Synthetic Aperture Radar Imaging with Motion Estimation and Autofocus

Transcription

1 Syntheti Aperture Radar Imaging with Motion Estimation and Autofous L. Borea 1, T. Callaghan 1, G. Papaniolaou 3 1 Computational and Applied Mathematis, Rie University, MS 134, 61 Main St. Houston, TX , USA 3 Department of Mathematis, Stanford University, Stanford CA 9435, USA borea@rie.edu, tsallaghan@rie.edu, and papaniolaou@stanford.edu Abstrat. We introdue from first priniples a syntheti aperture radar SAR imaging and target motion estimation method that is ombined with ompensation for radar platform trajetory perturbations. The main steps of the method are a segmentation of the data into properly alibrated small apertures, b motion or platform trajetory perturbation estimation using the Wigner transform and the ambiguity funtion of the data, in a omplementary way, ombination of small aperture estimates and onstrution of high resolution images over wide apertures. The analysis provides quantitative riteria for implementing the aperture segmentation and the parameter estimation proess. X-band persistent surveillane SAR is a speifi appliation that is overed by our analysis. Detailed numerial simulations illustrate the robust appliability of the theory and validate the theoretial resolution analysis. 1. Introdution. In syntheti aperture radar SAR we want to image the refletivity of a given surfae using an antenna system mounted on a platform flying over it, as illustrated in Figure 1. Information about the unknown refletivity is obtained by emitting periodially at rate s probing signals ft and reording the ehoes Ds, t, indexed by the slow time s of the SAR platform displaement and the fast time t of the probing signal. The slow time parametrizes the loation r p s of the platform at the instant it emits the signal, and the fast time t runs between two onseutive illuminations < t < s. An image is formed by superposing over a platform trajetory segment of ar length aperture a, the data Ds, t math-filtered with the time reversed emitted signal ft, and bak propagated with the round trip travel times τs, ρ I from the platform to the imaging points ρ I, Sa I ρ I = ds dt Ds, tf t τs, ρ I = Sa ωo+πb ω o πb dω Sa ds π fω Ds, ωe iωτs, ρ I. 1.1 Sa Here hat denotes Fourier transform, the bar stands for omplex onjugate, B and ω o are the bandwidth and entral frequeny of ft respetively, Sa is the slow time range over the aperture of length a, and τs, ρ I = r p s ρ I / 1.

2 z r p s r p s ρ I y x ρ I Figure 1. Setup for syntheti aperture imaging. is the round trip travel time from the platform to the imaging point ρ I. See Figure 1 for an illustration of the imaging setup. The basi theory of SAR imaging is presented in [9, 13, 8]. The imaging funtion 1.1 an be modified in the ase of large apertures by introduing a weight fator W s that aounts for the geometrial spreading of the wave field over the length of the aperture I W ρ I = ωo+πb ω o πb dω π Sa Sa ds W s fω Ds, ωe iωτs, ρ I. 1.3 The weight fator is often approximated by W s r p s ρ I. It an be alulated exatly in the ase of simple platform trajetories, so that the point spread funtion is as tight as possible [6, 15, 9]. Our main objetive in this paper is to image moving loalized refletors, or targets, that an be traked in real time by proessing data over suffiiently small sub-apertures. We introdue a proess based on the Wigner transform and the ambiguity funtion of properly segmented small aperture data, that an either estimate the target motion or ompensate the SAR platform trajetory perturbation. The latter proess is alled autofous. It is only after the motion estimation and autofous have been arried out over suessive, overlapping segments of the trajetory, that we an ompute the final image over an extended aperture, using an imaging funtion like 1.3. This approah an be applied to a wide range of SAR imaging systems, inluding X-band persistent surveillane SAR. The main steps in the analysis of SAR imaging with motion estimation and autofous are as follows. First we segment the data into small apertures and find analytially under what onditions phases an be linearized so that omputationally effiient loal Fourier transforms an be used. Next we desribe an approah that estimates inremental target motion relative to the SAR platform, using the loation in the phase spae of the peaks of the Wigner transform and the ambiguity funtion of the data. Moreover, we show that the Wigner transform and the ambiguity funtion provide omplementary information and that they have different resolution. Most of the analysis assumes data in a time window ontaining the ehoes from a single strong target, but we onsider the ase of multiple targets, as well. Note that it is only the relative motion between the target and the SAR platform that appears in the

3 problem. Thus, when imaging a single strong target, we annot estimate simultaneously its speed and the platform trajetory perturbations. To deouple the motion estimation from the autofous proess we need a omplex imaging sene, with multiple targets that move at different speeds. We onsider in this paper the ase of multiple strong stationary targets, and show under whih onditions we an arry the autofous proess. Beause the Wigner transform and the ambiguity funtion have multiple peaks in this ase, we use their entroids[, 3] to estimate the platform trajetory perturbations. The results extend easily to multiple targets that move in the same way, as a group. The ase of multiple targets in different motion is not disussed here, beause the phase spae approah alone does not work. It must be omplemented with a data pre-proessing filtering step that will be presented in a different paper. That the Wigner transform is a natural tool for detetion and imaging of moving targets has been known. For example, the estimation of target motion from the peaks of the Wigner transform is onsidered in [11, 1]. The robustness of the estimation to additive noise and extensions to ases with few targets are addressed in [3]. See also [16, 17] for experimental results. Alternative approahes to autofous are given in [14, 1, 4]. The main result of this paper is the theory of estimating target motion and ompensating for platform trajetory perturbation using in a omplementary way the Wigner transform and the ambiguity funtion. It inludes the analysis of robustness to unertainty of the flight path and of the initial target loations. The paper is organized as follows. We begin in Setion with a mathematial model of the SAR data and the range ompression proessing. The first appendix provides a summary of the relevant physial parameters that are typial in X-band persistent surveillane SAR. In Setion 3 we arry out a target motion estimation using the Wigner transform and the ambiguity funtion of the range ompressed data. In Setion 4 we apply the results to the autofous problem. These two setions show how by identifying the peaks in the Wigner transform and ambiguity funtion we an estimate parameters that an be used for either improving target motion estimation or for ompensation for SAR platform trajetory perturbations. The ase of multiple stationary targets is onsidered in Setion 5. In Setion 6 we present the results of detailed numerial simulations, first for target motion estimation alone, and then for autofous, without target motion. The latter inlude results for multiple targets. We end with a summary in Setion 7.. Mathematial model of the data. To study the resolution of motion estimation and autofous with phase spae methods based on Wigner transforms and ambiguity funtions, we use a simple model of the SAR data. It orresponds to the refletivity Rt, x of a small point-like moving target. The target trajetory is an arbitrary urve in the imaging surfae x = x, z = hx, with elevation z = hx. We suppose that it is smooth enough to approximate the motion loally, between two onseutive illuminations t, s, by translation from ρs = ρs, hρs, at speed us = us, hρs us.1 on the imaging surfae. For simpliity of the exposition, we take a flat surfae z = hx =, so that us = us, and we an write Rt, x = δx ρs tus, t, s.. The results extend easily to surfaes with known elevation z = hx and speed.1. 3

