SAR image formation toolbox for MATLAB

Transcription

1 SAR image formation toolbox for MATLAB LeRoy A. Gorham and Linda J. Moore Air Fore Researh Laboratory, Sensors Diretorate 2241 Avionis Cirle, Bldg 62, WPAFB, OH ABSTRACT While many syntheti aperture radar (SAR) image formation tehniques exist, two of the most intuitive methods for implementation by SAR novies are the mathed filter and bakprojetion algorithms. The mathed filter and (non-optimized) bakprojetion algorithms are undeniably omputationally omplex. However, the bakprojetion algorithm may be suessfully employed for many SAR researh endeavors not involving onsiderably large data sets and not requiring time-ritial image formation. Exeution of both image reonstrution algorithms in MATLAB is expliitly addressed. In partiular, a manipulation of the bakprojetion imaging equations is supplied to show how ommon MATLAB funtions, ifft and interp1, may be used for straight-forward SAR image formation. In addition, limits for sene size and pixel spaing are derived to aid in the seletion of an appropriate imaging grid to avoid aliasing. Example SAR images generated though use of the bakprojetion algorithm are provided given four publily available SAR datasets. Finally, MATLAB ode for SAR image reonstrution using the mathed filter and bakprojetion algorithms is provided. Keywords: SAR, image formation, mathed filter, bakprojetion algorithm 1. INTRODUCTION Over the past four deades, numerous image formation algorithms have been developed with varying levels of omplexity and auray. However, these algorithms have a steep learning urve for SAR novies. In this paper, we introdue two algorithms - mathed filter and bakprojetion - implemented in MATLAB, whih are very simple to understand. A benefit of mathed filter image reonstrution is that the method offers optimization of the signal-to-noise ratio (SNR). 1 Use of the mathed filter requires a hypothesis regarding a target s harateristis; these assumptions yield a speifially designed filter that will ideally math to the target. While the mathed filter algorithm is intuitive and straight-forward, the generation of a 2D SAR image using this tehnique requires O(N 4 ) operations. For most pratial situations, the omputational omplexity of the mathed filter algorithm renders use of the method unrealisti. However, understanding of this fundamental imaging algorithm is extremely benefiial, espeially for the SAR novie. While several tehniques exist for SAR image generation, two ommonly employed approahes worth noting here are the polar format and bakprojetion algorithms. These algorithms differ in regards to the tradeoff between omputational omplexity and image quality. 2,3 The polar format algorithm 4,5 for SAR image reonstrution is similar to omputer-aided tomography imaging tehniques. 6,7,8 Polar format reonstrution employs a diret 2D Fourier transform, whih provides the opportunity to take advantage of the extremely effiient fast Fourier transform. Polar format image reonstrution requires O(N 2 log N) operations given an NxN SAR image. While more omputationally effetive than the bakprojetion algorithm, the polar format imaging method involves an approximation of the reeived signal phase. This approximation introdues errors into the final reonstruted SAR image. Several orretion tehniques have been suggested as a means of improving the auray of SAR images generated via polar format reonstrution. 9,1 Another frequently used image formation tehnique is the bakprojetion algorithm. 7,11 A distint advantage of the bakprojetion algorithm is the ability to form SAR images as phase history is olleted, pulse by pulse, and to integrate newly obtained information into the SAR image as it beomes available. The bakprojetion algorithm is omputationally expensive, however, as it requires O(N 3 ) operations for an NxN SAR image. Fortunately, the bakprojetion algorithm lends itself naturally to parallel proessing. Bakprojetion has reently Further author information: ( leroy.gorham@wpafb.af.mil and linda.moore2@wpafb.af.mil)

