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Published in IET Radar, Sonar and NavigationReceived on 20th February 2012Revised on 26th April 2012doi: 10.1049/iet-rsn.2012.0062

ISSN 1751-8784

Ship detection based on morphological componentanalysis of high-frequency surface wave radar imagesS. Grosdidier A. BaussardENSTA Bretagne/Lab-STICC Laboratory (UMR CNRS 6285), 2 rue Francois Verny,29806 Brest cedex 9, FranceE-mail: [email protected]

Abstract: In this study, high-frequency surface wave radar (HFSWR) is considered for target detection. These systems,commonly used for oceanographic purposes, are of interest in maritime surveillance because of their long range detectioncapabilities compared with conventional microwave radar. Unfortunately, the received signals are strongly polluted bydifferent noises. In this contribution a target detection method based on morphological component analysis (MCA) isinvestigated. Basically, MCA is a source separation technique based on multiscale transforms and the sparsity representation.The authors goal is to extract the target signatures from the range-Doppler image and then to take the final decision through asimple rule. This study introduces the issue of ship detection from HFSWR images and gives an overview of the MCAapproach. Then, the algorithm used for target detection is depicted. Comparisons with a classical constant false-alarm rate(CFAR) detection method, the so-called greatest of cell averaging-CFAR, are given through receiver operating characteristiccurves computed from simulated data.

1 Introduction

High-frequency surface wave radar (HFSWR) has beenefficiently used these last three decades for the remotemeasurement of oceanographic parameters. They providesurface currents, wave spectra, wind intensities anddirections. Recently, these systems have proved to bepotentially useful in target detection and tracking [1, 2].The main interest is that they can cover an area of up to200 nautical miles, which is a long range compared withstandard microwave radar. Moreover, this range valuecorresponds to the exclusive economic zone (EEZ).Continuous maritime surveillance of activities within theEEZ is a key question for civilian and military applications.Unfortunately, the spatial and temporal resolutions are weakand the received signal includes significant backgroundnoise, different kinds of clutter and interference. That iswhy target detection by HFSWR is a challenging problemand requires an adapted detection method.

Various methods have already been proposed for targetdetection through range-Doppler (RD) images such asconstant false-alarm rate (CFAR)-based techniques [3, 4] oroptimal thresholding estimation [5, 6]. Wavelet-basedprocessing methods have also been proposed in this area ofapplication [7, 8]. In this contribution, a method basedon morphological component analysis (MCA) [9, 10] isinvestigated. This undetermined blind source separationtechnique, based on the use of sparse representations, dealswith the different morphologies of the sources in the image.Indeed, each physical phenomenon (electromagneticinteractions with target, sea surface and so on) and the post-signal processing (range processing, beamforming and

IET Radar Sonar Navig., 2012, Vol. 6, Iss. 9, pp. 813–821doi: 10.1049/iet-rsn.2012.0062

Fourier transform) used to construct the RD images lead toa specific morphological signature (see Fig. 1). That is whythe MCA approach seems to be useful in this application.Our goal is to extract the target signatures from the RDimages to enhance the capabilities of detection. This meansthat MCA is used as a preprocessing technique to separatethe input image into two images one of them being(hopefully) only made of target signatures. The finaldetection can then be reached thanks to simple thresholding.

The performance of the proposed approach is evaluatedusing simulated data since with real data it is not possibleto know the position of all the boats in the radar area andthe sea state exactly. However, some first results obtainedagainst real data will be discussed. Finally, some receiveroperating characteristic (ROC) curves are depicted in orderto compare common CFAR techniques with our detectionmethod which uses MCA processing.

In the following, Section 2 introduces the HFSWR set-up.Section 3 gives an overview of the MCA method. Then, inSection 4, more details are given about the proposedapproach together with some illustrative results usingsimulated data. The potentialities of this approach againstreal data are also discussed. In Section 5, the detectionmethod is detailed and comparisons with a classical CFARtechnique are proposed. Finally, Section 6 gives someconcluding remarks.