4 Our analysis is based on several approximations that are motivated by speifi SAR data regimes. One suh regime arises in the GOTCHA Volumetri SAR Data Set [7] for X-band persistent surveillane SAR, desribed in Appendix A, and used as an illustration throughout the paper..1. The basi data model and range ompression. We assume hereafter that us and the platform speed Vs = r ps = V s ts.3 satisfy us V s and ω o V s τs, ρs V s τs, ρs = π 1..4 λ o This says that we are in a high frequeny regime, where the platform moves a distane omparable to λ o over the very short travel time duration τs, ρs, at speed of light. The platform speed is along the unit vetor ts, tangential to the trajetory at r p s. The range sale L gives the typial distane from the SAR platform to the imaging sene and we suppose that it is similar to the radius Rs of urvature of the platform trajetory. We also assume a relatively narrow bandwidth B, satisfying V s B ν o 1..5 The bandwidth is narrow B ν o, but B is suffiiently large so that over the duration O1/B of the range ompressed signal, the platform moves a small distane with respet to λ o. The following model of the data Ds, t is derived in Appendix B. Proposition 1. Let ft = e iωot f B t be the probing signal, given by a base-band waveform f B t modulated by a arrier frequeny ν o = ω o /π. We have Ds, ω = dt e iωt Ds, t ω o/ f B ω ω o 4π r p s ρs eiωoψs+iωτs, ρs,.6 where ψs is the Doppler phase ψs = τs, ρs Vs us ms, ms = r ps ρs r p s ρs..7 Instead of working diretly with the reorded data Ds, t, we ompress them by onvolution with the omplex onjugate of the time reversed emitted pulse f t, offset by the travel time τs, ρ o to a referene point ρ o in the imaging plane, D r s, t = dt Ds, t f t t τs, ρ o, ρs ρ o L..8 We shall make the approximation f B ω ω o f B 1 [ωo,πb]ω,.9 where 1 [ωo,πb]ω is the indiator funtion of interval [ω o πb, ω o + πb]. This holds for linear frequeny modulated hirps [13] or for pulses ft e iωot sinπbt. We also set the Doppler phase ψs to the onstant ψ over the small apertures a used in our data proessing, beause for s Sa = a/v, we suppose ω o [ψs ψ] ω o s ψ ω osv 4 av 1..1 λ o

5 The model of the range ompressed data follows from.6,.8 and.9, D r s, ω ω f B 1 o [ωo,πb]ω 4π r p s ρs exp {iω oψ + iω [τs, ρs τs, ρ o ]}..11 It has the advantage that no matter how long is the waveform ft, D r s, t behaves like a short pulse with support O1/B. Another advantage is that sine D r s, ω has smaller phases than Ds, ω, it is more onvenient in numerial omputations. Remark 1. In the GOTCHA Volumetri SAR data set desribed in Appendix A, the assumptions above are justified as follows. Apertures a = 14m, of one degree on the irular planar path of the platform, with radius R = 7.1km, are spanned by the airraft at speed V = 5km/h 7m/s in 1.8s. In suh short time, the target motion an be approximated by translation at speed u 1km/h 8m/s V. This gives πτv =.9m, whih is similar to λ o = 3m, so.4 holds. Moreover, V / = B/ν = 6.5 1, so.5 holds. The target range is L 1km, whih is similar to R, and av/λ o = , so.1 holds. 3. Motion Estimation with the Wigner transform and ambiguity funtion. We desribe a phase spae approah to target motion estimation. It is based on the Wigner transform and the ambiguity funtion of the data over small sub-apertures. We give quantitative riteria for the aperture segmentation, and explain how the Wigner transform and the ambiguity funtion give omplementary information about the target motion. We also ompare their resolution, and study their robustness to unertainty of the flight path and of the initial target loation. We assume hereafter that the platform speed V s = Vs along the tangent ts of the flight path, the urvature Rs and the target speed us = us, are suffiiently smooth funtions of s, to approximate them by onstants over the short time intervals that define the sub-apertures. We also introdue the notation u = u Motion estimation with the Wigner transform. The Wigner transform of the range ompressed SAR data is given by Ω S Ws, Ω, ω, T = d ω d s D r s + s, ω + ω Ω S D r s s, ω ω e i sω i ωt, 3.1 where Ω and T are frequeny and time variables, dual to the variables of integration s and ω. The ω integral limit is Ω = πb ω ω o, so that ω ± ω/ remains in the support of D r for all ω [ Ω, Ω]. The time offset interval s [ S, S] is hosen in terms of an aperture a measured along the trajetory, S = a/v. The enter of the aperture is indexed by the slow time s. The Wigner transform takes the following form, as proved in Appendix C. Proposition. Consider apertures a so that the Fresnel number a /λ o L satisfies a λ o L 4ν ov Bu, and a λ o L 8LV au. 3. Then, the Wigner transform evaluated at ω = ω o has the form Ws, Ω, ω o, T f B 4 { [ ]} 4πa Ω sin{πb [T τs]}sin r p s ρs 4 ω o V Φs, λ o