2 been implemented on graphis proessing units (GPUs), 12,13 whih provide inexpensive platforms for parallel omputing. Several methods have been proposed as a means of reduing the omputational omplexity of the bakprojetion algorithm while retaining the advantages of this image formation approah. 14,15,16,17 Many of these fast bakprojetion algorithms are flexible, providing manners in whih image quality may be aeptably sarified in order to further improve data proessing time. These image formation algorithms, several of whih are favorable given wide bandwidths and large syntheti apertures, offer dereased omputational omplexities of O(N 5/2 ) 14 and O(N 2 log N). 15,16,17 Non-optimized bakprojetion imaging may be impratial for exeptionally large data sets with time-ritial image formation requirements; however, knowledge of this simple SAR imaging tehnique is undoubtedly valuable for many researh endeavors. The remainder of this paper is outlined as follows. In Setion 2, we define our terminology and the SAR signal model. We also derive limits for sene size and pixel spaing to aid in seleting an imaging grid that avoids aliasing. In Setion 3, we derive and desribe the mathed filter algorithm for image formation. In Setion 4, we derive a bakprojetion algorithm, whih is based on the mathed filter algorithm. We desribe how to utilize basi MATLAB funtions like ifft and interp1 to implement the algorithm. In Setion 5, we apply the bakprojetion algorithm to four publily available SAR datasets. Finally, MATLAB ode for the two imaging algorithms, mathed filter and bakprojetion, are supplied in Appendix A.1 and A.2. The authors will be glad to provide opies of the MATLAB ode used to generate the figures provided in this paper in response to requests via SIGNAL MODEL The following SAR signal model 18 is used for the development of MATLAB implementations of the mathed filter and bakprojetion imaging algorithms. The SAR sensor travels along a flight path suh that the antenna phase enter has a three-dimensional spatial loation denoted r a (τ) suh that r a (τ) = [x a (τ), y a (τ), z a (τ)] T (1) where τ denotes the syntheti aperture, or slow time, domain. All oordinates are defined with respet to a sene origin, whih orresponds to the SAR motion ompensation point. The distane from the antenna phase enter to the origin, r a (τ), is denoted by A target is speified at a loation d a (τ) = x 2 a(τ) + y 2 a(τ) + z 2 a(τ). (2) r(τ) = [x(τ), y(τ), z(τ)] T. (3) In general, this target an have any arbitrary motion, but in this paper, we will assume that the target is stationary. Thus, the dependene on τ is dropped for the remainder of the derivation. The target has a radar ross setion whih is dependent on frequeny and aspet angle. The distane from the antenna phase enter to the target is denoted by d a (τ) = (x a (τ) x) 2 + (y a (τ) y) 2 + (z a (τ) z) 2. (4) At periodi intervals, the radar transmits a pulse that reflets off satterers in the sene. Some of the energy is then reeived by the radar. In a given syntheti aperture, there are N p pulses used to form the image. The transmission time of eah pulse is denoted by the sequene {τ n n = 1, 2,..., N p }. The output of the reeiver at a given time, τ n, is a sequene of band-limited frequeny samples delayed with respet to the time of pulse transmission by the round-trip time to the target. There are K frequeny samples per pulse, and the assoiated frequeny values are represented by the sequene {f k k = 1, 2,..., K}. The reeiver output from the target loated at r is j4πfk R(τ n ) S(f k, τ n ) = A(f k, τ n ) exp. (5)

3 The amplitude, A(f k, τ n ), is related to the radar ross setion of the target while the phase is dependent on the frequeny of eah sample and on the differential range, R(τ n ), given by R(τ n ) = d a (τ n ) d a (τ n ). (6) Equation (5) assumes that eah pulse is motion ompensated suh that a satterer at the sene origin will have zero phase for all f k and τ n. The atual reeiver output is thus the sum of the ontributions of all satterers in the sene. The frequeny samples, {f k }, have a minimum value denoted by f 1, a median value denoted by f, a maximum value denoted by f K, and a step size denoted by f. The frequeny step size is inversely related to the maximum alias-free range extent of the image, W r, by W r = 2 f. (7) Thus, the frequeny step size is hosen to math the size of the sene to be imaged. The total bandwidth, B, of the reeived pulse is B = (K 1) f. The range resolution is thus δ r = 2B = 2(K 1) f. (8) In a similar manner as above, the azimuth angle traversed during the syntheti aperture determines the ross-range