2 HFSWR: set-up and model

In this paper, the WEllen RAdar (WERA) is consideredto be the measurement system [11]. Fig. 2 illustrates thecorresponding HFSWR set-up and introduces the main

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& The Institution of Engineering and Technology 2012

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parameters. From the measured data, the usual processing isto construct an RD image. This image gives the spectraldensity of the power backscattered by scatterers located in

Fig. 2 HF radar set-up

ur is the radar look angle, R is the radar distance, DR is the radar rangeresolution, uW is the wind direction, U10 is the wind intensity, V t is thetarget velocity vector and Rt the target range

Fig. 1 RD image obtained from a WERA system

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the red area in Fig. 2, according to the range and theDoppler frequency. In this area the scatterers are essentiallyocean waves and targets.

In a previous study [12], a simulator for generating realisticRD images was developed. It takes into account the sea clutter(through the sea spectrum, the significant height of the seawaves, dominant wave direction and so on), the targetparameters [speed, range and radar cross section (RCS)]and a given background noise level. Moreover, signalprocessing effects which appear on the real data havebeen added.

Fig. 3 shows RD images generated using our model [12]and obtained from real data (the WERA radar used arelocated in France along the coast of Brittany). The radarparameters used to obtain the simulated image are the sameas the experimental parameters and the sea parameters aresimilar to the estimated parameters during the acquisition.The first main difference is the power level close to theradar which is owing to range filtering (first 18 range cells,i.e. 27 km [12]) applied to the real data to prevent the radarfrom being dazzled by close direct scattering. The seconddifference is because of the local sea parameter variationswhich are not taken into account in the model. It isimportant to note that, for the real data, we just have accessto a global estimate of the sea state in the area covered bythe radar. That is why in the model we do not take intoaccount these local variations (the authors refer to [12] formore details).

3 MCA overview

In this section the MCA method [10], firstly proposed forsource or texture separation and inpainting applications, isbriefly introduced. Then, the MCA overall procedure isdepicted.

3.1 Basic MCA approach

This method assumes that if a proper dictionary is chosen foreach signal source, separation can be driven by sparsity. It isbased on the following concepts:

† Let S be a signal of dimension N. S is assumed to be alinear combination of (in this contribution and for

Fig. 3 RD images generated using our model [12] and obtained from real data

a Simulated RD image. The radar parameters are the same as the experimental parameters: the integration time Ti ¼ 1 min, the bandwidth B ¼ 100 kHz, the chirpduration Tc ¼ 0.26 s. The radar look direction ur ¼ 3158 and the wind direction uW ¼ 458. The wind speed U10 ¼ 4 m/sb RD image constructed from real data


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simplicity) two different signals S1 and S2

S = S1 + S2 (1)

† Each signal Sk can be written as the superposition of (afew) elementary functions f, called ‘atoms’, such that

Sk = Fkak =∑Lk

i=1

ak,ifk,i

Fk is an [N × Lk] matrix corresponding to the overcompletedictionary allocated to the source k. Notice that in practiceLk ≫ N.† The problem is now: how to separate S1 and S2 from S byusing the sparsity approach (i.e. obtain a good representationwith only very few non-zero coefficients). This can be doneby assuming the following holds true:

† For every possible signal Sk, there exists a dictionary Fk

so that

ak = Argmina

||a||0 subject to Sk = Fka

leads to a very sparse solution.† For every possible signal Sl, solving for k = l

al = Argmina

||a||0 subject to Sl = Fka

leads to a non-sparse solution. This requirement suggests thatthe dictionary distinguishes between the different types ofsignals to be separated. Thus, it plays the important role ofdiscriminant between the different content types.

‖.‖0 stands for the ℓ0-norm, which in fact counts thenumber of non-zero entries (or coefficients). In practice, wereplace the ℓ0-norm with an ℓ1-sparsity measurement [13],thus leading to a solvable optimisation problem.