6 where τs = τs, ρs τs, ρ o and Φs = u V ms ts ms m o s. 3.4 Here, symbol stands for approximate, up to a multipliative onstant, ms is the unit vetor in.7, and m o s = r ps ρ o r p s ρ o is the unit vetor from the platform to the referene point ρ o in the imaging plane. Remark. Among the two onditions on the Fresnel number in 3., the first one is more restritive, beause typially L νo a B 1. It allows us to linearize the phases in the integral over s and obtain therefore a simple expression of the Wigner transform W. If we relaxed the ondition to a λ o L 4ν ov Bu, we would get a more ompliated formula for W, involving Fresnel integrals instead of the sin funtions. However, the onlusions drawn below remain similar, beause they use estimates of the peaks Ω W s and T W s of W. These peaks are not affeted by the quadrati phases in the Fresnel integrals. In GOTCHA, the seond ondition in 3. says a 39m for u = 1km/h. The relaxed ondition 3.6 says a 19m, so we an work with a 1m. Now let us use 3.3 to relate the peaks Ω W s and T W s of W to the travel time τs, ρs and the target speed u projeted along ms, u V ms = ΩW s ω o V + λo ts ms m o s + O, 3.7 a and τs, ρs = T W s + τs, ρ o + O. 3.8 B Thus, we an use the peak T W to estimate the travel time, and therefore the distane to the target. There is only one peak if there is only one target. In general, there will be many peaks, assoiated to different targets [3], and the Wigner transform may be used to selet a time window ontaining the eho from a single target that we trak. This assumes a separation in range between the target and the remainder of the imaging sene. The analysis in this setion is based on this assumption, but we address in Setion 5 the ase of multiple strong targets whose pulse ompressed ehoes arrive at almost the same time. To estimate u, we need an initial estimate of the target loation in the sub-aperture. This estimate is then adjusted from one sub-aperture to another, using the estimated speed. To simplify the formulas, we assume in the analysis an estimate of ρs i.e. of ms at time s orresponding to the middle of the subaperture. The results an be obviously modified to initial loation estimates by letting ρs ρs u S/. Equation 3.7 determines only one omponent of u, along ms. To get u = u,, we also need its ross-range projetion Ps u, where Ps is the orthogonal projetion Ps = I ms ms T. 3.9 The latter an be determined, in priniple, from additional sub-apertures with enters slightly shifted at s + δ s, by small δ s. For example, we ould use the estimated u ms + δ s and u ms to write V ts u u [ ms + δ s ms] δ s u m s = δ s r p s ρs Ps u. 6

7 However, suh differentiation will not work well with noisy data. We show next that the ambiguity funtion omplements the Wigner transform, as it provides a more robust and diret estimate of Ps u. 3.. Motion estimation with the ambiguity funtion. The ambiguity funtion of the range ompressed SAR data is given by As, Ω, s, T = ωo+πb ω o πb dω S S ds D r s + s + s, ω D r s + s s, ω e isω iω ωot. 3.1 The slow time window enter s and the offset s are independent variables, and Ω and T are the dual variables of s and ω ω o, respetively. The expression of 3.1, in our setup, is given in the next proposition. Its derivation is very similar to that of the Wigner transform, and we do not repeat it here. Proposition 3. If we hoose S = a/v, with a satisfying 3., the ambiguity funtion evaluated at s = a/v is given approximately by A s, Ω, a V, T f B 4 e { [ ]} [ iωoa Φs πba T aω r p s ρ 4 sin a + Φs sin V + πa Φ ] s, 3.11 λ o r p s ρs where denotes approximate up to a multipliative onstant, Φs is the same as in 3.4, Φ s = ts Ps u P o s ts r p s ρs V r p s ρ o t s V ms m os, 3.1 and P o s is the orthogonal projetion P o s = I m o s m o s T. Thus, the ambiguity funtion allows us to estimate both the projetion of u on ms, and the norm of the projetion of the relative speed V u on the plane orthogonal to ms. As already noted, in 3.11 we evaluate the ambiguity funtion at the extreme value of the slow time offset s = a/v. This is to get the tightest seond sin funtion, whih gives the sharpest estimate of the peak Ω A s in Ω. The peak T A of the ambiguity funtion determines u A V ms = T s SV + ts ms m o s + O Ba We already have the estimate 3.7 of u ms, from the Wigner transform. In fat, estimate 3.13 is less preise, with resolution Ba = ν oλ o Ba λ o a. It is the other peak of A that gives the omplementary information, ts Ps u = P o s ts r p s ρs + r ps ρs V r p s ρ o V [ ] t s m o s ms ΩA s λo L + O ω o sv a, 3.14 without any need to differentiate, as was the ase with the Wigner transform. This equation, in onjuntion with 3.7, determine the target speed u, up to an ambiguity of the orientation of t u/v in the diretion orthogonal to m in the imaging plane. Here t = t, t z and m = m, m z. 7

8 Remark 3. Note that beause of our aperture limitation assumption 3.6, the estimate 3.14 is useful if we have a large Fresnel number 1 a λ o L 4ν ov Bu This means that the aperture a annot be too small if the motion estimation is to work. It must be large enough, but still limited by 3.6. For example, in GOTCHA, the Fresnel number is a /λ o L = 49 for one degree aperture, and 4ν o V/Bu = Sensitivity of target motion estimation to knowledge of its initial loation. The estimation of u requires the vetor ms in.7, and the target range r p s ρs. Negleting the SAR platform trajetory perturbations, the range is determined with resolution /B from the travel time τs, ρs, whih is estimated from the Wigner transform. The target loation annot be determined diretly from the data, before forming an image, but we suppose that we know an estimate ρ e of ρs, either from diret observations, or from traking the target at previous times. That is, we approximate ms by m e s = r ps ρ e s r p s ρ e s, with ρ e in the imaging plane, at distane /τs, ρs from r p s. Now we quantify the sensitivity of motion estimation to the error ρ ρ e. Let u = u, be the error of the estimated veloity. We obtain from 3.7 that and therefore u + u V From 3.14 we have P u + u e t V m e = ΩW ω o V + t m e m o = u V me + u V m e = t u λo m e m + O = O V a t P u = r p ρ V V ρ ρe = O L t u m e m + O V ρ ρe L t λo L m m e + O λo L + λ ol a = O λo a λo + O a a + BL a, 3.17 beause t = V/R V/ r p ρ. The seond term in the error dominates the first, provided that ρ ρ e a, beause 3.6 and u V give ρ ρ e L a L ν oλ o Ba = Ba λ ol a. We onlude from 3.17 that the ross-range veloity estimation is insensitive to the hoie of the referene point ρ e, as long as ρ e ρ a. However, 3.16 shows that to maintain the resolution λ o /a in the estimation of u m with the Wigner transform, we need a more preise estimate of the target loation, with ρ ρ e omparable to the ross range resolution λ o L/a. 8

