CONVOLUTION BACK-PROJECTION IMAGING ALGO- RITHM FOR DOWNWARD-LOOKING SPARSE LINEAR ARRAY THREE DIMENSIONAL SYNTHETIC APER- TURE RADAR

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1 Progress In Eletromagnetis Researh, Vol. 129, , 2012 CONVOLUTION BACK-PROJECTION IMAGING ALGO- RITHM FOR DOWNWARD-LOOKING SPARSE LINEAR ARRAY THREE DIMENSIONAL SYNTHETIC APER- TURE RADAR X. M. Peng *, W. X. Tan, Y. P. Wang, W. Hong, and Y. R. Wu Siene and Tehnology on Mirowave Imaging Laboratory, Institute of Eletronis, Chinese Aademy of Sienes, Beijing , China Abstrat General side-looking syntheti aperture radar (SAR) annot obtain sattering information about the observed senes whih are onstrained by lay over and shading effets. Downward-looking sparse linear array three-dimensional SAR (DLSLA 3D SAR) an be plaed on small and mobile platform, allows for the aquisition of full 3D mirowave images and overomes the restritions of shading and lay over effets in side-looking SAR. DLSLA 3D SAR an be developed for various appliations, suh as ity planning, environmental monitoring, Digital Elevation Model (DEM) generation, disaster relief, surveillane and reonnaissane, et. In this paper, we give the imaging geometry and dehirp eho signal model of DLSLA 3D SAR. The sparse linear array is omposed of multiple transmitting and reeiving array elements plaed sparsely along ross-trak dimension. The radar works on time-divided transmitting-reeiving mode. Partiularly, the platform motion impat on the eho signal during the time-divided transmitting-reeiving proedure is onsidered. Then we analyse the wave propagation, along-trak and ross-trak dimensional eho signal bandwidth before and after dehrip proessing. In the following we extend the projetion-slie theorem whih is widely used in omputerized axial tomography (CAT) to DLSLA 3D SAR imaging. In onsideration of the flying platform motion ompensation during timedivided transmitting-reeiving proedure and parallel implementation on multi-ore CPU or Graphis proessing units (GPU) proessor, the onvolution bak-projetion (CBP) imaging algorithm is proposed for DLSLA 3D SAR image reonstrution. At last, the fousing apabilities Reeived 13 May 2012, Aepted 7 June 2012, Sheduled 26 June 2012 * Corresponding author: Xueming Peng (yaotoufengshan2007@163.om).

2 288 Peng et al. of our proposed imaging algorithm are investigated and verified by numerial simulations and theoretial analysis. 1. INTRODUCTION Owing to side-looking geometry in general SAR systems, shading and lay over effets by trees, buildings, speial terrain shapes will hide essential information of the observed areas, partiularly in urban areas and deep valleys in mountain areas [1 3]. DLSLA 3D SAR aquires full 3D mirowave images by wave propagation dimensional pulse ompression, along-trak dimensional aperture synthesis with flying platform movement and ross-trak dimensional aperture synthesis with a sparse linear array [4]. And DLSLA 3D SAR observes nadir areas whih means it overomes restritions of shading and lay over effets in side-looking SAR. Downward-Looking Imaging Radar (DLIR) [5] was first introdued by Gierull in He took advantage of aperture synthesis by platform movement along along-trak dimension and linear array aperture synthesis along ross-trak dimension to form a two dimensional aperture and transmitted a single frequeny signal to obtain the two dimensional image. Nouvel et al. in ONERA [6 8] made use of hirp signal instead of single frequeny signal and developed the onept of downward-looking 3D-SAR and developed a Ka band linear array 3D-SAR system. Researhers in FGAN-FHR designed an Airborne Radar for Three-dimensional Imaging and Nadir Observation (ARTINO) [4, 9]. The system plaes a thinned linear array along ross-trak dimension, it obtains three dimensional resolution by wave propagation pulse ompression, aperture synthesis along the fly dimension and beam-forming along ross-trak dimension. The imaging algorithms of the two systems are not illustrated in detail and the 3D imaging results of the two systems have not been publily reported yet [10]. DLSLA 3D SAR requires pretty fast A/D sampling devies and pretty high apaity data olletion devies for eho data reording as there are more transmitting-reeiving hannels in DLSLA 3D SAR than in side-looking SAR. Sparse linear array with the timedivided transmitting-reeiving is often applied to the 3D SAR whih helps to redue the omplexity of the system. The motion of the platform during the time-divided transmitting-reeiving proedure degrades the reonstruted image and the motion effet should be ompensated in the imaging algorithm. Most of the published imaging algorithms suppose the eho signal without arrier frequeny and be digitized diretly, and take no onsideration of A/D sampling

