Wavelet–RELAX feature extraction in radar images

L. Vignaud

Abstract: In classical high-frequency radar imaging, like synthetic aperture radar imaging, man-made target contributions are often well described by decomposing their signature into a set ofbright points. The basic model supposes that these elementary reflectors are independent of therelative angular aspect and of the observation frequencies. Many feature extraction methods, suchas CLEAN/RELAX-based algorithms, are built on this hypothesis and so it is currently used inassisted/automatic target recognition algorithms (ATR). However, this simple model cannotdescribe the variability of the signatures one can observe in image databanks. The authors proposeextending the target feature extraction capacities of the CLEAN/RELAX algorithm to dispersivescatterers using generalised wavelets.

1 Introduction

In classical radar imaging, the basic assumption lie in theideal bright points model (or canonical peaks) ofelementary scatterers that compose the scene to be imaged;the discrete data model is usually written as a set ofK complex sinusoids with ‘frequencies’ ðxk; ykÞk ¼1;...;K andamplitude Ak in noise e

HðKx;KyÞ ¼XK

k¼1

Ak expf jðKx xk þ KyykÞg þ eðKx;KyÞ ð1Þ

and the image Iðx; yÞ is then the 2-D spectrum ofthe formatted data HðKx;KyÞ with Kx ¼ 4p f sin �=c andKy ¼ 4p f cos �=c; with frequency f in the radar bandwidthDf and observation angle � within D�: In synthetic aperturemeasurements, the angle � is linked to the observation timeT through the relative motion of the target and the radar.We note DT the integration time over which a coherentimage is formed.

Under this strong assumption, the extraction problemmatches the signal processing formulation of super-resol-ution. We can make use of a signal model (complexsinusoids) with autoregressive methods like Burg’s algor-ithm or work on the signal autocorrelation (MUSIC,PISARENKO, ESPRIT) to separate the elementary reflectorswith a better resolution than the Nyquist boundary (Fourierresolution cell).

An extended analysis of the various spectral estimationmethods that can be used is given, from a signal processingpoint of view in [1] and applied to SAR imaging in [2].However, most of the methods require an a priori knowledgeor estimation of the number of reflectors, which deeply limitstheir application to SAR imaging and feature extraction.

Unlike these, the RELAX algorithm [3, 4] is anasymptotically statistically efficient estimator that

minimises the following nonlinear least squares criterion:

Cðxk; yk;Ak;K Þk¼1;...;K ¼ jH � HHj2 ð2Þ

with HH ¼PK

k¼1 Akexpf jðKx xk þ KyykÞgOne advantage is that the number of elementary sources

is estimated within the algorithm. RELAX belongs to afamily of iterative methods which, within each iteration,search for the biggest peak in a complex image and subtractits complex contribution in its corresponding Fourierspectrum. It extends the CLEAN algorithm [5] with arelaxation step. (See [3, 4] for a more detailed description ofthe algorithm and its implementation.)

Unfortunately, the ideal bright point model is valid only ifthe measurement frequency bandwidth is narrow and theangular sector is small (or the integration time is short).Otherwise, phase shifting points, motion through resolutioncells and nonuniform blurring may occur, and the extractionmethods need to be adapted. We propose to extend the targetfeature extraction capacities of the CLEAN/RELAXalgorithm to dispersive scatterers. We will implement along-term (or very wide angle) and large frequencybandwidth coherent extraction using a wavelet-basedmodel of the scatterers behaviour in combination with theRELAX architecture.

