American Institute of Mathematical Sciences

Advanced
June  2018, 12(3): 697-731. doi: 10.3934/ipi.2018030

SAR correlation imaging and anisotropic scattering

Kaitlyn (Voccola) Muller ,

Department of Mathematics and Statistics, Villanova University, Villanova, PA 19085, USA

Corresponding author: k.muller@villanova.edu

Received  August 2017 Revised  December 2017 Published  March 2018

Fund Project: The author was supported by AFOSR grant FA9550-14-1-0124

In this paper we investigate the ability of correlation synthetic-aperture radar (SAR) imaging to reconstruct isotropic and anisotropic scatterers. SAR correlation imaging was suggested by the author previously in [34]. Correlation imaging algorithms produce an image of a second-order quantity describing an object an interest, for example, the reflectivity function squared. In the previous work [34] it was argued that the effects of volume scattering clutter on the image can be minimized by choosing which pairs of collected data to correlate prior to applying a backprojection-type reconstruction algorithm. This choice of pairs for the correlation process is determined by what is known as the memory effect of scattering of waves by random scatterers [42,43,40,41,7,14]. It is the goal of this current work to determine the different imaging outcomes for an isotropic or point scatterer versus an anisotropic or dipole scatterer. In addition we aim to determine if removing contributions to the image due to the memory effect is necessary for diminishing the contributions of anisotropic or clutter scatterers to the scene of interest. Finally we extend the analysis of [34] to the polarimetric SAR case to determine whether the additional data provided by this modality contributes to decreasing the effects of clutter on the SAR image.

Keywords: Synthetic aperture radar, polarimetric radar, correlation imaging, clutter mitigation.
Mathematics Subject Classification: Primary: 78A46, 94A13; Secondary: 35Q60.
Citation: Kaitlyn (Voccola) Muller. SAR correlation imaging and anisotropic scattering. Inverse Problems & Imaging, 2018, 12 (3) : 697-731. doi: 10.3934/ipi.2018030
References:
[1]

J. G. Berryman, L. Borcea, G. Papanicolaou and C. Tsogka, Statistically stable ultrasonic imaging in random media, J. Acoust. Soc. Am., 112 (2002), 1509-1522. doi: 10.1121/1.1502266.

[2]

W. M. Boerner, M. B. El-Arini, C. Y. Chan and P. M. Mastoris, Polarization dependence in electromagnetic inverse problems, IEEE Trans. on Antennas and Propagation, 29 (1981), 262-271. doi: 10.1109/TAP.1981.1142585.

[3]

L. Borcea, G. Papanicolaou and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Problems, 22 (2006), 1405-1436. doi: 10.1088/0266-5611/22/4/016.

[4]

L. Borcea, G. Papanicolaou and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), 1419-1460. doi: 10.1088/0266-5611/21/4/015.

[5]

L. Borcea, M. Moscoso, G. Papanicolaou and C. Tsogka, Synthetic aperture imaging of direction and frequency dependent reflectivities, SIAM J. Imaging Sci., 9 (2016), 52-81. doi: 10.1137/15M1036063.

[6]

L. Borcea, G. Papanicolaou and C. Tsogka, Theory and applications of time reversal and interferometric imaging, Inverse Problems, 19 (2003), S139-S164. doi: 10.1088/0266-5611/19/6/058.

[7]

T.-K. Chan, Y. Kuga and A. Ishimaru, Subsurface detection of a buried object using angular correlation function measurement, Waves Random Media, 7 (1997), 457-465. doi: 10.1080/13616679709409809.

[8]

R. D. Chaney, M. C. Burl and L. M. Novak, On the Performance of Polarimetric Target Detection Algorithms, IEEE International Radar Conference, 1990. doi: 10.1109/RADAR.1990.201114.

[9]

M. Cheney and B. Borden, Fundamentals of Radar Imaging, SIAM, Philadelphia, 2009.

[10]

S. R. Cloude and E. Pottier, A review of target decomposition theorems in radar polarimetry, IEEE Trans. on Geoscience and Remote Sensing, 34 (1996), 498-518. doi: 10.1109/36.485127.

