Sparsity based Ground Moving Target Imaging via Multi-Channel SAR

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1 Sparsity based Ground Moving Target Imaging via Multi-Channel SAR Di Wu, Mehrdad Yaghoobi and Mike Davies Shool of Engineering University of Edinburgh UK, EH9 3JL {D.Wu, m.yaghoobi-vaighan, Abstrat State-of-the-art Ground Moving Target Indiator GMTI shemes inlude the Displaed Phase Center Antenna DPCA and Along Trak Interferometry ATI whih are ommonly used image-based dual-hannel tehniques for moving target detetion. In the present paper, we provide a different perspetive for solving GMTI tasks by generalising the ground moving targets imaging as an parameter estimation and optimisation problem. A sparsity based ground target imaging approah is desribed to improve the image quality for moving targets and estimate their states. By exploiting the fat that moving targets are highly sparse in the observed sene and feasible veloity spae, the proposed method onstruts a veloity map for the illuminated region, and ombines this map with a sparsity based optimisation algorithm to realise the image formation. The performane of the presented method is demonstrated through GOTCHA airborne SAR data set. Index Terms SAR, GMTI, sparsity, ompressed sensing, veloity map I. INTRODUCTION One of the basi priniples of SAR is to assume a stationary sene. Therefore moving targets will indue displaement and blurring in the image. Ground Moving Target Indiator GMTI aims to detet the moving targets in the SAR image and estimate their veloities whih are of great value for the battlefield awareness and information gathering. The entire SAR/GMTI systems often onsist of several signal proessing steps suh as pre-proessing, range ompression, lutter anellation, target detetion, geo-loation, motion para estimation and SAR image formation [1][2][3]. Displaed Phase Center Antenna DPCA, Along Trak Interferometry ATI and Spae-time Adaptive Proessing STAP are the multi-hannel methods whih have been widely used in the SAR ommunity. DPCA and ATI [4][5] attempt to expose the moving targets from the SAR images of different hannels with magnitudes and interferene phases respetively. However the target radial veloities an be derived from the ATI results diretly and DPCA is not apable of doing that. STAP [6] is well known to be omputationally expensive, and it is hampered by the false alarm rates espeially in non-homogeneous urban environments. Also in 212, Prunte proposed to indiate moving targets based on multi-hannel SAR using ompressed sensing [7]. The proposed method is utilising the sparsities pixel-wise in the image domain. In this paper, we present a model to link up ertain parts of the proessing hain in multi-hannel SAR/GMTI and represent them as a generalised optimisation problem. By introduing the sparsities in the proposed model, i.e. the number of moving targets is reasonably small ompared to the size of the observed sene, ground moving targets an be mapped and their motion para an be estimated. We hope this novel model an provide an alternative perspetive for viewing SAR/GMTI problems. The remainder of this paper is organized as follows. Setion two desribes the signal model in a typial dual-hannel SAR system. In setion three, the sparsity based moving target imaging model is presented. We first disuss the hannel balaning tehniques to o-register different hannels. Then we present the mehanism of traditional SAR imaging and lutter anellation algorithms, and finally introdue the s- parse regularized optimisation model to realise simultaneous SAR/GMTI. In setion four, the real airborne SAR data is utilised to demonstrate the performanes of the proposed model. Conlusions and future work are presented in setion five. II. SIGNAL MODELING The proposed framework is presently demonstrated through a typial dual-hannel SAR system in this paper, but it an be generalised to the senarios with more than two hannels. The geometry with a moving target in the observed sene is shown in Fig. 1. The phase entres of the fore-antenna and aftantenna are spatially separated by a distane d on the flight path of the platform. The Cartesian veloity omponents of the moving target