9 3.4. Motion estimation with SAR platform trajetory perturbations. The SAR platform trajetory is known only approximately. We therefore onsider perturbations µs of r p s and set r p s r p s + µs in the data model.11. We assume that the SAR platform trajetory perturbations vary on the slow sale s and are small, so that only the unperturbed platform speed appears in the Doppler phase ψ in.7. This is beause ω o ψ 1 by assumption.4, and therefore perturbations of ψ are negligible. The effet of µs on the amplitude of D r s, ω is also negligible Assumptions about the SAR platform trajetory perturbations. Let us begin with the observation that the platform trajetory perturbations indue the following perturbations of the travel time r ps + µs ρs τs, ρs = µs µs ms + O, 3.18 L with negligible residual under the assumption µs λ o L When ms µs is large enough, we an estimate it diretly from the perturbations of the arrival time of the pulse ompressed SAR data. Sine the ompressed pulse support is O1/B, we an estimate and ompensate ms µs from the arrival times when ms µs O. B By ompensation, we mean that one we estimated ms µs, we an replae in all the subsequent proessing the ideal platform loations r p s by r p s + ms µs ms. ompensation has been made. We assume from now on that suh a We now introdue some simple, onservative bounds on the norm of the trajetory perturbation µs and its derivatives with respet to the slow time s. They are onservative in the sense that they are suffiient for the sensitivity analysis presented here and for the autofous analysis presented in Setion 4. We assume in 3.19 that the norm of the trajetory perturbations is bounded by µs µ λ o L a, 3. with the seond inequality being a onsequene of the smallness of the inverse Fresnel number λ o L/a 1 in The speed µ s should be a small fration of V. We sale it using a dimensionless parameter δ, so that S µ s λ ol aδ a for S = a/v, δ λ ol a. and δ µ a. 3.1 This implies µ s V, and it says that over the short time interval S defining the sub-aperture, the deviation of the platform from the unperturbed trajetory is muh smaller than a, but it may be larger than the spot size λ o L/a, depending on δ. The first assumption on δ implies, along with 3.6, that δ B/ν o for target speeds u V. We also take δ µ/a. We assume next that the aeleration of the trajetory perturbation satisfies S µ s λ o γ λ ol a for S = a/v, 3. where we have introdued another dimensionless parameter γ a/l. Moreover, S 3 µ s λ o 3.3 9

10 so that we an approximate µs + s by a seond degree polynomial in s over the small time interval s S used in the motion estimation, µs + s µs + s µ s + s µ s. 3.4 It is onvenient in the sensitivity analysis to assume that γ δ so that we an work with the single dimensionless parameter ε = min{1, γ, δ}, whih aording to our assumptions above satisfies ε λ ol a, ε B ν o and ε µ a. 3.5 This allows us to linearize phases and express the Wigner transforms and the ambiguity funtions in terms of sins, rather than Fresnel integrals. Remark 4. Let us illustrate the assumptions made in this setion in the GOTCHA regime desribed in Appendix A: Assumption 3.19 says that µs 17m. At one degree aperture a = 14m, we have λ o L/a =., B/ν o =.6 and so we an hoose in 3.5 ε.3. This gives in 3.1 µ /V.1, that is µ 5km/h 7m/s and in 3. µ 648km/h =.5m/s. These are very onservative bounds. In fat, when we look in detail at the alulations in Appendix D, we see that if we keep trak of all the multipliative onstants, the results stated below would hold for aelerations that an be about twenty times larger than this. The same applies to the assumption 3.3 on µ Sensitivity of motion estimation to SAR platform trajetory perturbations. With the saling above, we obtain the following results proved in Appendix D. Proposition 4. The Wigner transform of the range ompressed data takes the form Ws, Ω, ω o, T f B 4 r p s ρs 4 sin { πb [ T + δt W τs ]} { [ 4πa Ω + δω W ]} sin Φs, 3.6 λ o V ω o where denotes approximate, up to a multipliative onstant, τs and Φs are given in 3.4 and δt W µs ms δω W =, = µ s ms + ts u Ps µs V ω o V V r p s ρs. The ambiguity funtion evaluated at s = a/v has the form a A s, Ω, V, T f B 4 e iωoa [ r p s ρs 4 where Φ s is the same as in 3.1 and δt A = a V δω W ω o and δt A a +Φs δω A ω o ] { πba sin [ aω + δω A sin = V a V [ T + δt A + t u V r p s ρs Ps µ s V a ]} + Φs πa Φ ] s, 3.7 λ o r p s ρs + µ s V ms. The results stated in this proposition are quite intuitive. When we ombine the expressions of δt W and δω W with those of τs and Φs, given by 3.4, we see that equation 3.6 is the Wigner transform of the range ompressed eho from a target loated at ρs µs at the instant orresponding to the enter of the sub-aperture, and with veloity u µ s. Similarly, 3.7 is the ambiguity funtion for target speed u µ s and aeleration µ s. Sine this is a relative motion between the target and the platform, it 1