2 Wavelet CLEAN/RELAX

2.1 Introduction

Special modes commonly used in SAR imaging, such asspotlight observation where the radar is continuouslysteered to illuminate a small patch of terrain, need a verylarge integration angle for which the dispersivity of thescatterers cannot be ignored. Distortions appear in the imagewhen integration time is increased or when the image isformed over a wider angular sector and/or a largerfrequency bandwidth. This is because of the dispersivenature of real scatterers (anisotropy, unstationarityand frequency dependency) that leads to phase shifts inthe ðKx;KyÞ space, and of the residual uncompensatedmotion of the scene (due to the finite precision of relativepositioning systems or errors in autofocus) and residualerrors of the imaging algorithm. The angular dispersivity ofreal scatterers can be clearly spotted by building successive

q IEE, 2003

IEE Proceedings online no. 20030763

doi: 10.1049/ip-rsn:20030763

The author is with Departement ElectroMagnetisme et Radar, ONERA,Chemin de la Huniere, 91761 Palaiseau Cedex, France

Paper first received 18th December 2002 and in revised form 30th June2003

IEE Proc.-Radar Sonar Navig., Vol. 150, No. 4, August 2003242

looks of a SAR image (classical images computed forsuccessive mean angle of observation). See Fig. 1.

Classical algorithms are then useless to solve the featureextraction problem because real reflector behaviour does notfit the classical model. The spectral analysis spotted in (1)then depends on time (or angle) and frequency, and not onlyon the position of the individual scatterers. Moreover, acoupled interaction of the 4-D variables ðx; y; f ; �Þ occurs.

2.2 Generalised wavelet 4-D radar imaging

Evolution of classical radar imaging with a model offrequency-coloured and nonisotropic bright points has beenproposed using the concept of hyper-imaging (or 4-Dimaging). The reader should refer to [6] and [7] to get adeeper view of this subject.

The hyper-image Iðx; y; f ; �Þ can be viewed as adistribution of localised states with density built on thewavelet transform of the backscattering coefficient Hð f ; �Þ

Cðx; y; f ; �Þ ¼Z Z

Hð f 0; � 0ÞF0ð f 0=f ; � 0 � �Þ

� e4ipf 0ðx cos � 0þy sin � 0Þ=c df 0 d� 0

Iðx; y; f ; �Þ ¼ jCðx; y; f ; �Þj2 ð3Þ

with the mother wavelet F0ð f ; �Þ; which can be seen as thebackscattering function of an elementary dispersive scat-terer located at ðx ¼ 0 y ¼ 0Þ: The choice of the motherwavelet is restrained by admissibility rules (see [6] for moredetails), but the wavelet function has to remain ‘concen-trated’ on a spotted frequency and angle whatever frequencyscaling and angle rotation is applied to it. Here, for instance,we can choose an admissible wavelet that separatelycontrols the analysis spreading in frequency and angle viatwo parameters ðlf ; l�Þ

F0ð f ; �Þ ¼ f 2plf exp½�2plf f ð4Þ

Resolutions are given by uncertainty relations

�f ¼ fffiffiffiffiffiffiffiffiffiffi4plf

p ; �� ¼ffiffiffiffiffiffil�4p

r; �x � c

2f

ffiffiffiffiffiffilf

4p

r; �y � 1

fffiffiffiffiffiffiffiffiffiffi4pl�

p$ %

ð5ÞWe see that this enables us to study the frequency andangular dependencies of the reflectors while controlling thecoupling interaction between all the variables.

A strong property of the generalised wavelet decompo-sition is the ability to reconstruct the analysed back-scattering function by coherently summing the waveletcoefficients over the entire ðx; y; f ; �Þ space. It means that allthe information included in the data Hð f ; �Þ is entirelydistributed in the wavelet decomposition

Hðf ;�Þ/ZZ ZZ

Cðx;y; f 0;�ÞF0ð f=f 0;��� 0Þ

�exp½�4ipf 0ðxcos� 0þysin� 0Þ=c dxdydf 0 d� 0

ð6ÞWe can see (6) as the possibility to represent the radar dataHðf ;�Þ in an infinite base of wavelets representing thevarious behaviours of elementary dispersive scatterers.

