Simultaneous Navigation and Synthetic Aperture Radar Focusing

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1 Simulaneous Navigaion and Synheic Aperure Radar Focusing Zoran Sjanic and Fredrik Gusafsson Linköping Universiy Pos Prin N.B.: When ciing his work, cie he original aricle. Zoran Sjanic and Fredrik Gusafsson, Simulaneous Navigaion and Synheic Aperure Radar Focusing, 1, IEEE Transacions on Aerospace and Elecronic Sysems, (1),, hp://dx.doi.org/.19/taes IEEE. Personal use of his maerial is permied. However, permission o reprin/republish his maerial for adverising or promoional purposes or for creaing new collecive works for resale or redisribuion o servers or liss, or o reuse any copyrighed componen of his work in oher works mus be obained from he IEEE. hp://ieeexplore.ieee.org/ Posprin available a: Linköping Universiy Elecronic Press hp://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-194

2 1 Simulaneous Navigaion and Synheic Aperure Radar Focusing Zoran Sjanic and Fredrik Gusafsson Fellow, IEEE Absrac Synheic Aperure Radar (SAR) equipmen is a radar imaging sysem ha can be used o creae high resoluion images of a scene by uilising he movemen of a flying plaform. Knowledge of he plaform s rajecory is essenial o ge good and focused images. An emerging applicaion field is real-ime SAR imaging using small and cheap plaforms, where esimaion errors in navigaion sysems imply unfocused images. This conribuion invesigaes a join esimaion of he rajecory and SAR image. Saring wih a nominal rajecory, we successively improve he image, by opimizing a focus measure and updaing he rajecory accordingly. The mehod is illusraed using simulaions using ypical navigaion performance of an unmanned aerial vehicle. One real daa se is used o show feasibiliy, where he resul indicaes ha in paricular he azimuh posiion error is decreased as he image focus is ieraively improved. Keywords: Opimisaion, navigaion, Synheic Aperure Radar, auo-focusing Simulaed SAR image (a) Focused SAR image of poin arges. Simulaed SAR image wih Ampliude = 1 m Simulaed SAR image wih Ampliude =. m (b) Unfocused SAR image of poin arges wih sinusoidal rajecory, Ampliude =. m. Simulaed SAR image wih Ampliude = 1. m I. INTRODUCTION A general mehod for creaing high resoluion radar images from low resoluion radar daa, or real aperure images, is o use relaive moion beween radar anenna and he imaged scene and inegrae all he parial real aperure images aken along he flown rajecory [1]. Tradiionally, his operaion is performed in he frequency domain using FFT-like mehods, [] [], due o heir compuaional efficiency. The common denominaor of hese mehods is ha hey assume ha he aircraf s (or anenna s) flown pah is linear, i.e. wihou cross-rack velociy and acceleraion and wih consan alongrack velociy, and ha is generally no he case in pracice. If he rajecory is no linear he inegraion will resul in an unfocused image. I is possible o parly correc for he deviaion from he nonlinear rajecory bu hen he mehods become compuaionally inefficien. Anoher approach is o perform inegraion in ime domain by means of solving he back-projecion inegral [11]. Even in his process i is assumed ha he radar anenna s flown pah is linear wih consan aliude and heading, bu he mehod can be exended o non-linear racks as well. However exac inversion is no guaraneed. More deails abou his mehod will be provided in Secion II. The main disadvanage of his mehod is he large amoun of operaions required o creae an image, where he complexiy is proporional o O(NKM) for K M pixels image using an aperure wih N posiions. Here, O( ) denoes in he order of. However, by means of coordinae ransformaion, an approximaion o Zoran Sjanic and Fredrik Gusafsson are wih he Division of Auomaic Conrol, Deparmen of Elecrical Engineering, Linköping Universiy, SE Linköping, Sweden, {zoran, fredrik}@isy.liu.se (c) Unfocused SAR image of poin arges wih sinusoidal rajecory, Ampliude = 1 m. (d) Unfocused SAR image of poin arges wih sinusoidal rajecory, Ampliude = 1. m. Figure 1: Example SAR images wih differen perurbed rajecories, where cross-rack deviaion from he linear rajecory is one and a half period sinus wih differen ampliudes. exac back-projecion can be performed, which is called Fas Facorised Back-projecion, see [1]. The complexiy of his algorihm is proporional o KM log N operaions, which for large N implies an imporan saving. Wih his faser algorihm i should be possible o creae images in real ime, possibly in dedicaed hardware. Since back-projecion algorihms are dependen on exac knowledge of he anenna s posiion in order o ge focused images, he image focus can be measured and used for esimaion of he rajecory. An example of his is depiced in Figure 1, where poin arges are imaged. In Figure 1a, a linear pah is simulaed, which resuls in a perfecly focused image. In he oher hree images he variaion in cross-rack posiion was added as A sin(πnk/n) where A = {., 1, 1.}[m], n = 1., k = 1 : N and he images are creaed wih an assumpion ha he pah was linear. The range o he scene is 18 m. This gives unfocused images as depiced. Tradiional mehods for auo-focusing are mosly open-loop ype mehods where eiher SAR images or raw radar daa are used, [13] [9]. The significan common denominaor for all hese mehods is ha he image is creaed wih assumpions on linear fligh rajecory and focusing is