3 Progress In Eletromagnetis Researh, Vol. 129, rate limitation, eho data voxel volume and flying platform motion effet during the time-divided transmitting-reeiving proedure. These algorithms mainly get the wave propagation and along-trak dimension ompression first with Range Doppler algorithm, Chirp Saling algorithm and Range Migration algorithm like general two dimensional SAR image reonstrution, then obtain the ross-trak dimension ompression via SPECAN, beam-forming, CS (Compressed Sensing), et. [11 16]. These algorithms may be suited for multi-baseline 3D SAR whose eho signal is omposed of two dimensional general SAR eho signal along multi-baseline. Multi-baseline 3D SAR systems put less burden on A/D sampling devies and data olletion devies while require quite a long time to obtain the 3D eho signal, and the sampling interval between the multi-baseline is not uniform [13, 15, 16], whih restrits the imaging algorithms on the basis of uniform Nyquist sampling theorem. What s more, any hange of the imaging sene destroys the oherene of the imaging sene, while the oherene of the imaging sene is very important for mirowave imaging [2]. The dehirp signal proessing tehnique is mainly used in spotlight mode SAR systems whih helps to redue the eho signal bandwidth. Dehirp signal proessing tehnique mixes the eho signal with a referene funtion whih is the eho signal from APC (Antenna Phase Center) to SC (Sene Center) [2]. The redued bandwidth is only related to the range extension of the imaging sene. DLSLA 3D SAR operates in nadir observation. And in wave propagation dimension, the range gate is the undulation range of the imaging sene. This range gate is several hundred meters in urban areas or plane areas, and the orresponding range gate time delay is very small. While, the pulse of the transmitting signal is often larger than the range gate time delay in order to obtain enough eho power and eho SNR. So the dehirp signal proessing tehnique is very suitable for DLSLA 3D SAR for whih helps to redue the A/D sampling bandwidth and eho data volume. Our work below is based on dehirp proessing tehnique. Projetionslie theorem was introdued and used in CAT first [17]. Later, some researhers extended projetion-slie theorem to spotlight mode SAR image reonstrution [18, 19]. Knaell and Cardillo [20] disussed the radar tomography for the generation of three-dimensional images, but they negleted the enter frequeny shift of the onvolution kernel, and they did not give the ross-trak dimensional imaging geometry and the three dimensional point spread funtion. What s more, the algorithm they introdued takes no onsideration of the platform motion effet during the time-divided transmitting-reeiving proedure whih means it annot be used for DLSLA 3D SAR imaging. In this paper, we offer a new point of view to DLSLA 3D SAR eho signal aquisition on the

4 290 Peng et al. basis of projetion-slie theorem. And, the CBP imaging algorithm whih ompensates the flying platform motion effet during the timedivided transmitting-reeiving proedure and ontains the advantage of parallel implementation [21, 22], is introdued for DLSLA 3D SAR. The struture of this paper is organized as follows. Downwardlooking sparse linear array 3D SAR imaging geometry, eho signal model, platform motion effet during time-divided transmittingreeiving proedure and eho signal bandwidth before and after dehirp are established in Setion 2; then, in Setion 3, projetion-slie theorem and its appliation in DLSLA 3D SAR are desribed in detail; CBP imaging algorithm with platform motion ompensation during time-divided transmitting-reeiving proedure and the parallel implementation of the algorithm is disussed in Setion 4; Setion 5 gives some omputer simulations and result analysis. Finally, onlusion is provided in Setion DOWNWARD-LOOKING SPARSE LINEAR ARRAY THREE DIMENSIONAL SAR IMAGING GEOMETRY AND ECHO MODEL This setion is splitted into four subsetions. First, we give the DLSLA 3D SAR imaging geometry and illustrate why DLSLA 3D SAR an gather three dimensional satter information that avoided shading and lay over effets. Then the eho signal model with dehirp signal proessing tehnique is introdued. Next the platform motion effet during the time-divided transmitting-reeiving proedure is analyzed. At last, the eho signal bandwidth in every dimension before and after dehirp proessing is analysed Downward-looking Sparse Linear Array Three Dimensional SAR Imaging Geometry As is shown in Fig. 1. X-axis is parallel to the flight path of the platform, Y -axis is parallel to the ross-trak dimensional sparse linear array, Z-axis is perpendiular to the XY plane, the origin of the oordinate O is the enter of the three dimensional imaging sene (O is also used as the referene point for dehirp signal proessing). Q is the Antenna Phase Center (APC), Q is the projetion point of Q on XY plane, P is the target in the imaging sene. APC flight path QX is parallel to X-axis. QO is the instantaneous referene range from APC to SC with the distane of R 0, OP is the instantaneous range from SC to target with the distane of R, QP is the instantaneous slant

5 Progress In Eletromagnetis Researh, Vol. 129, (a) Figure 1. DLSLA 3D SAR imaging mode and imaging geometry. (a) DLSLA 3D SAR imaging mode. (b) DLSLA 3D SAR imaging geometry. range from APC to target with the distane of R t. QP = QO + OP (R t = R 0 + R) on the hypothesis of planar wave front. ψ is the angle between OQ and XY plane, alled as grazing angle. ϕ is the angle between OQ and Z-axis, alled as inident angle. θ is the angle between OQ and X-axis, alled as slant angle. γ 1 is the angle between QX and QP, alled as along-trak dimension Doppler one angle. γ2 is the angle between QQ and QP, alled as ross-trak dimension Doppler one angle. The SAR proessor needs three kinds of information to reonstrut an image from sensor signals. First, it needs positioning information to loate the soure of the return ehoes in the imaging sene. Seond, it needs spatial or angular resolution information to differentiate among return ehoes from separate satters or sene areas. Finally, it needs the intensity information of the signal assoiated with eah other or sene areas in the target field. In the ase of radar, the intensity is the radar ross setion of individual satter or the radar baksatter oeffiient of distributed areas [2]. As shown in Fig. 1, general sidelooking SAR measures range from APC to target and along-trak Doppler one angle γ 1. It an not distinguish targets on the yellow (b)

