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December  2019, 13(6): 1283-1307. doi: 10.3934/ipi.2019056

Microlocal analysis of Doppler synthetic aperture radar

1. 

School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA

2. 

Department of Mathematics and Statistics and Health Research Institute, University of Limerick, Limerick, V94 T9PX, Ireland

3. 

Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

* Corresponding author: Raluca Felea

Received  November 2018 Revised  May 2019 Published  October 2019

We study the existence and suppression of artifacts for a Doppler-based Synthetic Aperture Radar (DSAR) system. The idealized air- or space-borne system transmits a continuous wave at a fixed frequency and a co-located receiver measures the resulting scattered waves; a windowed Fourier transform then converts the raw data into a function of two variables: slow time and frequency. Under simplifying assumptions, we analyze the linearized forward scattering map and the feasibility of inverting it via filtered backprojection, using techniques of microlocal analysis which robustly describe how sharp features in the target appear in the data. For DSAR with a straight flight path, there is, as with conventional SAR, a left-right ambiguity artifact in the DSAR image, which can be avoided via beam forming to the left or right. For a circular flight path, the artifact has a more complicated structure, but filtering out echoes coming from straight ahead or behind the transceiver, as well as those outside a critical range, produces an artifact-free image. We show that these results are qualitatively robust; although initially derived under an approximation widely used for range-based SAR, they are either structurally stable or robust with respect to a more accurate model.

Citation: 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
References:
[1]

G. AmbartsoumianR. FeleaV. P. KrishnanC. Nolan and E. T. Quinto, A class of singular Fourier integral operators in synthetic aperture radar imaging, Jour. Func. Analysis, 264 (2013), 246-269.  doi: 10.1016/j.jfa.2012.10.008.

[2]

G. AmbartsoumianR. FeleaV. P. KrishnanC. J. Nolan and E. T. Quinto, Singular FIOs in SAR imaging Ⅱ: Transmitter and receiver with different speeds, SIAM J. Math. Analysis, 50 (2018), 591-621.  doi: 10.1137/17M1125741.

[3] G. Arfken, Mathematical Methods for Physicists, Academic Press, New York-London, 1966. 
[4]

V. I. Arnol'd, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, I: The Classification of Critical Points, Caustics and Wave Fronts, Monographs in Mathematics, 82. Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5154-5.

[5]

B. Borden and M. Cheney, Synthetic-aperture imaging from high-Doppler-resolution measurements, Inverse Problems, 21 (2005), 1-11.  doi: 10.1088/0266-5611/21/1/001.

[6]

T. Bröcker and L. Lander, Differentiable germs and catastrophes, London Math. Soc. Lect. Note Series, Cambridge Univ. Press, Cambridge-New York-Melbourne, (1975).

[7]

M. Cheney and B. Borden, Theory of waveform-diverse moving-target spotlight synthetic-aperture radar, SIAM J. Imaging Sci., 4 (2011), 1180-1199.  doi: 10.1137/100808320.

[8]

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, PA, 2009. doi: 10.1137/1.9780898719291.

[9]

S. L. Coetzee, C. L. Baker and H. D. Griffiths, Narrow band high resolution radar imaging, in 2006 IEEE Conference on Radar, (2006), 622–625. doi: 10.1109/RADAR.2006.1631865.

[10]

J. J. Duistermaat, Fourier Integral Operators, Progress in Mathematics, 130. Birkhäuser Boston, Inc., Boston, MA, 1996.

[11]

R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Comm. Partial Differential Equations, 30 (2005), 1717-1740.  doi: 10.1080/03605300500299968.

[12]

R. Felea, Composition of Fourier Integral Operators with Fold and Blow-Down Singularities, Thesis (Ph.D.)-University of Rochester, 2004. 62 pp.

[13]

R. Felea, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519-1531.  doi: 10.1088/0266-5611/23/4/009.

[14]

R. FeleaR. Gaburro and C. Nolan, Microlocal analysis of SAR imaging of a dynamic reflectivity function, SIAM J. Math. Analysis, 4 (2013), 2767-2789.  doi: 10.1137/120873571.

[15]

R. Felea and A. Greenleaf, Fourier integral operators with open umbrellas and seismic inversion for cusp caustics, Math. Res. Lett., 17 (2010), 867-886.  doi: 10.4310/MRL.2010.v17.n5.a6.

[16]

R. Felea and C. Nolan, Monostatic SAR with fold/cusp singularities, Jour. Fourier Analysis Appl., 21 (2015), 799-821.  doi: 10.1007/s00041-015-9387-0.

[17]

R. Felea and E. T. Quinto, The microlocal properties of the local 3-D SPECT operator, SIAM J. Math. Anal., 43 (2011), 1145-1157.  doi: 10.1137/100807703.