3 Radar IMU GPS Vision IMU GPS Vision Radar IMU GPS Vision Radar NAV NAV SAR Sensor Fusion Navigaion Sae Navigaion Sae SAR image SAR Sensor Fusion Navigaion Sae SAR image SAR image Figure : Top: SAR archiecure where navigaion daa is used in an open-loop manner. Middle: proposed SAR archiecure where navigaion and SAR daa is used ogeher in a decenralised sensor fusion framework. Boom: alernaive SAR archiecure where navigaion and SAR daa is used ogeher in a cenralised sensor fusion framework. done aferwards in an open-loop way discarding evenual fligh pah informaion. This is a consequence of he off-line image generaing process where he rajecory is no longer ineresing. In he seup where SAR images are generaed on-line an idea is o use informaion from he image focus and navigaion sysem, like measured acceleraions, ogeher and in a sensor fusion framework ry o obain he bes soluion o boh image focusing and navigaion simulaneously. This approach can be paricularly useful in he Unmanned Aerial Vehicles (UAV) applicaion where navigaion sysem has lower qualiy. This implies ha he esimaed rajecory has larger error giving unfocused SAR images. In he view of his approach for focusing of SAR images and esimaing he rajecory, he problem is relaed o Inverse Synheic Aperure Radar (ISAR) where he radar is saionary and he ask is o image a moving arge [] [3]. Even in his case, he arge s moion is arbirary and, in addiion, usually no oher measuremens of he moion are available, bu he focusing problem is in principle he same [33] [3]. Also, anoher closely relaed and well known problem is Simulaneous Localisaion and Mapping (SLAM), [36], [37], where a map of he unknown environmen is esimaed joinly wih he plaform s posiion. The SLAM problem has been well sudied during recen years and many differen soluion mehods have been proposed. One mehod ha has been quie successful is o solve he SLAM problem in a sensor fusion framework. In he SAR applicaion, he map of he environmen from SLAM, is he unknown scene ha is imaged and can be seen as he wo dimensional map of poin reflecors. The problem of posiioning he plaform is he same in SLAM. However, he main difference is ha we consider a nonparameric SAR image raher han a parameric map of poin reflecors, ha would be a oo resricive assumpion in SAR imaging. Tha is, hough here are many concepual similariies of join navigaion and mapping, he sae of he ar algorihms canno be applied here. This conribuion applies a sensor fusion framework, where he SAR image ogeher wih a focus measure is inerpreed as a sensor ha conains informaion abou he posiion of he plaform. The image creaing and auo-focusing mehods described above can be illusraed as in Figure. The mehod based on sensor fusion can be implemened in a cenralised or decenralised manner. In his work we focus on he decenralised manner. The ouline is as follows. Secion II summarizes noaion and makes a high-level mahemaical formulaion of he approach. Secion III inroduces he navigaion framework and sysem and measuremen models used. Secion IV describes he image focus measures ha will be used in he auofocus procedure. In Secion V an opimisaion framework and mehods are inroduced and heir usage is explained. Numerical examples for he simulaed images are covered in Secion VI and for he real SAR daa in Secion VII. Finally, conclusions and fuure work are discussed in Secion VIII. II. NOTATION AND PROBLEM FORMULATION The main noaion used in he paper is defined in Table I. Le he complex range compressed raw radar daa be denoed z (R ), also called Real Aperure Radar (RAR) image, where is he ime index, and where z (R ) denoes he reurned radio energy a ime corresponding o disance R o he scene from he radar. The range R is calculaed as R = 1 c τ where c is he speed of ligh and τ is he radar pulse oal propagaion ime o he scene. Noe ha and τ are differen imes, usually called slow and fas ime respecively. Furher, le x denoe he sae vecor of he plaform, which includes posiion and velociy (radar pose). The backprojecion mehod of producing he images, see Figure 3, can be expressed as inegraion per image pixel. For each pixel (i, j) in a complex valued image I, he oal energy from each radar pulse is inegraed by summing all he values from he raw daa given he range from he plaform o he poin in he scene corresponding o he pixel (i, j). This can be expressed as I = N z ( p s ) (1) =1 where p is he 3D posiion of he radar and s is he coordinae of he scene poin ha is mapped o he pixel (i, j) in he image. Now, for a SAR sysem on he UAV plaform, he pose canno be assumed o be known. Insead, we have access o an esimaed posiion ˆp, and he esimaed SAR

