University of Limerick, Ireland, 2009.

[7]

R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Comm. Partial Differential Equations, 30 (2005), 1717-1740.

[8]

R. Felea, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519-1531. 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), 45-77. doi: 10.1080/03605300701318716.

[10]

A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240. 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), 202-232. 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), 443-466.

[13]

A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry, in "Integral Geometry and Tomography" (Arcata, CA, 1989), 121-136 Contemporary Math., 113 Amer. Math. Soc., Providence, RI, 1990.

[14]

V. Guillemin, "Cosmology in $(2 + 1)$-Dimensions, Cyclic Models, and Deformations of $M_{2,1}$," Annals of Mathematics Studies, 121, Princeton University Press, Princeton, NJ, 1989.

[15]

V. Guillemin and S. Sternberg, "Geometric Asymptotics," Mathematical Surveys, No. 14, American Mathematical Society, Providence, RI, 1977.

[16]

V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267. doi: 10.1215/S0012-7094-81-04814-6.

[17]

L. Hörmander, Fourier integral operators, I, Acta Mathematica, 127 (1971), 79-183. doi: 10.1007/BF02392052.

[18]

A. M. Horne and G. Yates, "Bistatic Synthetic Aperture Radar," Proc. IEEE Radar Conf., 2002, 6-10.

[19]

L. Hörmander, "The Analysis of Linear Partial Differential Operators, I-IV," Springer-Verlag, New York, 1983.

[20]

A. K. Louis and E. T. Quinto, Local tomographic methods in SONAR, in "Surveys on Solution Methods for Inverse Problems" (Vienna/New York) (eds. D. Colton, H. Engl, A. Louis, J. McLaughlin and W. Rundell), Springer, Vienna, 2000, 147-154.

[21]

R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.

[22]

C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221-235. 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), 133-148. 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), 331-346. doi: 10.1090/S0002-9947-1980-0552261-8.

[25]

F. Tréves, "Introduction to Pseudodifferential and Fourier Integral Operators," Vol. 1 and 2, Plenum Press, New York, 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), 84-93. doi: 10.1109/TIP.2007.911812.

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