THE time-domain backprojection (BP) algorithm has already

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1 1460 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 014 A Fast BP Algorithm With Wavenumber Spetrum Fusion for High-Resolution Spotlight SAR Imaging Lei Zhang, Member, IEEE, Hao-lin Li, Zhi-jun Qiao, Member, IEEE, and Zhi-wei Xu Abstrat This letter presents the aelerated fast bakprojetion (AFBP) algorithm for high-resolution spotlight syntheti aperture radar (SAR) imaging. In onventional fast bakprojetion (FBP) algorithms, image-domain interpolation is employed in the subaperture (SA) fusion. However, in AFBP, by using a unified polar oordinate (UPC) system, the interpolation-based fusion is substituted by fusing the SA spetra in the wavenumber (WN) spetrum domain. The WN-domain SA fusion is effiiently implemented by fast Fourier transform and irular shifting. In this letter, an aurate impulse response funtion and the WN spetrum expression of the bakprojeted image in the UPC are expliitly derived, and furthermore, the implementations of AFBP are investigated in detail. Compared with onventional FBP algorithms, the AFBP an preisely fous on high-resolution SAR data with dramatially improved effiieny. Both simulation and real-measured data experiments validate the superiorities of AFBP by omparing it with the fast fatorization bakprojetion (FFBP) algorithm. Index Terms Aelerated fast BP (AFBP), fast bakprojetion (FBP), fast fatorized bakprojetion (FFBP). I. INTRODUCTION THE time-domain bakprojetion (BP) algorithm has already been reognized as an ideal approah for highresolution syntheti aperture radar (SAR) imaging 1] 3]. Rooted in the basi priniple of wideband beamforming with the -D oherent integral, the BP algorithm offers more advantages than traditional SAR image formations, suh as the preise ompensation of urve wave-front effet for any SAR onfiguration optimal aommodation of topography and aperture-dependent motion ompensation 4] 7]. However, the intensive omputational burden of the original BP algorithm makes it unaeptable in many real-time imaging senarios. Hene, great efforts have been devoted to enhaning the effiieny of the BP integral. Several novel fast BP (FBP) algorithms have been reently developed 8] 10], for instane, the fast fatorized bakprojetion (FFBP) algorithm 9] is one of them. The key idea of the FFBP algorithm is to split the Manusript reeived September 16, 013; revised Deember, 013 and Deember 10, 013; aepted Deember 13, 013. This work was supported in part by the Fundamental Researh Funds for the Central Universities under Grant K and in part by the National Natural Siene Foundation of China under Grant , Grant , and Grant L. Zhang, H. Li, and Z. Xu are with the National Laboratory of Radar Signal Proessing, Xidian University, Xi an , China ( leizhang@xidian.edu.n; lihaolin3@163.om; zwxu1990@gmail.om). Z. Qiao is with the Department of Mathematis, University of Texas Pan Amerian, Edinburg, TX USA ( qiao@utpa.edu). Color versions of one or more of the figures in this paper are available online at Digital Objet Identifier /LGRS long syntheti aperture into short subapertures (SAs), eah of whih generates a oarse angular resolution image in a loal polar oordinate (LPC) system. The oarse images are then reursively fused by image-domain interpolation. In a pyramid omputational arhiteture, the FFBP dramatially redues the omputational burden of the original BP algorithm, whih is lose to the theoretial effiieny of onventional frequenydomain algorithms. In the FFBP, however, the LPC system is adopted for an SA image, where the SA enter is defined as the origin. Hene, oordinates of a ertain target vary from one SA image to another. As a result, in the SA fusion of the FFBP, interpolation in both range and azimuth is neessary to fuse all SA images. The -D pixel-by-pixel interpolation brings two outstanding negative effets: 1) the main soure of the omputational burden of the FFBP lies in the -D interpolation; and ) the interpolation also brings inevitable errors in eah FFBP reursion, and those errors would be transferred and aumulated during the reursions, whih lead to foal degradation of the final SAR image. In partiular, with the derease in the SA length, the foal degradation beomes serious, whih instintively yields a tradeoff between preision and effiieny in FFBP imaging. Although some optimal interpolation kernels 11] are able to improve the performane of the FFBP, the negative effets indued by the -D interpolation are still distintive. In this letter, the aelerated FBP (AFBP) is proposed for the high-resolution spotlight SAR imaging. Different from the reursive arhiteture of the FFBP, the AFBP only ontains two stages: SA image formation using a unified polar oordinate (UPC) and SA fusion in the -D wavenumber (WN) domain. By the formation of all SA images in the UPC, the interpolation-based SA fusion in the FFBP is substituted by a 1-D spetrum fusion in the -D WN domain. Based on the rigid derivation of the impulse response funtion (IRF) in the BP image under the UPC, implementation of the AFBP is investigated in detail. The SA fusion of the AFBP is effiiently realized by fast Fourier transform (FFT) and irular shifting. As the -D interpolation within the FFBP SA fusion is avoided, the AFBP obtains signifiant improvements in both effiieny and preision. Experimental omparisons for the FFBP and the AFBP are presented by using real-measured SAR data set. II. DERIVATION OF THE AFBP ALGORITHM A. Introdution of the FFBP Algorithm Here, we introdue the signal model and the basi priniple of the BP integral. The spotlight SAR geometry in the polar oordinate system is shown in Fig. 1, where the SAR platform moves along a straight-line flight trak with a onstant veloity v, and a syntheti aperture of length L is generated. The straight X 014 IEEE. Personal use is permitted, but republiation/redistribution requires IEEE permission. See for more information.