[11]

A. J. Devaney, The inverse problem for random sources, J. Math. Phys., 20 (1979), 1687-1691. doi: 10.1063/1.524277.

[12]

R. L. Dilsavor and R. L. Moses, Fully-Polarimetric GLRTs for Detecting Scattering Centers with Unknown Amplitude, Phase, and Tilt Angle in Terrain Clutter, in SPIE's International Symposium on Optical Engineering in Aerospace Sensing, Orlando, FL, 1994.
[13]

D. E. Dudgeon, R. T. Lacoss, C. H. Lazott and J. G. Verly, Use of Persistent Scatterers for Model-Based Recognition, Proc. SPIE 2230, Algorithms for Synthetic Aperture Radar Imagery, 1994.
[14]

A. E. El-Rouby, A. Y. Nashashibi and F. T. Ulaby, Application of frequency correlation function to radar target detection, IEEE Trans. Aero. Elec. Sys., 39 (2003), 125-139. doi: 10.1109/TAES.2003.1188898.

[15]

E. Ertin, L. C. Potter and R. L. Moses, Enhanced imaging over complete circular apertures, Signals, Systems, and Computers, (2006). doi: 10.1109/ACSSC.2006.355025.

[16]

S. Feng, C. Kane, P. A. Lee and A. D. Stone, Correlations and fluctuations of coherent wave transmission through disordered media, Phys. Rev. Lett., 61 (1988), 834-837. doi: 10.1103/PhysRevLett.61.834.

[17]

L. Ferro-Famil, A. Reigber, E. Pottier and W. M. Boerner, Scene characterization using subaperture polarimetric SAR data, IEEE Trans, on Geoscience and Remote Sensing, 41 (2003), 2264-2276. doi: 10.1109/TGRS.2003.817188.

[18]

A. Freeman and S. L. Durden, A Three-Component Scattering Model for Polarimetric SAR data, IEEE Trans. Geosci. Remote Sensing, 36 (1998), 963-973. doi: 10.1109/36.673687.

[19]

A. C. Frery, H. J. Muller, C. C. F. Yanasse and S. J. S. Sant'Anna, A model for extremely heterogeneous clutter, IEEE Tran. Geosci. Remote Sensing, 35 (1997), 648-659. doi: 10.1109/36.581981.

[20]

I. Freund, Correlation Imaging through multiply scattering media, Phys. Lett. A., 147 (1990), 502-506. doi: 10.1016/0375-9601(90)90615-U.

[21]

J. Garnier and K. Solna, Coherent interferometric imaging for synthetic aperture radar in the presence of noise, Inverse Problems, 24 (2008), 055001, 23 pp.

[22]

A. Grigis and J. Sjostrand, Microlocal Analysis for Differential Operators: An Introduction, London Mathematical Society Lecture Note Series, 196. Cambridge University Press, Cambridge, 1994.

[23]

M. Gustafsson, Multi-static Synthetic Aperture Radar and Inverse Scattering, Technical Report LUTEDX, TEAT-7123, 2004.
[24]

J. R. Huynen, Phenomenological theory of radar targets, Electromagnetic Scattering, (1978), 653-712. doi: 10.1016/B978-0-12-709650-6.50020-1.

[25]

J. Jackson and R. Moses, Clutter model for VHF SAR imagery, Proc. SPIE 5427, Algorithms for Synthetic Aperture Radar Imagery Ⅺ, 271 (September 2, 2004).
[26]

P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27(2011), 035004, 22pp.

[27]

A. Mahalanobis, A. Forman, M. Bower, N. Day and R. Cherry, Multi-class SAR ATR using shift-invariant correlation filters, Patter Regonition, 27 (1994), 619-626. doi: 10.1016/0031-3203(94)90041-8.

[28]

L. M. Novak, G. J. Owirka and C. M. Netishen, Radar target identification using spatial matched filters, Pattern Regonition, 27 (1994), 607-617. doi: 10.1016/0031-3203(94)90040-X.

[29]

H. L. Royden, Real Analysis, Pearson, New York, 2010.
[30]

M. Soumekh, Synthetic Aperture Radar Signal Processing, John Wiley and Sons Inc., New York, 1999.
[31]

M. E. Taylor, Pseudodiferential Operators, Princeton University Press, Princeton, NJ, 1981.

[32]

L. C. Trintinalia, R. Bhalla and H. Ling, Scattering center parametrization of wide-angle backscattered data using adaptive Gaussian representation, IEEE Trans. Ant. Prop., 45(1997), 1664-1668.
[33]

L. Tsang, G. Zhang and K. Pak, Detection of a buried object under a single random rough surface with angular correlation function in EM wave scattering, Microw. Opt. Technol. Lett., 11 (1996).

