Image-Quality Focusing of Rotating SAR Targets

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750 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 5, NO. 4, OCTOBER 2008

Image-Quality Focusing of Rotating SAR TargetsBrian D. Rigling,Member, IEEE

AbstractSynthetic aperture radar (SAR) is a popular tool forlong-range imaging of stationary ground objects. Moving targetsin the imaged scene will have a mismatch to the matched filter inthe image formation process, thus degrading target image quality.In this letter, the impact of uncompensated target rotation in SARimagery is studied. An efficient algorithm for image-quality-basedtarget rotation correction is proposed.

Index TermsAutofocus, image entropy, moving target focus-ing, synthetic aperture radar (SAR).

I. INTRODUCTION

S

POTLIGHT synthetic aperture radar (SAR) [1] has long

been the sensor of choice in all-weather air-to-ground

surveillance. The physical principles that serve as the phenom-

enological foundation for SAR imaging are identical to those

of range-Doppler radar in general. Indeed, the motion of the

data collection platform in a direction nominally orthogonal

to the radar line-of-sight imparts a unique Doppler shift to the

return from a target displaced in crossrange. Provided that every

reflecting object in the scene remains stationary throughout

the coherent integration period, SAR processing to exploit the

range-Doppler information contained in the recorded phase

histories will yield a 2-D or even 3-D reconstruction of the

observed scene.

Moving objects within the radars beam will emit returns

with a phase mismatch, an extra pulse-to-pulse Doppler and/or range shift that is inconsistent with platform motion. This

mismatch will cause the objects image to be degraded by de-

focus, distortion, displacement, and range walk, thus reducing

the images interpretability by a commensurate amount. As

moving targets in imaged scenes are nonetheless of interest,

if not greater interest, it has been an ongoing challenge to

develop postprocessing algorithms to focus moving targets in

SAR imagery.

Past work in moving target focusing has been extensive in

both its breadth and depth. Successful approaches to translating

target focusing include subaperture imaging [2], [3], phase

estimation [4][6], keystone imaging [7], and inverse SARprocessing [8]. To image a target that is exhibiting both trans-

lational and rotational behavior, most effective algorithms (see,

e.g., [9][11]) have made use of parametric target assumptions.

However, these are typically only weak assumptions that the

target response contains a minimal number of discrete returns

Manuscript received June 9, 2008; revised July 23, 2008. Current versionpublished October 22, 2008. This work was supported by the United StatesAir Force Research Laboratory (AFRL/SNA) under a subcontract through SETCorporation.

The author is with the Department of Electrical Engineering, Wright StateUniversity, Dayton, OH 45435 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/LGRS.2008.2004792

that are at least piecewise observable throughout the syntheticaperture.

Significant work on nonparametric algorithms for motion

compensation has also been underway for many years. Meth-

ods based on numerical optimization of image-quality metrics

[12][14], such as entropy, contrast, and sharpness, are notably

robust to scene structure when used in autofocus and range

alignment. Similarly, entropy-based rotation corrections [15]

have been suggested. Rotation correction by the means de-

scribed in [15] can be computationally intensive, as each step

in the optimization requires a 2-D Fourier transform into the

phase history domain, a 2-D resampling of phase history data to

incrementally correct rotation errors, a 2-D Fourier transform to

realize the new image, and calculation of the 2-D image entropy

to feed the next optimization step. Repeated resampling of the

phase history may also introduce undue interpolation artifacts

while also being the most computationally intensive task.

In this letter, we present a more limited but much faster

implementation of the rotation correction that is described in

[15]. We first analyze the image effects of target rotation. These

analytical results provide the basis for a more efficient image-

quality-based rotation correction algorithm. This technique will

seek to simultaneously correct for target rotation and general

crossrange defocus by estimating a range-dependent crossrange

phase correction via image-quality optimization. This approach

will be applicable to smaller rotation errors than the algorithmin [15], but it will eliminate the phase history resampling at

each iteration, will eliminate the range fast Fourier transforms

(FFTs) at each iteration, and will correct rotational and trans-

lational errors in the same manner as a conventional autofocus

algorithm.

The remainder of this letter is organized as follows.

Section II introduces our models for SAR data collection and

target motion. Section III examines the image aberrations that

are introduced by uncompensated target motion. Section IV

presents a more efficient entropy-based algorithm for target

rotation correction, and Section V summarizes our results and

conclusions and outlines areas for future work.