† In practice, real signals are always corrupted by noise, thesources are not necessary linearly added (because of theobservation process), and the dictionaries cannot beperfectly adapted to the morphology of the sources. Then,one assumes these imperfections can be linked through anadded Gaussian noise so that S ¼ S1 + S2 + B (i.e. Bstands for the error model). Thus, and finally, fixing F1 andF2, the separation task can be summed up by the followingminimisation problem

{a1, a2} = Argmin{a1,a2}

||a1||1 + ||a2||1 +1��l

√ ||S −F1a1

−F2a2||22 (2)

The unknown components, S1 and S2, are deduced from theirrespective representation (S1 = F1a1 and S2 = F2a2).1/

��l

√stands for the regularisation parameter.

From this short overview, one can notice that the firstdifficulty is to find appropriate dictionaries. Starck et al.propose the use of the redundant wavelet families becauseof the multi-resolution decomposition, their efficiency insparsity representation, and the morphological diversitiesfrom one family to another. For instance, and for an image,the ridgelet transforms the linear forms into a sparse vectorand the undecimated discrete wavelet transform (UDWT)can do the same with Gaussian-shaped objects (or


isotropic). A description of those dictionaries and otherscan be found in [9, 14, 15].

To achieve the minimisation in (2), it is proposed in[9] to use the block coordinate relaxation (BCR) method.It is mainly based on an iterative thresholding of therepresentation vectors ak in order to ‘drive’ the sparsity.The threshold is decreased from an iteration to thefollowing, the final value is fixed at l. At each iteration,representation vectors are thresholded independently:when a1 is thresholded, a2 is assumed to be fixed to theprevious result (and vice versa). The most classicalthreshold technique is hard thresholding. Basically, at theith iteration the coefficients wn of the representation vectorak is given by

wn = wn if |wn| . li

0 otherwise

{(3)

where li is the threshold value at iteration i. For more detailsabout BCR, the readers can refer to [16].

3.2 MCA algorithm

In this contribution, S is an image ([n × n] matrix) made of alinear combination of S1, essentially the Bragg peaks and theionospheric and co-channel interferences, and S2, the targetsignatures. Suppose that we are given dictionaries Fk

(k ¼ 1, 2), also called synthesis operators, each associatedwith a linear transform (analysis) operator Tk (k ¼ 1, 2).

The usual MCA algorithm is given by the pseudo-code inFig. 4. The output of this algorithm S1 and S2 are expectedto contain morphologically different components.

4 Target signature extraction from RD image

In this section, an MCA-based method is applied to simulatedand real data in order to illustrate the potentialities of theproposed approach. Our purpose is to separate the targetcomponents from the others in the RD image.

In what follows, some more details about the proposedMCA approach are given. Then, numerical illustrativeexamples are proposed against simulated data and real data.

4.1 How to fix the parameter l

In (2), if it is assumed that B is a Gaussian noise with standarddeviation s, then l is generally fixed at 3s. This valuecorresponds to a probability of 0.27% that a noisycoefficient is allocated to a source after the separationprocess. Notice that for simulated images, s is generallyknown. But for real RD images, an estimation of s is oftennecessary.

4.2 Choice of dictionary

The choice of suitable dictionaries is an important step. Froma theoretical point of view, dictionaries have to respect thesparsity condition. From a practical point of view, thiscondition can be translated as a correlation condition:dictionaries must have at least one atom well-correlatedwith their associated sources but no atoms well-correlatedwith the other sources.

In RD images, targets produce generally point signatures orslightly extended signatures (for instance, owing to frequencyand radial resolution of the radar system or owing to shipmaneuvering during integration time). The signature can be

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Fig. 4 Usual MCA pseudo-code

assumed to be isotropic (at least in comparison to the noise).According to Starck et al. [9], UDWT is adapted to this kindof sources (atoms are well-correlated with isotropicsignature). That is why the UDWT is selected as the F2

dictionary to extract target signatures.The sea clutter components (Bragg peaks, second-

harmonic peaks and corner peaks and so on) can beconsidered as lines or line-shaped components. Moreover,the secondary lobes along the Doppler frequency-axiswhich are owing to a Fourier transform, and most of theinterference can be associated with this component. As oftheir properties, the second generation of discrete curvelettransform [17] seems to be adapted to the extraction ofthese components and it is assigned to be the dictionary F1

(curvelet atoms are well-correlated with sea cluttersignature). Notice that for the source S1 the classical hardthresholding (3) is used in the BCR algorithm.

