A New Three-Dimensional Imaging Algorithm

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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 1, JANUARY 2011 153

A New Three-Dimensional Imaging Algorithmfor Airborne Forward-Looking SAR

Xiaozhen Ren, Jiantao Sun, and Ruliang Yang, Member, IEEE

AbstractIn this letter, a new 3-D imaging algorithm is pro-posed for forward-looking synthetic aperture radar based on theimaging geometry and the characteristic of the echo signal. Thekey point of the proposed algorithm is the introduction of the non-linear frequency modulation scaling in along-track processing toobtain accurate focusing. As the method needs only Fourier trans-form and multiplication operations, it is computationally efficient.Simulations with point scatterers are used to validate the method.

Index TermsForward-looking, nonlinear frequency modula-tion scaling, synthetic aperture radar (SAR), 3-D imaging.

I. INTRODUCTION

T RADITIONAL synthetic aperture radar (SAR) systemscan reconstruct 2-D images of the investigated area withweather independence and all-day operation capabilities. Theyhave been widely used in civil and military applications. How-ever, 2-D images could not meet the requirements in manyapplications, and 3-D images are required. Conventional in-terferometric SAR (InSAR) can measure the elevation of theterrain patch, but the distribution of the scatterers in heightis underdetermined by a single-baseline measurement. As theextension of conventional InSAR, multibaseline SAR tomog-raphy adds multiple baselines in the direction perpendicular tothe azimuth and to the line of sight and forms an additional

synthetic aperture in the height direction. Therefore, it has a re-solving capability along this dimension. Unfortunately, for thecurrent SAR tomography, it is almost impossible to avoid an un-even track distribution in repeat-pass data acquisition, which isjust the main reason for the strong ambiguity in height [1], [2].

A forward-looking SAR system works in an innovative imag-ing mode [3][6]. Dr. Regiber from the German AerospaceCenter (DLR) first proposed that the forward-looking SAR sys-tem can be used to realize 3-D imaging of the interested region,and it can be done through one pass only [7]. Consequently,the 3-D imaging technique of forward-looking SAR became anew research direction, which can avoid the height ambiguityproblem in SAR tomography caused by the uneven track dis-

tribution. Later on, the 3-D imaging principle and resolutionsof airborne forward-looking SAR was analyzed in [8], and an

Manuscript received September 10, 2009; revised April 7, 2010 and May9, 2010; accepted June 3, 2010. Date of publication August 3, 2010; date ofcurrent version December 27, 2010.

X. Ren is with the College of Information Science and Engineering,Henan University of Technology, Zhengzhou 450052, China (e-mail: [email protected]).

J. Sun and R. Yang are with the Institute of Electronics, Chinese Academyof Sciences, Beijing 100190, China.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LGRS.2010.2055035

Fig. 1. Geometry and transmitting and receiving orders of forward-lookingSAR. (a) Geometry of forward-looking SAR. (b) Transmitting and receivingorders of forward-looking SAR.

accurate focused algorithm was proposed based on backprojec-tion, which suffered from severe computational complexity.

The main topic of this letter is to introduce an accurateand efficient 3-D imaging algorithm for forward-looking SAR.According to the analysis of the spatial geometry and the echosignal model, we divide the 3-D imaging process of forward-looking SAR into two steps. By using the principle of nonlinearfrequency modulation scaling, an along-track-variant filteringprocessing is introduced to compensate the along-track depen-dence of the along-track frequency modulation rate. The phasecompensation factors and algorithm realization procedure aredemonstrated in detail.

This letter is organized as follows. Section II presentsthe geometric and signal model in forward-looking SAR. InSection III, a novel 3-D imaging approach is deduced indetail based on nonlinear frequency modulation scaling. Theperformance of the method is investigated by simulated data inSection IV. Finally, Section V gives a brief conclusion.

II. FORWARD-L OOKING SA R

The geometry of forward-looking SAR is shown in Fig. 1(a).x, y, and r denote the along-track, azimuth, and slant-rangedirections, respectively. The radar platform flies along thex-direction with velocity v at height H. The receivingantennas are centered at the y-axis with spacing d, and asingle transmitting antenna is h below the center receivingantenna. Fig. 1(b) shows the transmitting and the receivingorder of each antenna for a forward-looking SAR. During thewhole flight, signals are transmitted by the single transmittingantenna, and the backscattered signals are received by eachindividual receiving antenna sequentially with a fixed switchvelocity. Suppose that the switch velocity is v

s, and then, a

1545-598X/$26.00 2010 IEEE


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154 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 1, JANUARY 2011

virtual receiving antenna moving with the same velocity alongthe azimuth direction can be utilized to form an approximatesynthetic aperture in the same direction. Therefore, the azimuthambiguity can be avoided [3].