[18]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1973.

[19]

A. Greenleaf and G. Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. Jour., 58 (1989), 205-240.  doi: 10.1215/S0012-7094-89-05811-0.

[20]

V. Guillemin, On some results of Gel'fand in integral geometry, in Pseudodifferential Operators and Applications (Notre Dame, Ind., 1984), Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 43 (1985), 149–155. doi: 10.1090/pspum/043/812288.

[21]

V. Guillemin, Cosmology in (2 + 1)-Dimensions, Cyclic Models, and Deformations of $M_{2, 1}$, Annals of Math. Studies, 121. Princeton Univ. Press, Princeton, NJ, 1989. doi: 10.1515/9781400882410.

[22]

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

[23]

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

[24]

L. Hörmander, The Analysis of Linear Partial Differential Operators, III: Pseudodifferential Operators, Grundlehren der Mathematischen Wissenschaften, 274. Springer-Verlag, Berlin, 1985.

[25]

L. Hörmander, The Analysis of Linear Partial Differential Operators, IV: Fourier Integral Operators, Grundlehren der Mathematischen Wissenschaften, 275. Springer-Verlag, Berlin, 1985.

[26]

R. B. Melrose and G. A. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.  doi: 10.1002/cpa.3160320403.

[27]

D. Mensa, High Resolution Radar Imaging, Artech House, Dedham, MA, 1981.

[28]

D. L. MensaS. Halevy and G. Wade, Coherent Doppler tomography for microwave imaging, Proceedings of the IEEE, 71 (1983), 254-261.  doi: 10.1109/PROC.1983.12563.

[29]

D. Mensa and G. Heidbreder, Bistatic synthetic-aperture radar imaging of rotating objects, IEEE Trans. Aerosp. and Electronic Sys., 18 (1982), 423-431.  doi: 10.1109/TAES.1982.309249.

[30]

C. J. Nolan and W. W. Symes, Global solution of a linearized inverse problem for the wave equation, Comm. Partial Differential Equations, 22 (1997), 919-952.  doi: 10.1080/03605309708821289.

[31]

C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221-235.  doi: 10.1088/0266-5611/18/1/315.

[32]

C. J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging, Jour. Fourier Analysis and Appl., 10 (2004), 133-148.  doi: 10.1007/s00041-004-8008-0.

[33]

E. T. Quinto and H. Rullgård, Local singularity reconstruction from integrals over curves in $\mathbb R^3$, Inverse Probl. Imaging, 7 (2013), 585-609.  doi: 10.3934/ipi.2013.7.585.

[34]

B. D. Rigling, Intrinsic processing gains in noise radar, IEEE Conference on Waveform Diversity & Design Conference, (2006). doi: 10.1109/WDD.2006.8321462.

[35]

M. S. Roulston and D. O. Muhleman, Synthesizing radar maps of polar regions with a Doppler-only method, Applied Optics, 36 (1997), 3912-3919.  doi: 10.1364/AO.36.003912.

[36]

P. Stefanov and G. Uhlmann, Is a curved flight path in SAR better than a straight one?, SIAM J. Appl. Math., 73 (2013), 1596-1612.  doi: 10.1137/120882639.

[37]

H. B. Sun, H. C. Feng and Y. L. Lu, High resolution radar tomographic imaging using single-tone CW signals, in 2010 IEEE Radar Conference, (2010). doi: 10.1109/RADAR.2010.5494477.

[38]

J. H. Thomson and J. E. B. Ponsonby, Two-dimensional aperture synthesis in lunar radar astronomy, Proc. R. Soc. London Ser. A, 303 (1968), 477-491. 

[39]

S. V. Tsynkov, On the use of start-stop approximation for spaceborne SAR imaging, SIAM J. Imaging Sci., 2 (2009), 646-669.  doi: 10.1137/08074026X.

[40]

L. Wang and B. Yazici, Bistatic synthetic aperture radar imaging using ultranarrowband continuous waveforms, IEEE Trans. Image Process., 21 (2012), 3673-3686.  doi: 10.1109/TIP.2012.2193134.

[41]

L. Wang and B. Yazici, Bistatic synthetic aperture radar imaging of moving targets using ultra-narrowband continuous waveform, SIAM J. Imaging Sci., 7 (2014), 824-866.  doi: 10.1137/130906714.

[42]

M. C. Wicks, B. Himed, J. L. E. Bracken, H. Bascom and J. Clancy, Ultra narrow band adaptive tomographic radar, in 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005, (2005). doi: 10.1109/CAMAP.2005.1574177.