6 292 Peng et al. region as they have the same range and the same along-trak Doppler one angle [3]. And grazing angle in side-looking SAR is small, espeially in the far range gate unit, the red region is shaded by other targets, and SAR sensor an not obtain sattering information of this region [3]. DLSLA 3D SAR measures range from APC to target, alongtrak dimension Doppler one angle γ 1, ross-trak dimension Doppler one angle γ 2 and the radar ross setion of the imaging sene whih means it aquires full 3D mirowave image of the observed areas. And, DLSLA 3D SAR operates nadir observation, therefore, layover and shading effets an be overome Downward-looking Sparse Linear Array Three Dimensional SAR Imaging Eho Model The sparse linear array is omposed of multiple transmitting array elements and multiple reeiving array elements plaed sparsely along the ross-trak dimension and works on the time-divided transmittingreeiving mode. Aording to the equivalent phase enter theorem, the sparse linear array with multiple transmitting array elements and multiple reeiving array elements on the basis of time division an be equivalent to a uniform linear array that every equivalent array element transmits and reeives signal by itself [12]. All the equations below are based on the equivalent phase enter theorem as the range from APC to target is large enough that the phase error generated by equivalent phase enter theorem an be ignored. The sparse linear array we design is based on ARTINO array distribution [3] as the length of equivalent array with this method is longer than other methods [23, 24]. What s more, ARTINO array distribution is quite easy for hardware implementation [25]. Suppose that there are L 1 transmitting array elements, L 2 reeiving array elements. The equivalent phase enter number is L 1 L 2 aording to ARTINO array distribution [3]. A hirp signal with arrier frequeny f, hirp rate K r, pulse width T p is transmitted and reeived. The transmitted signal an be written as ( ) ˆt S(m, n, t) = ret exp { j [ 2πf t + πk rˆt 2]}, (1) T p where m is the ross-trak dimensional equivalent array element number, n is the along-trak dimensional pulse number and ˆt is fast time. Here we are going to develop an equation for the signal phase reeived by DLSLA 3D SAR system from a single satter objet P at sene oordinate (x, y, z). This development assumes an ideal point satter objet with radar ross setion σ whose amplitude and phase harateristis do not vary with frequeny and aspet angle. For simpliity, the reeiving signal model ignores antenna gain, amplitude

7 Progress In Eletromagnetis Researh, Vol. 129, effets of propagation on the signal and any additional time delays due to atmospheri effets. The signal reeived from target P at array element number m and pulse number n is (ˆt t d S r (m, n, t) = a t ret T p ) exp { j [2πf (t t d )+πk r (ˆt t d ) 2 ]}, (2) where a t = σ, t d = 2R t is the dual time delay from APC to target P. It is appropriate to view the reeived signal as a three dimensional signal in the oordinates m, n and ˆt. The referene reeiving signal from APC to referene point at SC, O, is { ) ]} 2 S ref (m, n, t) = exp j [2πf (t t 0 ) + πk r (ˆt t 0, (3) where t 0 = 2R 0 is the dual time delay from APC to referene point at sene enter O. The radar reeiving signal is the video frequeny signal generated by mixing the reeived signal from target P with referene reeived signal from referene point O. It is onvenient to write the video frequeny eho signal in the form where S if (m, n, t) = S r (m, n, t) Sref (m, n, t) ) (ˆt t d = a t ret exp { jφ ( m, n, ˆt )}. (4) The phase term Φ(m, n, ˆt) in Eq. (4) an be written as, T p Φ ( m, n, ˆt ) = KR + 4πK r 2 R 2, (5) R = R t R 0. In Eq. (5), K = 4πKr ( f K r +ˆt 2R 0 ) and exp { j 4πKr R 2} is the Residual 2 Video Phase (RVP) term. The RVP term is the onsequene of dehirpon-reeive approah. It an be ompletely removed from the radar eho signal by a preproessing operation whih is illustrated in Appendix A Platform Motion Effets During Time-divided Transmitting-reeiving Proedure Now, we run bak over the eho data olletion proedure. The sparse linear array maintains multiple transmitting array elements and multiple reeiving array elements and the sparse array works in timedivided mode. As shown in Fig. 2, There are L 1 transmitting array elements and L 2 reeiving array elements. The transmitting array elements work sequentially, array element T 1 transmits signal first,

