Solution of inverse problems in imaging requires the use of a mathematical model of the observation process. However such models often involve errors and uncertainties themselves. The application of interest in this paper is synthetic aperture radar (SAR) imaging, which particularly suffers from motion-induced model errors. These types of errors result in phase errors in SAR data which cause defocusing of the reconstructed image. Mostly, phase errors vary only in cross-range direction. However, in many situations, it is possible to encounter 2D phase errors, which are both range and cross-range dependent. We propose a sparsity-driven method for joint SAR imaging and correction of 1D as well as 2D phase errors. This method performs phase error correction during the image formation process and provides focused, high-resolution images. Experimental results show the effectiveness of the approach

Cetin, Mujdat

Onhon, Ozben Naime

Çetin, Müjdat

Önhon, Özben Naime

English

Sabanci University Research Database

JOINT SPARSITY-DRIVEN INVERSION AND MODEL ERROR CORRECTION FOR RADARIMAGINGN. ¨Ozben ¨Onhon and Mu¨jdat C¸etinFaculty of Engineering and Natural Sciences, Sabancı University, Orhanlı, Tuzla, 34956 Istanbul, TurkeyABSTRACTSolution of inverse problems in imaging requires the use of a math-ematical model of the observation process. However such modelsoften involve errors and uncertainties themselves. The applicationof interest in this paper is synthetic aperture radar (SAR) imaging,which particularly suffers from motion-induced model errors. Thesetypes of errors result in phase errors in SAR data which cause defo-cusing of the reconstructed image. Mostly, phase errors vary only incross-range direction. However, in many situations, it is possible toencounter 2D phase errors, which are both range and cross-range de-pendent. We propose a sparsity-driven method for joint SAR imag-ing and correction of 1D as well as 2D phase errors. This methodperforms phase error correction during the image formation processand provides focused, high-resolution images. Experimental resultsshow the effectiveness of the approach.Index Terms— Motion errors, phase errors, autofocus, regular-ization, synthetic aperture radar, sparsity1. INTRODUCTIONIn many imaging systems model errors are one of the important prob-lem sources. This type of errors generally appears due to inaccuratemeasurement of the motion or position of the sensing platform or theobserved object. Furthermore, various environmental effects resultin similar type of errors. A well-known and widely-studied exam-ple of this type of errors appears in synthetic aperture radar (SAR)imaging. In SAR imaging, the time required for the transmitted sig-nal to propagate to the field center and back may not be measuredexactly. One of the reasons of this imperfection is the difficulty ofdetermining the distance between the SAR platform and the fieldcenter extremely accurately. In addition, environmental effects suchas atmospheric turbulence may induce random delays in the trans-mitted signal [1], which cause errors in the measured propagationtime. These uncertainties appear as phase errors in the SAR data.The effects of phase errors are seen as defocusing (blurring) and lossof contrast in the reconstructed image [1]. Since in every apertureposition a new signal is transmitted and the error for each signal isdifferent, the most widely encountered phase errors are only in thecross-range direction, which means that the phase error is a functionof the aperture position (cross range). However, in low frequencyUWB SAR systems, severe propagation effects may appear throughthe ionosphere, including Faraday rotation, dispersion, and scintilla-tion [2] which cause 2D phase errors, defocusing the reconstructedimage in both range and cross-range directions. Moreover, in 3DSAR imaging, phase errors are both range and cross range depen-dent. To remove the phase errors many techniques have been devel-This 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.oped which are called autofocus techniques. Most of them are basedon post-processing of the conventionally reconstructed SAR image[3–7]. However, we know that conventional imaging does not per-form well in sparse aperture scenarios or when the data are incom-plete. On the other hand, regularization based image reconstructionhas succesfully been applied to SAR imaging and it is shown thatit has many advantages over conventional imaging [8]. These tech-niques can alleviate the problems in the case of incomplete data orsparse aperture. Moreover, they produce images with increased res-olution, reduced sidelobes, and reduced speckle by incorporation ofprior information about the features of interest and imposing vari-ous constraints (e.g., sparsity, smoothness) about the scene . How-ever, they assume that there are no uncertainties in the observationmodel. Motivated by these observations and considering that in SARimaging, the underlying field has most of the time a sparse structure,we propose a sparsity-driven technique for joint SAR imaging andphase error correction by using a non-quadratic regularization basedframework. We presented a limited version of this idea, only for 1Dcross-range phase errors in [9]. The algorithm is an iterative algo-rithm, which is based on minimization of a cost function of both thefield and the phase error. In the first step of every iteration. an esti-mate of the field is found and using the field estimate in the secondstep, phase error is estimated and compensated. Depending on thedimension and structure of the phase error, error estimation step isperformed differently. We have implemented the proposed methodfor 2D separable and non-separable phase errors. Here we show re-sults for SAR imaging but this idea can be implemented for otherobservation systems, in which similar types of errors occur.2. SAR IMAGING2.1. SAR Observation ModelSAR is generally used for imaging of the ground from a plane orsatellite. On its flight path, a SAR sensor transmits pulses to theground and then receives the reflected signals. In most SAR applica-tions, chirp signals are transmitted. A chirp signal has the followingform:s(t) = Re{exp[j(ω0t+ αt2)]} (1)where ω0 is the center frequency and 2α is the so-called chirp-rate.The received signal qθ(t) at a certain aperture position θ involves theconvolution of the transmitted chirp signal with the projection pθ(u)of the field at that observation angle.qθ(t) = Re{∫pθ(u) exp[j[ω0(t− τ0 − τ(u)) + (2)α(t− τ0 − τ(u))2]]du}If we let the distance from the SAR platform to the center of thefield be R, τ0 + τ(u) is the delay for the returned signal from thescatterer at the range position R + u. Here, τ0 is the time requiredfor the transmitted signal to propagate to the scene center and back.The data used for imaging are obtained after a pre-processing step.In particular, the returned signal is first multiplied with delayed in-phase and quadrature versions of the transmitted chirp signal andthen the output is low-pass filtered. After this process, the relationbetween the field f(x, y) and the pre-processed SAR data rθ(t) be-comesrθ(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 returned sig-nals from all observation angles constitute a patch from the two di-mensional spatial Fourier transform of the corresponding field. Thecorresponding discrete model including all returned signals is as fol-lows. rθ1rθ2rθM︸ ︷︷ ︸r=Cθ1Cθ2CθM︸ ︷︷ ︸Cf (5)where rθm is the vector of observed samples, Cθm is discretized ap-proximation to the continuous observation kernel at the observationangle θm and f is a vector representing the unknown sampled reflec-tivity image. The data r are the phase histories. If we consider thatthere is also measurement noise, the observation model becomesg = Cf + v (6)where v stands for measurement noise which is assumed to be whiteGaussian noise and g is the noisy observation. Since the phase his-tory data are two dimensional spatial Fourier transform of the field,conventional imaging (polar format algorithm) for SAR is based onthe 2D inverse Fourier transform.2.2. Phase ErrorsAny error on τ0 (defined in Section 2.1), causes phase errors in theSAR data. The delay in every aperture position is usually assumedto be constant, which means that phase error varies only in the cross-range direction. However, ionospheric effects on the transmitted sig-nal, especially for low frequency UWB SAR systems cause 2D phaseerrors. Moreover, waveform errors such as ferequency jitter frompulse to pulse, transmission line reflections and waveguide disper-sion effects [10] may cause defocus in both range and cross-rangedirection. If there is a 2D phase error, it means that every samplepoint of the phase history data is perturbed with a different phase er-ror. The relation between the erroneous and error-free data for everyphase history sample can be expressed asr(s) = ejφ(s)r(s) (7)where r(s) and φ(s) denote s-th sample of the phase history data andthe corresponding phase error, respectively. In terms of observationmodel, the same relationship can be expressed asCs(φ(s))f = ejφ(s)Csf (8)Here, Cs represents the part of the model matrix C for the s-th datasample.3. SPARSITY-DRIVEN IMAGE RECONSTRUCTIONRegularization based image reconstruction techniques provide stableand reasonable estimates of the field f by incorporation of the priorknowledge about the field into the image formation process. Thesetechniques formulate image formation as an optimization problem.The cost functional is composed of a least-squares data fidelity term,as well as a side constraint related to the features of interest. In par-ticular, this side constraint incorporates information about the struc-ture of the scene (sparsity, smoothness etc.) into the optimizationproblem. In many imaging problems, the field of interest admits asparse representation in some domain. In particular, in the context ofSAR imaging of man-made objects, the underlying scene, dominatedby strong metallic scatterers, is sparse, i.e. there are few nonzeropixels. In such a case, a solution with great energy concentration isneeded. To provide it, a non-quadratic side constraint is appropriate.There are a variety of non-quadratic choices to use as the side con-straint. The general family of lp-norms is one of them. In spectralanalysis, lp-norm constraints, where p < 2, have been shown to re-sult in higher resolution spectral estimates compared to the l2-normcase. Moreover, smaller value of p implies less penalty on large pixelvalues as compared to larger p [8]. Based on these observations, lp-norm constraints with p < 2 are good choices to obtain sparse solu-tions. Here we consider one of many specific cases. Image formationis performed solving the following optimization problem.f̂ = argminf‖g − Cf‖22 + λ ‖f‖pp(9)The first terms enforces data fidelity, whereas the second term enfo-ces sparsity of the field. The scalar parameter λ is known as the reg-ularization parameter which determines the relative weight of thesetwo terms in the solution.4. PROPOSED METHODWe propose a sparsity-driven technique for joint SAR imaging andphase error correction. This technique is capable of handling 1D aswell as 2D phase errors. We consider two cases of 2D phase errorsas separable and non-separable. What we mean by ‘separable’ isthat the 2D phase error is composed of a range varying and cross-range varying 1D phase error functions. In non-separable case the2D phase error cannot be separated into two 1D error functions. Thealgorithm is an iterative algorithm, which cycles through steps ofimage formation and phase error estimation and compensation. Itis based on the minimization of the following cost function, withrespect to φ and f using coordinate descent technique.J(f, φ) = ‖g − C(φ)f‖22 + λ ‖f‖1 (10)In the first step of every iteration the cost function J(f, φ) is mini-mized with respect to f . This is the image formation step and samefor all type of phase errors.f̂ (n+1) = argminfJ(f, φ(n)) = (11)argminf∥∥∥g − C(φ(n))f∥∥∥22+ λ ‖f‖1where n denotes the iteration number. Note that C(φ(n)) denotesthe model matrix corresponding to the phase error obtained in then-th iteration. Second step is the phase error estimation step wherea different procedure is implemented for different types of phase er-rors.4.1. 1D Phase ErrorsIf the phase error is a 1D cross-range varying phase error, given thefield estimate, the following cost function is minimized for everyaperture position [9]∆̂φ(n+1)m = arg min∆φmJ(fˆ (n+1),∆φm) = (12)arg min∆φm∥∥∥gm − exp (j∆φm)Cm(φˆ(n)m )fˆ (n+1)∥∥∥22for m = 1, 2, ....,Mwhere ∆̂φ(n+1)m denotes the incremental phase error estimate for them-th aperture position in the iteration (n + 1). In (12), gm andCm(φm) stand for the part belonging to the m-th aperture positionof the SAR data and the model matrix, respectively. The optimiza-tion problem in (12) is solved in closed form for every aperture po-sition. The solution of the problem in (12) results in the followingexpression∆̂φ(n+1)m = − arctan(−IR) (13)whereR = Re{fˆHCm(φˆ(n)m )Hgm} I = Im{fˆHCm(φˆ(n)m )Hgm} (14)Using the incremental phase error estimate, the model matrix is up-dated as in (15) and the algorithm turns back to the image formationstep of the next iteration.Cm(φˆ(n+1)m ) = exp (j∆̂φ(n+1)m )Cm(φˆ(n)m ) (15)4.2. 2D Separable Phase ErrorsIf the phase error is a 2D separable phase error then in the phaseerror estimation step of every iteration, first, the phase error estimatefor cross-range is found as in (12). After updating the model matrix,the incremental phase error estimate for the range direction is foundrepeating the same procedure as in cross-range direction, this timefor every range position. Then the model matrix is updated using therange dependent incremental phase error and the algorithm passes tothe next iteration.4.3. 2D Non-separable Phase ErrorsIn a more general case in which we consider 2D non-separable phaseerrors, phase error estimation is done for every sample of the phasehistory data, since all sample points are perturbed with different andpotentially independent phase errors. Therefore, using the samepoint of view as in the previous two cases, in the phase error esti-mation step, the following cost function is minimized.∆̂φ(n+1)s = argmin∆φsJ(fˆ (n+1),∆φs) = (16)argmin∆φs∥∥∥gs − exp (j∆φs)Cs(φˆ(n)s )fˆ (n+1)∥∥∥22for s = 1, 2, ...., Swhere ∆̂φ(n+1)s denotes the incremental phase error estimate for thes-th element of the phase history data. Here gs and Cs(φs) standfor the s-th element of the SAR data and the corresponding row ofthe model matrix, respectively. This step is solved in closed formsimilar to that in (13). The solution of the optimization problem in(16) is as follows.∆̂φ(n+1)s = − arctan(−IR) (17)whereR = Re{fˆHCs(φˆ(n)s )Hgs} I = Im{fˆHCs(φˆ(n)s )Hgs} (18)If we let the number of cross-range positions be M and the numberof range positions be K, then we can say that in a 1D cross-rangephase error case, we solve the problem for M unknowns, in a 2Dseparable phase error case for M +K unknowns, and in a 2D non-separable phase error case for M ×K unknowns. Hence, correctingfor 2D non-separable phase errors is a much more difficult problemthan the others.5. EXPERIMENTAL RESULTSWe present results on two public SAR data sets provided by the U.S.Air Force Research Laboratory (AFRL): the ‘Slicy’ data, part of theMSTAR dataset [11]; and the ‘Backhoe’ data [12]. On Slicy datawe have applied a 2D non-separable phase error which is uniformlydistributed in [−pi,+pi). In Figure 1(a) and (b) the conventional im-age and the image formed by sparsity-driven image reconstructionfor the error-free case are presented, respectively. Figure 1(c) showsthe conventional image reconstructed from data with phase errors,whereas Figure 1(d) shows the result of the proposed method. Asseen from the figures, the proposed method provides the advantagesof the sparsity-driven imaging while removing the defocus effectof the phase error effectively. Another dataset on which we presentresults is the Backhoe dataset. To deal with the wide-angle observa-tions in the Backhoe dataset, we incorporate the subaperture-basedcomposite imaging approach of [13] into our framework. The Back-hoe dataset is perturbed with a 2D separable phase error whichis composed of two 1D phase error functions. One of these errorfunctions is only range dependent where the other is only cross-range dependent. These 1D errors are uniformly distributed in[−3pi/4,+3pi/4]. Figure 2(a) and (b) show the conventional imageand the image reconstructed by sparsity-driven imaging, respec-tively, when there is no phase error. The conventional image and theresult of the proposed method in the case of phase errors are shownin Figure 2(c) and (d), respectively. These results demonstrate theeffectiveness of the proposed method.6. CONCLUSIONIn this study, we have proposed a sparsity-driven technique for jointSAR imaging and phase error correction. The method can handle1D phase errors as well as 2D separable and non-separable phase er-rors. The method corrects the phase errors during the image forma-tion process while it produces high resolution focused SAR images,thanks to its sparsity enforcing nature. Since the current formulationof the proposed method does not include any prior information ofthe phase error, especially for large phase errors it is possible to endup with focused but shifted images due to inherent ambiguities. Todeal with this issue, the method can be extended incorporating someprior knowledge about the phase error or some region of support in-formation about the scene.(a) (b)(c) (d)Fig. 1. a) Conventionally reconstructed image from data withoutphase errors b) Image reconstructed by sparsity-driven imaging fromdata without phase errors c) Conventionally reconstructed imagefrom data corrupted by phase errors d) Image reconstructed by theproposed method from data corrupted by phase errors.7. REFERENCES[1] C.V. Jakowatz, D.E. Wahl, P.H. Eichel, D.C. Ghiglia, and P.A.Thompson, Spotlight-Mode Synthetic Aperture Radar: A Sig-nal Processing Approach, Springer, 1996.[2] D.W. Warner, D. Ghiglia, A. FitzGerrell, and J. Beaver, “Two-dimensional phase gradient autofocus,” Proc. SPIE, vol. 4123,2000.[3] D.E. Wahl, P.H. Eichel, D.C. Ghiglia, and C.V. Jakowatz,“Phase Gradient Autofocus - A robust tool for high resolutionSAR phase correction,” IEEE Trans. Aerosp. Electron.Syst.,vol. 30, no. 3, 1994.[4] C.E. Mancill and J.M. Swiger, “A map drift autofocus tech-nique for correcting higher-order sar phase errors,” Proc. 27thAnnual Tri-Service Radar Symp., 1981.[5] L. Xi, L. Guosui, and J. Ni, “Autofocusing of ISAR imagesbased on entropy minimization,” IEEE Trans. Aerosp. Elec-tron. Syst., vol. 35, no. 10, 1999.[6] F. Berizzi and G. Corsini, “Autofocusing of inverse syn-thetic aperture radar images using contrast optimization,” IEEETrans. Aerosp. Electron. Syst., vol. 32, no. 7, 1996.[7] J.R. Fienup and J.J. Miller, “Aberration correction by maxi-mizing generalized sharpness metrics,” J. Opt. Soc. Amer. A,vol. 20, no. 4, 2003.[8] M. C¸etin and W.C. Karl, “Feature-enhanced synthetic apertureradar image formation based on nonquadratic regularization,”IEEE Trans. Image Processing, pp. 623–631, 2001.(a) (b)(c)  (d)Fig. 2. a) Conventionally reconstructed image from data withoutphase errors b) Image reconstructed by sparsity-driven imaging fromdata without phase errors c) Conventionally reconstructed imagefrom data corrupted by phase errors d) Image reconstructed by theproposed method from data corrupted by phase errors.[9] N. ¨O. ¨Onhon and M. C¸etin, “A non-quadratic regularizationbased technique for joint SAR imaging and model error cor-rection,” Proc. SPIE, vol. 7337, 2009.[10] W.G. Carrara, R.M. Majewski, and R.S. Goodman, Spot-light Synthetic Aperture Radar: Signal Processing Algorithms,Artech House, 1995.[11] MSTAR, Air Force Research Laboratory, Model BasedVision Laboratory, Sensor Data Management Sys-tem,http://www.mbvlab.wpafb.af.mil/public/sdms/datasets/mstar/.[12] Backhoe Data Sample & Visual-D Challenge Problem, AirForce Research Laboratory, Sensor Data Management System,https://www.sdms.afrl.af.mil/main.htm.[13] M. C¸etin and R. L. Moses, “SAR imaging from partial-aperturedata with frequency-band omissions,” Proc. SPIE, vol. 5808,2005.

Joint sparsity-driven inversion and model error correction for radar imaging

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Joint sparsity-driven inversion and model error correction for radar imaging

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