[34]

K. Voccola, Synthetic aperture radar correlation imaging, SIAM J. Imaging Sci., 8 (2015), 299-330. doi: 10.1137/14096921X.

[35]

K. Voccola, M. Cheney and B. Yazici, Polarimetric synthetic-aperture inversion for extended targets in clutter, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/5/054003.

[36]

K. Voccola, Statistical and Analytical Techniques in Synthetic Aperture Radar Imaging, Ph. D. Thesis, Dept. Math. Sciences, Rensselaer Polytechnic Institute, Troy, NY, 2011.

[37]

T. Webster, Scalar and Vector Multistatic Radar Data Models, Ph. D. Thesis, Dept. Math. Sciences, Rensselaer Polytechnic Institute, Troy, NY, 2012.

[38]

J. L. Wong, A model for the radar echo from a random collection of rotating dipole scatterers, IEEE Trans. Aerosp. Electron. Syst., 3 (1967), 171-178. doi: 10.1109/TAES.1967.5408739.

[39]

B. Yazici, M. Cheney and C. E. Yarman, Synthetic-aperture inversion in the presence of noise and clutter, Inverse Problems, 22 (2006), 1705-1729. doi: 10.1088/0266-5611/22/5/011.

[40]

G. Zhang, L. Tsang and Y. Kuga, Application of angular correlation function of clutter scattering and correlation imaging in target detection, IEEE Trans. Geosci. Remote Sensing, 36 (1998), 1485-1493.

[41]

G. Zhang, L. Tsang and K. Pak, Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional rough surface, J. Opt. Soc. Am. A, 15 (1998), 2995-3002. doi: 10.1364/JOSAA.15.002995.

[42]

G. Zhang, L. Tsang and Y. Kuga, Numerical studies of the detection of targets in clutter by using angular correlation function and angular correlation imaging, Microw. Opt. Technol. Lett., 17 (1998), 82-86. doi: 10.1002/(SICI)1098-2760(19980205)17:2<82::AID-MOP3>3.0.CO;2-E.

[43]

G. Zhang, L. Tsang and Y. Kuga, Studies of the angular correlation function of scattering by random rough surfaces with and without a buried object, IEEE Trans. Geosci. Remote Sensing, 35 (1997), 444-453. doi: 10.1109/36.563283.

show all references

References:
[1]

J. G. Berryman, L. Borcea, G. Papanicolaou and C. Tsogka, Statistically stable ultrasonic imaging in random media, J. Acoust. Soc. Am., 112 (2002), 1509-1522. doi: 10.1121/1.1502266.

[2]

W. M. Boerner, M. B. El-Arini, C. Y. Chan and P. M. Mastoris, Polarization dependence in electromagnetic inverse problems, IEEE Trans. on Antennas and Propagation, 29 (1981), 262-271. doi: 10.1109/TAP.1981.1142585.

[3]

L. Borcea, G. Papanicolaou and C. Tsogka, Adaptive interferometric imaging in clutter and optimal illumination, Inverse Problems, 22 (2006), 1405-1436. doi: 10.1088/0266-5611/22/4/016.

[4]

L. Borcea, G. Papanicolaou and C. Tsogka, Interferometric array imaging in clutter, Inverse Problems, 21 (2005), 1419-1460. doi: 10.1088/0266-5611/21/4/015.

[5]

L. Borcea, M. Moscoso, G. Papanicolaou and C. Tsogka, Synthetic aperture imaging of direction and frequency dependent reflectivities, SIAM J. Imaging Sci., 9 (2016), 52-81. doi: 10.1137/15M1036063.

[6]

L. Borcea, G. Papanicolaou and C. Tsogka, Theory and applications of time reversal and interferometric imaging, Inverse Problems, 19 (2003), S139-S164. doi: 10.1088/0266-5611/19/6/058.

[7]

T.-K. Chan, Y. Kuga and A. Ishimaru, Subsurface detection of a buried object using angular correlation function measurement, Waves Random Media, 7 (1997), 457-465. doi: 10.1080/13616679709409809.