[43]

C. E. Yarman, L. Wang and B. Yazici, Doppler synthetic aperture hitchhiker imaging, Inverse Problems, 26 (2010), 065006, 26 pp. doi: 10.1088/0266-5611/26/6/065006.

[44]

B. Yazici, I.-Y. Son and H. C. Yanik, Doppler synthetic aperture radar interferometry: A novel SAR interferometry for height mapping using ultra-narrowband waveforms, Inverse Problems, 34 (2018), 055003, 28 pp. doi: 10.1088/1361-6420/aab24c.

show all references

References:
[1]

G. AmbartsoumianR. FeleaV. P. KrishnanC. Nolan and E. T. Quinto, A class of singular Fourier integral operators in synthetic aperture radar imaging, Jour. Func. Analysis, 264 (2013), 246-269.  doi: 10.1016/j.jfa.2012.10.008.

[2]

G. AmbartsoumianR. FeleaV. P. KrishnanC. J. Nolan and E. T. Quinto, Singular FIOs in SAR imaging Ⅱ: Transmitter and receiver with different speeds, SIAM J. Math. Analysis, 50 (2018), 591-621.  doi: 10.1137/17M1125741.

[3] G. Arfken, Mathematical Methods for Physicists, Academic Press, New York-London, 1966. 
[4]

V. I. Arnol'd, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, I: The Classification of Critical Points, Caustics and Wave Fronts, Monographs in Mathematics, 82. Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5154-5.

[5]

B. Borden and M. Cheney, Synthetic-aperture imaging from high-Doppler-resolution measurements, Inverse Problems, 21 (2005), 1-11.  doi: 10.1088/0266-5611/21/1/001.

[6]

T. Bröcker and L. Lander, Differentiable germs and catastrophes, London Math. Soc. Lect. Note Series, Cambridge Univ. Press, Cambridge-New York-Melbourne, (1975).

[7]

M. Cheney and B. Borden, Theory of waveform-diverse moving-target spotlight synthetic-aperture radar, SIAM J. Imaging Sci., 4 (2011), 1180-1199.  doi: 10.1137/100808320.

[8]

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, PA, 2009. doi: 10.1137/1.9780898719291.

[9]

S. L. Coetzee, C. L. Baker and H. D. Griffiths, Narrow band high resolution radar imaging, in 2006 IEEE Conference on Radar, (2006), 622–625. doi: 10.1109/RADAR.2006.1631865.

[10]

J. J. Duistermaat, Fourier Integral Operators, Progress in Mathematics, 130. Birkhäuser Boston, Inc., Boston, MA, 1996.

[11]

R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Comm. Partial Differential Equations, 30 (2005), 1717-1740.  doi: 10.1080/03605300500299968.

[12]

R. Felea, Composition of Fourier Integral Operators with Fold and Blow-Down Singularities, Thesis (Ph.D.)-University of Rochester, 2004. 62 pp.

[13]

R. Felea, Displacement of artefacts in inverse scattering, Inverse Problems, 23 (2007), 1519-1531.  doi: 10.1088/0266-5611/23/4/009.

[14]

R. FeleaR. Gaburro and C. Nolan, Microlocal analysis of SAR imaging of a dynamic reflectivity function, SIAM J. Math. Analysis, 4 (2013), 2767-2789.  doi: 10.1137/120873571.

[15]

R. Felea and A. Greenleaf, Fourier integral operators with open umbrellas and seismic inversion for cusp caustics, Math. Res. Lett., 17 (2010), 867-886.  doi: 10.4310/MRL.2010.v17.n5.a6.

[16]

R. Felea and C. Nolan, Monostatic SAR with fold/cusp singularities, Jour. Fourier Analysis Appl., 21 (2015), 799-821.  doi: 10.1007/s00041-015-9387-0.

[17]

R. Felea and E. T. Quinto, The microlocal properties of the local 3-D SPECT operator, SIAM J. Math. Anal., 43 (2011), 1145-1157.  doi: 10.1137/100807703.

[18]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1973.

[19]

A. Greenleaf and G. Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. Jour., 58 (1989), 205-240.  doi: 10.1215/S0012-7094-89-05811-0.

[20]

V. Guillemin, On some results of Gel'fand in integral geometry, in Pseudodifferential Operators and Applications (Notre Dame, Ind., 1984), Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 43 (1985), 149–155. doi: 10.1090/pspum/043/812288.

[21]

V. Guillemin, Cosmology in (2 + 1)-Dimensions, Cyclic Models, and Deformations of $M_{2, 1}$, Annals of Math. Studies, 121. Princeton Univ. Press, Princeton, NJ, 1989. doi: 10.1515/9781400882410.