are denoted as v x, v y, v z whih an be onverted to v az, v r in the azimuth and radial diretions. Let the azimuth time of the transmitted pulses be τ n where n = {1, 2,..., N} is the pulse number. Then rτ n is the position of the target at τ n, r t i τ n and r o i τ n represent the distanes from the moving target to the i th antenna and the distane from the sene origin to the orresponding antenna position. Also the platform veloity within a short sub-aperture is approximated by a onstant v p, and f P RF is the pulse repetition frequeny PRF. In the remainder of the paper, we apply the stop-and-hop approximation whih assumes that the targets and platform have onstant positions during the RF propagation of one pulse unless otherwise stated. After the de-hirping proess, in whih the platform motion has been ompensated with

2 y O target x hannel balaning tehniques, we attempt to retrieve the same responses for stationary targets between different hannels. Sine the baseline d in the SAR system is along the platform trak, the reeived eho of the aft-antenna an be viewed as the delayed reeived signal of the fore-antenna if the observed region does not hange over time. We an illustratively assume that r 1 τ n = r 2 τ n + where r i τ n denotes the loation of the i th antenna at τ n and = d/v p represents the delay between two hannels. Here τ n + may not orrespond to an exat pulse time. Then the differential range u 2 τ n + an be rewritten as Radar hannels Platform trak Fig. 1. The geometry of a dual-hannel SAR system with a moving target in the observed sene. referene to the sene origin, the hannel 1 fore-antenna and 2 aft-antenna disrete reeived signals of a single moving target loated at rτ n are: Y i f k, τ n = A i σrτ n exp j4πf ku i τ n where {f k k = 1, 2,..., K} denotes the range frequenies; A i is the nominal fator of the reeived signal for the i th hannel whih aounts for the beam pattern and energy loss; σrτ n is the omplex refletivity of this moving target; represents the speed of light and u i τ n denotes the differential range r t i τ n r o i τ n. We assume that the moving target has onstant veloity during the sub-aperture time, then the instantaneous spatial loation of the target rτ n an be expressed as x +τ n v x, y + τ n v y, z + τ n v z where x, y, z stands for the initial position of the target. The goal of SAR/GMTI is to estimate the target states inluding the geoloation rτ n and veloities v x, v y, v z given the reeived signals Y 1 f k, τ n and Y 2 f k, τ n. III. SPARSITY BASED MOVING TARGET IMAGING Based on the dual-hannel SAR system we desribe the proposed sparsity based moving target imaging approah and its relationship with ommonly used SAR/GMTI tehniques. A. Pre-proessing and Channel Balaning For subtrative GMTI methods suh as ATI and DPCA, the differenes between hannels are exploited to detet targets and suppress the lutter. Thus it is ruial to equalise the hannel transfer funtions in the pre-proessing step. With d 1 u 2 τ n + =r t 2 τ n + r o 2 τ n + = r 2 τ n + rτ n + r 2 τ n + = r 1 τ n rτ n + r 1 τ n 2 If the targets are isotropi and their refletivities keep the same over time then we an derive from 1 that Y 1 f k, τ n = A 1 σrτ n exp j4πf k r 1 τ n rτ n r 1 τ n 3 Y 2 f k, τ n + = A 2 σrτ n exp j4πf k r 1 τ n rτ n + r 1 τ n 4 where Y 2 f k, τ n + is the time-shifted version of Y 2 f k, τ n whih an be estimated by the multipliation in the frequeny domain with a -indued phase shift term. We denote this as Ỹ 2 f k, τ n = Y 2 f k, τ n +. It an be seen that the veloity and position information of the moving targets are fully desribed by the last exponential terms in 3 and 4. We also need to balane the hannels to equalise A 1 and A 2 as done for example in [8]. In this way, Y 1 f k, τ n and Ỹ2f k, τ n are supposed to be the same for stationary targets. We will employ this hannel balaning tehnique to pre-proess the SAR data in the remainder of the paper. B. SAR Imaging and Clutter Canelation Numerous algorithms have been proposed for SAR image formation by investigating the trade-off between omplexity and image quality. For simpliity we will fous on mathed filter based tehniques. Let the disrete grid on whih the image is formed be G = x m, y l, and d n = r 1 τ n G r 1 τ n, where m = {1, 2,..., M} and l = {1, 2,..., L}. Then the observed sene an be viewed as the olletion of the targets in this grid. Here the elevation is assumed to be zero for the onveniene of subsequent analysis. The formed