[8]

R. D. Chaney, M. C. Burl and L. M. Novak, On the Performance of Polarimetric Target Detection Algorithms, IEEE International Radar Conference, 1990. doi: 10.1109/RADAR.1990.201114.

[9]

M. Cheney and B. Borden, Fundamentals of Radar Imaging, SIAM, Philadelphia, 2009.

[10]

S. R. Cloude and E. Pottier, A review of target decomposition theorems in radar polarimetry, IEEE Trans. on Geoscience and Remote Sensing, 34 (1996), 498-518. doi: 10.1109/36.485127.

[11]

A. J. Devaney, The inverse problem for random sources, J. Math. Phys., 20 (1979), 1687-1691. doi: 10.1063/1.524277.

[12]

R. L. Dilsavor and R. L. Moses, Fully-Polarimetric GLRTs for Detecting Scattering Centers with Unknown Amplitude, Phase, and Tilt Angle in Terrain Clutter, in SPIE's International Symposium on Optical Engineering in Aerospace Sensing, Orlando, FL, 1994.
[13]

D. E. Dudgeon, R. T. Lacoss, C. H. Lazott and J. G. Verly, Use of Persistent Scatterers for Model-Based Recognition, Proc. SPIE 2230, Algorithms for Synthetic Aperture Radar Imagery, 1994.
[14]

A. E. El-Rouby, A. Y. Nashashibi and F. T. Ulaby, Application of frequency correlation function to radar target detection, IEEE Trans. Aero. Elec. Sys., 39 (2003), 125-139. doi: 10.1109/TAES.2003.1188898.

[15]

E. Ertin, L. C. Potter and R. L. Moses, Enhanced imaging over complete circular apertures, Signals, Systems, and Computers, (2006). doi: 10.1109/ACSSC.2006.355025.

[16]

S. Feng, C. Kane, P. A. Lee and A. D. Stone, Correlations and fluctuations of coherent wave transmission through disordered media, Phys. Rev. Lett., 61 (1988), 834-837. doi: 10.1103/PhysRevLett.61.834.

[17]

L. Ferro-Famil, A. Reigber, E. Pottier and W. M. Boerner, Scene characterization using subaperture polarimetric SAR data, IEEE Trans, on Geoscience and Remote Sensing, 41 (2003), 2264-2276. doi: 10.1109/TGRS.2003.817188.

[18]

A. Freeman and S. L. Durden, A Three-Component Scattering Model for Polarimetric SAR data, IEEE Trans. Geosci. Remote Sensing, 36 (1998), 963-973. doi: 10.1109/36.673687.

[19]

A. C. Frery, H. J. Muller, C. C. F. Yanasse and S. J. S. Sant'Anna, A model for extremely heterogeneous clutter, IEEE Tran. Geosci. Remote Sensing, 35 (1997), 648-659. doi: 10.1109/36.581981.

[20]

I. Freund, Correlation Imaging through multiply scattering media, Phys. Lett. A., 147 (1990), 502-506. doi: 10.1016/0375-9601(90)90615-U.

[21]

J. Garnier and K. Solna, Coherent interferometric imaging for synthetic aperture radar in the presence of noise, Inverse Problems, 24 (2008), 055001, 23 pp.

[22]

A. Grigis and J. Sjostrand, Microlocal Analysis for Differential Operators: An Introduction, London Mathematical Society Lecture Note Series, 196. Cambridge University Press, Cambridge, 1994.

[23]

M. Gustafsson, Multi-static Synthetic Aperture Radar and Inverse Scattering, Technical Report LUTEDX, TEAT-7123, 2004.
[24]

J. R. Huynen, Phenomenological theory of radar targets, Electromagnetic Scattering, (1978), 653-712. doi: 10.1016/B978-0-12-709650-6.50020-1.

[25]

J. Jackson and R. Moses, Clutter model for VHF SAR imagery, Proc. SPIE 5427, Algorithms for Synthetic Aperture Radar Imagery Ⅺ, 271 (September 2, 2004).
[26]

P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27(2011), 035004, 22pp.

[27]

A. Mahalanobis, A. Forman, M. Bower, N. Day and R. Cherry, Multi-class SAR ATR using shift-invariant correlation filters, Patter Regonition, 27 (1994), 619-626. doi: 10.1016/0031-3203(94)90041-8.