4 3 Symbol/Operaor Definiion Time index denoing plaform s ime, used for indexing moion saes like posiion or velociy and azimuh direcion of he RAR image [s] [X, Y, Z ] T Plaform s 3D posiion [m] [v X, vy, vz ]T Plaform s 3D velociy [m/s] [a X, ay, az ]T Plaform s 3D acceleraion [m/s ] x Plaform s saes (consising of posiions, velociies and acceleraions) P Plaform s saes covariance [w X, wy, wz ]T Process noise [m/s ] Q Proces noise covariance y Vecor of measuremens e Vecor wih measuremen noise V Covariance of he measuremen noise T s Sampling ime [s] f( ), h( ) Funcions describing sysem dynamics and measuremens respecively F, G Marices describing sysem dynamics H θ z (R ) I I R Marix describing measuremens Vecor of esimaed parameers Complex raw radar daa (range compressed) (Complex) SAR image Pixel (i, j) in he SAR image Slan range from he plaform a ime o some poin in he imaged scene (usually midpoin) used for indexing range direcion of he RAR image [m]. This range is calculaed from each radar pulse as R = 1 c τ, where c is he speed of ligh and τ is he radar pulse oal raveling ime F General image focus measure E 1 Enropy 1 image focus measure E Enropy image focus measure γ Weighing coefficien ˆx Esimae of x I i Ideniy i i marix i j i j marix wih all zero elemens O(N) Big Ordo noaion, in he order of N N (µ, P ) Gaussian (normal) disribuion wih mean µ and covariance P arg min x A(x) Minimising argumen of A(x) wr x f(x)/ x parial derivaive of f(x) wr x f(x) Gradien of scalar funcion f wr vecor x x -norm of vecor x, x T x x P 1 P 1 -weighed -norm of vecor x, x T P 1 x x Absolue value of x Table I: Definiion of he mahemaical symbols and operaors. image becomes Î = N z ( ˆp s ). () =1 This esimaed SAR image will be ou of focus, since all he conribuions from raw daa will now be scaered due o he error in posiion esimae, see Figure 1 for an example of his. The key idea in his conribuion is o perform a parameric focusing. To enable his, we will make use of a focus measure F(Î), wih he propery ha F(Î) > F(Io ), Î I, (3a) x 1:N = arg min x 1:N F(Î) (3b) R 1 z ( R ) 1 1 where I denoes he rue SAR image and x 1:N he rue sae sequence. We will, however, no opimize he focus w.r.. he pose blindly. We will opimize focus joinly wih he filering probp p p 1 3 p p 4 R 1 R R Scene R 3 + I R 3 s (i,j) SAR Image R 4 R 4 z ( R ) Figure 3: Illusraion of he back-projecion mehod for creaing synheic aperure radar (SAR) images. Real aperure radar (RAR) image, z (R ), where R denoes he range and is he ime he radar pulse is sen. In his work, he posiion (in paricular he azimuh) of he radar a ime, p, is unknown and par of he problem formulaion. Figure is no o scale. lem, in ha he saes obey he sae dynamics and observaions as well as possible. As already noed, building up he image of size KM pixels wih an aperure of N ime poins requires a huge compuaional effor (O(NKM)). I may seem ha an ouer loop ha performs focusing will increase he compuaional burden a leas an order of magniude more. However, we will show ha he gradien of he focus measure can be compued efficienly. Firs, le θ denoe he parameers used o describe he sae rajecory. I may be he whole posiion rajecory, θ = p 1:N, or he iniial saes only, θ = x, as wo special cases. Then, define I, = z (R ) and, as before, R = p s, so ha (1) can be wrien I = N =1 I,. Then, using he chain rule, we ge F(θ) θ = F I, I, R R R θ F I, R I, R θ The approximaion ha allows he dependence of on he range o be removed is based on small scene assumpion: he range R o he middle of he scene does no vary much over he whole scene indexed by. There are several advanages of using his approximaion: The firs and hird parial derivaives can be derived analyically. The second one can be compued almos for free from any SAR algorihm, and requires only KM exra memory cells and one or wo more SAR image compuaions, R z (4)

5 4 depending on how he numerical gradien I/ R is approximaed. More concree formulaions of (4) come laer when he focus measures are inroduced. In order o evaluae he performance of he esimaion mehods, some performance measures are needed. A popular measure for he parameer esimae is Roo Mean Square Error (RMSE) defined as N k=1 RMSE(ˆθ) = (ˆθ k θ ) () N where ˆθ 1,..., ˆθ N are he unbiased esimaes of he rue scalar parameer θ. To assess he qualiy of he obained SAR images, he power of he error image can be used. This can be defined as K M i=1 j=1 P = Î I (6) KM where Î is he K M complex SAR image obained wih he esimaion procedure and I is he perfec focused SAR image, i.e. creaed wih he rue rajecory. Parameer Meas. Error (1-σ) Sa. Error (1-σ) Posiion 3 m.93 m Velociy.4 m/s.1 m/s Acceleraion.6 m/s.1 m/s Table II: Measuremen error and saionary measuremen error for he navigaion parameers where T s is he sampling ime in seconds, X is he posiion in azimuh direcion and Y is he posiion in range direcion in meers, v X and v Y are he velociies in he X- and Y - direcions in meers per second respecively and a X and a Y are he acceleraions in X- and Y -direcions in meers per second squared respecively. This model is used for he whole rajecory. Since his model is linear and ime invarian he saionary Kalman filer can be used o esimae x giving ˆx and is corresponding covariance P. B. Navigaion Performance Due o he fac ha he sysem is ime invarian and linear, he covariance of he esimae will converge o he saionary covariance P. This covariance can be calculaed as III. NAVIGATION FRAMEWORK An Inerial Navigaion Sysem (INS) in an aircraf inegraes acceleromeer and gyroscope daa and correcs he sae wih aiding sensors such as baromeer and GPS using a general dynamics and measuremen equaions x +1 = f(x, w ) y = h(x ) + e (7a) (7b) where x are saes of he sysem, w denoes he process noise wih variance Q, e is measuremen noise wih variance V and y are he measuremens. Usually an Exended Kalman filer is applied o esimae he saes, see e.g. [38]. In his work, a simplified, ye useful, model of he dynamics will be assumed which will give simpler expressions