resolution, and the azimuth angle from pulse to pulse determines the maximum alias-free ross-range extent of the image, W x. Given an azimuth step size of θ, W x = λ min 2 θ where λ min is the minimum wavelength suh that λ min = /f K. The total azimuth angle, θ a, traversed during the syntheti aperture is θ a = (N p 1) θ. Thus, the ross-range resolution, δ x, is given by (9) δ x = λ λ = 2θ a 2(N p 1) θ (1) where λ is the enter wavelength suh that λ = /f. To utilize the imaging algorithms desribed in this paper, one an selet any arbitrary pixel loations. However, one should be areful when seleting these loations to avoid aliasing of the image or the frequeny support. The pixel spaing should be finer than the resolution defined in Equations (8) and (1) and the overall sene size should be less than the maximum sene size defined in Equations (7) and (9). 3. MATCHED FILTER ALGORITHM The most straightforward method for forming a SAR image is to perform a mathed filter. One an build the mathed filter to any kind of satterer, but here we will assume an isotropi point satterer. The reeived signal from a point satterer at loation r is given in Equation (5). An isotropi satterer will have a onstant amplitude and thus A(f k, τ n ) = A. Therefore, the mathed filter response, denoted by I(r), at loation r is given by I(r) = 1 N p K assuming a single satterer in the sene. N p n=1 +j4πfk R(τ n ) S(f k, τ n ) exp = A, (11) To form an image using this method, Equation (11) is applied for eah pixel in the image. This requires alulation of the differential range, R(τ n ), for every pixel for every pulse. The algorithm has a omputational omplexity of O(N 4 ) for 2D images, whih makes it impratial for most appliations. However, Equation (11) forms the basis for the derivation of the bakprojetion algorithm in Setion 4. MATLAB ode for the mathed filter algorithm is provided in Appendix A.1.

4 MATLAB Type Symbol Desription Units data.deltaf salar f Frequeny step size Hz data.minf N p vetor f 1 Minimum frequeny for every pulse Hz data.nfft salar N fft Length of FFT (bakprojetion only) data.x mat Arbitrary r X oordinate of every pixel in the image m data.y mat Arbitrary r Y oordinate of every pixel in the image m data.z mat Arbitrary r Z oordinate of every pixel in the image m data.antx N p vetor r a Antenna loation for every pulse m data.anty N p vetor r a Antenna loation for every pulse m data.antz N p vetor r a Antenna loation for every pulse m data.r N p vetor d a (τ n ) Distane to the mo-omp point m data.phdata (KxN p ) array S(f k, τ n ) Complex phase history data, Eq. (5) Table 1. Neessary fields in the MATLAB data struture for image formation y (m) 25 y (m) x (m) (a) mathed filter image x (m) (b) bakprojetion image Figure 1. Comparison of SAR images (in db) of 3 point targets, loated at (,,), (-3,2,), and (1,4,), generated using the (a) mathed filter and (b) bakprojetion algorithms. 5 In order to use the MATLAB funtion, mfbasi, the inputs are provided in a struture alled data. The fields of this struture are defined in Table 1. The image pixel loations, r, are speified in the pixel loation matries data.x mat, data.y mat and data.z mat whih ontain the (x, y, z) loations of eah pixel. The output image is stored in data.im final, whih has the same dimensions as the pixel loation matries. While these pixels an have arbitrary positions, it is often desirable to form an image onto a regular 2D or 3D grid. In this ase, the MATLAB funtion meshgrid is useful to build the pixel loation matries. In all the image examples provided in this paper, meshgrid was used to build data.x mat and data.y mat, while data.z mat was filled with zeros. If a Digital Elevation Map (DEM) is available, data.z mat an be filled with interpolated height values for every pixel to improve image quality. Figure 1(a) shows an example of an image formed using the mathed filter algorithm. Phase history data was simulated for 3 point targets using Equation (5) where A(f k, τ n ) = 1. Here, N p = 128 pulses were simulated with K = 512 frequeny samples per pulse, a enter frequeny of 1 GHz and a 6 MHz bandwidth. A irular flight path was used with a 3 degree depression angle and a slant range of 1 km. A 3 degree integration angle was used with a enter azimuth angle of 5 degrees. The sene extent was 1 m x 1 m with 2 m pixel spaing in eah dimension.