2 ZHANG et al.: FBP ALGORITHM WITH WN SPECTRUM FUSION FOR HIGH-RESOLUTION SPOTLIGHT SAR IMAGING 1461 Fig. 1. Geometry of spotlight SAR in the polar oordinate geometry. trajetory is defined as the X-axis with aperture entered at origin O. During data aquisition, the radar beam always illuminates the sene enter. When the antenna phase enter loates at X = vt L/,L/) (where t denotes the slow time), the instantaneous range from the radar to a point target P in the polar oordinates (r p,θ p ) is given by R(X;r p,θ p )= rp +X r p X sin θ p, L X<L. (1) Defining the azimuth angular oordinate p =sinθ p, we an rewrite the range history as R(X;r p, p )= rp + X r p X p, L X<L. () Assuming that the point target has unit sattering oeffiients, then after demodulation, we have the ehoed signal of the target, i.e., ] X s(τ,x) = ret s T (τ Δt p ) (3) L where Δt p =R(X; r p, p )/ orresponds to the time delay, is the speed of light, and s T (τ) is the transmitted waveform with bandwidth B and arrier frequeny f. By the mathedfiltering in range, the range-ompressed signal is given by ] X s M (τ,x)=sinb (τ Δt p )] ret L exp jk r R(X; r p, p )] (4) where the sin funtion is defined by sin(u) =sin(πu)/πu, symbol K r =4π/λ denotes the range WN enter, and λ is the wavelength orresponding to the arrier frequeny. The time-domain BP algorithm is implemented by a oherent integral along the range history in the range-ompressed and azimuth time domain. The bakprojeted pixel at the oordinates (r p, p ) is obtained by a BP integral, i.e., L/ S(r p, p )= s M τ = R(X; r ] p, p ),X L/ exp jk r R(X; r p, p )] dx. (5) For eah pixel in the BP image, a oherent integral along a speifi integral trajetory should be alulated in the -D range-ompressed and azimuth time domain. In a pixel-bypixel manner, BP imaging is inherently ineffiient. Inheriting the preision merits of the BP algorithm, FBP algorithms 8], 9] are apable of aelerating the BP integral. One of suh FBP algorithm is the FFBP, whih divides the BP integral into SA BP integrals and ombines the SA images oherently. In the first stage, the syntheti aperture is divided into a set of SAs, and eah SA data is bakprojeted into an SA BP image with oarse angular resolution in an LPC system. The full-resolution SAR image is ahieved by fusing all oarse images reursively, Fig.. Polar oordinates for SA imaging. (a) LPC. (b) UPC. whih usually ontains several SA fusion stages. In eah fusion stage, the LPC is established by setting the SA enter as the oordinate origin. As shown in 8] and 9], BP imaging in the LPC is benefiial to avoid spetrum aliasing and ambiguity. In the SA fusion of the original FFBP, a -D interpolation is required to ombine all SA images in different LPCs together. For larity, we use Fig. (a) to illuminate the LPC system in the SA imaging of the FFBP. In the following, we develop the AFBP algorithm. Without using the -D interpolation, the SA fusion of the AFBP relies on the spetrum onnetion implemented by the FFT and irular shifting. The key of the AFBP is that a UPC is utilized in the generation of SA images. Herein, we first interpret the onept of the UPC and ompare it with the LPC. Let us split the full aperture into N sub SAs with equal length l = L/N sub. An SA image is generated with the BP integral in the first stage. However, all SAs are bakprojeted onto the UPC defined by (r, ). The oarse angular resolution SA image an be formed by the BP integral over the nth SA, i.e., l/ I u(r, ) = s M τ = R(X + x ] u; r, ),X + x u l/ exp jk r R(X + x u ; r, )] dx (6) where x u =(u (N sub /) (1/)) l denotes the uth SA enter. For our onveniene, the UPC is set as the full-aperture polar oordinate shown in Fig. (b). Thus, all SA images are bakprojeted through using the UPC, namely, all SA images are mapped under the same polar oordinate system with the origin at O. B. Aelerated FBP Algorithm Here, we fous on the mathematial derivation and implementation of the AFBP algorithm. In order to failitate the mathematial desription, the IRF and its orresponding -D WN spetrum of the BP image in the UPC will be derived first. The IRF for the point loated at (r p, p ) in the UPC an be obtained through alulating the neighborhood pixels around the grid in the SA BP image. The IRF of the uth SA BP image is given as follows: l/ ( I u (r, ) = s M τ = R(X + x ) u; r, ),X + x u l/ exp jk r R(X + x u ; r, )] dx. (7) The magnitude of the IRF on grid (r, ) is ahieved as a oherent integral of the range-ompressed signal along the