8 294 Peng et al. all the reeiving array elements reeive eho signal simultaneously and L 2 equivalent array elements are obtained, the time interval of this transmitting and reeiving proedure is T. Then array element T 2 transmits signal, and the proedure loops until array element T L1 finishes transmitting signal. Finally the radar obtains L 1 L 2 equivalent array elements. The ross-trak dimensional equivalent element number m [1, L 1 L 2 ] in the video frequeny eho signal S if (m, n, t). The eho data olletion proedure is shown in Fig. 2. As the radar works in the time division mode, the whole transmitting and reeiving proedure of the sparse linear array is not short enough to be treated under stop-and-go approximation. The platform motion during the time-divided transmitting-reeiving proedure should be onsidered in the video frequeny eho signal S if (m, n, t). As shown in Fig. 3, O is the SC with oordinates (0, 0, 0). The platform movement is x = l V T, where l means array element T l transmits signal, V is flying platform veloity. The oordinates of APC is (x i + x, y i, z i ). The oordinates of target P is (x, y, z). The range between sene enter and APC is R 0, the range between target and APC is R t, where R 0 = (x i + x) 2 + r0 2 [(x i + x) x] 2 + rt 2, (6) R t = and where r 0 = yi 2 + z2 i, r t = (y y i ) 2 + (z z i ) 2, then making the substitution (without onsidering RVP) and taking along-trak Figure 2. Time divided MIMO eho data olletion.

9 Progress In Eletromagnetis Researh, Vol. 129, (a) (b) Figure 3. Referene and instantaneous range with platform motion. (a) Referene range. (b) Instantaneous range. dimensional Fourier transform, we obtain F x {S if (m, n, t)} = S if (m, n, t) exp { jk x x i }dx i x = A 0 exp { jkr} exp { jk x x i } dx i x { = A 1 exp {jk x x} exp j } K 2 Kx(r 2 t r 0 ) jk x x, (7) where R = R t R 0 = rt 2 + [(x i + x) x] 2 r0 2 + (x i + x). and A 0 and A 1 are the orresponding amplitudes before and after Fourier transform. Equation (7) desribes the platform motion during the timedivided transmitting-reeiving proedure auses an along-trak dimensional phase term, this phase term should be ompensated in image reonstrution proedure.

10 296 Peng et al Eho Signal Bandwidth Before and after Dehirp Proessing As shown in Fig. 4, the AD sampling begins at the same time as the dehirp starts. The video signal is digitized by the AD onverter. Corresponding to the return of the enter of the pulse from the quantized referene range R 0, the AD trigger for the first sample steps bak by T p /2 to the start of the pulse and steps bak T s /2 to the start of the range window. The total sampling time is T p + T s. The AD onverter samples the video signal at fast time ˆt s (m, n, k): ˆt s (m, n, k) = k F s + 2R 0 T p 2 T s 2. (8) Eq. (8) desribes the value of a sample of the array element m, alongtrak pulse n and wave-propagation dimension sample k. F s is the sampling frequeny, T s = 2R = 4 R is the time interval assoiated with the wave-propagation dimensional range gate R(R [ R, R]). The phase of the signal from APC sample ell (m, n, k) to target P is Φ(m, n, k) = 4πK ( r f + k T p K r F s 2 T ) s R + 4πK r 2 2 R 2. (9) Figure 4. Timing diagram for AD samples.

11 Progress In Eletromagnetis Researh, Vol. 129, Figure 5. Eho signal bandwidth before and after dehirp proessing. The three-dimensional digitized eho signal beomes N e 1 S d = a t m=0 N a 1 n=0 N r 1 k=0 exp {jφ(m, n, k)}. (10) whih enompasses N r samples per pulse over N a along-trak pulses and N e ross-trak equivalent phase entres. The eho signal bandwidths along wave-propagation dimension, along-trak dimension and ross-trak dimension before/after dehirp are ompared and shown in Fig. 5. The bandwidth along wave-propagation dimension in wavenumber domain before/after dehirp is shown in Eq. (11) and Eq. (12) B r1 = 2π K rt p (11) B r2 = 2π K r (2w r /), (12) where K r is the hirp rate and T p the pulse width of the transmitted signal. w r is the wave-propagation dimensional range gate of the imaging sene. Bandwidth B r2 is only related to the wave-propagation dimensional range gate of the imaging sene. The bandwidth along along-trak dimension in wave-number