5 4. BACKPROJECTION ALGORITHM The bakprojetion algorithm offers an intuitive imaging tehnique for SAR novies. While the bakprojetion algorithm is omputationally expensive, its implementation is not unreasonable for many SAR researh ventures. In this setion, bakprojetion imaging equations are manipulated in order to illustrate how the algorithm an be exeuted through use of ommon MATLAB funtions. 4.1 Derivation of Effiient Calulation of Range Profiles The mathed filter response shown in Equation (11) an be used to ompute the target response at a disrete range bin, m. Given SAR phase history, S(f k, τ n ), olleted by N p pulses over a range of K frequenies, the range profile at range bin m given a reeived pulse at slow time τ n is s(m, τ n ) = +j4πfk R(m, τ n ) S(f k, τ n ) exp. (12) By substituting the frequeny values f k = (k 1) f + f 1 into Equation (12), the range profile may be rewritten as +j4π((k 1) f + f1 ) R(m, τ n ) s(m, τ n ) = S(f k, τ n ) exp ( +j4π f R(m, τn )(k 1) = S(f k, τ n ) exp + +j4πf ) 1 R(m, τ n ) +j4πf1 R(m, τ n ) = S(f k, τ n ) exp[φ( R(m, τ n )) (k 1)] exp where phase funtion Φ( R(m, τ n )) = (+j4π f R(m, τ n ))/. In order to implement the bakprojetion algorithm in MATLAB, Equation (13) must be rewritten in terms of MATLAB s inverse disrete Fourier transform (ifft) funtion. In MATLAB, the definitions of the disrete Fourier transforms between X(k) and x(m) are given as 19 (13) X(k) = M m=1 x(m) = 1 K x(m) ω (m 1)(k 1) K X(k) ω (m 1)(k 1) K (14) where ω K = exp( j2π/k). Therefore, the inverse disrete Fourier transform of X(k) is given as x(m) = ifft(x(k)) = 1 K +j2π(m 1) X(k) exp (k 1) = 1 K K X(k) exp (Θ(m) (k 1)) (15) where Θ(m) = (j2π(m 1))/K. In order to write Equation (13) in terms of MATLAB s ifft funtion, the previously defined phase funtion, Φ( R(m, τ n )), from Equation(13), must equal Θ(m) from Equation (15). To satisfy this requirement, the following equality must hold: R(m, τ n ) = (m 1) K 2 f = (m 1) K W r. (16) Note that R(m, τ n ) represents a sampling aross range and the maximum unambiguous range ours when (m 1) = K (reall Equation (7)).

6 Inserting the result from Equation (16) into Equation (13) yields +j4π f(k 1) +j4πf1 R(m, τ n ) s(m, τ n ) = S(f k, τ n ) exp R(m, τ n ) exp ( ) +j4π f(k 1) m 1 = S(f k, τ n ) exp K +j4πf1 R(m, τ n ) exp 2 f +j2π(m 1) +j4πf1 R(m, τ n ) = S(f k, τ n ) exp (k 1) exp. K (17) Finally, realling the expression for the inverse disrete Fourier transform from Equation (15), the range profile may be expressed in terms of MATLAB s ifft. +j4πf1 R(m, τ n ) s(m, τ n ) = K ifft(s(f k, τ n )) exp (18) The onstant K appears in Equation (18) beause by definition, the inverse disrete Fourier transform ontains a saling fator of 1/K. In addition, it s important to note that the MATLAB implementation of the inverse disrete Fourier transform assumes that K is even. However, for values of K that are powers of 2, the omputational omplexity of the inverse disrete Fourier transform is optimized. By inserting the expression for R(m, τ n ) from Equation (16), an alternative expression for the implementation of a range profile in MATLAB is given by j2πf1 (m 1) s(m, τ n ) = K ifft(s(f k, τ n )) exp. (19) K f One final onsideration must be addressed. Eah pulse is motion ompensated suh that a satterer at the sene origin has zero phase and will appear in the zero frequeny bin in the range profile. This bin orresponds to the enter of the range profile omputed in Equation (19). By default, the funtion ifft omputes the values from 1 m K where m = 1 orresponds to the zero frequeny bin. To put m = 1 at the enter of the range profile, the ommand fftshift is applied to the output of the ifft ommand. This operation ensures that the output is orretly ordered