3 146 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 014 range history R(X + x u ; r, ), and by replaing X + x u with X, (7) is simplified as I u (r, ) = l/+xu l/+x u exp jk r ΔR(X; r, )] dx. (8) In (8), ΔR(X; r, ) stands for the differene between R(X; r p, p ) and R(X; r p,). To derive the analyti expression of the azimuth IRF, ΔR(X; r, ) is extended into the seond-order Taylor series, i.e., ΔR(X;r p,)=r(x;r p, p ) R(X;r p,) = rp +X Xr p p rp +X Xr p =( p ) X + ( p) X +σ(x 3 ) r p X l, l ) + x u ( p ) X (9) where σ(x 3 ) represents the high-order terms, whih usually take a fration of ΔR(X; r p,). In general, the vast majority of energy of IRF is intensively onentrated within a very small region around = p, whose size is assoiated with the angular resolution. For the syntheti aperture of length L, the angular resolution is given by Δ = λ/l 1]. Given a ertain aperture length, the magnitude of the quadrati term in (9) inreases as ( p ) deviates from zero. Assuming that is in the neighborhood of p that = p + δ, we an determine the effet of the quadrati term on the IRF as follows: ( p ) X p ( p +δ) ] = X δ p X. r p r p r p (10) The quadrati term auses a quadrati phase error (QPE) in the BP integral in (8) as QPE =4π δ p λr p X. (11) The magnitude of QPE is diretly proportional to δ. One the magnitude of QPE is small enough, its effet on the BP integral an be negleted, suh as when QPE is onstrained within π/8 1]. As the shape of IRF is mainly determined by its magnitude within the range p δ, p + δ], equivalently, the QPE at the boundary determines the shape of IRF. Substituting δ = Δ into (11) and onstraining the value of QPE below π/8, we have 4π( p /L r p ) (L/) (π/8), whih leads to p L (r p /4). It indiates that, for the BP IRF, the first-order approximation in (9) is preise enough if the value of p L is smaller than a fourth of r p. This onstraint is easy to be ahieved in real SAR senarios. In light of the azimuth IRF analysis, one an see that a nominal quadrati term makes the SA BP integral in (8) be simplified as I u () = l/+xu l/+x u exp j K r ΔR(X; r p,)] dx l/+xu l/+x u exp j K r X ( p )] dx. (1) To see the AFBP learly, we define the angular WN variables as follows: { K = K r X u = K r l (13) K u = K r x u. Apparently, the angular WN K, interval u, and enter K u are speifially assoiated with the azimuth oordinates X, SA length l, and SA enter x u, respetively. These variables reveal that there is an inherent orrespondene between the azimuth time domain and the angular WN domain in the BP framework. By (13), (1) beomes u /+K u I u () = u /+K u exp j K ( p )] dk ΔA, ΔA ] (14) where ΔA is the predefined angular range of the SA image, whih also determines the azimuth size of the final image. Obviously, the Fourier transform relationship between K and yields a sin expression of the azimuth IRF. This relationship paves a way for us to establish the SA fusion in the AFBP by seamlessly ompositing the SA WN spetra. The expression of the angular WN spetrum is aessible via applying an inverse Fourier transform to I u (), i.e., I u (K )= ΔA u /+A u ΔA u /+A u I u () exp(jk )d ] K K u = ret exp(jk p ) a K, ) + K u (15) where A u denotes the predefined oordinate enter of the uth SA image in the UPC. The expression of I u (K ) in (15) is easy to understand: The window funtion determines the position and the width of the angular WN spetrum, and the exponential term orresponds to the azimuth grid of the target in the UPC. Redefining Δ as the interval of the angular grid, we note that the Nyquist sampling riteria an be satisfied only if the grid of the SA image is restrited by Δ λ/l. A ruial phenomenon should be noted: If ( /) + K u < (π/δ) or ( /) + K u > (π/δ), the angular WN spetrum would be folding, whih is no matter to the SA fusion in the AFBP. For the uth SA, one an irularly shift the enter of its angular WN spetrum from zero to K u. Furthermore, (15) also indiates that, in the SA fusion of AFBP, eah SA ontributes an individual WN spetrum path, whose WN enter and width orrespond to the SA enter and its length, respetively. Based on above analysis, we an perform the SA fusion in the AFBP with the spetrum onnetion: Eah angular WN spetrum is irularly shifted in the azimuth diretion to move the WN enter from zero to the WN enter K u, and they are united as a long vetor, as illuminated in Fig. 3. The full-resolution SAR image is diretly ahieved by transforming the omplete WN bak into the image domain with a Fourier transform. In order to implement the AFBP for high-resolution SAR imagery, we need to extend the WN onnetion to the -D ase.

4 ZHANG et al.: FBP ALGORITHM WITH WN SPECTRUM FUSION FOR HIGH-RESOLUTION SPOTLIGHT SAR IMAGING 1463 Fig. 3. SA fusion with angular WN spetrum onnetion. The analyti -D WN spetrum an be diretly extended from the 1-D spetrum expression in (1) through replaing the WN variables in (13) by { K = K r X u = K r lk r πb, ] πb + Kr (16) K u = K r x u where K r is the range WN. By substituting (16) into (14) and (15), we arrive at the -D WN spetrum of the uth SA BP image, i.e., ] Kr K r I u (K r,k ) =ret exp( jk r r p )K r u, ) u + K u ] K K u ret exp(jk p )K r a r, ] r + K r (17) Fig. 4. Fig. 5. SA fusion with -D WN spetrum onnetion. (a) FFBP and (b) AFBP results. where r =4πB/ is the extent of range WN. From (16), we find that the WN enter K u is a linear funtion of range WN K r and that the -D WN spetrum is distributed in trapezoid form. With the variation of angular enter with K r, the angular WN spetrum onnetion in Fig. 3 an be extended immediately to the -D one, whih implements the AFBP algorithm. Fig. 4 presents the spetrum onnetion proessing of the AFBP. We may give a detailed proedure of the AFBP imaging in the following steps: 1) onstruting SA images with oarse angular resolution in the UPC by using the BP integral; ) applying the range FFT and azimuth inverse FFTs (IFFTs) to eah SA image to transform them into the -D WN domain; 3) diretly onneting the SA WN spetra to ahieve the omplete WN spetrum; and 4) the range IFFT and azimuth FFT are followed to obtain the full-resolution SAR image. III. REAL DATA EXPERIMENTS Let us first evaluate the performane of the AFBP. A omparison with the original FFBP algorithm is also provided. The first stage of the FFBP is idential to that of the AFBP exept in the seletion of polar oordinates, while its SA fusion is implemented by -D interpolation reursively. For a fair omparison between the AFBP and the FFBP, the BP integral with an idential SA length is performed in the first stages of AFBP and FFBP. Full-resolution SAR images are obtained via Fig. 6. Magnified subsene images of (a) FFBP and (b) AFBP. spetrum onnetion in the AFBP and reursive interpolationbased fusion in the FFBP. The tested data set was olleted by an X-band experimental SAR working at a spotlight mode. Meanwhile, the pulse repetition frequeny was 100 Hz, and the transmitting signal bandwidth was 1.16 GHz. In our experiment, the syntheti aperture time was 7.8 s, aquiring pulses with eah pulse of range bins. The losest operating range to the referene point was 10.5 km. The generated image was about km in the range and azimuth diretions with a nominal range and azimuth resolution of about m. We produe high-resolution and wide-swath SAR imageries by both FFBP and AFBP, as presented in Fig. 5(a) and (b). Idential SA images are generated in the first stage of both FFBP and AFBP, where the syntheti aperture is evenly divided into 104 SAs, eah with length 16 pulses. Using the spetrum onnetion for the SA fusion, the AFBP is over two times faster than the FFBP in this experiment. Both FFBP and AFBP ahieve high-quality imageries. In fat, their differene is not distinguishable in vision from the

5 1464 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 11, NO. 9, SEPTEMBER 014 Fig. 7. Comparison of (a) FFBP and (b) AFBP images for points 1 and. () Azimuth IRF plots of points 1 and. Fig. 8. Comparison of AFBP and FFBP images of point 3. (a) FFBP image of points 1 and. (b) AFBP image of point 3. () Azimuth IRF plots of point 3. full-sene image. In order to aess their performane omparison, a subsene in the imaging swath is magnified, as shown in Fig. 6, for a omparison. Partiularly, the subsene ontains some lose-loated trihedral orner refletors, i.e., points 1 3 irled in Fig. 6(b). Although those point-like targets are optimally foused in both FFBP and AFBP images, those from the AFBP