Resolving (2) using these dictionaries enables theextraction of the target signature in S2 from the originalimage S. In what follows, we will call S2 the solution of theMCA processing. Note that most of these transforms andthe basic MCA algorithm are available in the MCAlabtoolbox [http://www.greyc.ensicaen.fr/:jfadili/].

4.3 New considerations in solving minimisation

In order to enhance the separation process, the hardthresholding step is replaced, for the UDWT-baseddictionary, by another thresholding process. The flowchartin Fig. 5 gives a brief reminder, in one-dimension (1D), ofthe UDWT transform.

In the 1D-UDWT, both v1 and x1 have the same size x0. x1

is then split into xeven2 , veven

2 , xodd2 and vodd

2 .v2 = {veven

2 , vodd2 } contains the wavelet coefficients at the

second scale, and is also of the same size as x0. Notice thath and g are the filters associated with the undecimatedwavelet transform [9].

An equivalent sketch is defined for images. In this case x0

is split into x1, vH1 , vV

1 and vD1 and so on. Then, UDWT

atoms differ in location, scale and orientations (H, V and D

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stand for horizontal, vertical and diagonal). The goal of theproposed modification in the thresholding step is to force S2

to be composed only with ‘isotropic’ signatures.Let wm

j [k, l] be the coefficient of the representation vectorat scale j, at spatial position (k, l ) and for direction m(m ¼ H, V and D), the classical hard thresholding (3) isthen replaced by

wmj [k, l] = wm

j [k, l] if wmj [k, l] .

li

3for all m

0 otherwise

⎧⎨⎩ (4)

For a given scale and position, the coefficients associated tothe three directions have to be greater than li/3 otherwisethey are all set at zero.

4.4 Test against simulated data

In previous works, MCA applications deal with imagesshowing relatively ‘well-localised’ components. In our case,

Fig. 5 1D UDWT transform


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Fig. 6 Simulated RD image including a target with 12 dB RCS

a Simulated RD image. The radar frequency is 12.3 MHz. The integrated time is equal to 1 min. The bandwidth is 100 kHz. U10 ¼ 3.7 m/s. ur 2 uW ≃ 908b RD image after the MCA processing

Fig. 7 Simulated RD image and results after MCA processing and thresholding

a RD imageb Image obtained after MCA processingc Image obtained after thresholding image (Fig. 7b)

one can clearly see in the previously shown RD images thatthe different components are not well localised (i.e. diffuse,smooth). This is particularly true for the sea clutter. Indeed,along the range-axis, these peaks show discontinuities anddecreasing power, and along the frequency-axis the powersmoothly decreases (i.e. non-localised shape). Moreover,


some of the discontinuities along the Bragg peakscombined with the secondary lobes along the frequency-axis can look like target signatures. That is why ourextraction problem is really challenging.

Fig. 6 shows a simulated RD image including a target with12 dB RCS and the resulting image S2 (i.e. image expected

Fig. 8 Co-channel interference

a RD image with co-channel interferenceb Image obtained after MCA processing

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Fig. 9 Ionospheric noise

a RD image with ionospheric interferenceb Image obtained after MCA processingc and d Highlight some specific areas

to contain only the signature target) obtained after MCAprocessing.

Unfortunately, in Fig. 6b, the Bragg peaks are notcompletely removed at the very low range (0–10 km). Thisis probably because of the very high power level of theBragg peaks at this range. However, detection at such rangeis not of interest for HFSWR since this area can be coveredwith much more efficiency by classical microwave radar(both systems can be used in a maritime surveillancesystem of the EEZ including the near coast). In thefollowing, the image between 0 and 10 km will not beconsidered for detection purposes. Moreover, one can alsosee low power components along the Bragg peaks whichcould lead to false alarms.