II. DATAC OLLECTIONM ODEL

Consider the monostatic SAR geometry shown in Fig. 1. A

scene to be imaged is centered at the origin of the coordinate

system, and the data collection platform traverses some known

path nominally broadside to the radar line-of-sight. As shown

in Fig. 1, we will assume this path to be roughly perpendicular

to the x, z-coordinate plane. At regular intervals along thistrajectory, the system transceiver directs pulses of energy into

the scene. The pulses are assumed to have uniform power over

their bandwidth f[fcB/2, fc+B/2]. The transmitted1545-598X/$25.00 2008 IEEE


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RIGLING: IMAGE-QUALITY FOCUSING OF ROTATING SAR TARGETS 751

Fig. 1. Sensor measurement geometry for a point targetwith position (x,y,z)and velocity( x, y, z). Sensors are configured according to their aspect anglesin azimuth and elevation.

energy interacts with objects in the scene, and some of the

scattered energy is observed by the radar transceiver. After

receive signal processing [1], the measured data are indexed by

frequencyfand platform aspect in azimuth a and elevationa. The commonly used far-field assumption allows us to definethe frequency space coordinates: fx= fcos acos a, fy =fsin acos a, andfz =fsin a. Assuming that the responseof a target in the scene can be represented through the superpo-

sition of ideal point returns, the received signal model is then

s(fx, fy, fz) = m

Amexpj4

c [(xm+xc)fx

+ (ym+yc)fy+ (zm+zc)fz]

+w(fx, fy, fz) (1)

where the position (xm, ym, zm) of each point on the targetis defined relative to a target centroid at (xc, yc, zc). Thetransceiver aspect (a, a) varies across slow time, with oneaspect pair being recorded for each sample in slow time. For

a stationary target, the centroid and offsets {(xm, ym, zm)}areassumed to be constant throughout the synthetic aperture. In the

case of a moving target, the centroid and offsets may both be

time varying. In (1), the complex scattering coefficient of each

point response isAm, and the phase history data are corruptedby additive white Gaussian noisew(fx, fy, fz).The recorded phase histories may be processed to form im-

ages through any number of techniques, all of which amount to

matched filters to the received signal model in (1). To facilitate

the analysis contained in subsequent sections, we will assume

that imaging is done via polar format algorithm, thus requiring

a polar-to-rectangular resampling [1] (i.e., (f, ) (fx, fy))of the data. A ground-plane image is then computed as the 2-D

FFT of (1) with respect tofxand fy .Moving target analysis requires incorporating a time depen-

dence for the positions of the scatterers that constitute (1). Tar-

get translation may be modeled by defining the target centroid

coordinates (xc(), yc(), zc()) to be slow time-dependent(i.e.,-dependent) polynomials. Target rotation is incorporated

by defining time-dependent roll R, pitchP, and yawY forthe rigid configuration of point responses{(xm, ym, zm)}. Thetime-varying point offsets are thus

xm()ym()

zm()

= R (R(), P(), Y())

xmym

zm

(2)

where R(R, P, Y) is a conventional rotation matrix per-forming rotations ofR, P, and Y about the x-, y-, andz-axes, respectively. For simplicity, the target coordinate axesare assumed to be parallel to the corresponding scene coordi-

nate axes.

III. EFFECT OFT ARGETM OTION ONSAR IMAGE

Before presenting an algorithm for rotation correction, we

will first analyze in detail the image effects that result from

the filter mismatch induced by target rotation. To facilitate this

analysis, we will use the following simplifying assumptions:

1) fx= fccos a+fcx, such that fcx defines a band of fre-

quencies of widthBcos acentered at 0 Hz;2) fy fca()cos a fcacos a;3) fz fxtan a;4) aconstant.

While the first assumption simply defines fcx, the secondand third assumptions represent first-order approximations, and

the last assumption is a zeroth-order approximation. For a

nominal SAR system operating atX-band(fc= 10GHz)with1-ft resolution in range and crossrange, a roughly 3 angular

aperture is required. Combining this small angular requirement

with an assumption that the SAR aperture will not be at anunusually high grazing angle and that no significant aperture ir-

regularities will be attempted (e.g., to enable 3-D imaging), the

aforementioned assumptions are quite reasonable. Moreover,

for other operational parameters, these approximations will still

be effective to first order.