[28]

L. M. Novak, G. J. Owirka and C. M. Netishen, Radar target identification using spatial matched filters, Pattern Regonition, 27 (1994), 607-617. doi: 10.1016/0031-3203(94)90040-X.

[29]

H. L. Royden, Real Analysis, Pearson, New York, 2010.
[30]

M. Soumekh, Synthetic Aperture Radar Signal Processing, John Wiley and Sons Inc., New York, 1999.
[31]

M. E. Taylor, Pseudodiferential Operators, Princeton University Press, Princeton, NJ, 1981.

[32]

L. C. Trintinalia, R. Bhalla and H. Ling, Scattering center parametrization of wide-angle backscattered data using adaptive Gaussian representation, IEEE Trans. Ant. Prop., 45(1997), 1664-1668.
[33]

L. Tsang, G. Zhang and K. Pak, Detection of a buried object under a single random rough surface with angular correlation function in EM wave scattering, Microw. Opt. Technol. Lett., 11 (1996).

[34]

K. Voccola, Synthetic aperture radar correlation imaging, SIAM J. Imaging Sci., 8 (2015), 299-330. doi: 10.1137/14096921X.

[35]

K. Voccola, M. Cheney and B. Yazici, Polarimetric synthetic-aperture inversion for extended targets in clutter, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/5/054003.

[36]

K. Voccola, Statistical and Analytical Techniques in Synthetic Aperture Radar Imaging, Ph. D. Thesis, Dept. Math. Sciences, Rensselaer Polytechnic Institute, Troy, NY, 2011.

[37]

T. Webster, Scalar and Vector Multistatic Radar Data Models, Ph. D. Thesis, Dept. Math. Sciences, Rensselaer Polytechnic Institute, Troy, NY, 2012.

[38]

J. L. Wong, A model for the radar echo from a random collection of rotating dipole scatterers, IEEE Trans. Aerosp. Electron. Syst., 3 (1967), 171-178. doi: 10.1109/TAES.1967.5408739.

[39]

B. Yazici, M. Cheney and C. E. Yarman, Synthetic-aperture inversion in the presence of noise and clutter, Inverse Problems, 22 (2006), 1705-1729. doi: 10.1088/0266-5611/22/5/011.

[40]

G. Zhang, L. Tsang and Y. Kuga, Application of angular correlation function of clutter scattering and correlation imaging in target detection, IEEE Trans. Geosci. Remote Sensing, 36 (1998), 1485-1493.

[41]

G. Zhang, L. Tsang and K. Pak, Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional rough surface, J. Opt. Soc. Am. A, 15 (1998), 2995-3002. doi: 10.1364/JOSAA.15.002995.

[42]

G. Zhang, L. Tsang and Y. Kuga, Numerical studies of the detection of targets in clutter by using angular correlation function and angular correlation imaging, Microw. Opt. Technol. Lett., 17 (1998), 82-86. doi: 10.1002/(SICI)1098-2760(19980205)17:2<82::AID-MOP3>3.0.CO;2-E.

[43]

G. Zhang, L. Tsang and Y. Kuga, Studies of the angular correlation function of scattering by random rough surfaces with and without a buried object, IEEE Trans. Geosci. Remote Sensing, 35 (1997), 444-453. doi: 10.1109/36.563283.