in he algorihms. A. Aircraf Model In his seup, he following -DOF linear INS ime discree dynamics is used, [38], and i is assumed ha all saes are measured by GPS and inerial measuremens up o he ime synheic aperure is sared x +1 = F x + Gw F = I Ts T s I I I T s I I T 3 s 6 I G = T s I T s I (8a) (8b) (8c) x = [X Y v X v Y a X a Y ] T (8d) w = [w X w Y ] T (8e) y = I 6 x (8f) P = F P F T F P H T (H P H T + V ) 1 H P F T + GQG T (9) where F and G are defined above, and H is as menioned before chosen as H = I 6, since we assume ha all saes are measured by he navigaion sysem. For a ypical navigaion sysem used in an UAV, he accuracy for he posiion, velociy and acceleraion measured by he GPS and INS can be summarised according o Table II. Sysem noise covariance, Q, which represens disurbance on saes, like wind urbulence, can be aken as diag{.,.} [m /s 4 ]. Wih hese values, he saionary sandard deviaion is given in he hird column in Table II. IV. FOCUS MEASURES We here review and compare wo common focus measures. A. Two Enropy Measures One common focus measure in SAR or image processing lieraure is image enropy calculaed as 6 E 1 (I) = q k log (q k ) () k=1 where q k is an approximaed grey level disribuion of he K M grey-scale image I, where I is he complex-valued SAR image. I can be obained from he image hisogram calculaed as q k = {# of pixel values I } [k 1, k] KM k [1, 6]. (11a) (11b) The more focused he image is, he lower he enropy is, see for example [1]. Hisograms for he images in Figure 1 are given in Figure 4. Noe he log-scale on he y-axis. An alernaive

6 6 Hisogram of he focused SAR image Hisogram of he unfocused SAR image, A =. m 6 Simulaed SAR image Simulaed SAR image log(q k ) 4 3 log(q k ) Normalised inensiy (a) Hisogram of he focused image. Hisogram of he unfocused SAR image, A = 1 m Normalised inensiy (b) Hisogram of he unfocused image wih Ampliude =. m. Hisogram of he unfocused SAR image, A = 1. m 6 (a) SAR image of he srucured scene. (b) SAR image of he unsrucured scene. Figure : SAR images used for he illusraion of he Enropy 1 and focus measures. log(q k ) Normalised inensiy (c) Hisogram of he unfocused image wih Ampliude = 1 m. log(q k ) Normalised inensiy (d) Hisogram of he unfocused image wih Ampliude = 1. m. Figure 4: Hisograms for he images in Figure 1. Noe he log-scale on y-axis. Range direcion acceleraion error [m/s ] Enropy 1.. Azimuh direcion velociy error [m/s] (a) Enropy 1 focus measure for he srucured scene. Range direcion acceleraion error [m/s ] Enropy.. Azimuh direcion velociy error [m/s] (b) Enropy focus measure for he srucured scene. definiion of enropy, and more commonly used in he SAR conex, is, [1], [34], [39], K M E (I) = q ln(q ) (1a) q = i=1 j=1 B. Focus Measure Performance I K i=1 M j=1 I. (1b) Enropy 1 and focus measures are esed and compared on SAR images according o Figure and he resuls are depiced in Figure 6 where sandard deviaions 1 σ, σ and 3 σ are also drawn. These images are chosen since hey represen boh srucured and unsrucured scenes and hey are also small enough o saisfy he small scene assumpion needed for he gradien calculaion in (4). On op of ha, a small image is faser o calculae, giving he faser opimisaion calculaion in urn. As an addiional moivaion for using hese small arificial scenes, anoher, larger and more realisic scene is examined, see Figure 7a. The scene is creaed by using a real SAR image over par of Washingon DC and creaing raw daa from i. Afer ha he same simulaion is performed as for he small scenes and Enropy 1 and measures are given in Figure 7b and 7c. Since he principal form of he enropy measures is essenially he same, more horough examinaion will be done on he wo small scenes due o above menioned reasons. In all hese simulaions he sae noise in model (8) is se o zero, i.e. he rajecory is compleely deerminisic. This is done in order o illusrae he focus measure funcions F i in a wo dimensional plo, since he rajecory, and consequenly he focus measure, is hen only dependen on he iniial values. In all hese figures i can be seen ha enropy has a convex and prey nice behaviour around he rue value of Range direcion acceleraion error [m/s ] Enropy 1.. Azimuh direcion velociy error [m/s] (c) Enropy 1 focus measure for he unsrucured scene. Range direcion acceleraion error [m/s ] Enropy.. Azimuh direcion velociy error [m/s] (d) Enropy focus measure for he unsrucured scene. Figure 6: Focus measures for he image in Figure wih sandard deviaion ellipses. he iniial sae. However i looks very fla along he velociy direcion which indicaes ha i is very difficul o esimae ha paricular sae. The enropy 1 measure has, on he oher hand, a sharp minimum for he correc value of he iniial sae bu many local minima. This means ha he wo enropy measures complemen each oher perfecly, and can be used in combinaion o obain he global minimum of he focus measure. V. SEARCH METHODS As demonsraed in Secion IV-B, he enropy measure can be used as a coarse firs sep in he opimizaion o come close o he global minimum, and hen enropy 1 can be used o obain he global minimum. Noe ha a special srucure of he problem (8) allows for unconsrained soluion of he problem. This is due o he fac ha he consrains represening he rajecory can be aken ino accoun while calculaing he gradien of he cos funcion as will be demonsraed in Secion V-C.

