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Radio Science, Volume 30, Number 5, Pages 1435-1446, September-October 1995
Image domain ray tube integration formula for the shootingand
bouncing ray technique
Rajan Bhalla and Hao Ling
Department of Electrical and Computer Engineering, University of Texas at Austin, Austin
Abstract. A simple image domain ray tube integration formula is presented to
efficiently compute the inverse synthetic aperture radar (ISAR) image of a complex
target by the shootingand bouncing ray (SBR) technique. Contrary to the conventional
approach where the ISAR image is obtained by inverse Fourier transforming the
computed scattered field data over frequency and aspect, this image domain formula
gives the contribution of each ray to the overall ISAR image directly. Under the small
angle approximation and utilizing the bistatic-monostaticequivalence, the image
domain ray tube integration formula is determined in closed form. Simulation results
using the SBR-based code "Xpatch" show that the direct image domain method results
in good image quality and superior time performance when compared to the
conventional frequency aspect approach.
- Introduction
Geometrical ray tracing is a common technique
for predicting the radar signaturesof complex tar-
gets at high frequencies. One such implementation,
termed the shooting and bouncing ray (SBR) tech-
nique, was developed by Ling et al. [1989a], Baul-
dauf et al. [1991], Ling et al. [1991], and Kim and
Ling [1992] to simulate the radar returns from
realistic aerospace vehicles and the scattering by
complex media. The basic idea behind the SBR
method is very simple. Given the geometrical de-
scription of a target, a large set of geometrical
optics rays is shot toward the target (Figure 1).
Rays are traced according to the laws of geometri- cal optics as they bounce around the target. At the
exit point of each ray, a ray tube integration is
performed to sum up its contribution to the total
scattered field [Lee et al., 1988] (the ray tube
integration may also be carried out at every hit
point). While the basic idea behind the SBR meth-
odology is simple, when combined with computer-
aided design (CAD) tools for geometrical modeling
and fast ray-tracing algorithms developed in com-
puter graphics, this technique becomes a very gen-
eral tool for characterizing the scattering from
large, complex targets. One such development is
the general-purpose SBR code "Xpatch" [Lee,
Copyright 1995by the American GeophysicalUnion. Paper number 95RS02110. 0048-6604/95/95RS-02110508.
1992; Lee and Andersh, 1993; Andersh et al., 1994], which is currently used by the Air Force and the aerospace industry in programs related to target
identificationand low-observablevehicle design. In
this work we will present theoretical derivation and
simulation results of a direct image formation
scheme for generating the inverse synthetic aper-
ture radar (ISAR) imagery using the SBR tech-
nique.
The ISAR image of a target is a powerful visual-
ization tool that is often used to pinpoint key
scattering centers on a target in radar signature
studies [Mensa, 1981]. ISAR image formation is
typically achieved by utilizing the monostatic scat-
tered field data, obtained through either measure-
ment or numerical simulation, over a finite range of
look angles and frequencies. From a numerical simulation point of view, this is usually a rather time-consuming process. For SBR-based calcula-
tions the angular scan is a particularly expensive
operation, since for every new look angle on the target a new set of rays must be launched and traced
through the target. To alleviate this problem, we
have recently utilized bistatic scattered field data to
obtain the ISAR image [Bhalla and Ling, 1993].
Since ray tracing is performed once for the incident
direction and only the ray tube integration is needed
for every look angle, the bistatic image formation
schemeresults in time savings. In a separate devel-
opment, a closed form time domain ray tube inte-
gration formula was recently derived for the fast
computation of the time domain response(or range
1435
(2)
where the quantity in the parentheses is the total scattered field at frequency to and aspect 0 and is obtained through the summation over all exit rays. By interchanging the order of the inverse Fourier transform and the ray summation, the ISAR image can be formed by
O(X,Z)--Z Oi(x'Z) (3)
i rays
Figure 1. The shootingand bouncingray (SBR) method is a high-frequencysimulationtechniqueused for predict- ing the radar cross section of complex targets. Rays are launched from the incident direction toward the target. The scattered field is computed by summingthe exit ray contributions.