Figure 1.  Single point; standard reconstruction, correlation reconstruction with memory directions
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Figure 2.  Single point; correlation reconstruction without memory directions with $\tilde{s} = 1, 25, 75$ slow-time steps (recall $\tilde{s} = s-s'$)
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Figure 3.  Single dipole, orientation $\pi/4$, length$ = .3\lambda$; standard reconstruction, correlation reconstruction with memory directions
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Figure 4.  Single dipole, orientation $\pi/4$, length$ = .3\lambda$; correlation reconstruction without memory directions with $\tilde{s} = 1, 25, 75$ slow-time steps (recall $\tilde{s} = s-s'$)
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Figure 5.  Single dipole, orientation $\pi/4$, length$ = 3\lambda$; standard reconstruction, correlation reconstruction with memory directions
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Figure 6.  Single dipole, orientation $\pi/4$, length$ = 3\lambda$; correlation reconstruction without memory directions with $\tilde{s} = 1, 25, 75$ slow-time steps (recall $\tilde{s} = s-s'$)
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Figure 8.  Single dipole, orientation $\pi/4$, length$ = 30\lambda$; standard reconstruction, correlation reconstruction with memory directions
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Figure 9.  Single dipole, orientation $\pi/4$, length$ = 30\lambda$; correlation reconstruction without memory directions with $\tilde{s} = 1, 25, 75$ slow-time steps (recall $\tilde{s} = s-s'$)
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Figure 7.  Single dipole, orientation $\pi/4$, length$ = 3\lambda$; correlation reconstruction without memory directions with $\tilde{s} = 1, 25, 75$ slow-time steps (recall $\tilde{s} = s-s'$), second flight path
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Figure 10.  Single dipole, orientation $\pi/4$, length$ = 30\lambda$; correlation reconstruction without memory directions with $\tilde{s} = 1, 25, 75$ slow-time steps (recall $\tilde{s} = s-s'$), second flight path
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Figure 11.  Two point targets, dipole clutter, each dipole with orientation $\pi/4$, length$ = \lambda$, and Gaussian reflectivity; standard polarimetric reconstruction, HH, HV, VV images respectively
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Figure 12.  Two point targets, dipole clutter, each dipole with orientation $\pi/4$, length$ = \lambda$, and Gaussian reflectivity; correlation polarimetric reconstruction $\tilde{s} = 1$ (recall $\tilde{s} = s-s'$), HH-HH, HV-HV, VV-VV images respectively
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Figure 13.  Two point targets, dipole clutter, each dipole with orientation $\pi/4$, length$ = \lambda$, and Gaussian reflectivity; correlation polarimetric reconstruction $\tilde{s} = 50$ (recall $\tilde{s} = s-s'$), HH-HH, HV-HV, VV-VV images respectively
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Table 1.  Single point; peak image value at center pixel
Standard BP 1.7158 - 0.2106i
Correlation BP with memory directions 1.5112 - 0.0000i
Correlation BP with $\tilde{s}=1$ 1.4772 - 0.0000i
Correlation BP with $\tilde{s}=25$ 1.1470 - 0.0000i
Correlation BP with $\tilde{s}=50$ 0.8461 + 0.0000i
Correlation BP with $\tilde{s}=75$ 0.5869 + 0.0000i
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Standard BP 1.7158 - 0.2106i
Correlation BP with memory directions 1.5112 - 0.0000i
Correlation BP with $\tilde{s}=1$ 1.4772 - 0.0000i