7 6 Range direcion acceleraion error [m/s ] 6 7 Simulaed SAR image (a) An example of a larger SAR image, aken from a larger real SAR image over Washingon DC Enropy 1.. Azimuh direcion velociy error [m/s] (b) Enropy 1 focus measure for he srucured scene. Range direcion acceleraion error [m/s ] Enropy.. Azimuh direcion velociy error [m/s] (c) Enropy focus measure for he unsrucured scene. Figure 7: SAR image over more realisic scene and is Enropy 1 and focus measures wih sandard deviaion ellipses. SAR image over Washingon DC is from Sandia Naional Laboraories. A. Join Opimizaion of Trajecory and Focus Since he focus of he image depends on he unknown rajecory, one soluion is o solve he following minimisaion problem ˆθ = arg min g(θ) θ g(θ) =γ F E i (x :N )+ γ s ( N =1 subjec o θ = [x T, w T 1:N ] T y h(x )) V 1 x +1 = f(x, w ) + w Q 1 ) (13a) (13b) (13c) (13d) where γ F and γ s are he weighs (and γ F + γ s = 1) and measuremen equaion h(x) and sysem dynamics f(x, w) are defined as in Secion III. In Equaion (13b), E i (x :N ), i {1, }, is a funcion of he SAR image I creaed from he radar measuremens and is of he ype how focused is he image? according o Secion IV. B. Gradien Search Gradien search mehods will be exemplified here wih a couple of examples wih differen rajecories and errors in hem. Only wo saes and heir iniial values are considered (v X and a Y ), for illusraive purposes. In general, he minimisaion should be applied for hese saes for all or a leas some of he ime insans along he rajecory. Such an example will be sudied laer. A gradien search can, for he general problem (13a), be formulaed as θ k+1 = θ k + µ k H(θ k ) 1 g(θ k ) g(θ) = θ g(θ) (14a) (14b) where µ k is sep size wih µ = 1 and H(θ) is some (posiive definie) marix. The sep size is used o ensure ha each ieraion acually gives a decreasing value of he funcion. The iniial esimae, θ, can be aken as he usual esimae from he navigaion sysem. In he simples case H can be chosen as he ideniy marix and he procedure becomes a pure gradien search. The disadvanage of such procedure is he slow convergence, especially if he funcion o be minimised is ridge-like like enropy 1 focus measure. If H is chosen as he Hessian of g, he procedure becomes a Newon search. The Newon search has a fas convergence, and is o prefer if he Hessian is available. In many cases he Hessian is eiher no available or very difficul o obain, as in he case considered here, and some approximae mehods mus be applied. One opion is a quasi-newon mehod, and BFGS in paricular, where he Hessian is approximaed by uilising gradiens of he funcion during he search, see []. The general gradien search procedure is summarised in Algorihm 1. In all hese procedures i is essenial o obain he gradien of he loss funcion. Because of he special srucure of he focus measure funcion and he SAR processing algorihm, he complee analyical gradien is hard o obain. For example, for he enropy 1 measure i is hard o differeniae a hisogram of he image. In his case numerical mehods mus be used. However, for he enropy measure i is possible o obain analyical expressions for mos par of he gradien. Numerical gradien calculaion requires ha one new SAR image is creaed for each parameer and sep in he inner loop in Algorihm 1 in each ieraion. The analyical gradien requires only one new image in he inner loop execuion per ieraion. The SAR image creaion process is in his case main boleneck when i comes o compuaion ime, since gradien calculaion is very fas compared o i. How gradien for enropy is calculaed will be described in he nex subsecion. C. Calculaing he Gradien The calculaions o obain an analyical gradien of he enropy funcion will be presened. The key o doing his is he chain rule for gradien calculaion (see also Secion II and Equaion (4)), E θ = E q q I I R R θ. (1) In order o apply he chain rule, firs he decomposiion chain of he enropy focus measure will be considered and hen, all parial derivaives will be presened.

8 7 Algorihm 1 Gradien search procedure Inpu: Iniial value of he opimisaion parameers θ, raw radar daa, olerance hresholds ε 1, ε, ε 3 Oupu: Soluion ˆθ, focused SAR image X Xm Z Y Rm Plaform R k := repea Calculae gradien of he cos funcion, g(θ k ) Calculae (approximae) Hessian, H(θ k ) µ k := 1 repea θ k+1 := θ k µ k H(θ k ) 1 g(θ k ) µ k := µ k / unil g(θ k+1 ) < g(θ k ) k := k + 1 unil θ k θ k 1 < ε 1 or g(θ k 1 ) < ε or g(θ k ) g(θ k 1 ) < ε 3 The firs facor o be differeniaed is enropy focus measure E = K i=1 j=1 M q ln q = KM i=1 q i ln q i (16) where las equaliy is simply reformulaion of he double sum by vecorising he image. In he second facor, each q i is obained by q i = I i j I j (17) meaning ha q i is a funcion of he absolue value of he complex-valued SAR image. For he hird facor, we need o obain he derivaive I / R. In he creaion of he image, a back-projecion sum is evaluaed and all parial images are summed up. Each parial image is a funcion of one column in he RAR image and he range from he plaform o each pixel in he SAR image, see Secion II. Unforunaely i is no easy, if no impossible, o obain analyical expression for he derivaive