following the inverse disrete Fourier transform operation and that the zero frequeny omponent orresponds to the enter of the output vetor. As a result, range index, m, aquires values between K/2 + 1 and K/2. Therefore, R(m, τ n ) of Equation (16) has values between W r /2 and W r /2 W r /K and Equation (19) beomes j2πf1 (m 1) s(m, τ n ) = K fftshift{ifft(s(f k, τ n ))} exp. (2) K f 4.2 Image Formation Proess For implementation of the bakprojetion imaging algorithm in MATLAB, Equation (6) is used to ompute the differential range, R(τ n ), for eah pixel, for eah pulse, where the pixel (x, y, z) oordinates are inserted into Equation (4). In order to use Equation (2) to form a SAR image, an interpolation step is required due to the fat that the disrete values of R(m, τ n ) do not orrespond exatly to the R(τ n ) values alulated for every pixel. There are several methods for performing this interpolation, 2,11 but here we will simply use the MATLAB interp1 funtion to implement linear interpolation. Sine s(m, τ n ) is a band-limited signal, the ideal interpolator is a sin interpolator. This an be approximated by zero-padding the ifft omputation and then performing linear interpolation on the inverse disrete Fourier transform output. A good rule of thumb is that the length of the ifft, denoted N fft, should be 1 times the length of the data, K. Also, the ifft funtion is most effiient when N fft is a power of 2. In the bakprojetion algorithm, N fft is provided as an input to the imaging funtion, as shown in Table 1.

7 The first step in the image formation proess is to implement Equation (2) for every pulse, zero-padding the data suh that the length of the ifft is N fft. Thus, j2πf1 (m 1) s(m, τ n ) = N fft fftshift{ifft(s(f k, τ n ))} exp. (21) N fft f where S(f k, τ n ) = for all k > K. To find the image response for a pixel at loation r, given pulse n, R(τ n ) is alulated and used to find an interpolated value of s(m, τ n ). This is denoted as s int (r, τ n ). The final image response, I(r), is simply the summation of these values for every pulse. N p I(r) = s int (r, τ n ). (22) n=1 Figure 1(b) shows an example of an image formed using the bakprojetion algorithm. The image was generated using the same data and equivalent pixel loations as those employed in Figure 1(a). 5. IMAGE EXAMPLES In the last few years, the Air Fore Researh Laboratory (AFRL) has released several datasets of both syntheti and measured SAR data. In this setion, the bakprojetion algorithm introdued in Setion 4 is applied to data from the Bakhoe Data Dome, the 2D/3D Volumetri Challenge Problem, and the SAR-based Ground Moving Target Indiator (GMTI) Challenge Problem. In addition, the imaging ode was implemented on the Civilian Vehile Radar Data Domes syntheti data set from The Ohio State University. 5.1 Bakhoe Data Dome In 24, AFRL released a synthetially generated data dome of a bakhoe target. 21 The sattered field data for the bakhoe was omputed on a 2π-steradian data dome over the target for a frequeny bandwidth of 5.9 GHz at a enter frequeny of 1 GHz y (m) x (m) Figure 2. Bakprojetion image (in db) using the Bakhoe Data Dome syntheti dataset. The bakprojetion algorithm was performed on this dataset and a resulting image is given in Figure 2. The data imaged onsisted of all 36 aspet angles from a single elevation of 1 using VV polarization. The full 5.9 GHz bandwidth was used to form the image. The sene extent is 1 m x 1 m with 2 m pixel spaing, resulting in a 51 x 51 pixel image. No windowing was applied to the data before imaging. The alulated maximum sene size is W r = m; W x = 9.27 m. 7

