# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] M. Cheney, A mathematical tutorial on synthetic aperture radar, SIAM Rev., 43 (2001), 301-312. doi: 10.1137/S0036144500368859.  Google Scholar [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.  Google Scholar [4] J. Duistermaat, "Fourier Integral Operators," Birkhäuser Boston, Inc., Boston, Massachusetts, 1996. Google Scholar [5] J. Cohen and H. Bleistein, Velocity inversion procedure for acoustic waves, Geophysics, 44 (1979), 1077-1085. doi: 10.1190/1.1440996.  Google Scholar [6] T. Dowling, "Radar Imaging Using Multiply Scattered Waves," Ph.D. Thesis, University of Limerick, Ireland, 2009. Google Scholar [7] R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Comm. Partial Differential Equations, 30 (2005), 1717-1740.  Google Scholar [8] R. Felea, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519-1531. doi: 10.1088/0266-5611/23/4/009.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] V. Guillemin and S. Sternberg, "Geometric Asymptotics," Mathematical Surveys, No. 14, American Mathematical Society, Providence, RI, 1977.  Google Scholar [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.  Google Scholar [17] L. Hörmander, Fourier integral operators, I, Acta Mathematica, 127 (1971), 79-183. doi: 10.1007/BF02392052.  Google Scholar [18] A. M. Horne and G. Yates, "Bistatic Synthetic Aperture Radar," Proc. IEEE Radar Conf., 2002, 6-10. Google Scholar [19] L. Hörmander, "The Analysis of Linear Partial Differential Operators, I-IV," Springer-Verlag, New York, 1983. Google Scholar [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.  Google Scholar [21] R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.  Google Scholar [22] C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221-235. doi: 10.1088/0266-5611/18/1/315.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] F. Tréves, "Introduction to Pseudodifferential and Fourier Integral Operators," Vol. 1 and 2, Plenum Press, New York, 1980. Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [2] M. Cheney, A mathematical tutorial on synthetic aperture radar, SIAM Rev., 43 (2001), 301-312. doi: 10.1137/S0036144500368859.  Google Scholar [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.  Google Scholar [4] J. Duistermaat, "Fourier Integral Operators," Birkhäuser Boston, Inc., Boston, Massachusetts, 1996. Google Scholar [5] J. Cohen and H. Bleistein, Velocity inversion procedure for acoustic waves, Geophysics, 44 (1979), 1077-1085. doi: 10.1190/1.1440996.  Google Scholar [6] T. Dowling, "Radar Imaging Using Multiply Scattered Waves," Ph.D. Thesis, University of Limerick, Ireland, 2009. Google Scholar [7] R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Comm. Partial Differential Equations, 30 (2005), 1717-1740.  Google Scholar [8] R. Felea, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519-1531. doi: 10.1088/0266-5611/23/4/009.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] V. Guillemin and S. Sternberg, "Geometric Asymptotics," Mathematical Surveys, No. 14, American Mathematical Society, Providence, RI, 1977.  Google Scholar [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.  Google Scholar [17] L. Hörmander, Fourier integral operators, I, Acta Mathematica, 127 (1971), 79-183. doi: 10.1007/BF02392052.  Google Scholar [18] A. M. Horne and G. Yates, "Bistatic Synthetic Aperture Radar," Proc. IEEE Radar Conf., 2002, 6-10. Google Scholar [19] L. Hörmander, "The Analysis of Linear Partial Differential Operators, I-IV," Springer-Verlag, New York, 1983. Google Scholar [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.  Google Scholar [21] R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.  Google Scholar [22] C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221-235. doi: 10.1088/0266-5611/18/1/315.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] F. Tréves, "Introduction to Pseudodifferential and Fourier Integral Operators," Vol. 1 and 2, Plenum Press, New York, 1980. Google Scholar [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.  Google Scholar
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