Correlation BP with $\tilde{s}=25$ 1.1470 - 0.0000i
Correlation BP with $\tilde{s}=50$ 0.8461 + 0.0000i
Correlation BP with $\tilde{s}=75$ 0.5869 + 0.0000i
Table 2.  Peak Image Value at center pixel, single dipole target, orientation $\pi/4$, length$ = .3\lambda$
Standard BP 1.1491e-04 - 6.2788e-06i
Correlation BP with memory directions 0.6668e-08 - 0.0410e-08i
Correlation BP with $\tilde{s}=1$ 0.6575e-08 - 0.0410e-08i
Correlation BP with $\tilde{s}=25$ 0.5030e-08 - 0.0377e-08i
Correlation BP with $\tilde{s}=50$ 0.3642e-08 - 0.0290e-08i
Correlation BP with $\tilde{s}=75$ 0.2490e-08 - 0.0208e-08i
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Standard BP 1.1491e-04 - 6.2788e-06i
Correlation BP with memory directions 0.6668e-08 - 0.0410e-08i
Correlation BP with $\tilde{s}=1$ 0.6575e-08 - 0.0410e-08i
Correlation BP with $\tilde{s}=25$ 0.5030e-08 - 0.0377e-08i
Correlation BP with $\tilde{s}=50$ 0.3642e-08 - 0.0290e-08i
Correlation BP with $\tilde{s}=75$ 0.2490e-08 - 0.0208e-08i
Table 4.  Peak Image Value at center pixel, single dipole target, orientation $\pi/4$, length$ = 3\lambda$, second flight path
Standard BP 0.0090 + 0.0004i
Correlation BP with memory directions 0.4206e-04 + 0.0014e-04i
Correlation BP with $\tilde{s}=1$ 0.4072e-04 + 0.0014e-04i
Correlation BP with $\tilde{s}=25$ 0.2535e-04 + 0.0049e-04i
Correlation BP with $\tilde{s}=50$ 0.1363e-04 - 0.0087e-04i
Correlation BP with $\tilde{s}=75$ 0.0592e-04 - 0.0021e-04i
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Standard BP 0.0090 + 0.0004i
Correlation BP with memory directions 0.4206e-04 + 0.0014e-04i
Correlation BP with $\tilde{s}=1$ 0.4072e-04 + 0.0014e-04i
Correlation BP with $\tilde{s}=25$ 0.2535e-04 + 0.0049e-04i
Correlation BP with $\tilde{s}=50$ 0.1363e-04 - 0.0087e-04i
Correlation BP with $\tilde{s}=75$ 0.0592e-04 - 0.0021e-04i
Table 3.  Peak Image Value at center pixel, single dipole target, orientation $\pi/4$, length$ = 3\lambda$
Standard BP 0.0012 - 0.0002i
Correlation BP with memory directions 0.7152e-06 - 0.1448e-06i
Correlation BP with $\tilde{s}=1$ 0.6983e-06 - 0.1448e-06i
Correlation BP with $\tilde{s}=25$ 0.3869e-06 - 0.1183e-06i
Correlation BP with $\tilde{s}=50$ 0.2159e-06 + 0.0199e-06i
Correlation BP with $\tilde{s}=75$ 0.1153e-06 + 0.0139e-06i
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Standard BP 0.0012 - 0.0002i
Correlation BP with memory directions 0.7152e-06 - 0.1448e-06i
Correlation BP with $\tilde{s}=1$ 0.6983e-06 - 0.1448e-06i
Correlation BP with $\tilde{s}=25$ 0.3869e-06 - 0.1183e-06i
Correlation BP with $\tilde{s}=50$ 0.2159e-06 + 0.0199e-06i
Correlation BP with $\tilde{s}=75$ 0.1153e-06 + 0.0139e-06i
Table 5.  Peak Image Value at center pixel, single dipole target, orientation $\pi/4$, length$ = 30\lambda$
Standard BP 0.0010 - 0.0023i
Correlation BP with memory directions 0.3106e-05 - 0.0308e-05i
Correlation BP with $\tilde{s}=1$ 0.3053e-05 - 0.0308e-05i
Correlation BP with $\tilde{s}=25$ 0.2214e-05 - 0.0087e-05i
Correlation BP with $\tilde{s}=50$ 0.1554e-05 - 0.0159e-05i
Correlation BP with $\tilde{s}=75$ 0.1115e-05 - 0.0061e-05i
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Standard BP 0.0010 - 0.0023i
Correlation BP with memory directions 0.3106e-05 - 0.0308e-05i
Correlation BP with $\tilde{s}=1$ 0.3053e-05 - 0.0308e-05i
Correlation BP with $\tilde{s}=25$ 0.2214e-05 - 0.0087e-05i
Correlation BP with $\tilde{s}=50$ 0.1554e-05 - 0.0159e-05i
Correlation BP with $\tilde{s}=75$ 0.1115e-05 - 0.0061e-05i
Table 6.  Peak Image Value at center pixel, single dipole target, orientation $\pi/4$, length$ = 30\lambda$, second flight path
Standard BP 1.7528 + 0.2160i
Correlation BP with memory directions 1.6668 - 0.3576i
