May  2016, 10(2): 549-561. doi: 10.3934/ipi.2016011

The relationship between backprojection and best linear unbiased estimation in synthetic-aperture radar imaging

1. 

Colorado State University, Department of Mathematics, 1874 Campus Delivery, Fort Collins, CO 80523-1874, United States

Received  June 2015 Revised  October 2015 Published  May 2016

In this paper we investigate the relationship between two different techniques typically used in imaging and estimation problems. We focus on synthetic-aperture radar imaging and compare the methods of backprojection (standard for imaging) and best linear unbiased estimation (BLUE). We aim to reconstruct or estimate the reflectivity function of an object present in a scene of interest. We find that the estimate of the reflectivity (calculated using BLUE) and the reconstructed image (calculated using filtered backprojection) are the same when we utilize a criterion from microlocal analysis to define the optimal backprojection filter and assume the measured data is corrupted by zero-mean independently identically distributed (white) noise. In particular we show that the microlocal criterion for the optimal backprojection filter is equivalent to the unbiased constraint present in the BLUE technique.
Citation: 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
References:
[1]

H. Ammari, J. Garnier and K. Solna, A statistical approach to target detection and localization in the presence of noise, Waves in Random and Complex Media, 22 (2012), 40-65. doi: 10.1080/17455030.2010.532518.

[2]

G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform, J. Math. Phys., 26 (1985), 99-108. doi: 10.1063/1.526755.

[3]

M. Cheney and B. Borden, Fundamentals of Radar Imaging, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719291.

[4]

A. Grigis and J. Sjostrand, Microlocal Analysis for Differential Operators: An Introduction, in London Mathematical Society Lecture Note Series 196, Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511721441.

[5]

S. Kay and G. F. Boudreaux-Bartels, On the optimality of the Wigner distribution for detection, Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP‘85., IEEE, 10 (1985), 1017-1020. doi: 10.1109/ICASSP.1985.1168129.

[6]

C. J. Nolan and M. Cheney, Microlocal Analysis of Synthetic Aperture Radar Imaging, The Journal of Fourier Analysis and Applications, 10 (2004), 133-148. doi: 10.1007/s00041-004-8008-0.

[7]

H. V. Poor, An Introduction to Signal Detection and Estimation, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4757-3863-6.

[8]

D. C. Montgomery, E. A. Peck and G. G. Vining, Introduction to Linear Regression Analysis, Wiley-Interscience, New York, 2001.

[9]

S. Kay, Fundamentals of Signal Processing, vol. 1: Estimation Theory, Prentice Hall, Upper Saddle River, 1993.

[10]

D. J. Rossi and A. S. Willsky, Reconstruction from projections based on detection and estimation of objects - parts i and ii: performance analysis and robustness analysis, IEEE Trans. Acoustics, Speech, and Signal Processing, 32 (1984), 886-906. doi: 10.1109/TASSP.1984.1164405.

[11]

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

[12]

F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Vol. I & II, The University Series in Mathematics, Plenum Press, New York, 1980.

[13]

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

[14]

K. Voccola, B. Yazici, M. Cheney and M. Ferrara, On the Relationship between the Generalized Likelihood Ratio Test and Backprojection Method for Synthetic-Aperture Radar Imaging, SPIE Defense, Security, and Sensing, International Society for Optics and Photonics, (2009), 73350I-73350I.

[15]

H. C. Yanik, Analytic Methods for SAR Image Formation in the Presence of Noise and Clutter, Ph.D. Thesis, Rensselaer Polytechnic Institute in Troy, NY, 2014.

[16]

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.

show all references

References:
[1]

H. Ammari, J. Garnier and K. Solna, A statistical approach to target detection and localization in the presence of noise, Waves in Random and Complex Media, 22 (2012), 40-65. doi: 10.1080/17455030.2010.532518.

[2]

G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform, J. Math. Phys., 26 (1985), 99-108. doi: 10.1063/1.526755.

[3]

M. Cheney and B. Borden, Fundamentals of Radar Imaging, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719291.

[4]

A. Grigis and J. Sjostrand, Microlocal Analysis for Differential Operators: An Introduction, in London Mathematical Society Lecture Note Series 196, Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511721441.

[5]

S. Kay and G. F. Boudreaux-Bartels, On the optimality of the Wigner distribution for detection, Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP‘85., IEEE, 10 (1985), 1017-1020. doi: 10.1109/ICASSP.1985.1168129.

[6]

C. J. Nolan and M. Cheney, Microlocal Analysis of Synthetic Aperture Radar Imaging, The Journal of Fourier Analysis and Applications, 10 (2004), 133-148. doi: 10.1007/s00041-004-8008-0.

[7]

H. V. Poor, An Introduction to Signal Detection and Estimation, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4757-3863-6.

[8]

D. C. Montgomery, E. A. Peck and G. G. Vining, Introduction to Linear Regression Analysis, Wiley-Interscience, New York, 2001.

[9]

S. Kay, Fundamentals of Signal Processing, vol. 1: Estimation Theory, Prentice Hall, Upper Saddle River, 1993.

[10]

D. J. Rossi and A. S. Willsky, Reconstruction from projections based on detection and estimation of objects - parts i and ii: performance analysis and robustness analysis, IEEE Trans. Acoustics, Speech, and Signal Processing, 32 (1984), 886-906. doi: 10.1109/TASSP.1984.1164405.

[11]

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

[12]

F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Vol. I & II, The University Series in Mathematics, Plenum Press, New York, 1980.

[13]

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

[14]

K. Voccola, B. Yazici, M. Cheney and M. Ferrara, On the Relationship between the Generalized Likelihood Ratio Test and Backprojection Method for Synthetic-Aperture Radar Imaging, SPIE Defense, Security, and Sensing, International Society for Optics and Photonics, (2009), 73350I-73350I.

[15]

H. C. Yanik, Analytic Methods for SAR Image Formation in the Presence of Noise and Clutter, Ph.D. Thesis, Rensselaer Polytechnic Institute in Troy, NY, 2014.

[16]

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.

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