3 SAR images with the mathed filter method are X 1 m, l= Y 1 f k, τ n exp k=1 n=1 X 2 m, l= Ỹ 2 f k, τ n exp k=1 n=1 Typial lutter suppression methods suh as DPCA and ATI exploit the differenes between 5 and 6 to reveal the moving targets. Speifially, DPCA is ahieved by subtrating 6 from 5, and ATI is realised by multiplying 5 with the onjugate of 6. For DPCA, we have that X 1 m, l X 2 m, l = Y 1 f k,τ n exp 1 exp j4πf kw n k=1n=1 where w n = r 1 τ n r τ n + r 1 τ n r τ n. At azimuth time τ n, the position of the target whih was initially loated at x m, y l, is denoted as r τ n. w n an then be approximated by v r where vr is the radial veloity of this target. Also exp j4πf k w n / an be approximated with a onstant if the target remains in the same range resolution ell during the time interval [9]. Then equation 7 an be rewritten as: X 1 m, l X 2 m, l X 1 m, l 1 exp j4πf v r 8 where f is denoted as the entre frequeny of the transmitted signal hirps. Similarly for ATI we have that X 1 m, l X2 m, l X 1 m, l 2 j4πf v r exp 9 It an be seen that 8 and the phase of 9 are zero for stationary targets v r =. Partiularly the DPCA results are approximately the veloity-saled refletivities of the moving targets. Hene, lutter suppression tehniques suh as DPCA and ATI enable us to projet out stationary targets. Also the radial veloities {v r m = 1, 2,..., M; l = 1, 2,..., L} an be estimated diretly based on the phase of 9. C. Sparse Regularized SAR/GMTI There has been a number of SAR imaging algorithms with super-resolution effets by utilising the image sparsity [1][11]. The proposed approahes an mitigate the sidelobes in SAR images and reonstrut the bright image elements. In GMTI appliations the whole observed sene is not sparse but the moving targets are often sparse. In this way, if we an build up the linear projetion from the raw data to the moving targets and its inverse projetion, the tehniques whih are ommonly used in ompressed sensing [12] an then be leveraged to form the image of these sparse targets Let X i = {X i m, l} C M L be the SAR image of the i th hannel; Y 1 = {Y 1 f k, τ n } C K N and Ỹ 2 = Ỹ2f k, τ n C K N be the reeived phase history of the 1st hannel and the balaned phase history of the 2nd hannel respetively. Based on 1 we an write the signal model in matrix-vetor form as Y 1 = Φ F X 1 where Φ F is the forward projetion operator. Note that DPCA is essentially the linear projetion from Y 1 Ỹ 2 to X 1 X 2, thereby all stationary satterers are removed. Therefore a simple sparsity based GMTI ould exploit the following optimisation problem: min X Y 1 Ỹ 2 Φ F X 2 F + λ X 1 s.t. X C M L 1 where F represents the Frobenius norm, λ is a positive tuning parameter, and X is the sparsified image of moving targets and sparsity in X is enouraged through the inlusion of the L 1 norm penalty funtion. Note that this formulation still takes no aount of the veloity effet at this stage, and the moving targets will be displaed and blurred in the resulting X. A more advaned sparsity based SAR/GMTI formulation an also be derived that simultaneously takes into aount the sparsity of the moving targets and estimates their veloities. Let V = V x, V y, denote the veloity maps for a speifi observed sene X here X inludes stationary and moving targets where V x = {v x } RM L and V y = {v y } RM L are the veloity omponents of G in x and y diretions. The veloity map V an be interpreted as granting eah image element with speifi veloity omponents. With referene to the geometry Fig. 1, the veloity map an be diretly onverted to the orresponding azimuth veloity V az = {v az } R M L and radial veloity V r = {v r } RM L. In this model we make the reasonable assumption that there is only one dominant veloity for eah image element. This assumption makes the reonstrutions for a speifi physial position m, l to be 1-sparse along the V axis. Inorporating the veloities into the signal model it an be shown that: Y 1 f k, τ n = M L m=1l=1 Xm, l exp j4πf kd 1n Ỹ 2 f k,τ n M L = Xm, l exp j4πf kd 1n exp j4πf kv r m=1l= where d 1n = r 1 τ n G τ n v x, τ nv y, r 1 τ n. Given a veloity map V, we an denote the projetion operator from the physial spae X to the phase history Y 1 and Ỹ 2 as Φ V F and Φ V F respetively. The following model is hereby