12 298 Peng et al. domain before and after dehirp is shown in Eq. (13) and Eq. (14) B a1 = 2π 2w a λr t + 2π 2L a λr t (13) B a2 = 2π 2w a λr t, (14) where w a is the along-trak dimensional range gate, R t the instantaneous range form APC to target P, and L a the length of alongtrak dimensional syntheti aperture. Bandwidth B a2 is only related to the along-trak dimensional range gate of the imaging sene. The bandwidth of ross-trak dimension in wave-number domain before and after dehirp is shown in Eq. (15) and Eq. (16) B e1 = 2π 2w e λr t + 2π 2L e λr t (15) B e2 = 2π 2w e λr t, (16) where w e is the ross-trak dimensional range gate and L e the length of ross-trak dimensional linear array. Bandwidth B e2 is only related to the ross-trak dimensional range gate of the imaging sene. As DLSLA 3D SAR works in the mode of nadir observation, the undulation of the imaging sene is small, espeially in urban areas or plain areas. From Eq. (11) to Eq. (16), we an see the eho signal bandwidth in every dimension after dehirp proessing is only related with the size of the imaging sene, in this irumstane, the bandwidth is redued enormously, espeially in the wave-propagation dimension. Dehirp proessing is quite suitable for DLSLA 3D SAR whih possesses many reeiving hannels as it an redues burden on AD onverter and data olletion devie. 3. PROJECTION SLICE THEOREM AND ITS APPLICATION IN DOWNWARD-LOOKING SPARSE LINEAR ARRAY THREE DIMENSIONAL SAR Several years ago, it was shown that the mathematial struture of SAR image onstrution is similar to the image reonstrution problem in CAT [18]. The similarity in these two otherwise seemingly different imaging systems suggested that reonstrution algorithms used in CAT ould also be used in SAR [26]. In this paper we illustrate the three dimensional projetion-slie theorem suited for DLSLA 3D SAR. As shown in Fig. 6, QO = R0, OP = R, based on planar wave front hypothesis, range QP is Rt = R 0 + R. uv-plane is perpendiular to

13 Progress In Eletromagnetis Researh, Vol. 129, Figure 6. Three dimensional projetion-slie theorem. QP, and u-axis is parallel to OQ, v-axis is perpendiular to QP and u-axis. The dehirped 3D eho data of the whole imaging sene are obtained from the superposition of every point satter objet s eho information whih also means integrating the point satter eho obtained in Eq. (4) along u-axis, v-axis and R-axis. It an be written as S if (K, ϕ, θ) = S if (m, n, t)dvdudr R u v = f(x, y, z) exp { jkr} dudvdr, (17) u v R where f(x, y, z) is the omplex image that we want to reonstrut, and the RVP term in Eq. (4) is not onsidered. Aording to planar wave front hypothesis, Eq. (17) an be written as S if (K, ϕ, θ) = f(x, y, z) exp { jkr} dudvdr u v R = g(r, ϕ, θ)dr, (18) R

14 300 Peng et al. where g(r, ϕ, θ) = u v f(x, y, z)dudv. g(r, ϕ, θ) is the linear trae data along R-axis on uv plane. The dehirped 3D eho data of the whole imaging sene are the onedimensional Fourier transform of the linear trae data g(r, ϕ, θ) along R-axis dimension. As shown in Fig. 6, we an get that [ ] [ ] [ ] R osθsinϕ sinθsinϕ osϕ x u = sinθ osθ 0 y (19) v osθosϕ sinθosϕ sinϕ z where Substituting Eq. (19) to Eq. (17), we get S if (K, ϕ, θ) = f(x, y, z) exp { jkr} dudvdr u v R = f(x, y, z) exp { jφ 2 } J dxdydz, (20) x y z Φ 2 = KR = K(xosθsinϕ + ysinθsinϕ + zosϕ) = K x x + K y y + K z z. J is the Jaobi s transformation determinant (aording to Eq. (19), J = 1). Eq. (20) means the dehirp 3D eho data of the whole imaging sene is also the three dimensional Fourier transform of the reonstruted omplex image f(x, y, z) along x-axis, y-axis and z-axis dimension. This is the three dimensional projetion-slie theorem for DLSLA 3D SAR. The omplex image f(x, y, z) we want to reonstrut an be obtained from Eq. (20) with three dimensional Inverse Fourier transform. 4. CONVOLUTION BACK-PROJECTION ALGORITHM 4.1. Three Dimensional Image Reonstrution with Convolution Bak-projetion Algorithm Aording to Eq. (20), the omplex image f(x, y, z) an be reonstruted from video frequeny dehirp eho signal S if (K, ϕ, θ), while the flying platform motion effet during time-divided transmittingreeiving proedure should be ompensated in the imaging algorithm. The flying platform motion ompensation proedure an be written as S if (K, ϕ, θ) = F 1 x {F x {S if (K, ϕ, θ)} exp { jk x x}}, (21)

15 Progress In Eletromagnetis Researh, Vol. 129, where K x is the along-trak dimensional wave-number. Then, the three dimensional omplex image f(x, y, z) an be reonstruted from the flying platform motion ompensated eho signal f(x, y, z) = 1 8π 3 S if (K, ϕ, θ) J exp {jkr} dkdϕdθ, (22) where J = K 2 sinϕ is the Jaobi s transformation determinant. The inside integration an be seen as one dimensional inverse Fourier transform along R-axis. The inside integration an be written as Q (R, ϕ, θ) = S if (K, ϕ, θ) J exp {jkr} dk = Q(R, ϕ, θ) exp { jk R}, (23) take onsideration of sample interval T p 2 + 2(R 0 R) t T p 2 + 2(R 0 + R) K [K 1, K 2 ] is obtained, where K 1 = 4π λ 2πK ( r T p + 2 R ) K 2 = 4π λ + 2πK ( r T p + 2 R ). If we define K = 4π λ and K 0 = 2πKr (T p + 2 R ), then K [K K 0, K + K 0 ]. As K only takes part of the spae frequeny band whih means the dehirp eho signal is a narrow band signal, so the enter frequeny shifting should be onsidered when one dimensional inverse Fourier transform is taken [3, 19]. Q (R, ϕ, θ) is the one dimensional inverse Fourier transform of S if multiplying a filter with its spatial frequeny response is K 2 sinϕ. So, Q (R, ϕ, θ) an be seen as the onvolution in the spae domain. Then, the reonstruted omplex image f(x, y, z) an be written as f(x, y, z) = 1 ϕ1 θ1 4π 2 Q(R, ϕ, θ) exp { jk R} dϕdθ. (24) ϕ 1 θ 1 For digital omputer proessing, the reonstruted omplex image an be written as f(x, y, z) = 1 4π 2 N a 1 ii=0 N e 1 jj=0 Q(R, ϕ jj, θ ii ) exp { jk R} ϕ θ, (25) where N a is the along-trak dimensional pulse number and N e is the equivalent phase entre number.