11 is not onfined to the imaging plane, in general. In any ase, only projetions of the relative veloity and aeleration determine the loation of the peaks of the Wigner transform and ambiguity funtion of the data. We onlude from Proposition 4 that the platform perturbations indue an error of the order µs ms/ in the travel time estimation with the Wigner transform. As explained in the beginning of the previous setion, this perturbation is within the main lobe of the first sin funtion in W, with resolution O1/B. It has therefore a small effet on the travel time estimation. However, the target motion estimation is affeted by the platform trajetory perturbations. Expliitly, we have the errors δω W µ s ms = O + µs λo L = O V ω o V L a ε + µ L in the estimation of u/v ms, and errors δω A = O Ps µ s V aω o V + L µ s ms λo L V O a ε in the estimation of Ps ts u/v. Realling from Setion 3. that λ o L/a is the error bound on the estimated u/v with the unperturbed platform trajetory, we see that we an only approximate the target speed u by u I, with an error u λ ol V a ε, where u = ui u, 3.3 and ε is defined by Autofous with the Wigner transform and ambiguity funtion of the data. As stated in Setion 3.4, the SAR platform trajetory is known only approximately. The error in knowledge of the trajetory results in a shift and blur of the target image. The goal of the autofous proess desribed next is to estimate the phase errors, using the ambiguity funtion of data and its Wigner transform. One we have estimated the phase errors, we an go bak and use them in the image formation to get better resolution and to fous the image near the orret loation. While Setion 3.4 onsidered motion estimation in the more general situation of both target motion and platform perturbations, in this setion we will approah the autofous problem assuming that the target is stationary u =, and loated at ρ = ρ, Effet of platform perturbations on image. Let us rewrite the imaging funtion 1.1 in terms of the range ompressed data Iρ I = S S ds ωo+πb ω o πb dω π D r s, ωe iω r ps ρ I r ps ρ o, 4.1 where ρ I = ρ I, is the searh point in the horizontal image plane. We take for onveniene the origin of the slow time at the enter of the aperture. We also let, for simpliity, the referene point ρ o in the data range ompression be at the target. The data model is given by.11. The fousing of the preliminary image Iρ I is desribed in Proposition 5. Its proof is similar to that of Proposition 4 and we do not inlude it here. It involves approximation of phases by a seond degree polynomial in s, and areful justifiation of the approximations, using the saling in Setion 3.4.1, and the same onstraints on the aperture as in Proposition. 11

12 Hereafter, we use the onvention that when the arguments are missing, we evaluate the funtion at the slow time s = indexing the enter of the aperture. Expliitly, we let r p = r p, m = m = rp ρ r p ρ, and so on. Proposition 5. The image Iρ I is given by Iρ I ωo+πb ω o πb { iω dω exp r } p ρ r p ρ I S + ϕ S iω o sv t V m mi + { iωsv ds exp [ t m m I + ϕ 1 ] + P t P I t r p ρ r p ρ I + ϕ, 4. where the symbol denotes approximate, up to a multipliative onstant. The fousing of Iρ I is determined by the phases ϕ o = m µ, 4.3 ϕ 1 = m µ V + P µ t r p ρ, 4.4 ϕ = m µ t + µ V + V P µ r p ρ V. 4.5 Here we denote by ρ I = ρ I, the image peak and we let m I The Fresnel integral over s in 4. peaks when = rp ρ I r p ρ I, and PI = I m I m I T. t m m I = ϕ 1, and the integral over the bandwidth peaks when m I = r p ρ I r p ρ I, 4.6 r p ρ r p ρ I = ϕ. 4.7 Linearizing about the true target loation, we get that the peak ρ I ross-range by m ρ ρ I = Oϕ O B, of the image is shifted in range and t P ρ ρi r p ρ = Oϕ 1 λ ol a ε. 4.8 In the autofous proess we aim to estimate the phases ϕ o, ϕ 1, ϕ, and then ompensate the SAR platform trajetory perturbations to improve the image. The phases ϕ and ϕ 1 affet the loation of the peak of the image, as desribed in 4.8. The peak shift in range is small, within the resolution limits, but the shift in ross-range may be large. Beause ϕ appears in the quadrati part of the phase in 4., it only affets the spread of the image in ross-range, around its peak. The quadrati phase in 4. also depends on the shift ρ I ρ of the peak of the image from the true target loation. The larger the shift, the larger the phase and the more blur in the image. We explain next how to estimate the phases ϕ j, for j =, 1,, using the Wigner transform and ambiguity funtion. The autofous proess onsists in applying the orretion ] µ AF sv s = [ϕ o + sv ϕ 1 + ϕ m, 4.9 1

13 to the SAR platform trajetory and forming the image r p s r p s + µ AF s, 4.1 I AF ρ I = S S We annot estimate the trajetory perturbation ωo+πb dω ds ω o πb π D r s, ωe iω r ps+ µ AF s ρ I r ps ρ o µs µ + s µ + s µ from the three phases. It is is only the ombination of the perturbation loation, speed and aeleration that omes into play in and affets the image. Thus, we ompensate the effet of the trajetory perturbations with 4.9, as if we ould neglet the seond terms in Autofous with the Wigner transform and ambiguity funtion. Setting u = in Proposition 4, we obtain that the Wigner transform evaluated at ω = ω o takes the form where Ws =, Ω, ω o, T f B 4 r p ρ 4 sin { πb [ T + δt W]} { 4πaΩ + δω W } sin, 4.1 λ o V ω o δt W = µ m, δω W V ω o = m µ V + P µ t r p ρ We evaluate the Wigner transform at s =, beause that is the onvention for the enter of the sub-aperture assumed in this setion. If we have overlapping sub-apertures, then the origin of the slow time s should be shifted for eah aperture. Using 4.13 and , we have the following two estimates from the peaks Ω W, T W of the Wigner transform, in the phase spae frequeny-time plane Ω, T, ϕ o s = T W s + O, ϕ 1 s = λ o λo B 4πV ΩW s + O a Proposition 4 also gives that the ambiguity funtion evaluated at s = a/v has the form A s =, Ω, a V, T f B 4 e iωoδt A r p ρ 4 sin { πbt + δt A } sin [ aω + δω A V ], 4.15 where δt A = a V δω W ω o and δω A ω o = V a t u V r p ρ P µ V V m. + µ Using and letting Ω A, T A be the peaks of 4.15, we obtain that ϕ 1 s = a T A s + O, ϕ s = λ o ab πav ΩA s + O λo a Similar to what we have seen in the motion estimation problem in Setion 3., we get a redundant estimate of ϕ 1, with worse resolution than that in 4.13, beause ab λ o ω o a B λ o a. 13