Consider the data acquisition shown in Fig. 1(a). The positionof the nth receiving antenna is given by (xm, yn, H), whereyn

is the azimuth position of the nth receiving antenna andxm is the along-track position. For a point scatterer positionedat (xp, yp, zp), the transmitting and receiving paths RT,n andRR,n are given by

RT,n =

(xm xp)2 + y2p + (H h zp)2 (1)

RR,n =

(xm xp)2 + (yn yp)2 + (H zp)2

=R2m + (yn yp)

2 (2)

where Rm =

(xm xp)2 + (H zp)2.The linear-frequency-modulated pulse signal transmitted by

a radar is given by

so(t) = expj(2fct + t

2)

(3)

where fc is the carrier frequency, t is the range time, and isthe chirp rate.

For a point scatterer positioned at (xp, yp, zp), the echoreceived by the nth receiving antenna positioned at (xm, yn, H)can be written as

sraw(t, ynRm) =apst(RT,n +RR,n)/c

exp(j2fct) (4)

where ap is the radar scatter coefficient and c is the velocity oflight.

Substituting (1) and (2) into (4) yields

sraw(t, ynRm) =A0 expj

t2

R2m+(ynyp)

2/c

exp

j2

R2m+(ynyp)

2/

(5)

where A0 = ap exp(j2RT,n/) and is the signalwavelength.

According to (5), when all the receiving antennas havereceived echoes sequentially at a fixed sampling position xm(Rm is a constant now) for once, a virtual receiving antennamoving along the y-axis (the azimuth direction) can be utilizedto form an approximate synthetic aperture in the azimuth di-rection. Consequently, a SAR image, which is obtained after

azimuth and slant-range direction focusing, can be describedas [4], [5], [8]

s(t,yRm) =Asinc

fr(t2Rm/c)

sinc[Lf(yyp)/(Rm)]

exp(j4Rm/) (6)

where A is the amplitude of the focused target, fr is the rangebandwidth, and Lf is the effective azimuth aperture [3].

Since the backscattered signals are received by each individ-ual receiving antenna sequentially with a fixed switch velocity,a sequence of 2-D SAR images at the along-track position xmwith m [1,M] can be obtained as the SAR platform fliesalong the track continuously. M is the number of 2-D SAR

images [see Fig. 1(a)]. Suppose that all SAR images havebeen coregistered first, and then, in the areas illuminated by

Fig. 2. Two-dimensional focused images in the along-track direction offorward-looking SAR.

the beam, the azimuth positions of each scatterer in all SARimages are invariable, while only the distances between theradar and the scatterer decrease as the plane moves forward.Therefore, an along-trackslant-range section corresponding toone azimuth position can be focused at a time. Then, the 3-Dimage of forward-looking SAR can be obtained by processingall the sections with the same procedure.

III. THREE-D IMENSIONAL IMAGING ALGORITHM

FO R FORWARD-L OOKING SA R

According to the analysis of Section II, we can divide the3-D imaging process of forward-looking SAR into two steps.The 2-D images can be obtained using a classical chirp scalingmethod first, and then, height focusing is performed. As the firststep is a general 2-D imaging process, which can be found in[4] and [5], this letter focuses on the second step.

Supposing that all the 2-D focused images in the along-

track direction of forward-looking SAR have been obtainedand coregistered, as shown in Fig. 2, the along-trackslant-range section corresponding to azimuth position y = y0 can bewritten as

s(t, Rmy=y0) =Asinc

fr(t2Rm/c)

exp(j4Rm/)sinc[Lf(y0yp)/(Rm)]

=A1sinc

fr(t2Rm/c)

exp(j4Rm/)(7)

where A1 = Asinc[Lf(y0 yp)/(Rm)]. As A1 has no effecton the imaging process, it will be ignored in the followinganalysis. The slant range Rm of the point scatterer positioned

at (xp, yp, zp) in this section is given by

Rm(tm, rs) =

(xm xp)2 + (H zp)2

=r2s + v

2(tm tp)2 2rsv(tm tp)sin

(8)

xm [xp rs sin La/2, xp rs sin + La/2]

rs = (H zp)/ cos (9)

tp = (xp rs sin )/v (10)

where tm is the along-track time, is the forward-looking

angle, and La is the footprint of the beam in the along-trackdirection [8].