I / R. However, his value can simply be obained during image creaion by means of numerical derivaion. The cos for ha procedure is memory demand and execuion ime which boh are doubled. Bu his increase in cos is independen of he sae, and hus consan no maer how many parameers opimisaion is performed over. A sraighforward numerical gradiens would give a cos ha is increasing linearly wih he number of parameers. The las facor ha needs o be calculaed is he gradien of he range as a funcion of he saes, R(x ). To calculae an analyical expression of his funcion some SAR geomery preliminaries are needed. In order o express range as a funcion of he saes, he geomery seup as in Figure 8 can be considered. From he figure i can be seen ha he range R g Poin Targe Imaged Scene Figure 8: SAR geomery. The figure is no o scale. R can, wih help from Pyhagoras heorem, be expressed as R = (X m X ) + (R g Y ) + (Z Z ) (18a) R g = Rm Z (18b) i.e. as a funcion of he rajecory. Noe ha Z, he aliude of he plaform, is assumed o be known here. This can be achieved by measuring i wih, for example, baromeric sensors which is always done in he aircraf applicaions. Here he exac expression for he range along he rajecory is used, unlike in mos of he SAR lieraure, where approximae and linearised expressions are used, see for example [3]. This is due o he fac ha in low frequency SAR applicaion, as he one considered here, he raio beween range and rajecory lengh is no negligible due o he lobe widh. If approximae expressions are used, oo large errors would be inroduced in he beginning and he end of he rajecory. Nex, he dynamical model (8) can be used o express he posiion saes used in he range expression above as a funcion of any oher sae by using ha x = F k x k, > k (19) Noe ha he sae noise erm, w, is negleced in he following since i is equivalen o opimise over noise and over acceleraion saes, a, so he laer one is used here o simplify he expressions and reduce he amoun of variables in he problem. From (19) any posiion can be expressed explicily as X = X + T s ( k)vk X + T s ( k) Y = Y + T s ( k)vk Y + T s ( k) a X k a Y k (a) (b) If hese expressions are used in (18a), we can easily obain parial derivaives of he range wih respec o he velociies and acceleraions in arbirary ime poins. Now, we have everyhing needed o calculae he gradien of he focus measure wih respec o he rajecory saes. The

9 8 Parameer Radar cener frequency Aperure lengh (X m) Ground range o arge (R g) Nominal aliude (Z ) Nominal speed (v X) Value 3.1 MHz 77 m 18 m m m/s Table III: SAR parameers used for he simulaed daa. parial derivaives are, in urn (for > k), E = ln q i 1 (1a) q i q I j I I j I i i I j = (, i = j I ) I i (1b) (, i j I ) R vk X = (X m X )T s ( k) R (1c) R vk Y = (R g Y )T s ( k) R (1d) R a X k = (X m X )Ts ( k) R (1e) R = (R g Y )Ts ( k) R (1f) a Y k and I / R is numerically calculaed during image formaion. Now, a leas for enropy focus measure, we can calculae he gradien (semi-) analyically and use i in he minimisaion procedure. The second erm in (13b) is easy o differeniae, since i is a quadraic form and h(x) is a linear funcion in his case. VI. NUMERICAL EXAMPLES FOR SIMULATED IMAGES In order o demonsrae he behaviour of he gradien search for his seup, he SAR image from Figure a is used in wo differen experimens. In hese simulaions SAR parameers according o Table III are used. A. Two-Dimensional Opimizaion To be able o illusrae he convergence of he soluion, only wo opimisaion variables are considered here, θ = [v X, a Y ] T and he algorihm is iniiaed wih random saring poins θ based on he saionary covariance of he saes in he sysem. Those iniial values are [.,.] T, [99.99,.1] T, [99.99,.] T, [.,.1] T and [.,.3] T. The rajecories generaed wih hese iniial values are illusraed in Figure 9. In Figure a, he gradien search based enropy measure is illusraed and we can see ha he soluions converge o he fla ridge-like area close o he correc acceleraion, bu no necessarily o he correc velociy. In Figure b, he gradien search where he enropy 1 measure is used is depiced. In his case he algorihm is iniiaed wih he soluion from he enropy search. I can be seen ha his minimisaion sraegy works prey well, alhough one soluion is suck in a local minimum. In ha case he velociy error is he larges one of all errors. Noe also ha only he focus measure is used o find esimae of he saes i.e. γ s is se o zero while γ F is se o one in Equaion (13b). Range direcion [m] Example rajecories T 1 1 Azimuh direcion [m] T1 T T3 Figure 9: Trajecory examples for differen iniial values of he velociy, v X and acceleraion, a Y. Range direcion acceleraion error [m/s ] Azimuh direcion velociy error [m/s] T4 Enropy measure and search rajecories T (a) Search rajecory for five differen values of x using enropy focus measure. Range direcion acceleraion error [m/s ] 3 x T1 T3 T T4 Enropy 1 measure and search rajecories T T1 T T3 T Azimuh direcion velociy error [m/s] (b) Search rajecory for enropy 1 focus measure wih x given by he enropy gradien search. Figure : Search rajecory for five differen values of x using wo differen enropy measures.

