American Institute of Mathematical Sciences

Advanced
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

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), 214-232. 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

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.  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

[1]

Kaitlyn (Voccola) Muller. SAR correlation imaging and anisotropic scattering. Inverse Problems & Imaging, 2018, 12 (3) : 697-731. doi: 10.3934/ipi.2018030
[2]

Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of Doppler synthetic aperture radar. Inverse Problems & Imaging, 2019, 13 (6) : 1283-1307. doi: 10.3934/ipi.2019056
[3]

Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1
[4]

Mikhail Gilman, Semyon Tsynkov. Statistical characterization of scattering delay in synthetic aperture radar imaging. Inverse Problems & Imaging, 2020, 14 (3) : 511-533. doi: 10.3934/ipi.2020024
[5]

Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems & Imaging, 2007, 1 (1) : 1-11. doi: 10.3934/ipi.2007.1.1
[6]

Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7
[7]

James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013
[8]

Lassi Roininen, Markku S. Lehtinen, Petteri Piiroinen, Ilkka I. Virtanen. Perfect radar pulse compression via unimodular fourier multipliers. Inverse Problems & Imaging, 2014, 8 (3) : 831-844. doi: 10.3934/ipi.2014.8.831
[9]

Jiying Liu, Jubo Zhu, Fengxia Yan, Zenghui Zhang. Compressive sampling and $l_1$ minimization for SAR imaging with low sampling rate. Inverse Problems & Imaging, 2013, 7 (4) : 1295-1305. doi: 10.3934/ipi.2013.7.1295
[10]

Kaitlyn Muller. The relationship between backprojection and best linear unbiased estimation in synthetic-aperture radar imaging. Inverse Problems & Imaging, 2016, 10 (2) : 549-561. doi: 10.3934/ipi.2016011
[11]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[12]

Josselin Garnier, George Papanicolaou. Resolution enhancement from scattering in passive sensor imaging with cross correlations. Inverse Problems & Imaging, 2014, 8 (3) : 645-683. doi: 10.3934/ipi.2014.8.645
[13]

Giovanni Bozza, Massimo Brignone, Matteo Pastorino, Andrea Randazzo, Michele Piana. Imaging of unknown targets inside inhomogeneous backgrounds by means of qualitative inverse scattering. Inverse Problems & Imaging, 2009, 3 (2) : 231-241. doi: 10.3934/ipi.2009.3.231
[14]

Ennio Fedrizzi. High frequency analysis of imaging with noise blending. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 979-998. doi: 10.3934/dcdsb.2014.19.979
[15]

Gary Froyland, Cecilia González-Tokman, Anthony Quas. Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools. Journal of Computational Dynamics, 2014, 1 (2) : 249-278. doi: 10.3934/jcd.2014.1.249
[16]

Jorge J. Betancor, Alejandro J. Castro, Marta De León-Contreras. Variation operators for semigroups associated with Fourier-Bessel expansions. Communications on Pure & Applied Analysis, 2022, 21 (1) : 239-273. doi: 10.3934/cpaa.2021176
[17]

Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2401-2416. doi: 10.3934/dcdsb.2014.19.2401
[18]

Patricio Felmer, Alexander Quaas. Fundamental solutions for a class of Isaacs integral operators. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 493-508. doi: 10.3934/dcds.2011.30.493
[19]

Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915
[20]

Ahmad Al-Salman. Marcinkiewicz integral operators along twisted surfaces. Communications on Pure & Applied Analysis, 2022, 21 (1) : 159-181. doi: 10.3934/cpaa.2021173

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy