Imaging of moving targets with multi-static SAR using an overcomplete dictionary

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1 Imaging of moving targets with multi-stati SAR using an overomplete ditionary Ivana Stojanovi, Member, IEEE, and William C. Karl, Senior Member, IEEE ECE Department, Boston University 8 St. Mary s St, Boston, MA ivanas@bu.edu and wkarl@bu.edu arxiv:94.82v [s.it] 5 Apr 9 Abstrat This paper presents a method for imaging of moving targets using multi-stati SAR by treating the problem as one of spatial refletivity signal inversion over an overomplete ditionary of target veloities. Sine SAR sensor returns an be related to the spatial frequeny domain projetions of the sattering field, we exploit insights from ompressed sensing theory to show that moving targets an be effetively imaged with transmitters and reeivers randomly dispersed in a multistati geometry within a narrow forward one around the sene of interest. Existing approahes to dealing with moving targets in SAR solve a oupled non-linear problem of target sattering and motion estimation typially through mathed filtering. In ontrast, by using an overomplete ditionary approah we effetively linearize the forward model and solve the moving target problem as a larger, unified regularized inversion problem subjet to sparsity onstraints. Index Terms Multi-stati SAR, imaging, regularization, sparsity I. INTRODUCTION Syntheti aperture radar (SAR) is a remote sensing system apable of produing high-resolution imagery of target senes independent of time of day, distane, and weather. Conventional SAR radars are monostati, with olloated transmit and reeive antenna elements. These SAR sensors oherently proess multiple, sequential observations of a sene under the assumption the sene is stati. When the sene hanges between these observations, as ours when objets move, and these hanges are ignored, blurring, defous, and other artifats are introdued into the reonstruted imagery. This is beause the Doppler shift of moving objets are then determined not only by their geometri loation but also by their veloity. Imaging of senes with moving targets has gained inreasing interest as the desire for persistent and urban sensing has grown. Moving target loalization has proven hallenging in the ase of single antenna onventional narrow-angle SAR utilizing onventional reonstrution methods, suh as the polar format and the filtered-bak-projetion algorithms [], [2], due to an inherent ambiguity in target geoloation and veloity. Consequentially, most tehniques for imaging moving targets with onventional SAR aim at fousing and deteting smeared targets in SAR imagery [], [2], [3], [4], [5]. In reent years, however, a number of tehniques to handle moving objets expliitly have been developed. Spae-time adaptive proessing (STAP) [6] exploits multiple-phase enter antennas to suppress lutter and produe a moving target indiation image. Veloity syntheti aperture radar (VSAR) [7] exploits the veloity information ontained in phases of a sequene of images formed at multiple reeive antenna elements. Dualspeed SAR [8] has the radar platform move sequentially at two different veloities during the radar data olletion time. Distributed antenna radars also have the potential to break the veloity-loation ambiguity due to multiple phase enters of the antenna, while at the same time substituting spatial diversity for onventional bandwidth resoures. Reent work on multi-stati [9] and the related MIMO [], [], [2] radar with oherent proessing has shown the potential for resolution improvement that an far exeed the limit suggested by onventional arguments based on the radar s waveform. The onventional approah to resolving the moving target loalization problem is to perform mathed filter reonstrution at every pixel for every possible veloity hypothesis independently, yielding a large spae-veloity ube [9], [4]. A target is then plaed at the loations of maximal energy fous in the spae-veloity ube. An approah based on the inversion of the forward operator is presented in [3], where a filltered-bakprojetion approah to imaging of moving targets takes the form of a weighted mathed filter. These approahes require the solution of many large inversions and result in a large, somewhat ambiguous output. In this paper we utilize an inverse problem formulation and insights from sparse signal representation and ompressed sensing for effetive imaging of dynami environments using distributed antenna SAR sensor geometries. The nonlinear problem of the oupled target loalization and veloity estimation is linearized by onstrution of an overomplete ditionary of veloity states. The resulting inversion leads to a non-onvex optimization problem whih we effiiently solve through onvex relaxation. In ontrast to the filteredbakprojetion and the mathed filtering approahes of [3] and [9], in our approah, veloity estimation is performed during the image formation proess and all veloity hypothesis are evaluated jointly in a single optimization framework. II. OBSERVATION MODEL We onsider a multi-stati system onsisting of widely separated transmit and reeive elements within a forward one positioned at the enter of a sene of interest. We assume that different transmitters send out probes in TDMA fashion, while the reeive unit oherently proesses signals of all

2 2 reeivers aross all snapshots, i.e. pulses. The sene of interest is modeled by a set of point satterers refleting impinging eletromagneti waves isotropially to all reeivers within the forward one, thus, allowing for the oherent proessing of all reeived signals. The refletion oeffiient of the point satterer is a omplex number with an unknown amplitude and a random phase [4]. We introdue a oordinate system with the origin in the enter of the area of interest and, for simpliity, model the sene as two dimensional. Fig. illustrates this set up. The relative size of the sene is assumed to be small ompared to distanes from the origin of the oordinate system to all transmitter and reeivers, suh that transmit and reeive angles would hange negligibly if the oordinate origin moved to any point in the sene. Furthermore, we neglet signal propagation attenuation. v x x y e k e l x k th transmitter l th reeiver similarly, the signal delay to the loation x in the diretion of the l-th reeiver is τ l (x) = xt e l. Thus, the propagation delay is determined by the projetion of a satterer s loation onto the kl-th transmit-reeiver pair s bi-stati range vetor. e kl = ek + e l. For extended senes, multiple satterers will have the same projetion onto the bistati range diretion. The olletion of suh satterers satisfies {x 2 xt e kl = ρ}. These satterers are simultaneously illuminated and have their olletive response q kl (ρ) registered at the reeive antenna with the same delay. The so-alled range profile q kl (ρ) is an aggregate response at eah delay or range and is given by: q kl (ρ) = s(x)δ (ρ 2 ) xt e kl dx. x L Under the far field and narrow-band transmit signal assumption [3], the overall reeived signal from the entire ground path is assumed to be a superposition of the returns from all the sattering enters and is given by: L ( r kl (t) = q kl (ρ)γ k t τ kl (x o ) + 2ρ ) dρ. (2) L In terms of the spatial refletivity funtion s(x) the reeived signal is given by: r kl (t) = s(x)γ k (t τ kl (x o ) τ kl (x)) dx, (3) x L where τ kl (x) is given in (). Fig.. Geometry of the kl-th transmit-reeive pair with respet to the sene of interest. All transmit and reeive pairs are restrited to lie within a forward one of the angular extent θ. A. Stationary sene model The omplex signal reeived by the l-th reeiver for the exitation from the k-th transmitter refleted from a point satterer at the spatial loation x = [x, y] T is given by r kl (t) = s(x) γ k (t τ kl (x o ) τ kl (x)), where s(x) is the refletivity of the satterer, γ k (t) is the transmitted waveform from the k-th transmitter, x o = [, ] T is the sene s origin and τ kl (x o ) is the sum of the signal propagation delay from the k-th transmitter to the sene s origin, τ k (x o ), and the propagation delay from the sene s origin to the l-th reeiver, τ l (x o ), so that τ kl (x o ). = τ k (x o )+τ l (x o ). Under the far field assumption, the propagation delay from the sene s origin to the satterer at the loation x is given by: τ kl (x). = τ k (x) + τ l (x) = xt (e k + e l ) = xt e kl, () where e k = [osφ k, sin φ k ] T and e l = [osφ l, sinφ l ] T are unit vetors in the diretion of the k-th transmitter and l-th reeiver respetively. The signal delay to the loation x in the diretion of the k-th transmitter is τ k (x) = xt e k and B. Moving sene model When moving satterers are present in the sene we are interested in produing a foused image of the spatial refletivity funtion at some referene time t ref. We assume that the satterer at loation x has an assoiated arbitrary onstant veloity vetor v x = [v x, v y ] T. Note that in the ase of non-onstant motion, the true satter motion an be well approximated as onstant when the time sale of the oherent proessing interval (CPI) is relatively small. We onsider the general ase when the CPI interval ontains multiple transmitted pulses. Let us first onsider the effet of the satterer motion during one pulse transmission following the analysis developed in [3]. In the ase of motion, the delay of the transmitted waveform is dependent on both the loation and the veloity of the satterer. The signal refleted from a single moving satterer will now be of the form r kl (t) = s(x)γ k (t τ kl (x o,x,v x )), where τ kl (x o,x,v x ) is the delay inluding the effets of motion. Next, we derive an expression for t τ kl (x o,x,v x ) in the ase of motion. Let t x,vx (t) denote the time when the transmitted wave reahing the l-th reeiver at time t interated with the satterer, that at the referene time τ k (x o ) was loated at x. Reall that τ k (x o ) is the propagation delay between the k-th transmitter

3 3 and the sene s origin. Thus, the satterer at time t x,vx (t) is atually loated at x + v x t x,vx (t). We an write [ [ x + vx t x,vx (t) ] ] T el t x,vx (t) = t τ l (x o ). (4) Solving the above equation for t x,vx (t) and noting [ that τ k (x; t x,vx (t)) = x + vx t x,vx (t) ] T ek and [ τ l (x; t x,vx (t)) = x + vx t x,vx (t) ] T el, the argument of the delayed pulse is derived to be: t τ kl (x o,x,v x ) = = t [ τ k (x o ) + τ k (x; t x,vx (t)) ] [ τ l (x o ) + τ l (x; t x,vx (t)) ] = τ k (x o ) + x T e k / + + vt xe k / ( t τl vxe T (x o ) + x T e l / ) l / t τ kl (x o ) τ kl (x) + v x T ( ) e kl t τ k (x o ) + xt e l, where we used the fat that v x /, suh that +v T x e k/ vx Te l/ + v x T e kl /. Extending the model to the ase of multiple probe transmissions during the oherent proessing interval (CPI), we assume that the sene is imaged at some arbitrary time t ref outside the given pulse interval. Then, the satterer loated at x at the referene time t ref, will be loated at x + v x (t k t ref ) at time t k that represents the time when the k-th transmitted pulse reahed the sene origin. Updating x of the previous equation with x + (t k t ref )v x, we obtain for the k-th transmitter pulse: t τ kl (x o,x,v x ) t τ kl (x o ) τ kl (x) (5) + v x T e kl (t + t k t ref + ǫ kl (x,v x )), where ǫ kl (x,v x ) = τ k (x o ) + [x + (t k t ref )v x ] T e l /. Finally, the forward observation model in the presene of motion beomes: r kl (t) = s(x)γ k (t τ kl (x o,x,v x ))dx, (6) x L with the argument t τ kl (x o,x,v x ) given in (6) and s(x) representing the spatial refletivity funtion at the referene time. Comparing this equation to the reeived signal model for the stationary sene given by (3), we see that the two models differ by an additional delay attributed to the satterers motion. The additional delay of a satterer present at the loation x at the referene time t ref is proportional to the projetion of the satterers veloity v x to the kl-th transmitreeive pair bistati range vetor e kl and the time interval in between the observation time and the referene time. For narrowband waveforms, defined by γ k (t) = γ k (t)e jω kt, where γ k (t) is the low-pass equivalent signal and ω k the arrier frequeny, (6) is approximated [3] by: r kl (t) e jω k(t τ kl (x o)) s(x) γ k (t τ kl (x o ) τ kl (x)) x L [ e jω k τ kl (x)+[t+t k t ref +ǫ kl (x,v x)] vt x e kl ] dx. (7) The additional phase shift of (7) is a funtion of the quantity vx ω T e kl k, whih is basially the Doppler shift. Thus, the reeived signal is in the familiar form of the superposition of time-delayed and Doppler-shifted replias. Notie that the Doppler shift is unique for eah transmit-reeive pair as it depends on the bistati range vetor e kl. C. Disrete model A disrete version of the model in (6) or (7) an be obtained by disretizing the spatial variable x and sampling in time whih, in the presene of reeiver noise n, beomes: P r = Φ p (v p )s p + n = Φ(V)s + n. (8) p= In this equation, r represents the observed, thus known, set of return signals at all reeivers aross time. Its elements are indexed by the tuple (k, l, t s ), with t s being the sampling times assoiated with the kl-th transmit-reeive pair. The refletivity of the p-th spatial ell or pixel is denoted by s p C and Φ p (v p ) is the vetor apturing the ontribution to the reeived signal of a refletor that was loated in the p-th pixel at the referene time t ref and moved with the onstant veloity v p throughout the oherent proessing interval. Stationary point refletors are inluded in this model by simply setting v p =. The reeived signal model desribed in (8), represents the observation model of unknown satterers refletivity oeffiients s p and their orresponding veloity vetors v p. While the sattering oeffiients s p enter the problem linearly, the unknown veloities v p do not, so the overall problem is nonlinear and oupled. When the veloities are known the remaining equation for s p is linear, however, and straight forward fousing and estimation of sattering oeffiients is possible. When the veloity is ignored (set to zero) or set to an inorret value, the resulting reonstrution exhibits defousing of the energy of the moving satterer [], [4]. These observations lead to one onventional approah to solving this problem. In partiular, reonstrutions are performed for every possible veloity yielding a large spae-veloity ube [9], [2], [4]. Veloity slies where the image is well foused are assumed to indiate the orret veloity at a pixel. This approah requires the deoupled solution of many inversion problems and results in a large, somewhat ambiguous output. Reently, an approah based solely on the inversion of the forward operator is presented in [3], where a filtered-bakprojetion approah to imaging of moving targets takes the form of a weighted mathed filtering whih still produes a large spaeveloity ube. In this paper we present a different approah based on reent results in sparse signal representation using overomplete ditionaries [5], sparsity based reonstrution, and ompressed sensing [6], [7], whih is desribed next.

4 4 III. OVERCOMPLETE DICTIONARY APPROACH Sparse signal representation aims at apturing a ompliated signal as a linear ombination of a few generating elements [5]. In partiular, a signal in a given lass of dimension M should be representable by a small subset of a olletion of Q generating elements. For the ase when M = Q and the elements are independent, the olletion is termed a basis. When Q > M the olletion is termed an overomplete basis or ditionary. To represent this problem mathematially, let r R M represent the signal, Φ R M Q the ditionary and s R Q represent the linear oeffiients, suh that r = Φs. Sine Φ has a non-zero null spae many solutions are possible. What is sought is a sparse solution with only a few non-zero elements. While optimal design of ditionaries Φ is a topi of general interest, in this work we assume the ditionary is given and fixed based on prior knowledge of the expeted veloities in a sene. The problem is then one of finding an optimally sparse solution. A diret formulation of this problem an be given as: min s s s.t. r = Φs, where denotes the l norm, whih ounts the number of non-zero elements of the argument. Unfortunately, this formulation is omputationally diffiult to solve, as it involves NP-hard enumerative searh whih is prohibitively expensive for even moderate sizes of Q. A number of alternative, indiret tehniques have been developed, based either on relaxation tehniques or iterative greedy algorithms. The onvex relaxation approah relies on the fat that besides the l norm, the l norm also promotes ( sparsity in Q ) a solution. The l norm is defined as s = i= (s) i, where (s) i is the i-th element of s. This norm is a onvex funtion of its arguments. The relaxed version of the problem then takes the form: min s s s.t. r = Φs, whih is essentially a linear program (LP). The use of this formulation has also been motivated by the fat that under ertain onditions on the overomplete basis Φ, the original problem and the relaxed version an be shown to have the same solution [8]. When the signal r is noisy, the signal representation problem beomes a signal approximation problem. The onvex relaxation formulation of the noisy signal approximation problem is given by: min s s s.t r Φs 2 2 δ where δ represents a small noise allowane. Instead of satisfying the relationship exatly, the solution oeffiient vetor s is allowed to satisfy the relationship approximately. This problem is known in the literature as noisy basis pursuit [9]. We make use of this formulation in what follows. Compressed sensing (CS) [6], [7], takes the sparse representation framework one step further by seeking to aquire as few measurements as possible about a sparse signal, and given these measurements, reonstrut the sparse signal either exatly or with provably small probability of error using formulations like the above. Most of the work in CS assumes that the projetions are drawn at random. However it is also known that Fourier measurements represent good projetions for ompressed sensing of sparse point like signals [7]. This result immediately onnets to radar measurements, as SAR sensors an be viewed as measuring samples of the stationary sattering field in the spatial frequeny domain. A. The new formulation We exploit the overomplete ditionary approah to signal representation to reate a new formulation of the dynami SAR inversion problem. In partiular, we first introdue an appropriate overomplete ditionary of veloity hypotheses by onstruting a grid of all possible satterer veloities at eah loation. This defines an over-omplete representation of the reeived signal r. This representation is ombined with the SAR observation equation resulting in an inverse problem that is linear in terms of an extended refletivity oeffiient vetor, but is now under-determined. We then apply a modified version of the sparsity seeking formulation of () to solve the resulting large, under-determined linear problem. Details are given next. Step : Ditionary Definition. We hypothesize that a satterer veloity v p, p belongs to one of a disrete set of veloities Ṽ: v p Ṽ = {ṽ =,ṽ 2,ṽ 3,...,ṽ N }, where N denotes the size of the veloity grid. The first veloity vetor is set to zero to allow for stationary targets. The original observation vetor Φ p (v p ) desribing the ontribution of the p-th pixel to the reeived signal r in terms of an unknown pixel veloity v p now beomes the matrix of [Φ p (ṽ ),Φ p (ṽ 2 ),...Φ p (ṽ N )] omposed of the ontribution of eah possible known veloity hypothesis at pixel p to the reeived signal. There are no unknowns in this matrix, in ontrast to the original observation vetor. Finally, by ombining this model at eah pixel we obtain an overall overomplete forward operator: ΦṼ = [Φ (ṽ ),...,Φ (ṽ N ),...,Φ P (ṽ ),...,Φ P (ṽ N )]. To math this hange, the refletivity oeffiient at pixel p is aordingly expanded to beome a refletivity oeffiient vetor: b p s p s b b p2 p = s p., b pn where the auxiliary variables b pn are onstrained to the binary set b pn {, } to represent a true or false hypothesis that the p-th spatial loation moves with n-th quantized veloity vetor ṽ n. Additionally, the variables b pn need to satisfy n b pn = for eah p to reover the model of (8). This additional onstraint speifies that at most one spatial refletor

5 5 is present within the p-th resolution ell at the referene time. Other, more ompliated, models are also possible. Staking up these single-pixel refletivity oeffiients yields an overall, extended refletivity vetor for the entire image: s b s b s b = 2.. s b P Combining this extended refletivity vetor with the overomplete forward operator yields our new overall linear motion- SAR forward model: r = ΦṼs b + n + q, (9) where s b is the extended refletivity oeffiient and q represents an additional noise term due to the veloity spae quantization. In this new model ΦṼ is ompletely speified and ontains no unknowns. We have essentially onverted the original, diffiult, non-linear problem to a seletion problem with a linear observation. Step 2: Image Formation. We now treat the problem of image formation as a problem of inverting (9). In priniple, we seek to find a sparse vetor s b that best desribes the reeived signal. In partiular, we seek a solution of the following optimization problem: min s b,b,s subjet to s b r ΦṼs b 2 2 δ (s b ) (p )N+n = s p b pn b pn = p =,...,P n b pn {, } n =,...N. where δ represents some noise allowane aimed at apturing both reeiver and quantization noise and the l norm ounts the number of nonzero omponents in its argument. Sine the oherene in typial SAR senes is ontained in the field magnitude (due to random phase), the l norm is applied expliitly to magnitudes of the extended omplex refletivity field s b. B. Solving the formulation The above optimization problem is a non-onvex, mixedinteger program, known to be NP hard to solve. Thus, we resort to a two step proedure based on onvex relaxation for its approximate solution. Step :We first solve the following relaxed optimization problem: ŝ b = argmin s b s b subjet to r ΦṼs b 2 2 ǫ, where s b = PN i= (R(sb ) i ) 2 + (I(s b ) i ) 2 and (s b ) i is the i-th element of the vetor s b. Note in this step we use the onvex l penalty and do not enfore the exlusivity of veloity hypotheses represented by the auxiliary binary variables b pn. Step 2:We then find the final refletivity and veloity vetor estimates by seleting the maximum magnitude response at the p-th pixel: (ŝ p, n) = argmax n (ŝ b) (p )N+n, v p = ṽ n. () This is onsistent with the assumption that only one moving refletor is present in the p-th pixel at the referene time. By onstruting the overomplete ditionary and applying onvex relaxation we have linearized the forward model at the expense of an inrease in the size of the problem. The minimization problem in Step is a seond-order one program, whih we solve by a speialized large sale interiorpoint method for omplex variables proposed first in []. The method is a speialized entral path interior-point method with an approximate searh diretion found through a preonditioned onjugate gradient method whih yields effiient solution of suh large problems. Our approah relies on two levels of sparsity. First, the introdution and use of an overomplete ditionary mandates that we seek a sparse solution vetor s b. In addition, ompressed sensing theory relates the number of measurements neessary for aurate reovery of s b to its underlying sparsity [6], [7], indiating that it may be possible to perform aurate reovery of sparse senes with relatively few transmitters and reeivers. We also want to emphasize that as long as the sene ontains a sparse set of refletors, there is no onstraint on the objet veloity the method an handle. The veloity grid resolution should simply be mathed to the oherent proessing interval and the maximal arrier frequeny of the transmitted signal in order to avoid phase wrapping of the exponent in (7). The phase error due to the veloity quantization is linear in v x, so the veloity grid need not be onstant: at smaller veloity magnitudes it an be oarser and at higher veloity magnitudes it an be finer. Finally, the disrete model of (8) implies that only a single satterer is present at the referene time in eah spatial pixel. For the ase when the spatial grid is suffiiently oarse to inlude multiple point refletors within the resolution ell, (8) an be written as: r = M(p) m= p= P Φ p (v p,m )s p,m + n, () where v p,m models the veloity of the m-th refletor in the p-th resolution ell, that ontains M(p) satterers out of whih one is allowed to be stationary. The algorithm aommodates the new model with a small hange. In ase of M(p) refletors per spatial grid ell, the onstraint n b pn =, should be replaed with n b pn = M(p). After onvex relaxation, () should be aordingly modified to retain M(p) maximal magnitude refletivity values and their orresponding veloity vetors. IV. TRANSMITTED WAVEFORMS We presented the overomplete ditionary reonstrution algorithm without speifying the transmitted signals. The

6 6 formulation we presented is quite general and an potentially work for many waveforms and sensor onfigurations. In this initial work, to demonstrate the algorithm we onsider multiple snapshot, non-overlaping transmissions of hirp and ultra-narrow band signals. Suh signals are known to lead to a simple Fourier relationship between the stationary refletivity field and the measurements for both monostati and multiple distributed antenna onfigurations [4], [9], [2]. The orresponding forward operators posses ompressed sensing properties, allowing for good reonstrution of sparse fields with few measurements [22], [7]. In the following, we disuss the speifi forward radar model resulting from transmission of suh waveforms. The hirp signal is the most ommon spotlight SAR pulse [4], given by { e jα kt 2 e jωkt, τ γ k (t) = 2 t τ 2 otherwise, where ω k is the enter frequeny and α k is the so-alled hirp rate of the k-th transmit element. The frequenies enoded by the hirp signal extend from ω k α k τ to ω k +α k τ, suh that the bandwidth of this signal is given by B k = α kτ π. Ultranarrow band waveforms are speial ases of the hirp signal obtained by setting α k =. We use this transmitted hirp signal in (6) and apply typial demodulation and baseband proessing. In partiular, the reeived signal is mixed with the transmitted signal referened to the origin of the sene e j[ω k(t τ kl (x o)+α k (t τ kl (x o)) 2 ], and then low-pass filtered. If the quadrati phase error is ignored [4], we obtain the following signal as the input to our algorithm: r kl (t) s(x) e jω kl(t)x T e kl x <L e jω kl(t)[t+t k t ref +ǫ kl (x,v x)]v T x e kl dx, (2) where Ω kl (t) = [ω k 2α k (t τ kl (x o ))], depends on the frequeny ontent of the transmitted waveform. The first exponential term depends on the stationary satters only. The seond exponential term depends on the moving satters only. For stationary senes with v x =, x, (2) represents the 2D Fourier transform of the spatial refletivity funtion, evaluated at the disrete set of the spatial frequeny vetors k. = [k x, k y ] T given by k k,l,t = Ω kl (t)e kl = Ω kl (t)(e k + e l ). (3) This equation an be used to desribe the spatial frequeny sampling of both the monostati and the multi-stati onfiguration [9]. Reall that e k and e l are the unit vetors in the diretion of the k-th transmitter and the l-th reeiver. For the monostati ase these two vetors oinide, i.e. e k = e l. For the monostati ase and fixed t, (3) represents an ar of the irle of the radius Ω kl (t) entered the origin of the spatial frequeny domain, with the length of the ar determined by the radial span of the vetor e k. Changing t, expands or shrinks the irular ars, leading to the familiar key-hole sampling [4] of the onventional monostati ase. For the multi-stati ase, the situation is a bit more ompliated. For simpliity, assume that Ω kl (t) = Ω(t). At fixed t and fixed e k, (3) also desribes an ar of a irle, this time passing through the origin and entered at Ω(t) along the diretion e k. The length of the ar and its orientation are determined by the radial span of the reeiver vetor e l. Changing t results in expansion or shrinkage of the irle passing through the origin with its enter sliding along the diretion e k. Thus, different overing patters of k-spae are possible with various sampling strategies in spae and time, while the resolution is primarily determined by the extent of the k-spae overing. The range and ross-range resolution of both the onventional monostati SAR system and the multistati, distributed antenna SAR with antenna elements onfined within the forward one of θ < π/2 are lower bounded by the bounding box of the annulus: ρ x 2B eq, ρ y 4(f o + B/2)sin( θ/2), (4) where B eq = (f +B/2) (f B/2)os( θ/2). We illustrate the k-spae sampling patterns used in our experiments in Fig. 2 and Fig. 3. Fig. 2 and Fig. 2 show the k-spae sampling of the onventional monostati SAR with bandwidth of B = 5MHz, entered at f =.5GHz, over a narrow syntheti aperture of θ = 5 deg and a wide syntheti aperture of θ = 45 deg, respetively. The k-spae overing of the narrow-angle monostati SAR is well approximated with the irumsribed retangle, while this is not the ase for the wideangle monostati SAR. A k-spae overing similar to the wide-angle mono-stati ase in Fig. 2 an be ahieved with a multi-stati, distributed transmit and reeive antenna using ontinuous wave transmission. Fig. 3 illustrates the k-spae overing for two multi-stati distributed antenna onfigurations utilizing ontinuous wave transmission with transmitters and reeivers positioned within the forward one of θ = 45 deg. Fig. 3 shows a multistati onfiguration where transmitters sequentially transmit a single tone of frequeny f =.5GHz. Fig. 3 shows a multi-stati onfiguration where different transmitters send out different ontinuous wave signals within a bandwidth of 5MHz around the enter frequeny f =.5GHz. This sampling an be ahieved either via sequential multi-stati transmission or via simultaneous MIMO transmission, as suh signals are easily separated at eah reeiver. Although the expeted resolutions in the last three ases are similar, it is important to notie that the k-spae overing of Fig. 3 for MIMO transmission is ahieved in the least amount of time, making it the most favorable onfiguration for imaging of moving objets. V. NUMERICAL EXPERIMENTS In this setion we outline several numerial experiments demonstrating the reonstrution apability of the overomplete ditionary approah for imaging senes that ontain both stationary and moving objets. First, we show results for a multi-stati, distributed antenna sensing onfiguration for ases orresponding to a stationary sene, a sene with moving

7 7 3 3 k y k y k x k x Fig. 2. Mono-stati SAR k-spae sampling for B = 5MHz, f =.5GHz with the forward one entered at deg Conventional, narrow-angle SAR with θ = 5deg, (ρ x 2.9m, ρ y.3m) and Wide-angle SAR with θ = 45deg, (ρ x.9m, ρ y.3m). 3 3 k y k y k x k x Fig. 3. Multi-stati, distributed antenna SAR k-spae sampling with reeive and transmit antenna elements positioned within the forward one of θ = 45 deg entered at deg. The ase where eah transmitter uses ontinuous waves of frequeny f =.5GHz, (ρ x.32m, ρ y.3m) and The ase where eah transmitter sends out ontinuous waves of different frequenies within the bandwidth of B = 5MHz entered at f =.5GHz, (ρ x.9m, ρ y.3m). objets whose motion is ignored, a sene with moving objets whose motion is expliitly handled by the overomplete ditionary reonstrution algorithm. Next, we show reonstrution results for the same ases, but for a wide-angle monostati onfiguration with the same lower bounds on the range and the ross range resolution. Finally, we ompare our approah with the mathed filtering approah desribed in [9] and [3]. For the multi-stati, distributed antenna onfiguration, all transmit and reeive elements of the syntheti aperture are positioned in the forward one of θ = 45 deg with its enter diretion aligned with the x-axis of the oordinate system, as illustrated in Fig.. We hoose a relatively narrow forward one in order to better aommodate the isotropi sattering assumption of realisti satterers. Eah antenna element transmits a distint ontinuous wave signal with a frequeny randomly hosen within the bandwidth of B = 5MHz around the enter frequeny of f =.5GHz. Similarly, for the monostati ase, the angular extent of the syntheti aperture is also θ = 45 deg with the enter aspet aligned with the x- axis. The transmit waveform is hosen to be the onventional hirp signal of B = 5MHz entered at f =.5GHz. The resulting k-spae sampling patterns are shown in Fig. 2 and Fig. 3 for the monostati and the multi-stati distributed antenna onfigurations, respetively. The sene of interest is of 32 32m in size, represented by pixels in the x and y diretion respetively, suh that the spatial ell size is ( x, y) = (,.25)m. The sene ontains two rigid objets in motion and one rigid stationary objet, as illustrated in Fig. 4, whih displays the ground truth images. An objet is defined as a set of lustered pixels refleting eletromagneti energy isotropially within the one of 45 deg. There are ative satterers in eah objet. The magnitudes of veloities of the moving objets are 32.5m/s and 4.7m/s in the diretion of π/6 and π/6 + π/2, respetively.

8 Fig. 4. The ground truth of the sene ontaining one stationary objet (upper left orner) and two rigid objets moving at 32.5m/s (lower left orner) and 4.7m/s (upper right orner): The refletivity magnitude and The veloity vetors assoiated with different pixels. The refletivity magnitude of the point satterers in the sene is shown in Fig. 4 and their orresponding veloity vetors are shown in Fig. 4. All measurements are orrupted by independent Gaussian reeiver noise suh that SN R = db, with SNR defined as SNR = log Φ(V)x 2 y Φ(V)x 2. First, we show a set of reonstrution results for the multistati, distributed antenna onfiguration with N rx = 4 reeive elements and N tx = transmit elements, eah sequentially transmitting a distint tone signal with a pulse repetition interval of P RI = 2ms. We start by onsidering what happens when there is no motion and when motion is ignored. In Fig. 5 the sene of interest is made ompletely stationary and our reonstrution is performed with the veloity ditionary Ṽ ontaining only the veloity vetor v =, orresponding to a stati sene. In this ase, we see that all objets are well foused in the reonstrution as would be expeted. These sparse multi-stati stationary sene results essentially extend those in presented in [23] for the ase of the onventional narrow-angle monostati SAR. In Fig. 5 the sene is now made dynami, as desribed by Fig. 4, but satterers veloities are ignored in the reonstrution, i.e. the ditionary Ṽ again ontains only the vetor v =. We see that when the motion of the objets is ignored, the stationary objet still ahieves reasonable fous, while the moving objets appear severely blurred. Next we demonstrate what happens when we use our overomplete ditionary approah to apture the objet veloities in dynami senes. In Fig. 5() we show the refletivity magnitude reonstrution when the veloity ditionary ontains veloities with a magnitude resolution of 3m/s in the range [, ]m/s and a resolution of m/s in the range [3, 4]m/s. The true objet veloities are not part of this ditionary. The whole dynami sene is re-foused with satterer loations orretly identified, as illustrated in Fig. 5(d). All targets appear foused and aurately loalized. In Fig. 6 we show the orresponding estimated target veloities. All reonstruted veloities are orretly estimated within the resolution grid error. The oarseness of the veloity grid is hosen to avoid phase wrapping and further, the phase deviation is minimized for shorter CPIs whih, in return, are easier to support with multi-stati and MIMO onfigurations. We now repeat these experiments for the wide-angle monostati onfiguration. Results are presented in Fig. 7 and Fig. 8, showing reonstruted magnitudes and veloities, respetively. Reall that eah