8 5.2 2D/3D Volumetri Challenge Problem In 27, AFRL released a hallenge problem for 2D/3D imaging of targets from a volumetri data set in an urban environment. 22 The data was olleted of a sene onsisting of numerous ivilian vehiles and alibration targets. The bakprojetion algorithm was performed on this dataset and a resulting image is given in Figure 3. The data imaged onsisted of Pass 1 with HH polarization. An integration angle, θ, of 4 entered at 4 azimuth was used. The sene extent is 1 m x 1 m with 2 m pixel spaing, resulting in a 51 x 51 pixel image. No windowing was applied to the data before imaging. The alulated maximum sene size is W r = m; W x = m and the alulated resolution is δr =.24 m; δx =.23 m y (m) x (m) Figure 3. Bakprojetion image (in db) using the 2D/3D Volumetri Challenge Problem dataset SAR-based GMTI Challenge Problem In 29, AFRL released a hallenge problem for SAR-based GMTI in urban environments. 23 This data onsists of a 71-seond portion of phase history data from a radar operating in irular SAR mode. The data has been range-gated around the known loation of a moving vehile, severely limiting the range swath of the data. However, during the first 25 seonds of the senario, the vehile is waiting in line to ross a busy intersetion. The bakprojetion algorithm was performed on this dataset and a resulting image is provided in Figure 4. The sene extent is 2 m x 2 m with 2 m pixel spaing, resulting in a 11 x 11 pixel image. The first 8 pulses (almost 4 seonds) were used, and no windowing was applied to the data before imaging. The alulated maximum sene size is W r = m; W x = 2, m, and the alulated resolution is δr =.23 m; δx =.25 m. The relatively small range extent aounts for the blak areas in the image. 5.4 Civilian Vehile Radar Data Domes In 21, The Ohio State University released a set of synthetially generated data domes of ivilian targets. 24 The sattered field data for the targets was omputed for full azimuth overage of elevation angles from 3 to 6. The data is full polarimetri with a frequeny bandwidth of 5.35 GHz at a enter frequeny of 9.6 GHz. The bakprojetion algorithm was implemented on the 1993 Jeep target, and a resulting image is shown in Figure 5. The data imaged onsisted of all 36 aspet angles from a single elevation of 3 using VV polarization. The full bandwidth was used to form the image. The sene extent is 1 m x 1 m with 2 m pixel spaing, resulting in a 51 x 51 pixel image. No windowing was applied to the data before imaging. The alulated maximum sene size is W r = 14.3 m; W x = m.

9 5 1 2 y (m) x (m) Figure 4. Bakprojetion image (in db) using the SAR-based GMTI dataset y (m) x (m) Figure 5. Bakprojetion image (in db) using the Civilian Vehile Radar Data Domes syntheti dataset CONCLUSIONS In this paper, we developed two simple SAR imaging algorithms: a mathed filter algorithm and a bakprojetion algorithm. The bakprojetion algorithm was derived from the mathed filter algorithm by utilizing a fast Fourier transform to ompute range profiles. Both algorithms were implemented in MATLAB, and image examples were provided using four publily available SAR datasets.

10 A.1 Mathed Filter Algorithm APPENDIX A. MATLAB CODE 1 funtion data = mfbasi(data) 2 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4 % This funtion performs a mathed filter operation. The following % 5 % fields need to be populated: % 6 % % 7 % data.deltaf: Step size of frequeny data (Hz) % 8 % data.minf: Vetor ontaining the start frequeny of eah pulse (Hz) % 9 % data.x mat: The x position of eah pixel (m) % 1 % data.y mat: The y position of eah pixel (m) % 11 % data.z mat: The z position of eah pixel (m) % 12 % data.antx: The x position of the sensor at eah pulse (m) % 13 % data.anty: The y position of the sensor at eah pulse (m) % 14 % data.antz: The z position of the sensor at eah pulse (m) % 15 % data.r: The range to sene