16 302 Peng et al Convolution Bak-projetion Algorithm Flow The three dimensional reonstruted omplex image is the bakprojetion of the onvolution term Q. And, this is the onvolution bak-projetion imaging algorithm for the DLSLA 3D SAR. For DLSLA 3D SAR, suppose the 3D imaging sene has N x N y N z voxel, and the oordinate of every voxel an be pre-omputed before image reonstrution. During the data olletion proedure, the eho signal is reorded at every digitized position of ϕ jj (jj = 1,..., Ne) and θ ii (ii = 1,..., Na), and its orresponding onvolution term an be omputed from Eq. (23). And in Eq. (24), the onvolution term of every voxel in the reonstruted formula should be interpolated from the digitized onvolution term of the reorded eho signal. And the the flow diagram of onvolution bak-projetion imaging algorithm for DLSLA 3D SAR is shown in Fig. 7. The image reonstrution proedure an be summarized as, Step 1. Selet the 3D imaging region and ompute the (x, y, z) oordinate of every voxel. Step 2. Read wave-propagation dimension eho data along every ross-trak dimension sample ϕ jj and every along-trak dimension sample θ ii. Step 3. Compensate the platform motion during the time-divided transmitting-reeiving proedure. Step 4. Filter the dehirp eho signal in spae frequeny K domain with K 2 sinϕ. Step 5. Center frequeny shift the eho signal and ompute the digitized onvolution term via inverse Fast Fourier transform of the filtered eho data obtained in Step 3. Step 6. Compute R of every voxel at ross-trak dimension sample ϕ jj and along-trak dimension sample θ ii. Step 7. Interpolate the onvolution term of every voxel from the digitized onvolution term obtained in Step 4. Step 8. Compute the bak-projetion data of every voxel at rosstrak dimensional sample ϕ jj and along-trak dimensional sample θ ii. Step 9. Write the binary 3D imaging result to hard disk and display the 3D imaging results. As an be seen from the flow diagram of the onvolution bakprojetion imaging algorithm, the main omputation operations of the CBP imaging algorithm are the FFT, the omplex multipliation and the interpolation. All these operations an be performed in parallel.

17 Progress In Eletromagnetis Researh, Vol. 129, Figure 7. The flow diagram of onvolution bak-projetion algorithm. The omputation of the bak-projetion proedure at every ross-trak dimension sample and every along-trak dimension sample an also be performed in parallel [21, 22]. Consequently, parallel implementation of the onvolution bak-projetion imaging algorithm an be implemented in parallel on multi-ore CPU platform via OpenMP or MPI (Message Passing Interfae), or on GPU (Graphis proessing units) platform via CUDA/OpenCL(Open Computing Language). 5. DOWNWARD-LOOKING SPARSE LINEAR ARRAY THREE DIMENSIONAL SAR COMPUTER SIMULATION In this setion, we present two numerial simulation examples based on downward-looking sparse linear array onfiguration to illustrate the performane of our algorithm. The parameters used in the simulation experiments are listed in Table 1. For simpliity, the losses due to the antenna pattern are not onsidered here. All the numerial simulations are implemented with Intel Math Kernel Library(MKL) and OpenMP on Intel(R) Core(TM) GHz CPU platform.

18 304 Peng et al. Table 1. Measurement parameters used in the simulations. Parameters Value Center Frequeny 37.5 GHz Transmitting Signal Bandwidth 150 MHz Platform Fly Height 2500 m Platform Fly Veloity 50 m/s Transmitting Signal Pulse Width 10 µs AD Sampling Rate 250 MHz Range Gate m PRF 1280 Hz Along-trak Dimension Sampling Interval 0.08 m Transmitting Array Elements 8 Reeiving Array Elements 32 Beam width of TR Array 14 (Cross-trak) Beam width of TR Array 14 (Along-trak) Equivalent Phase Center Number 256 Cross-trak Dimension Sampling Interval 0.04 m where TR means Transmitting and Reeiving. Figure 8. The onfiguration of the sparse linear array. Table 2. Cross-trak dimensional oordinates of array elements. Coordinates Value (m) Transmitting Array T i = i 0.04, i [1, 4] Transmitting Array T i = (i 4) 0.04, i [5, 8] Reeiving Array R i = i 0.12, i [1, 32] Equivalent Array T R i = i 0.04, i [1, 256]