[22]

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

[23]

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

[24]

L. Hörmander, The Analysis of Linear Partial Differential Operators, III: Pseudodifferential Operators, Grundlehren der Mathematischen Wissenschaften, 274. Springer-Verlag, Berlin, 1985.

[25]

L. Hörmander, The Analysis of Linear Partial Differential Operators, IV: Fourier Integral Operators, Grundlehren der Mathematischen Wissenschaften, 275. Springer-Verlag, Berlin, 1985.

[26]

R. B. Melrose and G. A. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.  doi: 10.1002/cpa.3160320403.

[27]

D. Mensa, High Resolution Radar Imaging, Artech House, Dedham, MA, 1981.

[28]

D. L. MensaS. Halevy and G. Wade, Coherent Doppler tomography for microwave imaging, Proceedings of the IEEE, 71 (1983), 254-261.  doi: 10.1109/PROC.1983.12563.

[29]

D. Mensa and G. Heidbreder, Bistatic synthetic-aperture radar imaging of rotating objects, IEEE Trans. Aerosp. and Electronic Sys., 18 (1982), 423-431.  doi: 10.1109/TAES.1982.309249.

[30]

C. J. Nolan and W. W. Symes, Global solution of a linearized inverse problem for the wave equation, Comm. Partial Differential Equations, 22 (1997), 919-952.  doi: 10.1080/03605309708821289.

[31]

C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221-235.  doi: 10.1088/0266-5611/18/1/315.

[32]

C. J. Nolan and M. Cheney, Microlocal analysis of synthetic aperture radar imaging, Jour. Fourier Analysis and Appl., 10 (2004), 133-148.  doi: 10.1007/s00041-004-8008-0.

[33]

E. T. Quinto and H. Rullgård, Local singularity reconstruction from integrals over curves in $\mathbb R^3$, Inverse Probl. Imaging, 7 (2013), 585-609.  doi: 10.3934/ipi.2013.7.585.

[34]

B. D. Rigling, Intrinsic processing gains in noise radar, IEEE Conference on Waveform Diversity & Design Conference, (2006). doi: 10.1109/WDD.2006.8321462.

[35]

M. S. Roulston and D. O. Muhleman, Synthesizing radar maps of polar regions with a Doppler-only method, Applied Optics, 36 (1997), 3912-3919.  doi: 10.1364/AO.36.003912.

[36]

P. Stefanov and G. Uhlmann, Is a curved flight path in SAR better than a straight one?, SIAM J. Appl. Math., 73 (2013), 1596-1612.  doi: 10.1137/120882639.

[37]

H. B. Sun, H. C. Feng and Y. L. Lu, High resolution radar tomographic imaging using single-tone CW signals, in 2010 IEEE Radar Conference, (2010). doi: 10.1109/RADAR.2010.5494477.

[38]

J. H. Thomson and J. E. B. Ponsonby, Two-dimensional aperture synthesis in lunar radar astronomy, Proc. R. Soc. London Ser. A, 303 (1968), 477-491. 

[39]

S. V. Tsynkov, On the use of start-stop approximation for spaceborne SAR imaging, SIAM J. Imaging Sci., 2 (2009), 646-669.  doi: 10.1137/08074026X.

[40]

L. Wang and B. Yazici, Bistatic synthetic aperture radar imaging using ultranarrowband continuous waveforms, IEEE Trans. Image Process., 21 (2012), 3673-3686.  doi: 10.1109/TIP.2012.2193134.

[41]

L. Wang and B. Yazici, Bistatic synthetic aperture radar imaging of moving targets using ultra-narrowband continuous waveform, SIAM J. Imaging Sci., 7 (2014), 824-866.  doi: 10.1137/130906714.

[42]

M. C. Wicks, B. Himed, J. L. E. Bracken, H. Bascom and J. Clancy, Ultra narrow band adaptive tomographic radar, in 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005, (2005). doi: 10.1109/CAMAP.2005.1574177.

[43]

C. E. Yarman, L. Wang and B. Yazici, Doppler synthetic aperture hitchhiker imaging, Inverse Problems, 26 (2010), 065006, 26 pp. doi: 10.1088/0266-5611/26/6/065006.

[44]

B. Yazici, I.-Y. Son and H. C. Yanik, Doppler synthetic aperture radar interferometry: A novel SAR interferometry for height mapping using ultra-narrowband waveforms, Inverse Problems, 34 (2018), 055003, 28 pp. doi: 10.1088/1361-6420/aab24c.

Figure 1.  Schematic of a circular flight path
Figure 2.  The curves of constant $ u $ (hyperbolas) and constant $ v $ (vertical lines) for a location on the flight path in which the flight velocity vector is along the vertical axis. The coordinate system is centered directly under the antenna
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