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REN et al.: NEW 3-D IMAGING ALGORITHM FOR AIRBORNE FORWARD-LOOKING SAR 155

To aid further analysis, Rm(tm, rs) is expanded into Taylorsseries

Rm(tm, rs) rs +dRmdtm

tm=tp

(tm tp)

+1

2!

d2Rm

dt2

mtm=tp

(tm tp)2

+1

3!

d3Rmdt3m

tm=tp

(tm tp)3 +

= rs

2fdc(tm tp)

4fr(tm tp)

2

12fr(tm tp)

3 + (11)

where

fdc = 2

dRmdtm

tm=tp

=2v

sin (12)

fr = 2

d2Rm

dt2m

tm=tp

= 2v2

rscos2 (13)

fr = 2

d3Rmdt3m

tm=tp

= 6v3 cos2 sin

r2s(14)

with fdc being the centroid of the along-track frequency, frbeing the along-track frequency modulation rate, and fr beingthe variation rate of the along-track frequency modulation rate.

From (11), we get that the distance between the radar and thescatterer decreases with a linear term because the SAR platformmoves forward in the along-track direction with a forward-looking angle . The linear term is called range walk, whichis usually greater than one range cell. Due to the coupling ofthe phase and envelope in the slant-rangealong-track plane,

the range migration effect should be removed first if a focusedimage is to be founded.

The amount of range walk to be corrected is given by thesecond term of the right side of (11)

R(tm) = v sin tm. (15)

Transform the signal expressed in (7) from the range-timealong-track time domain into the range-frequencyalong-tracktime domain, and correct the range walk in the range-frequencydomain. Then, the signal becomes

S(ft, tmy = y0) = rect (ft/(2fr)) exp(j4Rm/) exp[j4(Rm + R)ft/c] (16)

where ft is the range frequency. An along-track directionFourier transform is then performed on each range gate to trans-form the data into the range-frequencyalong-track frequencydomain. If the constant and higher order phase terms (higherthan third order) are ignored, the signal becomes

S1(ft,fay=y0)

=rect

ft

2fr

exp

j

4ftc

rs+vtp sin +

1

2Rs

(fafdc)

2v cos

2

expj

fr(fafdc)

2j fr3f3r

(fafdc)3j2fatp

(17)

where fa is the along-track frequency and Rs is the distancebetween the radar and the scene center. The correction functionfor 2-D coupling can be derived from (17)

H1(fa, ft) = exp

j

2

cRs

(fa fdc)

2v cos

2ft

. (18)

After the multiplication of the signal S1(ft, fa; y = y0) withthe phase function H1(fa, ft) in the range-frequencyalong-track frequency domain, the range trajectory of every pointscatterer will be corrected to a line approximately. Then, movethe corrected signal along the along-track frequency directioncircularly with a distance fdc, which will shift the center ofthe along-track frequency to zero. After a 2-D inverse Fouriertransform, the signal in the 2-D time domain is

s1(t, tmy = y0) = sinc

fr(t 2r/c)

expjfr(tm tp)

2 + jfr(tm tp)3/3

(19)

where r = rs + vtp sin . From (13) and (19), we can getthat the range in the along-track frequency modulation ratefunction fr becomes r = rs + vtp sin instead of rs afterthe range migration correction, which varies with the along-track position of the point scatterer. That is to say, the along-track frequency modulation rate is along-track dependent. Ifa reference function with a fixed frequency modulation rateis used to process the along-track signal, it will not be suf-ficient to focus all targets signatures that are positioned indifferent range cells in the along-track direction previously. Be-cause the along-track dependence of the along-track frequencymodulation rate in forwarding-looking SAR is similar to thecharacteristic of the range-frequency modulation rate in con-

ventional squint-looking SAR, which is range dependent, thenonlinear frequency modulation scaling algorithm is introducedin the along-track processing to tackle the problem in forward-looking SAR.

Based on the principle of the nonlinear frequency modulationscaling algorithm [9], a cubic phase filter is added first. Trans-form the signal (19) into the range-timealong-track frequencydomain

S2(t, fay = y0) = sinc

fr(t 2r/c)

exp

j

frf2a j

fr3f3r

f3a j2fatp

. (20)

Then, a multiplication of the signal with a small cubic phasefilter function H2(t, fa) is performed, where

H2(t, fa) = exp

j

2

3Y(t)f3a

.

Then, we get

S3(t, fay = y0) =S2 H2=sinc

fr(t 2r/c)

exp

j

frf2a + j

2

3Ymf

3

a j2fatp

(21)

where Ym=Yfr/2f3r is the modified cubic phase coefficient.