19 Progress In Eletromagnetis Researh, Vol. 129, The sparse linear array we used in the simulation is omposed of 8 transmitting array elements and 32 reeiving array elements. The onfiguration of the sparse linear array is shown in Fig. 8. A is the real array distribution, B is the equivalent array distribution. The oordinates of transmitting array elements, reeiving array elements and equivalent array elements is illustrated in Table 2. In the first numerial simulation, to illustrate the apability of the reonstruted image on different height with the proposed algorithm, 3 8 point satter objets are distributed on three irles at different height. The radius of the three irles is 40 m, 30 m, 20 m and the height of the three irles is 20 m, 40 m, 60 m, respetively. A sketh of three irles point targets in the 3D spae is shown in Fig. 9(a). The projetion of the three irles onto XY plane is shown in Fig. 9(b), the projetion of the three irles onto XZ plane is shown in Fig. 9() and, the projetion of the three irles onto Y Z plane is shown in Fig. 9(d). All the point satter objets in the simulated model have the same radar ross setion. The image reonstruted by the proposed CBP imaging algorithm is shown in Fig. 10(a) Fig. 10(d). The 19 db iso-surfae of the (a) (b) () Figure 9. Simulated model of three irles point targets. (a) Sketh of point targets. (b) Projetion onto XY plane. () Projetion onto XZ plane. (d) Projetion onto Y Z plane. (d)

20 306 Peng et al. (a) (b) () Figure 10. Reonstruted image of point targets with CBP algorithm. (a) 3D reonstruted image. (b) Projetion onto XY plane. () Projetion onto XZ plane. (d) Projetion onto Y Z plane. reonstruted 3D image of point satter objets with CBP algorithm is shown in Fig. 10(a), the projetion of the reonstruted 3D image ( 19 db) onto XY plane is shown in Fig. 10(b), the projetion of the reonstruted 3D image ( 19 db) onto XZ plane is shown in Fig. 10() and, the projetion of the reonstruted 3D image ( 19 db) onto Y Z plane is shown in Fig. 10(d). The refletivity values as well as the position of the point targets are all in perfet agreement with the simulated model shown in Fig. 9(a) Fig. 9(d). In the seond numerial simulation, an undulating hill onsisting of 546 unit amplitude point satter objets is onfined in a region of size 120 mm 120 m 120 m. Fig. 11(a) is a sketh of the undulating hill in the 3D spae. Fig. 11(b) is the projetion of undulating hill onto XY plane. Fig. 11() is the projetion of undulating hill onto XZ plane. Fig. 11(d) is the projetion of undulating hill onto Y Z plane. The undulating hill model is generated by a surf funtion in (d)

21 Progress In Eletromagnetis Researh, Vol. 129, MATLAB. The undulating hill 3D image reonstruted with CBP imaging algorithm is shown in Fig. 12(a) Fig. 12(d). The 22 db iso-surfae of the reonstruted 3D undulating hill is shown in Fig. 12(a), the projetion of the reonstruted 3D image of undulating hill ( 22 db) onto XY plane shown in Fig. 12(b), the projetion of the reonstruted 3D image of undulating hill ( 22 db) onto XZ plane shown in Fig. 12(), and the projetion of the reonstruted 3D image of undulating hill ( 22 db) onto Y Z plane shown in Fig. 12(d). In order to demonstrate the performane of the proposed imaging algorithm, the 3D resolution, the Peak Side Lobe Rates (PSLR) and the Integral Side Lobe Rates (ISLR) are all evaluated, as indiated in Table 3. The PSLR is the ratio between the height of the largest side Wave Projetion Dimension Wave Projetion Dimension Cross Trak Dimension undulating hill 0 (a) undulating hill Along Trak Dimension Along Trak Dimension () Cross Trak Dimension Wave Projetion Dimension undulating hill Along Trak Dimension (b) undulating hill Cross Trak Dimension Figure 11. undulating hill model. (a) Sketh of undulating hill. (b) Projetion onto XY plane. () Projetion onto XZ plane. (d) Projetion onto Y Z plane. (d)

22 308 Peng et al. (a) (b) () Figure 12. Reonstruted image of undulating hill with CBP algorithm ( 22 db). (a) The iso-surfae of the reonstruted 3D undulasting hill ( 22 db). (b) Projetion onto XY plane. () Projetion onto XZ plane. (d) Projetion onto Y Z plane. lobe A side and the height of the main lobe A main ; that is { } Aside PSLR = 20log 10. (26) A main The ISLR is often used to analyse the side lobe power of point spread funtion (PSF). Assume that the main lobe power is P main and the total power is P total, the ISLR is then { } Ptotal P main ISLR = 10log 10, (27) P main where the numerator is the total power of the side lobes. Usually, the null-to-null of the PSF an be defined as the main lobe width. Figure 13 shows wave-propagation, along-trak dimension, rosstrak dimension profiles of a point target in Fig. 10. The profiles are used for omputing PSLR and ISLR. (d)