enter (m) % 16 % data.phdata: Phase history data (frequeny domain) % 17 % Fast time in rows, slow time in olumns % 18 % % 19 % The output is: % 2 % data.im final: The omplex image value at eah pixel % 21 % % 22 % Written by LeRoy Gorham, Air Fore Researh Laboratory, WPAFB, OH % 23 % leroy.gorham@wpafb.af.mil % 24 % Date Released: 8 Apr 21 % 25 % % 26 % Gorham, L.A. and Moore, L.J., "SAR image formation toolbox for % 27 % MATLAB," Algorithms for Syntheti Aperture Radar Imagery XVII % 28 % 7669, SPIE (21). % 29 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 31 % Define speed of light (m/s) 32 = ; % Determine the size of the phase history data 35 data.k = size(data.phdata,1); % The number of frequeny bins per pulse 36 data.np = size(data.phdata,2); % The number of pulses % Determine the azimuth angles of the image pulses (radians) 39 data.antaz = unwrap(atan2(data.anty,data.antx)); 4 41 % Determine the average azimuth angle step size (radians) 42 data.deltaaz = abs(mean(diff(data.antaz))); % Determine the total azimuth angle of the aperture (radians) 45 data.totalaz = max(data.antaz) min(data.antaz); % Determine the maximum sene size of the image (m) 48 data.maxwr = /(2*data.deltaF); 49 data.maxwx = /(2*data.deltaAz*mean(data.minF)); 5 51 % Determine the resolution of the image (m) 52 data.dr = /(2*data.deltaF*data.K); 53 data.dx = /(2*data.totalAz*mean(data.minF)); % Display maximum sene size and resolution 56 fprintf('maximum Sene Size: %.2f m range, %.2f m ross range\n',data.maxwr,data.maxwx); 57 fprintf('resolution: %.2fm range, %.2f m ross range\n',data.dr,data.dx); % Initialize the image with all zero values 6 data.im final = zeros(size(data.x mat)); % Set up a vetor to keep exeution times for eah pulse (se) 63 t = zeros(1,data.np); % Loop through every pulse 66 for ii = 1:data.Np % Display status of the imaging proess 69 if ii > 1 7 t sofar = sum(t(1:(ii 1))); 71 t est = (t sofar*data.np/(ii 1) t sofar)/6; 72 fprintf('pulse %d of %d, %.2f minutes remaining\n',ii,data.np,t est); 73 else

11 74 fprintf('pulse %d of %d\n',ii,data.np); 75 end 76 ti % Calulate differential range for eah pixel in the image (m) 79 dr = sqrt((data.antx(ii) data.x mat).ˆ (data.anty(ii) data.y mat).ˆ (data.antz(ii) data.z mat).ˆ2) data.r(ii); % Calulate the frequeny of eah sample in the pulse (Hz) 84 freq = data.minf(ii) + (:(data.k 1)) * data.deltaf; % Perform the Mathed Filter operation 87 for jj = 1:data.K 88 data.im final = data.im final + data.phdata(jj,ii) * exp(1i*4*pi*freq(jj)/*dr); 89 end 9 91 % Determine the exeution time for this pulse 92 t(ii) = to; 93 end return A.2 Bakprojetion Algorithm 1 funtion data = bpbasi(data) 2 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4 % This funtion performs a basi Bakprojetion operation. The % 5 % following fields need to be populated: % 6 % % 7 % data.nfft: Size of the FFT to form the range profile % 8 % data.deltaf: Step size of frequeny data (Hz) % 9 % data.minf: Vetor ontaining the start frequeny of eah pulse (Hz) % 1 % data.x mat: The x position of eah pixel (m) % 11 % data.y mat: The y position of eah pixel (m) % 12 % data.z mat: The z position of eah pixel (m) % 13 % data.antx: The x position of the sensor at eah pulse (m) % 14 % data.anty: The y position of the sensor at eah pulse (m) % 15 % data.antz: The z position of the sensor at eah pulse (m) % 16 % data.r: The range to sene enter (m) % 17 % data.phdata: Phase history data (frequeny domain) % 18 % Fast time in rows, slow time in olumns % 19 % % 2 % The output is: % 21 % data.im final: The omplex image value at eah pixel % 22 % % 23 % Written by LeRoy Gorham, Air Fore Researh Laboratory, WPAFB, OH % 24 % leroy.gorham@wpafb.af.mil % 25 % Date Released: 8 Apr 21 % 26 % % 27 % Gorham, L.A. and