In synthetic aperture radar (SAR) imaging, the presence of moving targets in the scene causes phase errors in the SAR data and subsequently defocusing in the formed image. The defocusing caused by the moving targets exhibits space-variant characteristics, i.e., the defocusing arises only in the parts of the image containing the moving targets, whereas the stationary background is not defocused. Considering that the reflectivity field to be imaged usually admits sparse representation, we propose a sparsity-driven method for joint SAR imaging and removing the defocus caused by moving targets. The method is performed in a nonquadratic regular-ization based framework by solving an optimization problem, in which prior information about both the scene and phase errors are incorporated as constraints

Çetin, Müjdat

Önhon, Özben Naime

English

Sabanci University Research Database

SPARSITY-DRIVEN IMAGE FORMATION AND SPACE-VARIANT FOCUSING FOR SARN. ¨Ozben ¨Onhon and Mu¨jdat C¸etinFaculty of Engineering and Natural Sciences, Sabancı University, Orhanlı, Tuzla, 34956 Istanbul, TurkeyABSTRACTIn synthetic aperture radar (SAR) imaging, the presence ofmoving targets in the scene causes phase errors in the SARdata and subsequently defocusing in the formed image. Thedefocusing caused by the moving targets exhibits space-variant characteristics, i.e., the defocusing arises only in theparts of the image containing the moving targets, whereasthe stationary background is not defocused. Considering thatthe reflectivity field to be imaged usually admits sparse rep-resentation, we propose a sparsity-driven method for jointSAR imaging and removing the defocus caused by movingtargets. The method is performed in a nonquadratic regular-ization based framework by solving an optimization problem,in which prior information about both the scene and phaseerrors are incorporated as constraints.Index Terms— Motion errors, phase errors, space-variantfocusing, regularization, synthetic aperture radar, sparsity1. INTRODUCTIONIn synthetic aperture radar (SAR) imaging, uncertainties onthe position of the sensing platform or on the motion ofthe targets in the underlying scene cause artifacts in the re-constructed image. Due to the inexact knowledge about theposition of the SAR sensor, the time required for the transmit-ted signal to propagate to the scene center and back cannot bedetermined accuretely, which cause phase errors in the SARdata [1]. This type of phase errors cause space-invariant defo-cusing, i.e., the amount of the defocusing in the reconstructedimage is same for all points of the scene. Moving targetsin the scene cause defocusing in the reconstructed image aswell. However, this defocusing needs to be corrected witha space-variant refocus algorithm, since the defocusing ap-pears only around the positions of the moving targets whereasthe stationary background is not defocused [2]. Therefore,autofocus techniques developed for space-invariant focusingcannot handle the defocusing arising in the imaging of a sceneincluding multiple moving targets with different velocities.The cross-range component of the target velocity causes theThis work was partially supported by the Scientific and Technologi-cal Research Council of Turkey under Grant 105E090, and by a TurkishAcademy of Sciences Distinguished Young Scientist Award.image of the target to be defocused in the cross-range di-rection, whereas the range component causes shifting in thecross-range direction and defocusing in both cross-range andrange directions [3]. The image of a target that experiencessignificant vibration is defocused in the cross-range directionas well [4]. The common approach to space-variant focusingis to partition the image into smaller subimages such thatthe error on each subimage is approximately space-invariant[3, 5]. After each of the small subimages is focused in-dependently using one of the conventional space-invariantautofocus techniques, these subimages are combined togetherto obtain the focused image. These kinds of approaches arebased on post-processing of the conventionally reconstructedimage. However, we know that conventional imaging doesnot perform well in sparse aperture scenarios or when thedata are incomplete. On the other hand, regularization-basedimage reconstruction has succesfully been applied to SARimaging and it is shown that it has many advantages overconventional imaging [6]. These techniques can alleviate theproblems in the case of incomplete data or sparse aperture.Moreover, they produce images with increased resolution,reduced sidelobes, and reduced speckle by incorporation ofprior information about the features of interest and impos-ing various constraints (e.g., sparsity, smoothness) about thescene. Motivated by these observations and considering thatin SAR imaging, the underlying field usually exhibits a sparsestructure, we previosly proposed a sparsity-driven techniquefor joint SAR imaging and space-invariant focusing by us-ing a nonquadratic regularization based framework [7, 8].Here, extending this framework we propose a method forjoint sparsity-driven imaging and space-variant focusing forcorrection of phase errors caused by target motion. Thisnot only involves a nontrivial extension of the phase errorestimation piece of our previous framework, but it also pro-vides opportunities for incorporation of information aboutthe expected spatial structure of the motion errors as well.In particular, in the new approach presented here, we notonly exploit the sparsity of the reflectivity field, but we alsoimpose a constraint on the spatial sparsity of the phase errorsbased on the assumption that motion in the scene will belimited to a small number of spatial locations. The methodis based on minimization of a cost function of both the fieldand phase errors. The algorithm is iterative and each iterationinvolves two steps, first of which is for image formation andsecond is for phase error estimation. Successful results havebeen obtained in experiments involving synthetic scenes withsimulated multiple targets.2. SAR IMAGING MODELSAR is generally used for imaging of the ground from a planeor satellite. On its flight path, a SAR sensor transmits pulsesto the ground and then receives the reflected signals. In mostSAR applications, the transmitted signal is a chirp signal,which has the following form:s(t) = Re{exp[j(ω0t+ αt2)]} (1)Here, ω0 is the center frequency and 2α is the so-called chirp-rate. The received signal qm(t) at a certain aperture positionθ involves the convolution of the transmitted chirp signal withthe projection pm(u) of the field at that observation angle.qm(t) = Re{∫pm(u) exp[j[ω0(t− τ0 − τ(u)) + (2)α(t− τ0 − τ(u))2]]du}Here, τ0 represents the time required for the transmitted sig-nal to propagate to the scene center and back. τ0+ τ(u) is thedelay for the returned signal from the scatterer at the range po-sition d0+u, where d0 is the distance between the SAR sensorand the scene center. The data used for imaging are obtainedafter a pre-processing operation involving mixing and filter-ing steps. After this process, the relation between the fieldF (x, y) and the pre-processed SAR data rm(t) becomesrm(t) =∫ ∫x2+y2≤L2F (x, y) exp{−jU(x cos θ + y sin θ)}dxdy (3)whereU =2c(ω0 + 2α(t− τ0)) (4)and L is the radius of the illuminated area. All of the returnedsignals from all observation angles constitute a patch fromthe two dimensional spatial Fourier transform of the corre-sponding field. The corresponding discrete model includingall returned signals is as follows.r1r2rM︸ ︷︷ ︸r=C1C2CM︸ ︷︷ ︸Cf (5)Here, rm is the vector of observed samples, Cm is a dis-cretized approximation to the continuous observation kernelat the m− th cross-range position, f is a vector representingthe unknown sampled reflectivity image and M is the totalnumber of cross-range positions. The vector r is the SARphase history data of all points in the scene. It is also possibleto view r as the sum of the SAR data corresponding to eachpoint in the scene.r = Cclmn−1f(1)︸ ︷︷ ︸rp1+Cclmn−2f(2)︸ ︷︷ ︸rp2+..+ ..+ Cclmn−If(I)︸ ︷︷ ︸rpI(6)Here, Cclmn−i is the i−th column of the model matrixC and,f(i) and rpi represent the complex reflectivity at the i − thpoint of the scene and the corresponding SAR data, respec-tively. I is the total number of points in the scene. Targetsmoving in cross-range direction or vibrating targets cause de-focusing in the reconstructed image. The defocusing arisesdue to the phase errors in the SAR data of these targets. Letus assume the i− th point in the scene as a point target mov-ing in cross-range direction or vibrating without translation.The SAR data of this target can be expressed as:rpi1erpi2e...rpiMe=ejφi(1) rpi1ejφi(2) rpi2...ejφi(M) rpiM(7)Here, φi represents the phase error in the cross-range direc-tion caused by the motion of the target and, rpi and rpie arethe phase history data for the stationary and moving point tar-get, respectively. In a similar way, this relation can be ex-pressed in terms of model matrix as follows:Cclmn−i1(φ)Cclmn−i2(φ)...Cclmn−iM (φ)=ejφi(1) Cclmn−i1ejφi(2) Cclmn−i2...ejφi(M) Cclmn−iM(8)Here, Cclmn−i(φ) is the i-th column of the model matrixC(φ) that takes the movement of the targets into account andCclmn−im(φ) is the part of Cclmn−i(φ) for the m− th cross-range position. In the presence of additional observationnoise, the observation model for the overall system becomesg = C(φ)f + v (9)where, v is the observation noise. Here, the aim is to estimatef and φ from the noisy observation g.3. PROPOSED METHODIn the context of SAR imaging of man-made objects, theunderlying scene, dominated by strong metallic scatterers,is usually sparse, i.e., there are few nonzero pixels. Basedon that observation, we propose a sparsity-driven methodfor joint estimation of the field and phase errors caused bythe targets moving in the cross-range direction. The methodis based on a nonquadratic regularization-based frameworkwhich allows the incorporation of the prior sparsity informa-tion about the field and about the phase error into the problem.The phase errors are incorporated to the problem using thevector β , which includes phase errors corresponding to allpoints in the scene, for all cross-range positions.β =β1β2...βM(10)Here, βm is the vector of phase errors for the m − th cross-range position and has the following form:βm =[ejφ1(m), ejφ2(m), ...., ejφI(m)]T(11)The method is performed by minimizing the following costfunction with respect to the field and phase errors.argminf,βJ(f, β) = argminf,β‖g − C(φ)f‖22 + λ1 ‖f‖1 + λ2 ‖β − 1‖1s.t. |β(i)| = 1 ∀i(12)Here, 1 is a MI × 1 vector of ones. Since the number ofmoving points is much less than the total number of points inthe scene, most of the φ values in the vector β are zero. Sincethe elements of β are in the form of ejφ’s, when φ is zero,β becomes one. Therefore, this sparsity on the phase errorsis incorporated into the problem by using the regularizationterm ‖β − 1‖1. This problem is solved iteratively. In eachiteration, in first step, the cost function J(f, β) is minimizedwith respect to the field f .fˆ (n+1) = argminfJ(f, βˆ(n)) =argminf∥∥∥g − C(n)(φ)f∥∥∥22+ λ1 ‖f‖1 (13)This minimization problem is solved using the technique in[6]. Using the field estimate fˆ , in the second step, to estimatethe phase errors imposed by the moving targets, the followingcost function is minimized for each cross-range position:βˆ(n+1)m = argminβmJ(fˆ (n+1), βm) =argminβm∥∥∥gm − CmT (n+1)βm∥∥∥22+ λ2 ‖βm − 1‖1s.t. |βm(i)| = 1 ∀i (14)Here, T is a diagonal matrix, with the entries fˆ(i) on its maindiagonal, as follows:T (n+1) = diag{fˆ (n+1)(i)}(15)In (14), 1 is a I × 1 vector of ones. The constrained opti-mization problem in (14) is replaced with the following un-constrained problem that incorporates a penalty term on themagnitudes of βm(i)’s.βˆ(n+1)m = argminβm∥∥∥gm − CmT (n+1)βm∥∥∥22+ λ2 ‖βm − 1‖1 +λ3I∑i=1(|βm(i)| − 1)2= argminβm∥∥∥gm − CmT (n+1)βm∥∥∥22+ λ2 ‖βm − 1‖1 +λ3 ‖βm‖22 − 2λ3 ‖βm‖1m = 1, 2, ...,M (16)This optimization problem is solved by using the same tech-nique as in the field estimation step. Using the estimate βˆm,the following matrix is created,B(n+1)m = diag{βˆ(n+1)m (i)}(17)which is used to update the model matrix for the m−th cross-range position.C(n+1)m (φ) = CmB(n+1)m (18)After these phase estimation and model matrix update proce-dures have been completed for all cross-range positions, thealgorithm passes to the next iteration, by incrementing n androtating to (13).4. EXPERIMENTAL RESULTSWe show experimental results on two different syntheticscenes. To demonstrate the effectiveness of and highlightthe benefits specificly provided by the proposed method, forboth experiments, the images reconstructed by conventionalimaging (the polarformat algorithm [2]) and sparsity-drivenimaging [6] are presented as well. In the first experiment,there are multiple moving targets in the scene. To simulatedifferent motions and velocities of the targets, the phase his-tory data of each target are corrupted by a different phaseerror function. The phase histories of the three point targetsare corrupted by independent random phase error functionsuniformly distributed in [−pi/2, pi/2]. The phase historiesof the two bigger targets are corrupted by quadratic phaseerror functions of different peak values. In Figure 1, theresults of the first experiment are displayed. In the secondexperiment, the scene is constructed so that it involves manystationary point targets and a strongly vibrating rigid-bodytarget. To simulate it, the phase history data corresponding toeach point of this target are corrupted by independent randomphase error functions uniformly distributed in [−pi/2, pi/2].The results of the second experiment are displayed in Figure2. From the results for conventional imaging and sparsity-driven imaging without any phase error correction, the defo-cusing and artifacts in the reconstructed images caused by the(a) (b)(c) (d)Fig. 1. a) Original scene. b) Image reconstructed by con-ventional imaging. c) Image reconstructed by sparsity-drivenimaging. d) Image reconstructed by the proposed method.moving targets are clearly seen. On the other hand, imagesreconstructed by the proposed method are well focused andshow the advantages of sparsity-driven imaging such as highresolution, reduced speckle and sidelobes, as well as effectivecorrection of the phase errors due to target motion.5. CONCLUSIONWe proposed a sparsity-driven method for joint imaging andcorrection of space-variant defocusing in SAR. The methodeffectively produces high resolution images and removes thecross-range dependent phase errors caused by moving targets.Moreover, the estimated phase errors can be used to estimatethe velocity and characteristics of the motion. With slightextensions, the method is also applicable to range dependentphase errors imposed by moving targets. Our planned futurework involves experiments on real SAR data.6. REFERENCES[1] C. V. Jakowatz, Jr., D. E. Wahl, P. H. Eichel, D. C.Ghiglia, and P. A. Thompson, Spotlight-Mode Syn-thetic Aperture Radar: A Signal Processing Approach,Springer, 1996.[2] W. G. Carrara, R. M. Majewski, and R. S. Goodman,Spotlight Synthetic Aperture Radar: Signal ProcessingAlgorithms, Artech House, 1995.(a) (b)(c) (d)Fig. 2. a) Original scene. b) Image reconstructed by con-ventional imaging. c) Image reconstructed by sparsity-drivenimaging. d) Image reconstructed by the proposed method.[3] C. V. Jakowatz, Jr., D. E. Wahl, and P. H. Eichel, “Refo-cus of constant velocity moving targets in synthetic aper-ture radar imagery,” Algorithms for Synthetic ApertureRadar Imagery V, SPIE, 1998.[4] A. R. Fasih, B. D. Rigling, and R. L. Moses, “Analysisof target rotation and translation in SAR imagery,” Algo-rithms for Synthetic Aperture Radar Imagery XVI, SPIE,2009.[5] J. R. Fienup, “Detecting moving targets in SAR imageryby focusing,” IEEE Transactions on Aerospace and Elec-tronic Systems, vol. 37, no. 3, pp. 794–809, 2001.[6] M. C¸etin and W.C. Karl, “Feature-enhanced syntheticaperture radar image formation based on nonquadraticregularization,” IEEE Trans. Image Processing, vol. 10,no. 4, pp. 623–631, 2001.[7] N. ¨O. ¨Onhon and M. C¸etin, “A nonquadratic regulariza-tion based technique for joint SAR imaging and model er-ror correction,” Algorithms for Synthetic Aperture RadarImagery XVI, Proc. SPIE, vol. 7337, 2009.[8] N. ¨O. ¨Onhon and M. C¸etin, “Joint sparsity-driven in-version and model error correction for radar imaging,”IEEE Int. Conf. Acoustics, Speech, Signal Processing, pp.1206–1209, 2010.

Sparsity-driven image formation and space-variant focusing for SAR

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Sparsity-driven image formation and space-variant focusing for SAR

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