A Multiple Measurement Vector Approach To Synthetic Aperture Radar Imaging
PDF: A Multiple Measurement Vector Approach To Synthetic Aperture Radar Imaging
We study a multiple measurement vector (MMV) approach to synthetic aperture radar (SAR) imaging of scenes with direction dependent reflectivity and with polarization diverse measurements. The data are gathered by a moving transmit- receive platform which probes the imaging scene with signals and records the backscattered waves. The unknown reflectivity is represented by a matrix with row support corresponding to the location of the scatterers in the scene, and columns corresponding to measurements gathered from different sub-apertures, or different polarization of the waves. The MMV methodology is used to estimate the reflectivity matrix by inverting in an appropriate sense the linear system of equations that models the SAR data. We obtain a resolution analysis of SAR imaging with MMV, which takes into account the sparsity of the imaging scene, the separation of the scatterers and the diversity of the measurements. The results of the analysis are illustrated with some numerical simulations.
A MULTIPLE MEASUREMENT VECTOR APPROACH TO SYNTHETIC APERTURE RADAR IMAGING
LILIANA BORCEA AND ILKER KOCYIGIT ∗
Abstract. We study a multiple measurement vector (MMV) approach to synthetic aperture radar (SAR) imaging of scenes with direction dependent reflectivity and with polarization diverse measurements. The data are gathered by a moving transmit- receive platform which probes the imaging scene with signals and records the backscattered waves. The unknown reflectivity is represented by a matrix with row support corresponding to the location of the scatterers in the scene, and columns corresponding to measurements gathered from different sub-apertures, or different polarization of the waves. The MMV methodology is used to estimate the reflectivity matrix by inverting in an appropriate sense the linear system of equations that models the SAR data. We obtain a resolution analysis of SAR imaging with MMV, which takes into account the sparsity of the imaging scene, the separation of the scatterers and the diversity of the measurements. The results of the analysis are illustrated with some numerical simulations.
Key words. synthetic aperture radar imaging, convex optimization, multiple measurement vector, simultaneously sparse. AMS subject classifications. 35Q93, 58J90, 45Q05.
1. Introduction. Sparsity promoting optimization [26, 25, 22, 9, 11, 10, 12] is an important method- ology for imaging applications where scenes that are sparse in some representation can be reconstructed with high resolution. There is a large body of literature on this topic in synthetic aperture radar imaging [4, 32, 28], sensor array imaging [13, 14, 7], medical imaging [30], astronomy [6], geophysics [33], and so on.
We are interested in the application of synthetic aperture radar (SAR) imaging, where a transmit-receive antenna mounted on a moving platform probes an imaging scene with waves and records the scattered returns [20, 17]. This is a particular inverse problem for the wave equation, where the waves propagate through a homogeneous medium, back and forth between the platform and the imaging scene, and the unknown is modeled as a two-dimensional reflectivity function of location on a known imaging surface. Most SAR imaging is based on a linear model of the data, where the unknown reflectivity is represented by a collection of independent point scatterers [17]. The image is then formed by inverting approximately this linear relation, using filtered backprojection or matched filtering [17], also known as Kirchhoff migration [5]. Such imaging is popular because it is robust to noise, it is simple and works well when the linear model is a good approximation of the data. However, the resolution is limited by the extent of the aperture, the frequency and the bandwidth of the probing signals emitted by the moving platform [20, 17]. The promise of sparsity promoting optimization is that these resolution limits can be overcome when the unknown reflectivity has sparse support [4, 32, 28].
The modeling of the reflectivity as a collection of points that scatter the waves isotropically may lead to image artifacts. It is known that even if the scatterers are small, so that their support may be repre- sented by a point and the single scattering approximation (i.e., the linear data model) can be used, their reflectivity may depend on the frequency and the direction of illumination [2, Chapters 3, 5]. Moreover, the scatterers have an effective polarization tensor that describes their response to different polarizations of the probing electromagnetic waves [2, 3]. Thus, the reflectivity function depends on more variables than the two dimensional location vector assumed in conventional SAR, and depending on how strong this dependence is, the resulting images may be worse than expected. For example, a scatterer that reflects only within a narrow cone of incident angles cannot be sensed over most of the synthetic aperture, so its reconstruction with filtered backprojection will have low resolution. Direct application of sparse optimization methods does not give good results either, because of the large systematic error in the linear data model that assumes a scalar, constant reflectivity over the entire aperture.