10 9 Simulaed SAR image Simulaed SAR image Simulaed SAR image Simulaed SAR image (a) Image creaed wih error in velociy of. m/s and in acceleraion of.1 m/s. (b) Image creaed wih error in velociy of.14 m/s and in acceleraion of.3 m/s. Simulaed SAR image (a) SAR image of he srucured scene creaed wih noisy posiion daa. (b) SAR image of he unsrucured scene creaed wih noisy posiion daa. Figure 1: SAR images creaed wih noisy posiion daa. (c) Focused image as a reference. Figure 11: Resuling images from he minimisaion procedure wih saring poin [.,.1] T. I is ineresing o see how he image creaed wih he soluion ha is suck in he local minimum of he enropy 1 measure looks like compared o he unfocused image ha is iniialised wih. As illusraed in Figure 11, i can be seen ha he image creaed wih values from he minimisaion procedure is very close o he focused image and much beer han he unfocused images ha are iniialised wih. The probable explanaion for his comes from he fac ha small azimuh direcion velociy errors do no influence he final image much due o he quanisaion effecs. However he esimae of he navigaion saes is no correc. B. High-Dimensional Opimizaion In he second example a more realisic seup is done. The opimisaion problem o be solved is ˆθ = arg min θ subjec o γ F E 1, (x :N ) + γ s N =1 a my θ = [v X, a Y, a Y N/4, ay N/, ay 3N/4 ]T γ F =.99, γ s =.1 a Y V 1 (a) (b) (c) Parameer RMSE (op. wih E ) RMSE (op. wih E 1 ) ˆv X 7. 3 m/s m/s â Y m/s m/s â Y N/ m/s m/s â Y N/ m/s m/s â Y 3N/ m/s m/s Mean value of he error image power Table IV: RMSE for he esimaed parameers and he mean value for he error image power for he srucured scene. E 1, (x :N ) is eiher enropy or enropy 1, exacly as in he previous example. Here i is assumed ha a change in Y -direcion acceleraion will behave in a sep like manner only a few imes during he SAR image generaion and ha he ampliude of he sep is arbirary. I is also assumed ha he acceleraion in X-direcion will vary slowly due o he plaforms inheried ineria in his direcion, so i can be assumed o be zero. The meaning of P in (d) is ha here is no prior informaion abou he iniial values of he rajecory. Anoher spline-like inerpreaion of he seup in () is o find a bes rajecory by fiing he second order polynomials beween four poins evenly spaced along he rajecory. Boh scenes from Figure are used and Mone Carlo simulaions are performed in order o evaluae he performance of he esimaion procedure. The resuling RMSE of he parameers and he mean value of he error image power are presened in Table IV and Table V for boh srucured and unsrucured scene. Here, he acual acceleraion is presened insead of he process noise value, since i is more physically ineresing. I can be noiced ha x +1 = F x, [X, Y, v Y, a X ] T = 4 1, P = I (d) a X =, { : N } (e) a Y, { : N/4 1 } a Y a Y N/4 =, { N/4 : N/ 1 } a Y N/, { N/ : 3N/4 1 } (f) a Y 3N/4, { 3N/4 : N } where a my is he measured acceleraion in Y -direcion wih addiive whie Gaussian noise wih V =. m /s 4. Parameer RMSE (op. wih E ) RMSE (op. wih E 1 ) ˆv X m/s m/s â Y m/s.98 4 m/s â Y N/ m/s 6. 4 m/s â Y N/ m/s m/s â Y 3N/ m/s m/s Mean value of he error image power Table V: RMSE for he esimaed parameers and he mean value for he error image power for he unsrucured scene.

11 Simulaed SAR Image (a) Image of he srucured scene afer minimisaion wih enropy as focus measure. Simulaed SAR Image (c) Image of he unsrucured scene afer minimisaion wih enropy as focus measure. Simulaed SAR Image (b) Image of he srucured scene afer minimisaion wih enropy 1 as focus measure. Simulaed SAR Image (d) Image of he unsrucured scene afer minimisaion wih enropy 1 as focus measure. Figure 13: Resuling images from he gradien search minimisaion. Parameer Srucured scene Unsrucured scene ˆv X m/s.13 3 m/s â Y m/s m/s â Y N/ m/s m/s â Y N/ m/s. 4 m/s â Y 3N/ m/s m/s Error power image Table VI: Error in he esimaed parameers for he wo scenes afer enropy 1 minimisaion procedure. These are he parameer values used o creae images in Figure 13b and Figure 13d he improvemen of he RMSE afer furher minimisaion wih enropy 1 is no very big, i is in he magniude of. I suggess ha he exra sep of minimisaion wih enropy 1 can be skipped if a faser procedure is sough. In Figure 1, a noisy posiion (one of he noise realisaions) is used for he image generaion. We see ha boh images are unfocused and he image of he unsrucured scene is prey bad, all he dominan arges are blurred. In Figure 13 he images afer minimisaion wih enropy and 1 are depiced (for he same noise realisaion as above). Here i can be seen ha any improvemen in he image wih exra minimisaion wih enropy 1 is impossible o see wih he naked eye, i.e. he improvemen of he navigaion saes does no visibly improve he images. This could be expeced from he resuls from MC simulaions. The resuling esimaes of he parameers and error image power afer enropy 1 minimisaion for he wo scenes and his paricular realisaion of he noise are presened in Table VI. For hese simulaion examples he average amoun of ieraions were abou for he enropy case and 7 for Parameer Radar cener frequency Aperure lengh (X m) Ground range o arge (R g) Nominal aliude (Z ) Nominal speed (v X) Value.187 MHz 1187 m 88 m 1 m 18 m/s Table VII: SAR parameers used for he real daa. he enropy 1 case. Addiional sep size calculaions were abou per ieraion giving and 14 gradien calculaions in oal. As previously menioned, he main compuaional speed boleneck is he SAR image creaion in each ieraion and for he used images i is abou 1 minue. Tha gives an average ime of abou minues for enropy measure and 7 minues for he enropy 1 measure (since here are parameers in oal). This ime can be decreased if faser image creaion procedure is used. VII. EXAMPLE WITH REAL SAR IMAGE Here, we illusrae he esimaion resuls using daa from he CARABAS II sysem [41] colleced in wesern Sweden. The rajecory