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

Previous Article
A fast patch-dictionary method for whole image recovery
IPI Home
This Issue
Next Article
The factorization method for the Drude-Born-Fedorov model for periodic chiral structures

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

Kaitlyn Muller ^1,

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

Abstract
Related Papers

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.

Keywords: radar imaging, backprojection, best linear unbiased estimation, microlocal analysis., Synthetic-aperture radar.

Mathematics Subject Classification: Primary: 78A46, 94A13; Secondary: 35Q6.

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.

[1]	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
[2]	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
[3]	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
[4]	Seonho Park, Maciej Rysz, Kaitlin L. Fair, Panos M. Pardalos. Synthetic-Aperture Radar image based positioning in GPS-denied environments using Deep Cosine Similarity Neural Networks. Inverse Problems and Imaging, 2021, 15 (4) : 763-785. doi: 10.3934/ipi.2021013
[5]	T. Varslo, C E Yarman, M. Cheney, B Yazıcı. A variational approach to waveform design for synthetic-aperture imaging. Inverse Problems and Imaging, 2007, 1 (3) : 577-592. doi: 10.3934/ipi.2007.1.577
[6]	Bo Jiang, Yongge Tian. On best linear unbiased estimation and prediction under a constrained linear random-effects model. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021209
[7]	Mikhail Gilman, Semyon Tsynkov. A mathematical perspective on radar interferometry. Inverse Problems and Imaging, 2022, 16 (1) : 119-152. doi: 10.3934/ipi.2021043
[8]	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
[9]	Hanqing Jin, Shige Peng. Optimal unbiased estimation for maximal distribution. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 189-198. doi: 10.3934/puqr.2021009
[10]	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
[11]	Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of borehole seismic data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022026
[12]	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
[13]	Gang Bao, Jun Lai. Radar cross section reduction of a cavity in the ground plane: TE polarization. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 419-434. doi: 10.3934/dcdss.2015.8.419
[14]	Markku Lehtinen, Baylie Damtie, Petteri Piiroinen, Mikko Orispää. Perfect and almost perfect pulse compression codes for range spread radar targets. Inverse Problems and Imaging, 2009, 3 (3) : 465-486. doi: 10.3934/ipi.2009.3.465
[15]	Qiang Yin, Gongfa Li, Jianguo Zhu. Research on the method of step feature extraction for EOD robot based on 2D laser radar. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1415-1421. doi: 10.3934/dcdss.2015.8.1415
[16]	Simon Hubmer, Andreas Neubauer, Ronny Ramlau, Henning U. Voss. On the parameter estimation problem of magnetic resonance advection imaging. Inverse Problems and Imaging, 2018, 12 (1) : 175-204. doi: 10.3934/ipi.2018007
[17]	Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems and Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341
[18]	Lei Zhang, Luming Jia. Near-field imaging for an obstacle above rough surfaces with limited aperture data. Inverse Problems and Imaging, 2021, 15 (5) : 975-997. doi: 10.3934/ipi.2021024
[19]	Bernadette N. Hahn, Melina-Loren Kienle Garrido, Christian Klingenberg, Sandra Warnecke. Using the Navier-Cauchy equation for motion estimation in dynamic imaging. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022018
[20]	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

2021 Impact Factor: 1.483

Tools

Metrics

Other articles
by authors

on AIMS
Kaitlyn Muller
on Google Scholar
Kaitlyn Muller

[Back to Top]