14 The ambiguity funtion is useful for the estimation of ϕ, and thus omplements the Wigner transform in the autofous proess. As we remarked above, the Wigner transform and ambiguity funtion in 4.1 and 4.15 are for the sub-aperture entered at slow time s =. The autofous proess involves working with overlapping sub-apertures, with enters indexed by s. Eah sub-aperture gives an estimate of the three phases, ϕ j s, for j =, 1,, and they an be ombined to improve the ompensation 4.9 of the platform trajetory perturbations. We illustrate this in Setion 6, where we present numerial results. 5. Autofous with entroids of the Wigner transform and ambiguity funtion The peak seletion used in the previous setion for the autofous proess may be problemati in pratie, speially in noisy environments. It is also omputationally expensive beause aurate peak seletion requires a very fine sampling of the phase spae frequeny-time plane in whih we evaluate the Wigner transform and ambiguity funtion. We show in Setion 5. that the entroids of the Wigner transform and ambiguity funtion of data from an imaging sene with a strong target give in theory exatly the same information as the peaks. Thus, we an do the autofous with the entroids, whih are more robust than the peak seletion, beause they are given by smooth funtionals integrals of the Wigner transform and ambiguity funtion. When there are multiple strong targets at loations ρ k and similar range from the SAR platform, we annot separate their pulse ompressed ehoes by time windowing, as we have assumed in the previous setion. Then, the Wigner transform and ambiguity funtion have multiple peaks in the phase spae, even when we neglet multiple sattering between the targets and write the data model as N ω D f B e iωoψ o r s, ω 4π r p s ρ k e iω rps+ µs ρ k rps ρ o. 5.1 k=1 Roughly speaking, there is one peak per target at loations similar to those desribed in Setion 4., for ρ ρ k and k = 1,,..., N. There are also additional peaks oming from the ross-terms k k in the quadrati expressions of the Wigner transforms and ambiguity funtions. It is not lear in suh ases how to selet a partiular peak in the phase spae in order to arry out the autofous proess, but we may be able to use the entroids. We show in Setion 5.3 that the entroids are useful for estimating the platform trajetory perturbations in the ase of multiple targets that are not too spread out in the imaging plane. We also illustrate the performane of the autofous proess with numerial simulations in Setion Definition. The entroid of the Wigner transform is the point Ω W s = T W s = Ω W s, T W s in the phase spae with oordinates dω dt Ω Ws, Ω, ω o, T dω dt Ws, Ω, ω o, T, dω dt T Ws, Ω, ω o, T dω dt Ws, Ω, ω o, T. 5. With our notation onvention, the aperture is entered at s =, and we let Ω W = Ω W and T W = T W. The entroid Ω A, T A of the ambiguity funtion A a s =, Ω, V, Ω is defined similarly. 14

15 To relate the entroids to the phases that arise in the autofous proess, we need the following result, whih we state for an arbitrary enter s of the aperture, not just s = : Lemma 1. The entroid of the Wigner transform of the range ompressed data has the form [ i D Ω W s r s, ω o D r s, ω o D r s, ω o D ] s r s, ω o s = D, r s, ω o [ i D T W ω r s, ω D r s, ω o D r s, ω o D ] ω r s, ω o s = D. r s, ω o Similarly, the entroid of the ambiguity funtion satisfies, for s = a/v, [ i D Ω A s r s + s, ω o Dr s s, ω o + Dr s + s, ω D o s r s s, ω ] o s = D r s + s, ω o Dr s s, ω, o [ i D T A ω r s + s, ω o Dr s s, ω o + Dr s + s, ω D o ω r s s, ω ] o s = D r s + s, ω o Dr s s, ω. o These expressions follow from the definition 5. of the entroids and definitions 3.1 and 3.1 of the Wigner transform and ambiguity funtion. Expliitly, we have from 3.1 that Ω S dω dt Ws, Ω, ω o, T = 4π d ω d s D r s + s, ω o + ω Ω S dω π ei sω = 4π F s,ωo s, ω, δ s, ω. dt π e i ωt D r s s, ω o ω Here δ s, ω = δ sδ ω is the Dira delta distribution, ating on the test funtion F s,ωo s, ω of arguments s, ω, and parametrized by s and ω o. The test funtion is given by F s,ωo s, ω = χ s, ω D r s + s, ω o + ω D r s s, ω o ω with χ s, ω an arbitrary, smooth window funtion with support in [ S, S] [ Ω, Ω], that is equal to one in the viinity of the origin. Similarly, dω dt Ω Ws, Ω, ω o, T = 4π Ω Ω S d ω S, d s D r s + s, ω o + ω D r s s, ω o ω dω π Ω ei sω = 4iπ F s,ωo s, ω, δ s s, ω, dt π e i ωt where δ s s, ω = δ sδ ω. Thus, we have from the properties of the Dira delta distribution that Ω W = i sf s,ωo, F s,ωo,, and its expression given in Lemma 1 follows. The proof of the expressions of T W, Ω A and T A is similar. 15

16 5.. The ase of a single target. Lemma 1 and the model.11 of the range ompressed data give that Ω W = ω ov Φs δω W, T W = τs δt W, 5.3 where Φ and τ are given by 3.4 and δω W and δt W are defined in Proposition 4. The entroid of the ambiguity funtion satisfies Ω A s = πv aφ s V a λ o r p s ρs δωa + O λ o L T A s = a a Φs δt A 3 + O L,, 5.4 where Φ is given in 3.1 and δω A and δt A are defined in Proposition 4. These equations follow from Lemma 1 and the mean value theorem. The residuals ome from estimates of the seond derivatives of the range ompressed data model, for slow time offsets that do not exeed s = a/v. When we disussed the autofous in Setion 4, we assumed a stationary target and took for simpliity the referene point ρ o in the range ompression exatly at the target loation ρ. Then, the phases Φ and Φ vanish and we note that the entroids oinide with the peaks of the Wigner transform and ambiguity funtion given in Setion 4.. Naturally, the equality of the entroids and the loation of the peaks does hold for a moving target and for ρ o ρ, as follows from a straightforward alulation. The onlusion is that we an apply the results in the previous setion, with the peaks replaed by the entroids, to do the autofous of imaging senes with a strong target Autofous with Multiple Stationary Targets. We study here the entroids of the Wigner transform and ambiguity funtion in the ase of N stationary targets. They are at similar ranges from the SAR platform, and thus their ehoes annot be separated by time windowing. The data model is given by 5.1, and it neglets multiple sattering between the targets. The assumption is that either the multiply sattered signals are weak, or that they arrive at a later time and may be filtered out by time windowing. To study the entroids, we obtain from the data model that s D r s, ω o = ω D r s, ω o = N k=1 N k=1 iω o [ V ts m k s m o s + µ s m k s] Dk r s, ω o, i r ps + µs ρ k r p s ρ o D k r s, ω o, where D k s, ω o is the Fourier transform of the range ompressed eho from the k th target at loation ρ k, and m k s is the unit vetor pointing form the platform at r p s to ρ k. The results below are derived from this equation and Lemma 1 by expanding around the referene point ρ o. where We obtain that the entroid of the Wigner transform satisfies Ω W o, T W o Ω W s = Ω W o s + ω ov E 1, T W s = To W s + 1 B E, 5.1 is the entroid of the Wigner transform for a single target at ρ o, and the dimensionless 16