4 introdued to simultaneously estimate the target s states and form the SAR images: min X,V Y 1 Φ V F X 2 F + Ỹ 2 Φ V F X 2 F +λ V x +λ V y s.t. X C M L, V x R M L, V y R M L suppv x = suppv y 13 where λ is a positive tuning parameter. The resulting X represents the refletivities of the observed sene inluding both stationary and moving targets. Pratially solving 13 is hallenging and therefore in the next setion we will explore a solution based on a partial formulation of 13 that further leverages the ATI tehnique. IV. PROCESSING RESULTS OF THE GOTCHA DATA The sparsity method is demonstrated through the AFRL GOTCHA data set [13]. The desribed sope ontains an X- band SAR system with three phase enters and a number of moving vehiles in an urban environment. The ground truth data of one vehile is provided. To be speifi, the transmitted hirp is entred at 9.6 GHz, the phase history is olleted over 71 seonds interval, and the PRF is 2.17 khz. Furthermore, as presented in [13], the data was rangegated from 54 range samples to 384 sub bins to derease the required storage. The senario an be found in Fig. 2. Based on the ground truth data, the target trajetory is illustrated by the red path. With referene to the streets and roads, the target trajetory is displaed along the x diretion for about 2. These displaements result from the elevation variations of the observed terrain. In this setion, we onsider the data from first two hannels at the 46 th seond snapshot of the senario with 2 pulses about.1 seond interval around. y meter x meter Fig. 2. The reonstruted SAR image in db with partial data to show the GMTI senario. Here we assume a flat terrain and the aelerated version of 5 is employed. The red path stands for the target trajetory based on the ground truth data. To preproess the data, we first run the inverse operation of the the range gating and replae the unknown range gates with zero. The phase histories are reorganized into data matries Y 1 C 54 2 and Y 2 C We then apply the 2D hannel balaning tehnique [8] to the first two hannels and the alibrated phase histories are Y 1 and Ỹ 2. To derease the omputational omplexity, here we approximately rop the phased history in the 2D Fourier transform domain with retangular windows 49 points in range and 1 points in azimuth diretions to fous on the known target. The ropped phase histories are denoted as Y t 1 C 54 2 and Ỹ t 2 C The formed image of Y t 1 with 5 an be found in Fig Fig. 3. The orresponding SAR image of the ropped phase history. The image is displayed in -7 db. No veloity is assumed in this image. The target appears to be displaed and blurred. It is well known that the radial veloities an be estimated from the phase information using ATI. Therefore, to simplify the problem, based on ATI, we still estimate the radial veloity V r by Θ/4πf given that the phases of the ATI results are Θ. Here the baseline d =.238m an be estimated in the preproessing [2] and the radial veloity is hosen to be the mean value of nonzero V r. One we have V r, it is straight forward to map V r and V az to V based on the geometry, and we denote this as V = ΥV az, V r. We now have the following model: min Y t X,V az 1 Ỹ t 2 Φ V F X 2 F + λ X 1 s.t. X C M L, V az R M L, V = ΥV az, V r suppx = suppv az As Y t 1 and Ỹ t 2 are fousing on the target, the elements in V az an be assigned to a single v az throughout the image. Here a naive method to solve 14 is implemented. We first ompute 14 using five iteration FISTA algorithm H Y t 1 Ỹ t 2 with different v az [14] λ =.3 2 Φ V F independently. The objetive value of 14 with eah speifi v az is reorded. The optimal v az and X are hosen to orrespond with the minimised objetive value. The objetive value against v az plot is shown in Fig. 4. The estimated X gives the reloated and refoused image of the moving target whih is shown in Fig. 5.