2.3 CLEAN/RELAX wavelet extension

If we make the hypothesis that a backscattering function(or SAR data) of a man-made target is mainly composedby the contributions of a set of individual dispersivereflectors, we can have a second look at (6) anddecompose any target backscattering function into thefinite sum of elementary dispersive scatterers using thecombination of wavelets. This extends the classicalproblem formulation (1) to

Hð f ; �Þ ¼XK

k¼1

Ak Fxk ;yk ;fk ;�k;lf ;l�ð f ; �Þ þ eð f ; �Þ ð7Þ

where Fxk ;yk ; fk ;�k ;l f ;l� represents the backscattering functionof an elementary scatterer localised around the point ðxk; ykÞ

Fig. 1 Successive SAR looks of a target showing scatterer dispersivity

IEE Proc.-Radar Sonar Navig., Vol. 150, No. 4, August 2003 243

active around the fk frequency and the �k angle with ðlf ; l�Þcontrolling the spreading of the dispersivity in the ð f ; �Þspace (colouration and specular length)

Fxk ;yk ; fk ;�k ;lf ;l�ð f ;�Þ¼F0ð f=fk;���kÞ� exp ½4ipfkðxk cos�k þ ysin�Þ=c

ð8ÞThe problem is quite complicated to solve because it ishighly nonlinear and multidimensional. The criterion wechoose to minimise is the following:

Cðxk; yk; fk; �k; lf ; l�;Ak;KÞk¼1;...;K ¼ jH � HHj2 ð9Þ

We then use a combination of the RELAX architecture withstandard optimisation procedures. A similar procedure hasalready been used in [8] to discriminate trihedral fromdihedral reflectors.

The CLEAN/RELAX architecture remains essentially thesame except for the estimation step.

For each iteration k; we compute the following steps:

(i) We start with an automatic search for the strongest pixelin the ðx; yÞ space of the classical image Iðx; yÞ: This givesthe first estimation of the reflector position ðxxk; yykÞ:

(ii) A local transformation (windowed around ðxxk; yykÞÞ toð f ; �Þ; or simply to the Fourier space ðKx;KyÞ of the image,gives an estimate ð ffk; ��kÞ of its mean distribution in ð f ; �Þ:Note that a 2-D time–frequency analysis can be used forthis purpose, but this usually leads to heavy computationloads.(iii) The estimated parameters ðxxk; yyk; ffk; ��kÞ are used asinitial guess in the optimisation scheme. We use a standardoptimisation procedure (Matlab toolbox) of the criterion (9)

to obtain ðllf ; ll�Þ estimates and AAk as a subproduct.Parameters ðxxk; yyk; ffk; ��kÞ may be estimated onceagain through an optional loop of the previous stepsknowing ðllf ; ll�Þ: This leads to a better robustness butincreases computation load by a lot.(iv) After the optimisation converges, the estimatedcontribution AAk Fxxk;yyk ;ffk ;��k ;llf ;ll�

ð f ; �Þ is removed (cleaned)

from the data Hð f ; �Þ and another iteration begins.

Of course, iterations are embedded in the usual relaxationprocedure. Each previously extracted reflector is estimatedonce again each time another reflector is extracted in itsneighbourhood.

The application of wavelet–RELAX is given in Figs. 2and 3 on a real dataset (anechoic chamber measurement on areduced scale aircraft model). Here the extraction isconducted inside the annular region of measured data(1 GHz frequency bandwidth over 3608). Of course, SAR

Fig. 2 Results for reduced scale aircraft model

a Backscattering data domain of a reduced scale aircraft in an anechoicchamber ðKx ¼ Kr sinð�Þ; Ky ¼ Kr cosð�ÞÞb Reconstruction of the extracted spectra on the annular data domain andextrapolation on the whole domain by the wavelet–RELAX algorithm

Fig. 3 Results for reduced scale aircraft model

a Classical Fourier imageb Super-resolved image obtained by extrapolation of the wavelet–RELAXalgorithm

IEE Proc.-Radar Sonar Navig., Vol. 150, No. 4, August 2003244

data would only represent a portion of the 3608 illumination,but the principles remain nevertheless the same.