profile) of a conducting target [Jeng et al., 1993;
Ling and Bhalla, 1993a]. This formula gives the explicit contribution of each exit ray in the time domain. Therefore by summing the contribution from each ray in the time domain directly, the
overall time domain response of the target can be
obtained without resorting to any multifrequency calculations'
ES(t)= Z E•(t) (1)
i rays
whereE[(t) is the closedform time domainformula
for each ray given by Ling and Bhalla [1993a]. In
this work we shall show that the time domain
concept can be further extended, with the aid of the
bistatic imaging scheme of Bhalla and Ling [1993],
to directly compute,in closedform, the contribution
of each ray to the overall ISAR image of a target. To illustrate our idea we first note that in the
conventional image formation process, the scat-
tered field must be computed over a band of fre- quenciesand angles. Subsequently, by inverse Fou- rier transforming the resulting two-dimensional data, the ISAR image of the target, O(x, z), can be obtained [Farhat et al., 1984; Li and Farhat, 1989].
This conceptis describedby the followingexpression:
where Oi(x, z) is the contributionof each exit ray to
the ISAR image. It is Oi(x, z) for which we shall
derive a closed form expression. We should point out that the direct image domain
ISAR formation concept is not new; it has been
implemented and presented in the radar community
previously. We cite the work done under the Sim-
ulation of Radar Image (SRIM) program [Arnold et al., 1984; LaHaie, 1985; Burns and Herrick, 1992; Herrick, 1992a, b]. However, the aim of this paper is to derive, from first-principle electromagnetics, the image domain ray tube integration formula under the SBR context. Moreover, we shall exam- ine the image quality and simulation time of the image domain formulation when applied to both
canonical as well as realistic complex targets. The
detailed derivation of this formula will be presented
in section 2. In section 3 simulation results using the image domain method will be compared against the
conventionalfrequency aspect approachfor various
targets. Time performance between the two meth-
ods will also be discussed.
- Ray Tube Integration Formula Before deriving the image domain formula, we will first summarize the frequency domain ray tube
integration formula previously derived by Lee et al.
[1988]. The scattered far field from a target in the
frequency domain at an observation point (r, 0, •b)
can be written as
e -jkr ES(w,0, •b)=- (^) F (bAo+ •A4,) (4)
where k = to/c. When the SBR method is used, the
Bistatic Scattered Field
•kx
Figure 4. The shaded area S represents the Fourier space data collected under the bistatic scattering arrange- ment. The frequency is steppedfrom kmin to kmax,and the observationangle is steppedfrom -190 to 19o-
B• = (-$2E3 + $3E2+ $1goH3- $3goH1)
Assumingthe target is perfectly conducting, we can
factor out the explicit frequencydependencein B 0,
B4 asfollows:
Bo = ((-SlE• + s3E] - s2ZoH• + s3ZoH•)
+ O(SlZoH• - s2ZoHi))e-Jkdi (lO)
B• = ((-s2E [ + s3E• - SlZoH[ + s3ZoHi)
+ O(slE•- s2Ei))e-j•xli (11)
where di is the total distancetraveled by the ith ray
to the last hit point on the target, and the primed
field quantities have no frequency dependence.
From(10)and(11) we seethatB0, B4 takeon the form
Bo,•,= (a + Ol3)e-jka' (12) In the rest of the derivation we will use the form in (12) to representBo,4,,with the subscriptdenoting the appropriate polarization. With the explicit angular and frequency depen- dence of the scattered field in hand, we will now proceed to the imaging algorithm. The ISAR image and the bistatic scattered field are related through a two-dimensional Fourier transform [Bhalla and Ling, 1993]'
o0,0(x, z) =
(2,r)2 0o,4,(k,,,kz)e-Jkxxe-j•zzdkxdkz (13)
whereO0,,(x, z) is theISAR imageof thetargetand 0o,4,(k,•,kz) is the range-correctedscatteredfield given by 477'/' Oo,,(k•, kz)=_jke_J•:rEo,• (14)
Under the bistatic condition, the two-dimensional k-space integration in (13) is performed over the shaded area S shown in Figure 4. The k space is
accessedby stepping the frequency from kmin to
kmax and the observationanglefrom -0o to 0o. By
substituting (4), (5), and (14) in (13) we get (with S(0,. qb)set to 1)
o0,, (x, z) = 277 -
'Zi rays fsfBo,•(AA)exite3•'rAe
(15)
We will next evaluate the double integration in closed form under the small 0 approximation. As is shown in Figure 3, under the small-angle imaging
condition the area S is nearly rectangular, and k•
and kz can be approximatedas kx • koO,and kz •
2k, where ko = (kmaxq- kmin)/2.The differentials
can hencebe written as dkx = kodOand dkz = 2dk.