SAR imaging of frequency-dependent reflectivities has been studied in [16, 35, 34], using either Doppler effects, or data segmentation over frequency sub-bands. Data segmentation is a natural idea for imaging both frequency and direction dependent reflectivities that are regular enough so that they can be approximated as piecewise constant functions over properly chosen frequency sub-bands and cones of angles of incidence (i.e., sub-apertures). Images can be obtained separately from each data segment, but the question is how to fuse the information to achieve better resolution. The study in [8] uses the multiple measurement vector (MMV) methodology [31, 15, 39], also known as simultaneously sparse approximation [38, 37], for this purpose. The MMV framework fits here because the reflectivity is supported at the same locations in the imaging scene, for each data set. Only the values of the reflectivity change. In the discrete setting, this means that the unknown is represented by a matrix X with row support corresponding to the pixels in the image that contain scatterers, and with columns corresponding to the different values of the reflectivity for each frequency band and sub-aperture. The MMV methodology is used in [8] to reconstruct this unknown matrix under the assumption of sparse row support.
In this paper we pursue the ideas in [8] further, by studying the resolution of the MMV reconstructions and analyzing conditions under which the multiple views of the imaging scene improve the results. We also discuss the application of the MMV methodology to SAR imaging with polarization diverse measurements.
The paper is organized as follows: We begin in section 2 with the theoretical results, stated for a general linear system with unknown matrix X. They consist of a resolution theory of imaging with MMV, that takes into account the separation of the points in the support of the reflectivity. We also show that if the rows of X are almost orthogonal, we can expect better reconstructions than with sparsity promoting l1 optimization applied separately to each data set. This orthogonality condition may arise in SAR imaging, as explained in section 3, in the context of imaging direction dependent reflectivities. SAR with polarization diverse measurements is discussed in section 4. The proofs of the results are in section 5. We end with a summary in section 6.
2. Theory. We state here our main results on the resolution of imaging with MMV. We begin in section 2.1 with a brief discussion on MMV, and then give the results in section 2.2.
REFERENCES
[1] GOTCHA volumetric SAR data set. https://www.sdms.afrl.af.mil/index.php?collection=gotcha.
[2] Habib Ammari, Josselin Garnier, Wenjia Jing, Hyeonbae Kang, Mikyoung Lim, Knut Sølna, and Han Wang,
Mathematical and statistical methods for multistatic imaging, vol. 2098, Springer, 2013.
[3] Habib Ammari, Ekaterina Iakovleva, Dominique Lesselier, and Gae ̈le Perrusson, MUSIC-type electromagnetic imaging of a collection of small three-dimensional inclusions, SIAM Journal on Scientific Computing, 29 (2007),
pp. 674–709.
[4] Richard Baraniuk and Philippe Steeghs, Compressive radar imaging, in Radar Conference, 2007 IEEE, IEEE, 2007,
pp. 128–133.
[5] B. Biondi, 3D seismic imaging, Society of Exploration Geophysicists, 2006.
[6] Je ́roˆme Bobin, Jean-Luc Starck, and Roland Ottensamer, Compressed sensing in astronomy, IEEE Journal of
Selected Topics in Signal Processing, 2 (2008), pp. 718–726.
[7] Liliana Borcea and Ilker Kocyigit, Resolution analysis of imaging with l1 optimization, SIAM Journal on Imaging
Sciences, 8 (2015), pp. 3015–3050.
[8] Liliana Borcea, Miguel Moscoso, George Papanicolaou, and Chrysoula Tsogka, Synthetic aperture imaging of
direction-and frequency-dependent reflectivities, SIAM Journal on Imaging Sciences, 9 (2016), pp. 52–81.
[9] A. M. Bruckstein, D. L. Donoho, and M. Elad, From sparse solutions of systems of equations to sparse modeling of
signals and images, SIAM review, 51 (2009), pp. 34–81.
[10] Emmanuel J Cande`s, Justin Romberg, and Terence Tao, Robust uncertainty principles: Exact signal reconstruction
from highly incomplete frequency information, IEEE Transactions on information theory, 52 (2006), pp. 489–509. [11] E. J. Candes and T. Tao, Decoding by linear programming, Information Theory, IEEE Transactions on, 51 (2005),
pp. 4203–4215.
[12] Emmanuel J Candes and Terence Tao, Near-optimal signal recovery from random projections: Universal encoding
strategies?, IEEE transactions on information theory, 52 (2006), pp. 5406–5425.
[13] A. Chai, M. Moscoso, and G. Papanicolaou, Robust imaging of localized scatterers using the singular value decompo-
sition and 1 minimization, Inverse Problems, 29 (2013), p. 025016. 29
- [14] , Imaging strong localized scatterers with sparsity promoting optimization, SIAM Journal on Imaging Sciences, 7 (2014), pp. 1358–1387.
- [15] J. Chen and X. Huo, Theoretical results on sparse representations of multiple-measurement vectors, IEEE Transactions on Signal Processing, 54 (2006), pp. 4634–4643.