The super-resolution capacities of the algorithm areillustrated by extrapolating the extracted data outside themeasured dataset (Fig. 2b). It only means that the behaviourof the extracted reflectors is extrapolated, in no way shouldit be interpreted as the true extrapolation of the realbackscattering function of the target on a wider bandwidthsimply because no other scatterer is created and we cannotpredict what does not first exist!

The classical Fourier image and the wavelet–RELAXextrapolated image are given in Fig. 3. Though a lot of workmust still be done to stabilise the extraction, the resolutionenhancement is quite impressive. Furthermore, we have afully parametric description of the target with a lot moreinformation on the frequency and angle dispersivity of eachpoint than the conventional image could give.

We present another example on real SAR data of a groundtarget. Successive looks of the target were shown (Fig. 1) toillustrate the angular anisotropy of real scatterers. If wecombine these looks together, either coherently (fullaperture image) or not (multilook image), we see that theresidual motion causes an unacceptable blur in both images(see Fig. 4). We have then adapted the autofocusing strategyproposed in [9] for the RELAX algorithm to our wavelet-RELAX architecture. Figure 5 gives a well autofocused fullaperture image, which is a subproduct of our algorithm,and the dispersive feature extraction for which we knowthe precise orientation of each extracted scatterer (notrepresented here).

3 Conclusions

The CLEAN/RELAX architecture is a valuable tool forfeature extraction in radar images as it can be extendedand adapted to use various models of the elementaryreflectors. We have presented a wavelet-based extensionthat works on frequency directivity dispersive reflectors.The wavelet model was derived from hyper-imagingtheory that extends classical radar imaging by adding ajoint frequency and angular analysis. We showed howthe wavelet–RELAX algorithm can be implemented withstandard optimisation procedures, and successfullyapplied for feature extraction over very wide anglesand large-frequency bandwidths. The super-resolution andautofocusing possibilities that derive from this algorithmwere also illustrated for real data.

We think that automatic target recognition proceduresshould benefit from these improvements mainly becausethey take into account the target aspect angle variabilityproblem while delivering richer information on theindividual parts of the target.

4 References

1 Marple, S.L.: ‘Digital spectral analysis with applications’ (Prentice,Englewood Cliffs, NJ, USA, 1987)

2 De Graaf, S.R.: ‘SAR imaging via modern 2D spectral estimationmethods’, IEEE Trans. Image Process., 1998, 7, (3), pp. 729–761

3 Li, J., and Stoica, P.: ‘Efficient mixed-spectrum estimation withapplications to target feature extraction’, IEEE Trans. Signal Process.,1996, 44, (2), pp. 281–295

Fig. 4 Multi-look image and full aperture image with residualmotion compensation errors

Fig. 5 Full aperture image and dispersive feature extraction

a Full aperture image with autofocusb Full aperture dispersive scatterers extraction (wavelet–RELAX)

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4 Li, J.: ‘Implementation of the RELAX algorithm’, IEEE Trans. Aerosp.Electron. Syst., 1998, 24, (2), pp. 657–664

5 Tsao, J., and Steinberg, B.D.: ‘Reduction of sidelobe and speckleartifacts in microwave imaging: the CLEAN technique’, IEEE Trans.Antennas Propag., 1988, 36, (4), pp. 543–556

6 Bertrand, J., Bertrand, P., and Ovarlez, J.P.: ‘Dimensionalized wavelettransform with application to radar imaging’. Proc. IEEE ICASSP-91,1991

7 Ovarlez, J.P., Vignaud, L., Castelli, J., and Tria, M.: ‘Analysis of SARimages by multidimensional wavelet transform’, IEE Proc., Radar SonarNavig., 2003, 150

8 Bi, Z., Li, J., and Liu, Z.-S.: ‘Super resolution SAR imaging viaparametric spectral estimation methods’, IEEE Trans. Aerosp. Electron.Syst., 1999, 35, (1), pp. 267–281

9 Zheng, Y., and Bao, Z.: ‘Autofocusing of SAR images based onRELAX’. Proc. IEEE Int. Conf. Radar 2000, May 2000

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