and he SAR image obained by he proposed esimaion mehod are compared o he image creaed wih he GPS based rajecory, which is assumed o be he ground ruh. The SAR image used for he esimaion is illusraed in Figure 14a where he GPS based rajecory is used o generae he image. SAR parameers used for he real daa are given in Table VII. For he real daa case, he opimisaion problem o be solved is formulaed according o ˆθ = arg min θ subjec o γ F E (x :N ) + γ s N a m =1 θ = [v X, v Y, a X, a Y, a X i, a Y i ] T i { : 199}N/ γ F =.36, γ s =.64 a V 1 x +1 = F x, [X, Y ] T = 1, P = I 4 a, i { : N/ 1 } a N/, i { N/ : N/ 1 } a i =. a 199N/, i { 199N/ : N } where he variables are defined as a mx a = [a X, a Y ] T a m (3a) (3b) (3c) (3d) (3e) (3f) (4a) = [a mx, a my ] T (4b) V = diag{v X, V Y } = diag{.1,.1}[m /s 4 ] (4c) and a my are he acceleraions of he plaform in X- and Y -direcions measured by he navigaion sysem. Noe ha in his case only he enropy focus measure is used due o he compuaional load o calculae he numerical gradien for he enropy 1 measure. However, according o he resuls in Secion VI, he improvemen of he esimaes by using

12 11 Range pixels SAR image (real daa, GPS rajecory) Range pixels SAR image (real daa, esimaed rajecory) Cos funcion Toal cos funcion Focus measure Focus measure FM (esimaed) FM (GPS based) 6 1 Azimuh pixels (a) SAR image for he real daa case creaed wih he GPS based rajecory (assumed o be ground ruh). Error [m] Azimuh pixels (b) SAR image for he real daa case obained wih he proposed opimisaion procedure. Figure 14: SAR images for he real daa case. Error beween GPS and esimaed rajecory X posiion Y posiion 6 8 Time [s] (a) Error in posiion beween GPS based and rajecory esimaed wih he proposed opimisaion procedure. Error [m] 1 Error beween GPS and iniial rajecory X posiion Y posiion 6 8 Time [s] (b) Error in posiion beween GPS based and iniial (inerial only based) rajecory. Figure 1: Error in posiion for beween GPS-based and esimaed or iniial rajecory respecively. addiional opimisaion wih enropy 1 is small and herefore i is omied here. Also, he weighs and he covariance of he acceleraion measuremens are seen as uning parameers. Resuls from he opimisaion procedure, which akes five seps o converge, is illusraed in Figure 14b, where he esimaed rajecory is used o generae he image and Figure 1a where error beween GPS and esimaed rajecory is shown. Tha error can be compared o he error in he rajecory wih he iniial values of he parameers, θ, shown in Figure 1b. I can be seen from hese wo plos ha improvemen in Y - direcion is much less han improvemen in X-direcion. The oal loss funcion (3a) and enropy measure as a funcion of ieraion number defined hrough Algorihm 1 are depiced in Figures 16a and 16b, respecively. The image resuling form he esimaed rajecory is hard do disinguish from he GPS based rajecory image, excep ha i is shifed in range direcion. This ambiguiy is, unforunaely, unobservable in he auo-focusing process, i.e. he mehod is invarian o he ranslaion of he image. Compuaional ime in he real daa case is five ieraions and abou gradien calculaions in oal (including inner loop in he Algorihm 1). Time for processing he SAR image was minues which gives oal ime of abou minues. VIII. CONCLUSIONS AND FUTURE WORK An ieraive opimisaion mehod is presened based on a decenralised sensor fusion framework, which is inended o provide beer focused SAR images on fuure cheap and small SAR plaforms. The approach is based on joinly opimizing a focus measure and he error in he navigaion saes. As was concluded from simulaion examples of he simple scene and Ieraion number (a) Toal cos funcion, defined in Equaion (3a), as a funcion of he ieraion number Ieraion number (b) Enropy value, defined in Equaion (1), as a funcion of he ieraion number. Enropy value for he GPS-based image is also shown for a comparison. Figure 16: Error in posiion for beween GPS-based and esimaed or iniial rajecory respecively. real SAR daa, he mehod works fairly well, alhough no all he saes are observable and some errors in hese are sill presen. Neverheless, even if he whole navigaion sae vecor can no be correced, he resuling SAR image is much more focused afer opimizaion han he original one. An imporan heoreical conribuion, o reach he requiremens on compuaion complexiy, is an analyical expression for he gradien of a focus measure. The resul enables an implemenaion of gradien search algorihms ha adds only marginal complexiy o he SAR image creaing process. In he derivaion of he focus measure gradien needed for he opimisaion process, a small scene approximaion is used. Since SAR images can be quie large, an exension of using several small images from one large, each wih differen range, can be applied. This migh give more observabiliy of he parameers in he loss funcion based on enropy. ACKNOWLEDGMENTS The auhors would like o hank Lars Ulander and Anders Gusavsson from Swedish Defence Research Agency (FOI) in Linköping, Sweden, for providing real CARABAS II daa and help wih hese. Our hanks go also o Hans Hellsen from Saab Elecronic Defence Sysems in Göeborg, Sweden, for all help wih he SAR sysem heory. This work was suppored by he Indusry Excellence Cener LINK-SIC founded by The Swedish Governmenal Agency for Innovaion Sysems, VINNOVA and Saab AB. REFERENCES [1] L. J. Curona, W. E. Vivian, E. N. Leih, and G. O. Hall, A highresoluion radar comba-surveillance sysem, IRE Transacions on Miliary Elecronics, vol. MIL-, no., pp , April [] I. Cumming and J. R. Benne, Digial processing of Seasa SAR daa, in Acousics, Speech, and Signal Processing, IEEE Inernaional Conference on ICASSP 79., vol. 4, Apr 1979, pp [3] J. A. Fawce, Inversion of N-Dimensional Spherical Averages, SIAM Journal on Applied Mahemaics, vol. 4, no., pp , 198. [Online]. Available: hp:// [4] F. 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