Correlation BP with $\tilde{s}=1$ 1.5072 - 0.3576i
Correlation BP with $\tilde{s}=25$ 0.1084 - 0.1624i
Correlation BP with $\tilde{s}=50$ 0.0157 - 0.0024i
Correlation BP with $\tilde{s}=75$ 0.0045 - 0.0001i
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Standard BP 1.7528 + 0.2160i
Correlation BP with memory directions 1.6668 - 0.3576i
Correlation BP with $\tilde{s}=1$ 1.5072 - 0.3576i
Correlation BP with $\tilde{s}=25$ 0.1084 - 0.1624i
Correlation BP with $\tilde{s}=50$ 0.0157 - 0.0024i
Correlation BP with $\tilde{s}=75$ 0.0045 - 0.0001i
Table 7.  Input vs. ouput SCR in dB, scalar SAR, two point targets in dipole clutter
Input SCR Output SCR Standard BP Output SCR Correlation BP $\tilde{s}=1$ Output SCR Correlation $\tilde{s}=10$ Output SCR Correlation $\tilde{s}=50$
20 17.3087 51.3593 52.0820 51.9498
10 7.3087 31.3593 32.0820 31.9498
0 -2.6913 11.3593 12.0820 11.9498
-10 -12.6913 -8.6407 -7.9180 -8.0502
-20 -22.6913 -28.6407 -27.9180 -28.0502
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Input SCR Output SCR Standard BP Output SCR Correlation BP $\tilde{s}=1$ Output SCR Correlation $\tilde{s}=10$ Output SCR Correlation $\tilde{s}=50$
20 17.3087 51.3593 52.0820 51.9498
10 7.3087 31.3593 32.0820 31.9498
0 -2.6913 11.3593 12.0820 11.9498
-10 -12.6913 -8.6407 -7.9180 -8.0502
-20 -22.6913 -28.6407 -27.9180 -28.0502
Table 8.  Input vs. ouput SCR in dB, pol SAR standard BP, two point targets in dipole clutter
Input SCR Output SCR HH Output SCR HV Output SCR VV
20 18.0264 18.8373 18.9482
10 8.0264 8.8373 8.9482
0 -1.9736 -1.1627 -1.0518
-10 -11.9736 -11.1627 -11.0518
-20 -21.9736 -21.1627 -21.0518
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Input SCR Output SCR HH Output SCR HV Output SCR VV
20 18.0264 18.8373 18.9482
10 8.0264 8.8373 8.9482
0 -1.9736 -1.1627 -1.0518
-10 -11.9736 -11.1627 -11.0518
-20 -21.9736 -21.1627 -21.0518
Table 9.  Input vs. ouput SCR in dB, pol SAR correlation BP, two point targets in dipole clutter, $\tilde{s} = 1$
Input SCR Output SCR HH-HH Output SCR HV-HV Output SCR VV-VV
20 51.1293 54.0229 53.9465
10 31.1293 34.0229 33.9465
0 11.1293 14.0229 13.9465
-10 -8.8707 -5.9771 -6.0535
-20 -28.8707 -25.9771 -26.0535
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Input SCR Output SCR HH-HH Output SCR HV-HV Output SCR VV-VV
20 51.1293 54.0229 53.9465
10 31.1293 34.0229 33.9465
0 11.1293 14.0229 13.9465
-10 -8.8707 -5.9771 -6.0535
-20 -28.8707 -25.9771 -26.0535
Table 10.  Input vs. ouput SCR in dB, pol SAR correlation BP, two point targets in dipole clutter, $\tilde{s} = 10$
Input SCR Output SCR HH-HH Output SCR HV-HV Output SCR VV-VV
20 51.3130 54.0887 53.9973
10 31.3130 34.0887 33.9973
0 11.3130 14.0887 13.9973
-10 -8.6870 -5.9113 -6.0027
-20 -28.6870 -25.9113 -26.0027
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Input SCR Output SCR HH-HH Output SCR HV-HV Output SCR VV-VV
20 51.3130 54.0887 53.9973
10 31.3130 34.0887 33.9973
0 11.3130 14.0887 13.9973
-10 -8.6870 -5.9113 -6.0027
-20 -28.6870 -25.9113 -26.0027
Table 11.  Input vs. ouput SCR in dB, pol SAR correlation BP, two point targets in dipole clutter, $\tilde{s} = 50$
Input SCR Output SCR HH-HH Output SCR HV-HV Output SCR VV-VV
20 51.4206 53.9605 53.7934
10 31.4206 33.9605 33.7934
0 11.4206 13.9605 13.7934
-10 -8.5794 -6.0395 -6.2066
-20 -28.5794 -26.0395 -26.2066
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Input SCR Output SCR HH-HH Output SCR HV-HV Output SCR VV-VV
20 51.4206 53.9605 53.7934
10 31.4206 33.9605 33.7934
0 11.4206 13.9605 13.7934
-10 -8.5794 -6.0395 -6.2066
-20 -28.5794 -26.0395 -26.2066
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