For brevity, we restrict our analysis to examining the

effects of target yaw. We first exploit the spatially invari-

ant effect of target centroid displacement to assume that

(xc(), yc(), zc()) = (0, 0, 0). A point scatterer experi-encing a yaw rotationY()about thex-axis of a target wouldthus have displacements

xm() =xmcosY()ymsinY()

ym() =xmsinY() +ymcosY() (3)

where we have simply extracted the relevant entries from

R(R(), P(), Y()) in (2). The modeled phase of a ro-tating point scatterer is then

m=4

c [(xmcosY()ymsinY()) fx

+ (xmsinY() +ymcosY()) fy+ zmfz] . (4)

We simplify our analysis by considering first-order rotation

effects, meaningY() Y similar to the earlier simplify-ing assumption 2)a() afor the platform motion, and by


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TABLE IYAW-INDUCEDPHASEERRORS. NOTE : RD= RANGE-DEPENDENT

AN DCD= CROSSRANGE-DEPENDENT

TABLE IINOMINALSAR PARAMETERS FORSIMULATION EXAMPLES

making small angle approximations to the trigonometric func-

tions:cos Y() 1(1/2)Y()2 andsin Y() Y().These approximations allow more intuitive results to be reached

without severely reducing their application. In particular, exten-

sion to higher order phase errors is straightforward, and in cases

where small angle approximations are inappropriate, discrepan-

cies may be largely absorbed into a polynomial rotation model

of increased order. Applying the small angle approximations

and incorporating our earlier enumerated assumptions yield the

phase error terms summarized in Table I. Similar analysis may

be performed in the case of pitching and rolling rigid-body

motion.

Two simulations, using the SAR parameters in Table II, were

performed to validate the conclusions reached by analysis. In

the first, a constellation of point targets with range and cross-

range offsets were simulated using (1) without any unknown

rotational behavior. The ideal image is shown in Fig. 2 with

a35-dB Taylor weight. In the second simulation, a constantyaw rate ofY =1.5

/s was added to the point targets. Asshown in Fig. 3, the point responses atx = 2.5m experiencemain lobe broadening by roughly a factor of 3 with the Taylor

weighting, compared to a predicted broadening by a factor of

3.3 from Table I. The yawing of the target may improve or

degrade resolution depending on whether the target rotation

is in the same or the opposite direction of the data collectionplatforms trajectory with respect to the scene, as defined by the

crossrange shift terms in Table I. Targets that rotate in such

a fashion to synthesize a larger (or smaller) aperture will have

finer (or coarser) resolution. In this case, the radar is modeled

on a path in the positive y-direction such that a negative targetyaw rate will increase the subtended synthetic aperture size.

Resolution is thus improved, and when displayed on the same

pixel spacing, the image appears stretched in crossrange.

IV. ENTROPY-B ASEDM OVINGT ARGETF OCUSING

Having conducted an analysis of the image effects of un-

compensated target rotation, the stage is set to describe an

algorithm for target focusing based on image-quality metrics,

Fig. 2. Ideal of image point scatterers displaced in thex- and y-directions,with a phase history domain SNR of30 dB.

Fig. 3. Effect ofY = 1.5/s target yaw. As suggested in Table I, scat-

terers with range offsets are most sensitive to yaw-induced crossrange defocus,and scatterers with crossrange offsets are most sensitive to crossrange shifting.

such as entropy, contrast, or sharpness. Numerical optimization

of these metrics has been shown to correlate well with image

focus [13]. As the analysis of the previous section showed,

most forms of target rotation induce defocusing or increased

sidelobes. Numerically estimating corrections to compensate

these effects may also allow one to mitigate other effects, such

as crossrange shifting, that do not degrade image quality from

a heuristic point of view but that nonetheless distort the target

image and thereby affect its usefulness.

Target translation can be mitigated with algorithms for range

alignment [14] and autofocus [12], [13] that are commonly

employed to remove errors due to uncompensated collection

platform motion. In contrast, target rotation actually causes the

target phase histories to be incorrectly mapped into frequency


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RIGLING: IMAGE-QUALITY FOCUSING OF ROTATING SAR TARGETS 753

space during image formation. A warping of the frequency

space data collection manifold is thus required to place each

phase history at the correct aspect angle.