Similarly, the wave vector k can be approximated
ask = k[(•cos 4•+ psin40sin 0+ •cos 0]-•
koO•:+ k•. Consequently,we can rewrite k'rA •
koOxi.•q- kzi•:, where (xi, zi) are the coordinatesof
(a)
Einc
Ill
z
y
in
Figure 5. The ISAR images of a rhombic cube (VV polarization). The imageswere generated for a frequency scan of 11 to 16 GHz and a monostatic angular scan of -6 ø to 6ø. (a) Target geometry. (b) Conventional Fourier processingof monostatic frequency-aspect data. (c) Di- rect image domain ray tube integration formula.
(b) (c)
Figure 5. (continued)
the last hit point on the target. Substituting these approximations and (12) into (15) we get
O•,• (x, z) = (^) •2. (Z•[)exit e dk I S min
(16)
The above integrations can be easily carried out analytically. Denoting sinc (u) = sin (u)/u and sinc' (u) = d(sinc (u))/du = (-sinc (u) + cos (u))/u, we arrive at a closed form expression for the TSAR image of the target in terms of the exit ray field:
ko
00,4)(X,Z)= 2 Z (z•A)exit
i rays
.{Ake_Jt•o(2z+cli_zi)sinc[Ak(z+^ di 2 -z/)]}
ß{2aOo sinc [koOo(x- xi)] 2
- 2j130osinc' [koOo(x- xi)]} (17)
The above expression is in the form of (3) and gives the explicit contribution of each ray to the TSAR image. Several remarks are in order:
- We have utilized the bistatic configuration to arrive at the image domain formula. The monostatic TSAR image is often the more desirable result. However, for small-angle imaging, the equivalence between the monostatic and bistatic configuration is a simple one, that is, the ISAR image generated from (17) should be equivalent to the TSAR image generated using monostatic data with the same frequency bandwidth and half the angular window
from -00/2 to 00/2. Detailed discussionson the
image quality generated under the two configura- tions can be found in work by Bhalla and Ling [1993], Farhat et al. [1984], and Shaeffer [1992]. In section 3 we will make direct comparisons between the bistatic image generated from (17) and the monostatic image generated from conventional fre- quency aspect data.
- The quantity in the second parentheses in (17) contains both a sinc and a sinc' term. However,
the sinc term is of order 0o while the sinc' term is of
order 003. Forsmall-angleimagingthesinc'termcan be ignored. As a result, the image domain ray tube integration formula essentially consistsof the prod- uct of two sinc functions. In the -z (or down range) direction, the sinc function has its maximum at
--Z -- (di - zi)/2, which is half of the total distance
traveled by the ray to the far field measured with respect to the origin. The width of the sinc function
x
(a) Targetgeometry
45 ø! "x •0" square cross-section
(c) (d)
I ..l• ii i• : .... .•. •I ii
Figure 7. The ISAR imagesof a rectangularduct (HH polarization). The imageswere generatedfor a frequency scan of 9.5 to 10.5 GHz and a monostatic angular scan from -3 ø to 3ø. (a) Target geometry. (b) Conventional Fourier processingof monostaticfrequency aspect data generated using a modal technique. (c) Conventional Fourier processingof the monostatic frequency aspect data generated using the shooting and bouncing ray technique. (d) Direct image domain ray tube integration formula.
cross-range coordinate [Walker, 1980; Mensa, 1981], and this results in the above conclusion.