are foused with lower sidelobes, and their energy is more onentrated. The main reason for this phenomenon lies in that the interpolation error aumulation in the FFBP reursions is avoided in the AFBP. In subsene A, there are two lose-loated orner refletors (points 1 and ) with 0.15-m distane in azimuth and a single refletor (point 3), whih are irled in the upper region in Fig. 6(b). To demonstrate the optimal performane of the proposed approah, we give their images and azimuth IRFs with hamming windowing in Figs. 7 and 8. One an learly find that by both approahes, the two refletors are distintively separated, whereas the IRF distortion of the FFBP is obvious ompared with the AFBP IRF. In Fig. 8(), high sidelobes (up to 15 db) exist near the main lobe, whereas the AFBP ahieves muh lower sidelobes (below 5 db). We interpret that the high sidelobes in FFBP IRF an be redued with the inrease in SA length, i.e., 64 pulses for an SA, at a prie of signifiant inrease in omputational omplexity. On the ontrary, the AFBP an maintain its optimal performane with a short SA length, providing high effiieny simultaneously. IV. CONCLUSION This letter has proposed the AFBP algorithm for highresolution spotlight SAR imaging. In the AFBP, SA images with oarse angular resolution are generated by mapping SA data into a UPC system. Based on the analyti expressions of the IRF, the SA fusion in onventional FFBP with interpolation is simplified into the proessing of 1-D WN spetrum onnetion. Along with dramati effiieny improvement, the AFBP an also redue the undesired effets in the FFBP indued by the interpolation-based SA fusion. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable omments that helped improve the letter quality. REFERENCES 1] D. Munson, J. O Brien, and W. Jenkins, A tomographi formulation of spotlight-mode syntheti aperture radar, Pro. IEEE, vol. 71, no. 8, pp , Aug ] M. Desai and W. Jenkins, Convolution bakprojetion image reonstrution for spotlight mode syntheti aperture radar, IEEE Trans. Image Proess., vol. 1, no. 4, pp , Ot ] C. Jakowatz, D. Wahl, and D. Yoky, Beamforming as a foundation for spotlight-mode SAR image formation by bakprojetion, in Pro. SPIE Algorithms Syntheti Aperture Radar Imagery, Mar. 008, vol. 6970, p Q. 4] O. Frey, C. Magnard, M. Rüegg, and E. Meier, Fousing of airborne syntheti aperture radar data from highly nonlinear flight traks, IEEE Trans. Geosi. Remote Sens., vol. 47, no. 6, pp , Jun ] M. Rodriguez-assola, P. Parts, G. Krieger, and A. Moreira, Effiient time-domain image formation with preise topography aommodation for general bistati SAR onfigurations, IEEE Trans. Aerosp. Eletron. Syst., vol. 47, no. 4, pp , Ot ] M. Soumekh, Time domain non-linear SAR proessing, Dept. Elet. Eng. State Univ. New York, New York, NY, USA, Teh. Rep., ] C. V. Jakowatz and D. E. Wahl, Considerations for autofous of spotlight-mode SAR imagery reated using a beamforming algorithm, in Pro. SPIE Algorithms Syntheti Aperture Radar Imagery XVI, 009, vol. 7337, pp A A-9. 8] A. F. Yegulalp, Fast bakprojetion algorithm for syntheti aperture radar, in Pro. IEEE Radar Conf., Waltham, MA, USA, Apr. 0, 1999, pp ] L. M. H. Ulander, H. Hellsten, and G. Stenström, Syntheti aperture radar proessing using fast fatorized bak-projetion, IEEE Trans. Aerosp. Eletron. Syst., vol. 39, no. 3, pp , Jul ] D. E. Wahl, D. A. Yoky, and C. V. Jakowatz, An implementation of a fast bakprojetion image formation algorithm for spotlight-mode SAR, in Pro. SPIE, 008, vol. 6970, pp H 69700H ] P. O. Frölind and L. M. H. Ulander, Evaluation of angular interpolation kernels in fast bak-projetion SAR proessing, IEE Pro. Radar Sonar Navigat., vol. 153, no. 3, pp , Jun ] W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Syntheti Aperture Radar: Signal Proessing Algorithm M]. Boston, MA, USA: Arteh House, 1995, pp