However, if the target does not present scatterers at a range

of displacements from the target centroid, a spatially varying

focusing solution may not be feasible. This is an important

point to be emphasized. Target rotation effects can only becorrected to the degree that the target structure combines with

its rotational behavior to produce observable image degradation

(i.e., defocus and range walk). If only shifting is observed,

the target image has not been degraded heuristically, and these

effects will not be corrected.

Theentropy-based rotation correction described in [15, Ch. 7]

seeks to remap the phase history data by optimizing image

entropy. This approach thus requires costly 2-D interpolations

of the phase history data and 2-D FFTs for each function eval-

uation in the optimization process. A fine autofocus correction

to the interpolated phase histories must be simultaneously esti-

mated. To mitigate arbitrary target motion, this computationally

intensive algorithm is likely required.

For more limited conditions under which image degradation

is restricted to crossrange distortion and defocus, a more effi-

cient algorithm may be conceived. In examining the results of

Table I, we observe that the dominant image degradation due to

rotation will be range-dependent and height-dependent cross-

range defocusing. Given the layover relationship between scat-

terer height and imaged range, we can thus implement a rotation

correction as a range-dependent crossrange autofocus. Includ-

ing the need for a fine crossrange autofocus for residual trans-

lation effects, we seek a correction to the range-compressed

phase history data s(x, fy) =s(x, fy)exp{j(x, fy)} that

minimizes an image-quality metric [12]

J=x,y

[I(x, y)] (5)

where I(x, y) =|S(x, y)|2 is the image intensity. The range-compressed data to be corrected is represented by s(x, fy),

and S(x, y) is the complex-valued image equal to the DFTof s(x, fy) in the crossrange dimension. Assuming well-behaved target motion, a suitable polynomial form for the phase

correction

c(x, fy) =

Nl=0

Nk=2

l,kxlfky (6)

may be adopted. Above, N is the polynomial order of therange dependence, and N is the polynomial order of thephase correction. The image-quality metric (5) may then be

efficiently optimized via steepest descent as described in [12].

The gradient of the correction parameters may be computed

expediently as

Jl,k

= 2Nfy

xlfky Jc(x, fy)

(7)

Fig. 4. Focused moving target image with corrected scaling in crossrangecompares well with Fig. 2. The image peak is normalized to 0 dB.

for

J

c(x, fy)=

2

N

x

s(x, fy)

S(x, y)

(I(x, y))

I(x, y)

(8)

where ()represents the Fourier transform,()indicates theimaginary part,Nrepresents the number of samples in fy , and

() denotes complex conjugation.For the simulation example, MATLABs fminunc function

was employed. The final focused image is shown in Fig. 4.

This result compares favorably with the stationary target image

in Fig. 2 with the notable difference that the target rotation

has provided finer crossrange resolution. For this illustrative

example, there was no significant residual broadening of the

main lobe or increased sidelobes relative to the noise floor.

The estimated yaw rate may be extracted from the second-

order phase correction by equating it to the predicted crossrange

defocus in Table I and solving forY, thus allowing the focusedimage to be displayed with corrected scaling in crossrange

as shown in Fig. 4. However, the scaling correction can besensitive to errors in the estimated yaw rate.

V. CONCLUSION

In this letter, the impact of uncompensated target motion

on SAR image quality was analyzed for a rotating target. An

image-quality-based algorithm for rotating target focusing was

presented and demonstrated on simulated data. Provided suffi-

cient target observability to introduce defocusing effects, this

algorithm seems capable of compensating for unknown motion

of reasonable model order. For the narrow synthetic apertures

considered in this letter, the simplifying assumptions to arrive at

Table I had negligible impact on their applicability, based on the

simulation example presented. Necessarily, as the integration


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angle increases, the quality of these assumptions will degrade

for purposes of analysis, and higher order phase corrections

will be required to maintain algorithm performance. While

no benchmarking was performed, we note that the proposed

focusing algorithm is little more computationally complex than

earlier image-quality-based autofocus methods and may thus

have significant computational benefits over methods that re-sample the phase history data to correct for target rotation.

Future work will study algorithm performance (e.g., conver-

gence time and residual phase error) as a function signal-to-

noise and signal-to-clutter ratios and as a function of target

background.

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Image-Quality Focusing of Rotating SAR Targets

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