Moore, L.J., "SAR image formation toolbox for % 28 % MATLAB," Algorithms for Syntheti Aperture Radar Imagery XVII % 29 % 7669, SPIE (21). % 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Define speed of light (m/s) 33 = ; % Determine the size of the phase history data 36 data.k = size(data.phdata,1); % The number of frequeny bins per pulse 37 data.np = size(data.phdata,2); % The number of pulses % Determine the azimuth angles of the image pulses (radians) 4 data.antaz = unwrap(atan2(data.anty,data.antx)); % Determine the average azimuth angle step size (radians) 43 data.deltaaz = abs(mean(diff(data.antaz))); % Determine the total azimuth angle of the aperture (radians) 46 data.totalaz = max(data.antaz) min(data.antaz); % Determine the maximum sene size of the image (m) 49 data.maxwr = /(2*data.deltaF); 5 data.maxwx = /(2*data.deltaAz*mean(data.minF)); 51

12 52 % Determine the resolution of the image (m) 53 data.dr = /(2*data.deltaF*data.K); 54 data.dx = /(2*data.totalAz*mean(data.minF)); % Display maximum sene size and resolution 57 fprintf('maximum Sene Size: %.2f m range, %.2f m ross range\n',data.maxwr,data.maxwx); 58 fprintf('resolution: %.2fm range, %.2f m ross range\n',data.dr,data.dx); 59 6 % Calulate the range to every bin in the range profile (m) 61 data.r ve = linspae( data.nfft/2,data.nfft/2 1,data.Nfft)*data.maxWr/data.Nfft; % Initialize the image with all zero values 64 data.im final = zeros(size(data.x mat)); % Set up a vetor to keep exeution times for eah pulse (se) 67 t = zeros(1,data.np); % Loop through every pulse 7 for ii = 1:data.Np % Display status of the imaging proess 73 if ii > 1 74 t sofar = sum(t(1:(ii 1))); 75 t est = (t sofar*data.np/(ii 1) t sofar)/6; 76 fprintf('pulse %d of %d, %.2f minutes remaining\n',ii,data.np,t est); 77 else 78 fprintf('pulse %d of %d\n',ii,data.np); 79 end 8 ti % Form the range profile with zero padding added 83 r = fftshift(ifft(data.phdata(:,ii),data.nfft)); % Calulate differential range for eah pixel in the image (m) 86 dr = sqrt((data.antx(ii) data.x mat).ˆ (data.anty(ii) data.y mat).ˆ (data.antz(ii) data.z mat).ˆ2) data.r(ii); 89 9 % Calulate phase orretion for image 91 phcorr = exp(1i*4*pi*data.minf(ii)/*dr); % Determine whih pixels fall within the range swath 94 I = find(and(dr > min(data.r ve), dr < max(data.r ve))); % Update the image using linear interpolation 97 data.im final(i) = data.im final(i) + interp1(data.r ve,r,dr(i),'linear').* phcorr(i); % Determine the exeution time for this pulse 1 t(ii) = to; 11 end return REFERENCES [1] Soumekh, M., [Syntheti aperture radar signal proessing with MATLAB algorithms], Wiley-Intersiene (1999). [2] Jakowatz Jr, C., Wahl, D., Yoky, D., Bray, B., Bow Jr, W., and Rihards, J., Comparison of algorithms for use in real-time spotlight-mode SAR image formation, in [Proeedings of SPIE], 5427, 18 (24). [3] Jakowatz Jr, C. and Doren, N., Comparison of polar formatting and bak-projetion algorithms for spotlight-mode SAR image formation, in [Proeedings of SPIE], 6237, 6237H (26). [4] Carrara, W., Goodman, R., and Majewski, R., [Spotlight Syntheti Aperture Radar - Signal Proessing Algorithms], Norwood, MA: Arteh House (1995). [5] Jakowatz, C., Wahl, D., Eihel, P., Ghiglia, D., and Thompson, P., [Spotlight-mode syntheti aperture radar: a signal proessing approah], Kluwer Aademi Pub (1996). [6] Stark, H., Woods, J., Paul, I., and Hingorani, R., An investigation of omputerized tomography by diret fourier inversion and optimum interpolation, IEEE Transations on Biomedial Engineering, (1981). [7] Munson Jr, D., O Brien, J., and Jenkins, W., A tomographi formulation of spotlight-mode syntheti aperture radar, Proeedings of the IEEE 71(8), (1983).

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