Now let us consider a multi-target configuration. Theacquisition parameters to generate the data remained thesame as in the previous simulations. However, this time,four simulated targets are added in the RD image. Fig. 7shows the simulated RD images and the result after MCAprocessing. The approach clearly leads to the extraction ofthe four target signatures. Unfortunately, as noted in theprevious part, again, one can observe some unexpectedcomponents at very low ranges.

Here, it is proposed to already consider the detectioncapabilities (this point will be discussed in Section 5) fromthe so-called S2 image. When applying a threshold(see Section 5) on this image (Fig. 7b) one obtains thebinary image given in Fig. 7c. This image illustrate whatwas already discussed and expected: the four targets are

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detected but unfortunate detections appear at low ranges(,20 km).

4.5 Tests against real data

In this part, some first results using real data are proposed toillustrate the capabilities of the proposed approach. Note thatthe goal is not to characterise the potentiality of the proposedmethod for target detection but to show that this methodcan also efficiently deal with typical noises at HF such asionospheric or co-channel interference. Moreover, and

Fig. 10 Position of the selected cells needed for local-noise levelestimation


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unfortunately, with these real data, it is not really possible toknow for sure the position of all the boats in the area coveredby the radar (that is why they cannot be used for the detectionanalysis of the method).

The two considered RD images have been chosen so thatthey include different kinds of noise or interference. Fig. 8includes co-channel interferences and Fig. 9 includesionospheric noise. In these figures, the images S2 (targetsignatures) obtained at the output of the MCA process isalso given. They illustrate the good separation results whenusing real data. If we focus on the results in Fig. 9, andcompare Fig. 9c with Fig. 9d where some areas arehighlighted, one can observe that this method leads to(relatively) efficiently separate the noise signatures (sea


clutter, ionospheric interference and so on) and the targetsignatures, which was expected. From Fig. 9d (i.e. image S2

at the output of the MCA approach) one can clearly see acomponent, which can potentially be a target, just behindthe position where the ionospheric noise was. Moreover, atarget signature in the extension of the right Bragg peak isalso extracted. One can of course wonder if this targetsignature is a real target or a false alarm because of theBragg peak discontinuity.

However, upon closer inspection of the RD image, one canobserve that the energy of the Bragg peak reduces with therange and that the signature appears just after ‘the end’ ofthe peak (the energy of the Bragg peak is lower than thenoise) with a shape and energy that leads us to think that it

Fig. 11 ROC curves for several target positions in RD images

a, c and e Example of generated RD imagesb, d and f Corresponding ROC curves for the GOCA-CFAR and the MCA-based approach

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is really a target which has a Doppler frequency close to theBragg peak.

These results show the effectiveness of the proposedMCA processing to extract possible target signaturesespecially from real data. Of course, future work will focuson the detection of existing boats in the radar area thanks todata from the automatic identification system (AIS). In thiscase, we will only be able to tell which boats are detectedand not identify the false alarms since not all the boats usean AIS or sometimes the AIS can be disconnected orbreakdown. That is also why, once again, in the following,only the simulated data are used to evaluate detectionperformance through receiver operating characteristic curves.

5 Detection analysis and comparison with aclassical CFAR technique

In this section, the detection capability of our approachis studied through the comparison of two detectionalgorithms, using simulated data. The first is called greatestof cell averaging (GOCA)-CFAR and has been shown to beefficient on this kind of data [1]. Basically this algorithmconsists in estimating a local signal-to-noise ratio (LSNR)for each cell of the RD image. The local noise is estimatedover selected cells surrounding the cell under test. Thisprocess is illustrated in Fig. 10. In order to take intoaccount different noises with specific profiles (range profileand Doppler profile) two local noise levels are estimated(with respectively a range window and a Doppler window).The highest value of these two estimations is chosen to bethe local noise value. Thus, the target is detected if theLSNR is above the given threshold.