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156 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 1, JANUARY 2011

Transforming the filtered signal (21) into the range-time-along-track time domain, the signal becomes

s3(t, tmy = y0) = sinc

fr(t 2r/c)

expjfr(tm tp)

2 + j2Ymf3

r (tm tp)3/3

. (22)

Considering the along-track dependence of the along-track

frequency rate fr, we make an approximation that fr variesnearly linearly with the along-track time. Then, fr can bedescribed as

fr = fref + fsvtm

where fref is the reference along-track frequency rate and fs isthe variation slope of the along-track frequency rate

fs =dfrdtm

=dfrdRm

dRmdtm

=v sin

rfref

fref = 2v2 cos2

r.

When the signal passes the cubic phase filter, we perform

chirp scaling. The chirp scaling phase function used in thisalgorithm is described as

H3(t, tm) = expjq2(t)t

2

m j2q3(t)t3

m

.

Therefore, the signal after chirp scaling can be written as

s4(t, tmy = y0) = s3(t, tmy = y0)H3(t, tm).

Then, an along-track direction Fourier transform is per-formed for the scaled signalS4(t, tm; y = y0), and the principleof stationary phase is used to evaluate the Fourier transformintegral. Choose the proper Ym, q2, and q3 to remove thesecondary compression term and the nonlinear migration termthat are along-track dependent, and set also the coefficient of thelinear migration term to 1/, where is the scaling coefficientand close to one. Thus, the result of Ym, q2, and q3 can beobtained according to the aforementioned constrains, whichare Ym = fs( 0.5)/f

3

ref/( 1), q2 = fref(1 ), andq3 = fs(1 )/6.

Therefore, after the along-track direction Fourier transform,the phase of the signal s4(t, tm; y = y0) becomes

S4(t, fay = y0) = exp

j2

tpfa

exp

j

freff2a j

2q3 Ymf

3

ref

33f3ref

f3a

exp(j)

(23)

where = fref(1 1/)t2p + fs(1 1/)t

3p/3. The first

term in (23) gives the along-track position of the point target,and the second term corresponds to the compressed signal in thealong-tack direction, while the third term represents the residualerror.

From (23), the reference function used to compress in thealong-track direction can be calculated as

H4(t, fa) = exp

j

freff2a + j

2q3 Ymf

3

ref

33f3ref

f3a

.

The along-trackslant-range image domain data can beachieved by transforming the compressed data back to the

along-track time domain

s5(t, tmy=y0)=sincfr(t2r/c)

sinc [fa(tmtp)]. (24)

The along-track bandwidth fa is given by [8]

fa = 2v cos / (25)

where is the beamwidth in the along-track direction.From (24), we get that the slant-range position of the scat-

terer in the focused image is r = rs + vtp sin , which movesvtp sin in the slant-range direction. In order to correct theposition displacement in the slant-range direction, we transformthe signal expressed in (24) into the range-frequencyalong-track time domain and multiply it by the linear phase

H5(ft) = exp[j4vtp sin ft/c]. (26)

Then, an inverse Fourier transform is performed in the slant-range direction, and the signal becomes

s6(t, tmy=y0)=sinc

fr(

t2rs/c)

sinc [fa(tmtp)] . (27)

From (27) and (10), we get that the along-track position ofthe scatterer is vtp, which moves rs sin in the along-trackdirection. Therefore, the same displacement correction can bemade in the along-track direction as what is done in the slant-range direction. Transform the signal expressed in (27) into therange-timealong-track frequency domain, and multiply it bythe linear phase

H6(fa) = exp[j2rs sin fa/v]. (28)

Then, an inverse Fourier transform is performed in the along-track direction, and the signal becomes

s7(t, tmy=y0)=sinc

fr(t2rs/c)

sinc[fa(tmxp/v)](29)

where rs =

(xp vtp)2 + (H zp)2.Moreover, from (9), we get that

zp = H rs cos . (30)

Then, after geometry rotation and scale transformation with(30), an along-trackheight section image can be obtained as

s8(z, tmy=y0)=sinc[fh(zzp)]sinc[fa(tmxp/v)] (31)

where fh is the height bandwidth and the reciprocal of the

height resolution. Let us consider the resolution in the heightdirection. Choose two point scatterers positioned at (xp, zp) and(xo, zo), and let (xo, zo) = (xp, zp + z), where z is verysmall. Substitute them into (29), and we get [8]

s9(z, tmy=y0)=sinc[fa(tm xp/v)]

sinc

2frc

(xpvtp)2 + (Hzo)2

(xpvtp)2+(Hzp)2

sinc [fa(tmxp/v)] sinc2fr [z(Hzpz/2)]c

(xpvtp)2+(Hzp)2

sinc [fa(tmxp/v)] sinc{2fr cos z/c}. (32)


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REN et al.: NEW 3-D IMAGING ALGORITHM FOR AIRBORNE FORWARD-LOOKING SAR 157

TABLE IPARAMETERS USED FOR SIMULATION

Fig. 3. Real spatial position and final 3-D image. (a) Real spatial position ofpoint scatterers. (b) Final 3-D image of forward-looking SAR.