- Finally, while the conditionsunder which our
image domain formula is derived have already been
explicitly stated, a comment is in order on the
applicability of this formula in actual imaging sce-
narios. It is well known that the cross-range reso-
lution of an ISAR image is inversely proportional to
the product of the angular scan and the center
frequency. For example, at 10 GHz the angular window needed to achieve a 15-cm cross-range resolution is 6ø. At 2 GHz the needed angular window for the same resolution widens 30 ø. There-
fore the present image domain technique is well-
suited for high-frequency imagery at X band and above. At lower frequencies the wider angular
aperture of the imaging scenario may violate the
limit of the small-angleapproximation.
(a)
Ei•c
Figure 8. The ISAR imagesof a genericaircraft (VV polarization).The imageswere generatedfor a frequency scan of 9.5 to 10.5 GHz and a monostatic angular scan from -3 ø to 3ø. (a) Target orientation. (b) Conventional Fourier processingof monostaticfrequencyaspectdata. (c) Direct image domain ray tube integrationformula.
- Numerical Implementation and Results
Numerical implementationof (17) hasbeen incor-
porated into the SBR-based code Xpatch3. The
implementationstrategyis as follows. For a given
target we first choosea rangewindow [-R/2, R/2]
and a cross-rangewindow I-L/2, L/2]. Here the zero
rangeand cross-rangepoint correspondto the origin
of the target, and (R, L) are chosento be approxi-
matelytwice the dimensionsof the target.The image
planeis thendividedintoN equallyspacedrangebins
with bin width A r = R/N and M equally spaced
cross-rangebins with bin width Acr = L/M. The
imageplaneis discretiredwith grid points'
Xm = Acr(rn - 1) - L/2 rn = 1, 2, ... , M + 1 (18)
Zn = Ar(n - 1) - R/2 n = 1, 2, ..., N q- 1 (19)
The bin widths should be chosen so as to ensure
propersamplingin the rangeand cross-rangedirec-
tions. As a general guideline, the range bin width
should be no greater than (0.15 m)/(bandwidth in
GHz) and the cross-rangebin width should be no
greater than (0.15 m)/(center frequency in GHz)/
(0o). Using (18) and (19) in (17) we arrive at the
desiredexpressionfor numerical implementation:
ko O0,•b(Xm,Zn)'-qr2i raysE (•)exit
ß Ake-j!•ø(2zn+di-zi)sinc Ak Znq- 2
ß{2a0o sinc [koOo(xm- xi)]} (20)
In the implementationonce the total distancetrav-
eledby the ray di and the last hit point rA for the ith
ray is obtained following the ray tracing, we can
update the image plane according to the above
formula. In practiceit is not necessaryto updatethe
entire image plane, since the two-dimensionalsinc
Figure 8. (continued)
(a)
E•c
/
120 ø
Figure 9. The ISAR images generic aircraft (VV polar- ization). The images were generated for a frequency scan of 9.5 to 10.5 GHz and a monostatic angular scan -3 ø to 3ø. (a) Target orientation. (b) Conventional Fourier pro- cessing of monostatic frequency aspect data. (c) Direct image domain ray tube integration formula.
monostatic scattered field was computed with the
frequency scanned from 9.5 to 10.5 GHz in 133
steps. The angular look window was _+3ø in 124 stepscentered about 30øwith respect to the nose-on direction of the aircraft (see Figure 8a). The fre-
quency domain data were then zero-padded, refor-
matted, and processed to generate the 256 x 256 pixel image shown in Figure 8b. The range and cross-rangeresolutions of the image are both 15 cm. Figure 8c shows the ISAR image generated using the direct image domain ray tube formula on a 256 x 256 grid for the same frequency window. The
incident angle was 30ø with respect to nose-on and
the equivalent bistatic observation window was
_+6ø. We observe that the image generated by the direct image domain ray tube formula bears good fidelity to that produced by the conventional method, even for such a highly complex target. In
particular, the location of the large cloud on the
fight, which is due to the left inlet termination of the
aircraft, is quite faithfully reproduced in the image
domain approach. However, as was discussed in
the duct example, the image domain results cannot
exactly reproduce the diffused features in the ISAR
image with high fidelity due to its single-angle
assumption. Figures 9a-9c show a different look
angle on the same target at 120ø with respect to the
nose-on direction of the aircraft. For this aspect
angle the target is well described by a collection of nondiffusedpoint scatterers. As a result, the image generated by the direct image domain method agrees very well with that produced by the conven- tional approach.