23 Progress In Eletromagnetis Researh, Vol. 129, Along trak dimensional profile of 3D image 0 _ 20 _ 20 Cross trak dimensional profile of 3D image 0 _ 40 _ 40 _ 60 _ 60 _ (a) (b) _ 20 _ 40 Wave propagation dimensional profile of 3D image 0 _ Figure 13. (a) Along-trak. (b) Cross-trak. () Wave-propagation dimension profile of the 3D image of one target obtained in the simulation 1. () Table 3. The measured PSLR, ISLR in the along-trak dimension, ross-trak dimension and wave-propagation dimension. Measured Parameter Along-trak Cross-trak Wave-propagation PSLR (db) ISLR (db) CONCLUSION This paper presents a 3D imaging algorithm espeially tailored for aurately reonstruting the 3D image of the DLSLA 3D SAR with dehirp on reeive tehnique. The dehirp eho signal is the threedimensional Fourier transform of the omplex image that we want to reonstrut on the basis of three dimensional projetion-slie theorem. This is the kernel of projetion-slie theorem and basis of our CBP imaging algorithm. Then, the onvolution bak-projetion imaging algorithm with platform motion ompensation during the time-divided transmitting-reeiving proedure is dedued in detail, and the parallel implementation of the CBP imaging algorithm is disussed later. Then the algorithm is verified with two numerial simulations. The merits of the numerial reonstruted images onfirm the auray of the proposed algorithm with DLSLA 3D SAR. ACKNOWLEDGMENT The authors would like to thank the support from the National Natural Siene Foundation of China General Program (Grant No ), and the National Natural Siene Foundation of China Key Program (Grant No ).

24 310 Peng et al. APPENDIX A. REMOVAL OF RESIDUAL VIDEO PHASE From Eq. (4), an undesirable RVP term is a onsequene of the signal dehirp on reeive in the DLSLA 3D SAR. The RVP term may ause degrading and geometry distortion effets in the reonstruted image. And these effets are spae-variant [2]. Sub-path proessing proedure is one way to redue the impat of RVP while it annot remove the RVP term ompletely. In this appendix, we pass the dehirp video signal through the digital equivalent of a frequeny-dependent delay line and remove the RVP term ompletely. where S if (m, n, t) = S r (m, n, t) S ref (m, n, t) N e 1N a ) (ˆt t 0 = a t ret exp { jφ ( m, n, ˆt )}, (A1) Φ(m, n, ˆt) = 4πK r ii=0 jj=0 T p ( f +ˆt 2R ) 0 (R t R 0 )+ 4πK r K r 2 (R t R 0 ) 2. (A2) Assuming a positive linear FM rate, it is desirable to delay high intermediate frequenies in time relative to low video frequenies to align the signals from all point satter objets in time. This alignment requires a frequeny-dependent time delay to relate frequeny by t d = f K r. This delay removes the original eho time delay differene between signals from different ranges. Appliation of new delay introdues a multipliative term of phase πf 2 K r into the original signal and removes RVP term from the signal history. Taking the time derivative of Eq. (A2), the instantaneous frequeny is f = 1 dφ 2π dˆt. (A3) As R = R t R 0, relative to effets of time delay of t d = f K r, evaluating this derivative yields the approximation f 2K r R, (A4) is adequate. It is indiated that satters that are nearer than R 0 have a positive frequeny in this representation while those that are distant have a negative frequeny. The result of passing the signal of phase Φ

25 Progress In Eletromagnetis Researh, Vol. 129, Figure A1. The flow diagram of RVP term elimination. through the delay line with t d = f K r is the new phase Φ r = Φ πf 2 K r, whih, on substitution of the expression in Eq. (A4) for f, beomes Φ r (m, n, ˆt) = 4πK r ( ˆt 2R a + f K r ) R. (A5) This expression is the desired phase for a target R t and referene range R 0. The RVP term has been eliminated. Fig. A1 illustrates the blok diagram of the RVP term elimination proedure. The proedure involves a Fourier transform, a omplex multiply and an inverse Fourier transform. where { S (fˆt ) = exp j fˆt 2 }. (A6) K r REFERENCES 1. Soumekh, M., Syntheti Aperture Radar Signal Proessing with Matlab Algorithms, John Wiley & Son, Carrara, W. G., R. S. Goodman, and R. M. Majewski, Spotlight Syntheti Aperture Radar: Signal Proessing Algorithms, Arteh House, Jakowatz, C. V., D. E. Wahl, P. H. Eihel, D. C. Ghiglia, and P. A. Thompson, Spotlight-mode Syntheti Aperture Radar: A Signal Proessing Approah, Kluwer Aademi Publishers, Boston London Dordreht, Weib, M. and J. H. G. Ender, A 3D imaging radar for small unmanned airplanes-artino, EURAD, , Gierull, C. H., On a onept for an airborne downwardlooking imaging radar, International Journal of Eletronis and Communiations, Vol. 53, No. 6, , Nouvel, J. F., S. Roques, and O. R. Du Plessis, A low-ost imaging radar: DRIVE on board ONERA motorglider, IGARSS, , 2007.

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