Therefore

fh = 1/z = c/(2fr cos ). (33)

When all the along-trackslant-range sections correspondingto each azimuth position are processed following the procedurementioned earlier, the 3-D image of forward-looking SAR canbe achieved. From (6) and (31), we get that the 3-D point spreadfunction of forward-looking SAR can be denoted as

psf(x,y,z)sinc 2 cos

(xxp) sinc LfRm

(yyp)sinc

2fr cos

c(zzp)

. (34)

Consequently, the resolutions of forward-looking SAR in thealong-track, azimuth, and height directions can be calculatedfrom (34)

x =/(2 cos ) (35)y =Rm/Lf (36)z = c/(2fr cos ). (37)

IV. SIMULATION RESULTS

In this section, point-target simulation is carried out to verifythe validity of the proposed imaging algorithm. The mainparameters used for simulation are listed in Table I.

The distributions of the nine point scatterers used for simula-tion are shown in Fig. 3(a). After raw-data generation and 3-Dimaging processing using the proposed algorithm, the surfacesof the final 3-D image are shown at 3 dB in Fig. 3(b). Asexpected, the image is reconstructed in 3-D space, and thewhole space structure is very consistent with the real situationin Fig. 3(a).

In order to analyze the performance of the imaging results,Fig. 4 shows three sections of the final 3-D image of forward-looking SAR. Fig. 4(a) shows the 2-D image of the selected

along-trackheight section. Fig. 4(b) shows the 2-D imageof the selected azimuthalong-track section. Fig. 4(c) shows

Fig. 4. Two-dimensional image of the selected sections. (a) Two-dimensionalimage of the along-trackheight section. (b) Two-dimensional image of theazimuthalong-tracksection. (c) Two-dimensional image of the azimuthheightsection.

the 2-D image of the selected azimuthheight section. Theaforementioned imaging results show that the point scatterersare well focused in the three directions, confirming the validityof the proposed algorithm.

V. CONCLUSION

In this letter, a new 3-D imaging algorithm that is capableof focusing forward-looking SAR data has been proposed. Theprinciple behind the method is based on considering the along-

track dependence of the along-track frequency modulationrate in focusing. Moreover, the method needs only Fouriertransform and multiplication operations, making it suitable forpractical applications. The raw data of forward-looking SAR inX-band were simulated, and the 3-D image was achieved. Theresults of the simulated data confirm the effectiveness of theproposed method.

REFERENCES

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[3] T. Sutor, F. Witte, and A. Moreira, A new sector imaging radar for en-hanced visionSIREV, Proc. SPIE, vol. 3691, pp. 3947, 1999.

[4] G. Krieger, J. Mittermayer, M. Wendler, F. Witte, and A. Moreira, SIREV-sector imaging radar for enhanced vision, Aerosp. Sci. Technol., vol. 7,no. 2, pp. 147158, Mar. 2003.

[5] J. Mittermayer and M. Wendler, Data processing of an innovative forwardlooking SAR system for enhanced vision, in Proc. EUSAR, Munich,Germany, 2000, pp. 733736.

[6] Y. Venot and M. Younis, Compact forward looking mode SAR usingdigital beamforming on receive only, in Proc. EUSAR, Munich, Germany,2000, pp. 795798.

[7] A. Reigber, Airborne polarimetric SAR tomography, Ph.D. dissertation,Stuttgart Univ., Stuttgart, Germany, 2001.

[8] X. Ren, L. Tan, and R. Yang, Research of three-dimensional imagingprocessing for airborne forward-looking SAR, in Proc. IET Radar, Guilin,China, 2009.

[9] G. W. Davidson, I. G. Cumming, and M. R. Ito, A chirp scaling approachfor processing squint mode SAR data,IEEE Trans. Aerosp. Electron. Syst.,vol. 32, no. 1, pp. 121133, Jan. 1996.

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A New Three-Dimensional Imaging Algorithm