5 retrieved objetive value in eah FISTA run x azimuth veloity m/s Fig. 4. Resulting objetive values with respet to different v az based on the FISTA for 14. Fig. 5. The reonstruted image in db of the moving target. Five iteration FISTA is employed for the 46 th seond with 2 pulses. The red line indiates the ground truth data. TABLE I COMPARISONS BETWEEN THE GROUND TRUTH AND ESTIMATIONS Ground Truth Estimations x m y m z m -3.2 range distane m 1.38e4 1.39e4 v r m/s v az m/s overall v m/s The estimation results are onluded in Table I. It an been seen from Fig. 2 that the ground truth trajetory has about 2 deviation in x diretion due to the terrain elevation. The estimated target position is also shifted towards the negative x diretion in the image, and the range distane ground truth oinides with the estimation result. Note that all the estimations are based on a very short sub-aperture, a larger sub-aperture is likely to inrease the auray as long as the onstant veloity assumptions still hold. SAR/GMTI task an be generalised as an optimisation problem and how the sparsity an be used to help estimate the estimate targets state and form SAR images. The experimental results based on real GOTCHA GMTI data illustrate the effetiveness of the proposed model. In pratie, SAR/GMTI is likely to enounter the problems suh as the imperfetions in hannel balaning, the ambiguities in veloity estimations and the demand on omputational power. These will be investigated in the future work. ACKNOWLEDGMENT This work was supported by the Engineering and Physial Sienes Researh Counil EPSRC grants [EP/K14277/1]; and the University Defene Researh Collaboration UDRC. REFERENCES [1] R. Deming, M. Best, and S. Farrell, Simultaneous SAR and GMTI using ATI/DPCA, Pro. SPIE, vol. 993, pp. 9 93U 9 93U 19, 214. [2] B. Guo, D. Vu, L. Xu, M. Xue, and J. Li, Ground moving target indiation via multihannel airborne SAR, IEEE T. Geosiene and Remote Sensing, vol. 49, no. 1, pp , 211. [3] J. Yang, C. Liu, and Y. Wang, Detetion and imaging of ground moving targets with real SAR data, IEEE T. Geosiene and Remote Sensing, vol. 53, no. 2, pp , 215. [4] S. Chiu and C. Livingstone, A omparison of displaed phase entre antenna and along-trak interferometry tehniques for RADARSAT-2 ground moving target indiation, Canadian Journal of Remote Sensing, vol. 31, no. 1, pp , 25. [5] I. Sikaneta and C. Gierull, Ground moving target detetion for alongtrak interferometri SAR data, in Aerospae Conferene, 24. Proeedings. 24 IEEE, vol. 4, Marh 24, pp Vol.4. [6] J. Ward, Spae-time adaptive proessing for airborne radar, in Spae- Time Adaptive Proessing Ref. No. 1998/241, IEE Colloquium on, Apr 1998, pp. 2/1 2/6. [7] L. Prunte, GMTI from multihannel SAR images using ompressed sensing, in Syntheti Aperture Radar, 212. EUSAR. 9th European Conferene on, April 212, pp [8] C. Gierull, Digital hannel balaning of along-trak interferometri SAR data, in Tehnial Memorandum DRDC Ottawa TM Defene R&D, Ottawa, Canada, Marh 23. [9] R. W. Deming, Along-trak interferometry for simultaneous SAR and GMTI: appliation to gotha hallenge data, Pro. SPIE, vol. 851, pp. 8 51P 8 51P 18, 211. [1] S. Kelly and M. Davies, RFI suppression and sparse image formation for UWB SAR, in Radar Symposium IRS, th International, vol. 2, June 213, pp [11] M. Cetin and R. L. Moses, SAR imaging from partial-aperture data with frequeny-band omissions, Pro. SPIE, vol. 588, pp , 25. [12] M. Fornasier and H. Rauhut, Compressive sensing, in Handbook of Mathematial Methods in Imaging. Springer, 211, pp [13] S. M. Sarborough, C. H. Casteel, Jr., L. Gorham, M. J. Minardi, U. K. Majumder, M. G. Judge, E. Zelnio, M. Bryant, H. Nihols, and D. Page, A hallenge problem for SAR-based GMTI in urban environments, Pro. SPIE, vol. 7337, pp G 73 37G 1, 29. [14] A. Bek and M. Teboulle, A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems, SIAM Journal on Imaging Sienes, vol. 2, no. 1, pp , Mar. 29. V. CONCLUSION This paper presents a sparse regularized model for multi-hannel SAR/GMTI. Speifially we show how the