transmitted pulse is now a hirp signal with B = 5MHz at f =.5GHz. The number of transmitted probes within the angular extent of θ = 45 deg is 4, with eah pulse return sampled at frequenies. The pulse repetition interval is kept as before for onsisteny (we do not worry about the platform veloity required to transverse the angular extent within the oherent proessing interval). The results presented for reonstrution of the dynami sene are thus optimisti. We observe that these reonstrutions are inferior to those obtained in the multi-stati senario, as some features of objets are blurred. The additional diversity provided by the multi-stati onfiguration apparently translates into improved robustness and quality of the reonstrution. Finally, we show reonstrution results for the weighted mathed filtering approah desribed in [9] and [3] for the multi-stati, distributed antenna onfiguration. The refletivity magnitude is evaluated at eah pixel for every hypothesized veloity, leading to a large spae-veloity ube. Here we show the maximal value of the refletivity magnitude of eah pixel aross the veloity diretion in this spae-veloity ube. Fig. 9 shows the results obtained from mathed filtering when N rx = 4 reeivers and N tx = transmitters are used with ultra-narrow band waveforms, as before. This onfiguration results in 4 observations. Fig. 9a) is the slie of the mathed filter result at zero veloity, showing what happens when veloity is ignored. Fig. 9b) is the mathed filter reonstrution of the sene ignoring motion. Fig. 9) is the result obtained by taking the maximum response aross the spae-veloity ube and Fig. 9d) shows satter loalization, by thresholding the result of Fig. 9) at.2. For the same

9 () (d). Fig. 5. Refletivity magnitude reonstrution with our new, overomplete ditionary approah for the multi-stati, distributed antenna onfiguration at SNR = db with 4 measurements, (N tx, N rx) = (, 4): The reonstrution of the stationary sene assuming no motion. The reonstrution of the dynami sene when veloities are ignored, (Ṽ = {v = }). () The reonstrution of the dynami sene with the full overomplete veloity ditionary. (d) The orresponding loations of refletors in the reonstrution of () whose magnitudes are greater than.2. amount of data, the mathed-filter-based reonstrutions are muh worse than those provided by our new overomplete ditionary approah. To obtain mathed-filter-based reonstrutions that have similar quality to those produed by the overomplete ditionary approah, we an inrease the amount of data by using N rx = 4 reeivers and N tx = transmitters, but with the transmitters now emitting hirp waveforms of B = 5MHz at f =.5GHz with N f = 3 samples per waveform. Fig. shows the math-filter-based results for this onfiguration, whih orresponds to 2, observations. Fig. a) is the mathed filter reonstrution of the stati sene. Fig. b) is the mathed filter reonstrution of the sene ignoring motion. Fig. ) is the maximum magnitude refletively response of the mathed filter reonstrution and Fig. d) is the thresholded version of this result, showing satter loalization. We see that the mathed filter reonstrution an ome lose to reovering fine objet features, but at the expense of signifiant inrease in measurement data relative to the overomplete ditionary approah. In Table I we summarize these results by providing the per-pixel refletivity magnitude error for the different ases, where this error is defined as E = x x 2 2/P, with x the ground truth refletivity magnitude, x its estimate and P is the number of pixels in the sene. The over-omplete ditionary results are for 4 observations and the mathedfilter results are for the 4 observation ase and the expanded 2, observation ase. The estimates generated by our new overomplete ditionary method are the most aurate, produing over a several ten-fold and a 55% redution in the error generated by the mathed filter solution whih uses the same amount of data and 3 times as muh data, respetively. Further, the spatial diversity of the multi-stati, distributed antenna onfiguration provides good reonstrutions even in the ase of narrowband transmission waveforms.

10 () (d) Fig. 6. Veloity estimates with our new, overomplete ditionary approah for the multi-stati, distributed antenna onfiguration at SNR = db. The upper right part of the sene with objet moving at 4.7m/s: The true veloity field. The orresponding estimate. The lower left part of the sene with objet moving at 32.5m/s: () The true veloity field. (d) The orresponding estimate. multi-stati, OCD, mono-stati, OCD, multi-stati, FBP/MF, multi-stati, M = 4 M = 4 M = 4 M = 2, stationary dynami, motion ignored dynami FBP/MF, TABLE I PER PIXEL REFLECTIVITY MAGNITUDE ERROR OF DIFFERENT RECONSTRUCTION SCENARIOS WITH A DIFFERENT NUMBER OF MEASUREMENTS M. VI. CONCLUSION The radar imaging of senes that ontain motion has long been an interesting and hallenging researh topi. We have onsidered multi-stati, distributed antenna onfigurations for high-resolution loalization of senes ontaining both moving and stationary satterers. We have presented an overomplete ditionary inversion approah to simultaneous imaging of stationary and moving satterers. The non-linear, oupled problem of joint veloity and refletivity estimation is effetively linearized through introdution of an appropriately defined overomplete veloity ditionary. The resulting optimization problem is then approximately solved through onvex relaxation. Initial experimental results were presented showing the potential of the method for multi-stati onfigurations with narrow band transmissions. In ontrast to the existing mathedfiltering approahes that are onerned with the inversion of the forward operator only, our overomplete ditionary formulation expliitly and jointly enodes the sparse point sattering assumption of both the spatial and the veloity dimension, leading to foused imagery with more foused objet detail. Initial results suggest the new method exhibits improved robustness to data loss over existing approahes.