- [16] Margaret Cheney, Imaging frequency-dependent reflectivity from synthetic-aperture radar, Inverse Problems, 29 (2013), p. 054002.
- [17] Margaret Cheney and Brett Borden, Fundamentals of radar imaging, SIAM, 2009.
- [18] Albert Cohen, Wolfgang Dahmen, and Ronald Devore, Compressed sensing and best k-term approximation, J.Amer. Math. Soc, (2009), pp. 211–231.
- [19] S. F. Cotter, B. D. Rao, K. Engan, and K. Kreutz-Delgado, Sparse solutions to linear inverse problems with multiplemeasurement vectors, Signal Processing, IEEE Transactions on, 53 (2005), pp. 2477–2488.
- [20] John C Curlander and Robert N McDonough, Synthetic aperture radar, John Wiley & Sons New York, NY, USA,1991.
- [21] CVX Research, Cvx: matlab software for disciplined convex programming, version 2.0.http://cvxr.com/cvxhttp://cvxr.com/cvx, August 2012.
- [22] David L Donoho, Compressed sensing, IEEE Transactions on information theory, 52 (2006), pp. 1289–1306.
- [23] David L Donoho and Michael Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via 1minimization, Proceedings of the National Academy of Sciences, 100 (2003), pp. 2197–2202.
- [24] D. L. Donoho and X. Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inf. Theor., 47 (2006),pp. 2845–2862.
- [25] David L Donoho and Benjamin F Logan, Signal recovery and the large sieve, SIAM Journal on Applied Mathematics,52 (1992), pp. 577–591.
- [26] David L Donoho and Philip B Stark, Uncertainty principles and signal recovery, SIAM Journal on Applied Mathe-matics, 49 (1989), pp. 906–931.
- [27] Y. C. Eldar and M. Mishali, Robust recovery of signals from a structured union of subspaces, IEEE Transactions onInformation Theory, 55 (2009), pp. 5302–5316.
- [28] A. Fannjiang and H-C Tseng, Compressive radar with off-grid targets: a perturbation approach, Inverse Problems, 29(2013), p. 054008.
- [29] A. C. Fannjiang, T. Strohmer, and P. Yan, Compressed remote sensing of sparse objects, SIAM Journal on ImagingSciences, 3 (2010), pp. 595–618.
- [30] Michael Lustig, David Donoho, and John M Pauly, Sparse mri: The application of compressed sensing for rapid mrimaging, Magnetic resonance in medicine, 58 (2007), pp. 1182–1195.
- [31] D. Malioutov, M. Cetin, and A. S. Willsky, A sparse signal reconstruction perspective for source localization withsensor arrays, IEEE Transactions on Signal Processing, 53 (2005), pp. 3010–3022.
- [32] Lee C Potter, Emre Ertin, Jason T Parker, and Mu ̈jdat Cetin, Sparsity and compressed sensing in radar imaging,Proceedings of the IEEE, 98 (2010), pp. 1006–1020.
- [33] Fadil Santosa and William W Symes, Linear inversion of band-limited reflection seismograms, SIAM Journal onScientific and Statistical Computing, 7 (1986), pp. 1307–1330.
- [34] Paul Sotirelis, Jason Parker, Xueyu Hu, Margaret Cheney, and Matthew Ferrara, Frequency-dependent reflec-tivity image reconstruction, in SPIE Defense, Security, and Sensing, International Society for Optics and Photonics,
2013, pp. 874602–874602.
- [35] Paul Sotirelis, Jason T Parker, Michael Fu, Xueyu Hu, and Richard Albanese, A study of material identificationusing sar, in Radar Conference (RADAR), 2012 IEEE, IEEE, 2012, pp. 0112–0115.
- [36] J. A. Tropp, Greed is good: Algorithmic results for sparse approximation, Information Theory, IEEE Transactions on,50 (2004), pp. 2231–2242.
- [37] Joel A. Tropp, Algorithms for simultaneous sparse approximation: Part ii: Convex relaxation, Signal Process., 86(2006), pp. 589–602.
- [38] Joel A. Tropp, Anna C. Gilbert, and Martin J. Strauss, Algorithms for simultaneous sparse approximation: Parti: Greedy pursuit, Signal Process., 86 (2006), pp. 572–588.
- [39] E. van den Berg and M. P. Friedlander, Theoretical and empirical results for recovery from multiple measurements,IEEE Transactions on Information Theory, 56 (2010), pp. 2516–2527.
- [40] Yuehao Wu, Peng Ye, Iftekhar O Mirza, Gonzalo R Arce, and Dennis W Prather, Experimental demonstrationof an optical-sectioning compressive sensing microscope (csm), Optics express, 18 (2010), pp. 24565–24578.