Microlocal aspects of common offset synthetic aperture radar imaging

Previous Article
Explicit characterization of the support of non-linear inclusions
IPI Home
This Issue
Next Article
A fast algorithm for global minimization of maximum likelihood based on ultrasound image segmentation

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
Related Papers

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 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.

[1]	Kaitlyn (Voccola) Muller. SAR correlation imaging and anisotropic scattering. Inverse Problems and 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 and 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 and 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 and Imaging, 2020, 14 (3) : 511-533. doi: 10.3934/ipi.2020024
[5]	Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems and 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 and Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013
[8]	Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of borehole seismic data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022026
[9]	Lassi Roininen, Markku S. Lehtinen, Petteri Piiroinen, Ilkka I. Virtanen. Perfect radar pulse compression via unimodular fourier multipliers. Inverse Problems and Imaging, 2014, 8 (3) : 831-844. doi: 10.3934/ipi.2014.8.831
[10]	Jiying Liu, Jubo Zhu, Fengxia Yan, Zenghui Zhang. Compressive sampling and $l_1$ minimization for SAR imaging with low sampling rate. Inverse Problems and Imaging, 2013, 7 (4) : 1295-1305. doi: 10.3934/ipi.2013.7.1295
[11]	Kaitlyn Muller. The relationship between backprojection and best linear unbiased estimation in synthetic-aperture radar imaging. Inverse Problems and Imaging, 2016, 10 (2) : 549-561. doi: 10.3934/ipi.2016011
[12]	Josselin Garnier, George Papanicolaou. Resolution enhancement from scattering in passive sensor imaging with cross correlations. Inverse Problems and 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 and Imaging, 2009, 3 (2) : 231-241. doi: 10.3934/ipi.2009.3.231
[14]	Deyue Zhang, Yue Wu, Yinglin Wang, Yukun Guo. A direct imaging method for the exterior and interior inverse scattering problems. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022025
[15]	John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems and Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[16]	Ennio Fedrizzi. High frequency analysis of imaging with noise blending. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 979-998. doi: 10.3934/dcdsb.2014.19.979
[17]	Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete and 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 and 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 and 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 and Applied Analysis, 2022, 21 (1) : 159-181. doi: 10.3934/cpaa.2021173

2021 Impact Factor: 1.483

Tools

Metrics

Other articles
by authors

on AIMS
Venkateswaran P. Krishnan Eric Todd Quinto
on Google Scholar
Venkateswaran P. Krishnan Eric Todd Quinto

[Back to Top]