11 () (d). Fig. 7. Refletivity magnitude reonstrution with our approah for the wide-angle mono-stati onfiguration with SNR = db and 4 measurements: The reonstrution of the stationary sene assuming no motion. The reonstrution of the dynami sene when veloities are ignored, (Ṽ = {v = }). () The reonstrution of the dynami sene with the full overomplete veloity ditionary. (d) The orresponding loations of refletors in the reonstrution of () whose magnitudes are greater than.2. REFERENCES [] C. V. Jakowatz, D. E. Wahl, and P. H. Eihel, Refous of onstantveloity moving targets in syntheti aperture radar imagery, Algorithms for Syntheti Aperture Radar Imagery V, vol. 337, no., pp , 998. [2] M. J. Minardi, L. A. Gorham, and E. G. Zelnio, Ground moving target detetion and traking based on generalized SAR proessing and hange detetion (invited paper), Algorithms for Syntheti Aperture Radar Imagery XII, vol. 588, no., pp , 5. [3] J. Fienup, Deteting moving targets in SAR imagery by fousing, Aerospae and Eletroni Systems, IEEE Transations on, vol. 37, no. 3, pp , July. [4] J. K. Jao, Theory of syntheti aperture radar imaging of a moving target, Geosiene and Remote Sensing, IEEE Transations on, vol. 39, no. 9, pp , Sep. [5] C. Logan, H. Krim, and A. Willsky, An estimation-theoreti tehnique for motion-ompensated syntheti-aperture array imaging, Image Proessing, 998 International Conferene on, vol., pp. 9 3 vol., Ot 998. [6] R. Klemm, Spae-time adaptive proessing, priniple and appliations. London, UK, Institution of Eletrial Engineers, 998. [7] B. Friedlander and B. Porat, VSAR: a high resolution radar system for detetion of moving targets, Radar, Sonar and Navigation, IEE Proeedings -, vol. 44, no. 4, pp. 5 28, Aug 997. [8] G. Wang, X.-G. Xia, and V. C. Chan, Dual-speed SAR imaging of moving targets, Aerospae and Eletroni Systems, IEEE Transations on, vol. 42, no., pp , Jan. 6. [9] B. Himed, H. Basom, J. Clany, and M. C. Wiks, Tomography of moving targets (TMT), in Sensors, Systems, and Next-Generation Satellites, H. Fujisada, J. B. Lurie, and K. Weber, Eds., vol. 454, no.. Pro. SPIE,, pp [] A. M. Haimovih, R. S. Blum, and L. J. Cimini, MIMO radar with widely separated antennas, Signal Proessing Magazine, IEEE, vol. 25, no., pp. 6 29, 8. [] N. H. Lehmann, A. M. Haimovih, R. S. Blum, and L. Cimini, High resolution apabilities of MIMO radar, Signals, Systems and Computers, 6. ACSSC 6. Fortieth Asilomar Conferene on, pp. 25 3, Nov. 6. [2] J. Li, P. Stoia, and X. Zheng, Signal synthesis and reeiver design for MIMO radar imaging, Signal Proessing, IEEE Transations on, vol. 56, no. 8, pp , Aug. 8. [3] M. Cheney and B. Borden, Imaging moving targets from sattered waves, Inverse Problems, vol. 24, no. 3, Jun. 8. [4] C. V. Jakowatz, D. E. Wahl, P. S. Eihel, D. C. Ghiglian, and P. A. Thompson, Spotlight-mode Syntheti Aperture Radar: a Signal Proessing Approah. Norwell, MA: Kluwer Aademi Publishers, 996. [5] D. L. Donoho, M. Elad, and V. N. Temlyakov, Stable reovery of sparse overomplete representations in the presene of noise, IEEE

12 2 () (d) Fig. 8. Veloity estimates with our new, overomplete ditionary approah for the wide-angle, mono-stati onfiguration at SNR = db. The upper right part of the sene with objet moving at 4.7m/s: The true veloity field. The orresponding estimate. The lower left part of the sene with objet moving at 32.5m/s: () The true veloity field. (d) The orresponding estimate. Transations on Information Theory, vol. 52(), pp. 6 8, January 6. [6] D. L. Donoho, Compressed sensing, IEEE Trans. on Inf. Theory, vol. 52(4), pp , Apr 6. [7] E. J. Candes, J. Romberg, and T. Tao, Robust unertainty priniples:exat signal reonstrution from highly inomplete frequeny information, IEEE Trans. Inform. Theory, vol. 52(2), pp , Feb. 6. [8] D. L. Donoho and M. Elad, Optimally sparse representation in general (nonorthogonal) ditionaries via l minimization, Proeedings of the National Aademy of Siene, vol. (5), pp , Marh 4 3. [9] S. S. Chen, D. L. Donoho, and M. A. Saunders, Atomi deomposition by basis pursuit, SIAM Journal on Sientifi Computing, vol., pp. 33 6, 998. [] S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, An interiorpoint method for large-sale l-regularized least squares, Seleted Topis in Signal Proessing, IEEE Journal of, vol., no. 4, pp , De. 7. [2] M. C. Wiks, B. Himed, J. L. E. Braken, H. Basom, and J. Clany, Ultra narrow band adaptive tomographi radar, Computational Advanes in Multi-Sensor Adaptive Proessing, 5 st IEEE International Workshop on, pp , De. 5. [22] M. Lustig, D. Donoho, and J. M. Pauly, Sparse MRI: The appliation of ompressed sensing for rapid MR imaging, Magneti Resonane in Mediine, vol. 9999, no. 9999, 7. [23] M. Cetin, Feature-Enhaned Syntheti Aperture Radar Imaging. Boston University: Ph.D. Thesis, February.

13 () (d). Fig. 9. Refletivity magnitude reonstrution of the mathed-filtering/filtered bakprojetion approah for the multi-stati, distributed antenna onfiguration at SNR = db with 4 measurements, (N tx, N rx, N f ) = (, 4, ): The reonstrution of the stationary sene assuming no motion. The reonstrution of the dynami sene when veloities are ignored. () The maximum refletivity response in the estimated spae-veloity ube for the dynami sene. (d) The orresponding loations of refletors in the reonstrution of () whose magnitudes are greater than.2.

14 () (d). Fig.. Refletivity magnitude reonstrution of the mathed-filtering/filtered bakprojetion approah for the multi-stati, distributed antenna onfiguration at SNR = db with 2, measurements, (N tx, N rx, N f ) = (, 4, 3): The reonstrution of the stationary sene assuming no motion. The reonstrution of the dynami sene when veloities are ignored. () The maximum refletivity response in the estimated spae-veloity ube for the dynami sene. (d) The orresponding loations of refletors in the reonstrution of () whose magnitudes are greater than.2.