# American Institute of Mathematical Sciences

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

## SAR correlation imaging and anisotropic scattering

 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.

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

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##### References:
Single point; standard reconstruction, correlation reconstruction with memory directions
Single point; correlation reconstruction without memory directions with $\tilde{s} = 1, 25, 75$ slow-time steps (recall $\tilde{s} = s-s'$)
Single dipole, orientation $\pi/4$, length$= .3\lambda$; standard reconstruction, correlation reconstruction with memory directions
Single dipole, orientation $\pi/4$, length$= .3\lambda$; correlation reconstruction without memory directions with $\tilde{s} = 1, 25, 75$ slow-time steps (recall $\tilde{s} = s-s'$)
Single dipole, orientation $\pi/4$, length$= 3\lambda$; standard reconstruction, correlation reconstruction with memory directions
Single dipole, orientation $\pi/4$, length$= 3\lambda$; correlation reconstruction without memory directions with $\tilde{s} = 1, 25, 75$ slow-time steps (recall $\tilde{s} = s-s'$)
Single dipole, orientation $\pi/4$, length$= 30\lambda$; standard reconstruction, correlation reconstruction with memory directions
Single dipole, orientation $\pi/4$, length$= 30\lambda$; correlation reconstruction without memory directions with $\tilde{s} = 1, 25, 75$ slow-time steps (recall $\tilde{s} = s-s'$)
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
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
Two point targets, dipole clutter, each dipole with orientation $\pi/4$, length$= \lambda$, and Gaussian reflectivity; standard polarimetric reconstruction, HH, HV, VV images respectively
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
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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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
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
 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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