17 residuals are given by and E 1 = E = B N k,k =1 N k,k =1 ρ k + ρ k ρ o [ ] Ps ts + µ s/v r p s ρ o ρ k + ρ k ρ o m o s D k r s, ω o D k r s, ω o D r s, ω o. D k r s, ω o D k r s, ω o D r s, ω o, Let us denote by ρ the typial distane of a target from ρ o, and by ρ m o the typial distane along the range diretion m o. The estimates of the residuals are ρ ρ mo E 1 O, E O. 5. L /B Similarly, we an estimate the entroid of the ambiguity funtion. We only need the result for Ω A s, beause T A s gives basially the same information as T W s, at worse resolution. We get Ω A s = Ω A o s + ω ov ρ E 3, E 3 O, 5.3 L where Ω A o s is the frequeny oordinate of the entroid of the ambiguity funtion for a target at ρ o. We onlude that when the targets are not too spread out in the imaging plane, the entroids are approximately as those for a single target at ρ o. Expliitly, if we have ρ L µ m o, 5.4 V we an estimate µ m o from Ω W Ω W o. When we ompensate this term in the image formation, we orret the shift in ross-range. If in addition ρ L L µ m o V, 5.5 we an estimate µ m o from Ω A Ω A o, and then use the estimate to redue the blur in the image. When we reall the assumptions in Setion on µ and µ, we see that basially, the bounds in say that the targets should be ontained in a disk in the imaging plane of radius smaller than L, the range. The seond equations in 5.1 and 5. say that to determine the perturbation µ m o from T W, the projetion of ρ in the range diretion m o must not exeed the range resolution /B, ρ m o B. 5.6 This ondition may appear stringent, but if there were a larger range spread, we ould separate the targets by simply time windowing the pulse ompressed ehoes. Let us end this setion with the remark that the results presented here extend naturally to the ase of multiple targets that move the same way, as a group, with speed u. Then, it is the relative motion of the targets with respet to that of the SAR platform that is determined by the entroids of the Wigner transform and ambiguity funtion. 17

18 6. Numerial simulations. We present in Setion 6.1 results for target motion and no platform perturbations and in Setion 6. results for autofous. The data is generated with models.11, 5.1, with parameters hosen as in the GOTCHA Volumentri SAR regime desribed in Appendix A. The unperturbed SAR platform trajetory is irular, at altitude H, and with enter projeted at the origin, in the imaging plane. To ompute the Wigner transform W and ambiguity funtion A for eah s in a small a small aperture a, we need the range ompressed data over a larger aperture a e =.5a. We use the two-dimensional fast Fourier transform FFT to ompute W and A. To avoid aliasing, we pad with zeroes the N s N ω matrix with entries D s,ω s, ω = D r s + s, ω + ω D r s s, ω ω. Here s and ω are parameters, and N s = 118 and N ω = 44 are the number of samples in s and ω. The padding reates a larger matrix, of size 56 14, for eah s in the slow time range and ω in the bandwidth. The motion estimation and autofous is arried out either by seleting the peaks of the Wigner transform and ambiguity funtion, or by omputing their entroids. The preliminary images an be omputed diretly from 1.1, and the autofoused ones from However, beause of the small sub-apertures, we an ompute them more effiiently by linearizing the phases and using the two dimensional FFT of the range ompressed data Motion estimation without platform perturbations. We onsider here a target moving with onstant veloity u = 8/ 1, 1m/s. The aperture is a = πr/18 14m. It orresponds to a one degree ar on the irular trajetory of radius R = 7.1km. We show in Figure the Wigner transform and ambiguity funtion evaluated at s =. Note the large spread of A in the time variable T. This is onsistent with the resolution results in Setion 3.. We obtained there that the target speed projeted in the range diretion an be determined either form the peak Ω W of the Wigner transform or the peak T A of the ambiguity funtion, but the resolution of the Wigner estimate is better. This is illustrated in Figure by the tighter peak of the Wigner transform. We estimate the target speed u Us using equations 3.7 and We move the target from one enter s of a sub-aperture to another s ± s, with the estimated Us. We plot in Figures 3 and 4 the atual and estimated projetions of the veloity as in equations 3.7, 3.13 and The estimates are shown in solid lines. The dotted lines indiate the resolution limits. Beause we assume that over a single aperture the target has a onstant veloity, we average the multiple estimates Us, from the overlapping sub-apertures, to get u I. We use this estimated target speed to ompute an image using equation 4.1, with ρ I ρ I + su I,. The pixels are squares of side.1 m, and the image domain is 1m 1m. The image is ompared in Figure 6 with that omputed with the exat target veloity. We also show in Figure 7 the error between the true target trajetory and the estimated one. 18

19 x 1 8 x T se T se Hz Hz a Wigner transform W Ω, T b Ambiguity funtion AΩ, T Figure. The Wigner transform and Ambiguity funtion of the range ompressed ehoes from a single target Wigner Estimate Ambiguity Estimate Atual Wigner Resolution Ambiguity Resolution Relative Projeted Veloity Slow Time seonds Figure 3. Estimated veloity projeted onto ms. The Wigner estimate is in solid blue and the ambiguity one in solid green. The tue value is in solid red. The blue dotted line indiates the theoretial resolution limit for the Wigner estimates. The green dotted line shows the resolution limit for the estimates with the ambiguity funtion. 6.. Autofous with stationary targets. We begin with the ase of a single stationary target at known loation ρ = ρ,. We take ρ at the origin and use the simple SAR platform trajetory perturbation µs = 5λ s + λ s + λ s 3 6, 3λ, λ s λ s + λ s 3, 6 that satisfies our assumptions in Setion The autofous is done as desribed in Setion 4, exept that we use the entroids of the Wigner transform and ambiguity funtion to estimate the platform trajetory perturbation 4.9. The image before the autofous is shown on the left in Figure 8. The peak is shifted 19