The computation time to form each image using
the direct image-domain method for the genetic aircraft is approximately 4 hours on our Silicon
Graphics Iris Indigo R3000 workstation. This com-
putation time is obtained using a ray density of 10
rays per wavelength and 50 maximum bounces. It is
orders of magnitude faster than using the conven- tional image formation scheme. To generate the monostatic frequency aspect data needed in the
(c)
Figure 9. (continued)
conventional scheme, we estimate that it would
have taken over 300 hours of computationtime on
the Indigo. The actual data were generated on an
Intel iPSC/860 hypercubeat SandiaNational Labo-
ratory. There are two factors which give rise to
superior time performance of the direct image do-
main scheme. First, the bistatic approximation al-
lows the ray tracing to be performed for one inci-
dent angle only. This advantage has already been
explored by Bhalla and Ling [1993]. Second, with
the image domain formula, only a small portion of
the image plane near the peak of the ray spread
function needs to be updated for each ray. In the
complex aircraft example, to achieve a 30-dB dy-
namic range in the ISAR image, only a 20 x 20 grid
(correspondingto approximately 5 sidelobes)needs
to be updated in the image plane for each ray, as
oppose to the entire 256 x 256 grid. This is in contrast to the conventional data collection in the
frequency aspect plane where each ray contributes
to every frequency and every angle in general.
- Conclusion
In this paper we derived from first-principleelec-
tromagneticsan image domain ray tube integration
formula for the shooting and bouncing ray tech-
nique. It was shown that using this formula the
ISAR imagesof targetscan be generateddirectly in
the imagedomain without resortingto any multiple
frequency aspect calculations. This result was de-
rived based on the small-angle,bistatic approxima-
tion. The performance of the direct image domain
scheme was evaluated by comparing the ISAR
imagesgeneratedusing this formula againstthose
generatedusing the conventionalfrequency aspect
approachfor both canonical as well as realistic com-
plex targets.Good agreementwas observedbetween
the imagesobtainedfrom the image domain scheme
and the conventionalapproach,especiallyfor nondif-
fusedscatteringcenters.For diffusedscatteringin the
ISAR image due to highly multiple bounces, the
generallocationof the featurescanbe capturedby the
image domain approach.However, it cannotrepro-
ducewith high fidelity the exact shapeand amplitude
of the diffusedfeatures due to the single-angleas-
sumption.The simulationtime of the image domain
schemeis ordersof magnitudefasterthanthe conven-
tionalfrequencyaspectapproach.The reasonsfor the
time savingsare twofold. First, the image domain
formula takes advantageof bistaticdata and doesnot
require calculation of monostatic scattered field at
multiple look angles.Second, the image domain for-
mula is spatially limited in the range/cross-range
plane, and only the contributionnear the peak of the
ray spread function needs to be taken into account.
(Further reduction in computation time can be
achieved by implementing the image domain ap-
proachusinga fast algorithm;seework by Bhalla and
Ling [ 1995]).The presentimagegenerationschemeis
limitedto small-angleimagingscenariosandto perfect
conductingtargets. When these conditions are met, the tremendous time advantage makes this scheme
the much preferred method for ISAR image simula-
tion.
Acknowledgments. The authors are grateful to S. W. Lee for providingthe variousversionsof Xpatch and for many helpful discussions.The authors would also like to thank D. J. Andershfor providingthe monostaticXpatch simulation data and for his continued support of this work. This work was supportedby NASA grant NCC3- 273 and in part by the Joint ServicesElectronicsProgram under contract AFOSR F49620-92-C-0027.
References
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