20 Squared Norm of Alternate Relative Projeted Veloity Ambiguity Estimate Atual Ambiguity Resolution Slow Time seonds Figure 4. The estimated projeted veloity in the diretion orthogonal to ms, using the ambiguity funtion. The true value is in solid red and the estimate in solid green. The dotted green line indiates the resolution limits. 6.8 x 1! Wigner Estimate Atual Wigner Resolution Travel Time seonds !1!.8!.6!.4! Slow Time seonds Figure 5. The estimated travel time using the Wigner transform. The true value is in red and the estimate is in solid blue. The dotted blue line denotes the resolution limits. by 1m in ross-range from the target loation. The autofoused image is shown on the right in Figure 8. It peaks near the target loation, indiated with the blak dot. It is also better foused in ross-range. The improved resolution an be seen in Figure 9, where we plot ross-setions of the image in range and ross-range. As expeted from the theory, the range resolution is not affeted by the SAR platform trajetory perturbation. It is the fousing in ross-range that is improved signifiantly by the autofous proess. Let us onsider now a sene with 81 stationary satterers, and a sinusoidal trajetory perturbation that is more diffiult to estimate. We show the perturbation along the range diretion with the blak solid line

21 Range meters 1 1 Range meters Cross Range meters a Image with exat motion ompensation 4 4 Cross Range meters b Image ompensated with motion estimation Figure 6. Images of moving target omputed with the true and estimated veloity Distane from True Trajetory meters Slow Time Figure 7. Error in meters between the true trajetory of the moving target and the trajetory estimated from the data over a single small aperture. in the top left plot in Figure 11, and the imaging sene in the bottom left plot. The estimated trajetory perturbation is shown in blue in the top left piture. It is shifted from the true value, but this shift only affets the range fousing within the resolution limit /B, and it has little effet on the image. The Wigner transform and ambiguity funtion are shown in Figure 1. The entroid is indiated with a blak star. The images before and after autofous are on the top and bottom right in Figure 11. These images are omputed over an aperture of 1km, using the estimates from ten overlapping sub-apertures of 14m. Figure 1 displays the results from another omplex sene, reated from a low-resolution aerial piture of a Stanford flower garden. Eah pixel is treated as a point satterer with refletivity equal to its pixel value and the data is generated with the Born approximation. The aperture is as above, of 1km, onsisting of ten overlapping sub-apertures of 14m. The platform trajetory perturbation is less osillatory here, as shown with the blak solid line in the top left plot. The estimated perturbation is shown in blue. The perfet image is in the bottom left plot, and the unfoussed and autofoused images are in the top right and bottom right 1

22 plots, respetively. The perfet image is for no SAR platform trajetory perturbations Range meters 1 1 Range meters Cross Range meters Cross Range meters a Unfoused Image b Autofoused Image Figure 8. Images of a stationary target before and after autofous. The image windows are the same size but the left one is translated to be enter at the peak of the unfoused image, whih is not at the target loation. The autofoused image is on the right, and the tagret is indiated with a blak dot. 5 No Perturbation Perturbation Autofoused 5 No Perturbation Perturbation Autofoused 1 1 Image Intensity db 15 5 Image Intensity db Cross Range meters a Cross-range resolution Range meters b Range Resolution Figure 9. Cross-range and range resolution of images before and after autofous. The platform perturbations affet only the ross-range fous. The autofous improves the ross-range resolution and gives an image that is essentially idential to the ideal one, without any platform perturbations. 7. Summary and onlusions. We have introdued and analyzed in detail from first priniples a syntheti aperture radar SAR imaging and target motion estimation approah that is ombined with ompensation for radar platform trajetory perturbations. We have formulated an approah that implements the theory for a single target and a single small aperture and extends it to autofous for senes with multiple satterers. This approah an deal with either target motion estimation or with SAR platform trajetory perturbation, but not with both. Combined estimation requires multi-target senes and multiple, overlapping apertures where the basi

23 x 1 7 x T se T se Hz Hz a Wigner transform W Ω, T b Ambiguity funtion AΩ, T Figure 1. The Wigner transform and ambiguity funtion of the range ompressed ehoes from an 81 stationary satterer sene. The asterisk indiates the loation of the entroid. algorithm presented here is used as a omponent in a broader estimation and imaging strategy. In addition to providing detailed analytial error estimates for the approximations that we use, we verify that they are appropriate for the regime that arises in the GOTCHA Volumetri SAR data set. We also assess the performane of this approah with detailed numerial simulations, presented in Setion 6. Aknowledgement The work of L. Borea was partially supported by the National Siene Foundation, grants DMS-648, DMS , DMS-97746, by the Offie of Naval Researh grants N14919 and N , and by Air Fore - SBIR FA865-9-M-153. The work of T. Callaghan was partially supported by AFOSR grant FA The work of G. Papaniolaou was partially supported by AFOSR grant FA and by Air Fore - SBIR FA865-9-M-153. We thank Elizabeth Bleszynski, Marek Bleszynski and Thomas Jaroszewiz of Monopole Researh with whom we ollaborated in the Air Fore SBIR. Appendies Appendix A. The GOTCHA Volumetri SAR parameters. The entral frequeny of the probing signal ft is ν = 9.6Ghz and the bandwidth is B = 6MHz. The SAR platform trajetory is irular, at height H = 7.3km, with radius R = 7.1km and speed V = 5km/h or 7m/s. One irular degree of trajetory is 14m. The pulse repetition rate is 117 per degree, whih means that a pulse is sent every 1.5m, and s =.15s. A typial distane to a target is L = 1km, and we take u 1km/h or 8m/se. Then, we have from the basi image resolution theory [9], in ideal senarios with known target motion and SAR platform trajetory, that the range resolution of the image 1.1 is /B = 48m, and the ross range resolution is λ L/a =.5m, with one degree aperture a and entral wavelength λ = 3m. 3