The second algorithm is mainly based on the MCA. Thedecision is directly taken from the solution S2 obtained afterMCA processing. A simple cell-averaging from all pixels isused to compute an signal-to-noise ratio (SNR) for each cellunder test (an LSNR is unnecessary). A target is detected ifthe SNR of the tested cell is above the given threshold. Onecan note that the decision could also be taken by using theextracted representation vector a2 (the coefficient would becompared with a given threshold). However, here, theimage S2 is chosen for simplicity. Of course the efficiencyof this simple approach will depend on the extractionresults and consequently on the correct choice of dictionary.

The comparison of these two methods is given throughROC curves. Thanks to our model [12] which includesrandom elements, it is possible to generate severalrealisations of a RD image using the same (input)parameters. The GOCA-CFAR and the MCA approachesare applied on each RD image in order to determine, for alarge range of thresholds, the detection and false alarmrates. Then, after averaging results obtained from theseveral realisations (around 150), it is possible to plot theROC curves for each method.

For the three results, proposed in Fig. 11, the radarfrequency is fixed at 12.3 MHz, the integrated time is equalto 1 min, the bandwidth is 100 kHz, U10 ¼ 4 m/s andur 2 uW ≃ 908. The position of the target (with a 14 dBRCS) in the image is surrounded by a black circle. Fordetection purposes, Figs. 11a and e correspond to easysituations so both methods can be efficient as illustrated bythe ROC curves. The second configuration (Fig. 11c) ismore challenging since the target is close to the Braggpeaks. In this case the proposed approach leads to muchbetter results than the ‘classical’ CFAR.

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As one can expect, based on the previous results and theresults in Section 4.5, that the proposed method could leadto interesting results against real data and that the detectionresults could be better than classical CFAR techniques(especially for situations similar to that depicted in Fig. 9).Moreover, one can observe that in real data the Bragg peaksare not straight, which would appear to be a problem forthe classical CFAR. One can expect, once again, in thelight of the previous results and comments, that theproposed MCA-based approach will lead to better results inthis case.

6 Conclusion

The main purpose of our project is to propose a new methodfor target detection in RD images from HFSWR. This methodis based on a MCA of the image in order to extract the targetsignatures. Then a simple threshold is applied to achieve thedetection. In this contribution, we have tested the MCAmethod against simulated and real data to show itscapabilities and then plotted ROC curves to evaluate ourdetection method. Comparison with a ‘classical’ GOCA-CFAR, which is known to be efficient in this context, isalso proposed.

The ROC curves show that for challenging configurations,the proposed method can be efficient. However, for moresimple configurations our method is less effective. Usingreal data confirms the significant interest in applying suchan approach, particularly in its ability to remove typicalhigh-frequency noises such as ionospheric clutter andco-channel interference (not taken into account in thesimulated data). Thus, the MCA-based processing could bepotentially more effective than classical detection methodsfor the real data and for challenging configurations.

Future studies will focus on exploiting the real data and theavailable AIS data to evaluate the capabilities of our methodto detect, and later track, true boats. However, one promisingway to enhance the separation step is to define efficientdictionaries. This can be done following recent works ondictionary learning [18]. Our model (see [12]) could be ofstrong interest in this study since we can simulate a numberof configurations. Above all, we can generate the sea cluttersignature and the target signature separately and so create anumber of sub-figures for each morphological componentfor the learning process.

Finally, the inclusion of the decision step in the MCAprocessing could be considered, that is, the detection ofships. This can be done by directly processing thedecomposition coefficients. This solution should be moreeffective and is expected to provide better and more directresults.

7 Acknowledgments

The authors would like to thank the French Institute SHOMand ACTIMAR for providing the real data. This work wassupported by ‘Region Bretagne’ (ARED–DECIMER) andENSTA Bretagne.

8 References

1 Gurgel, K.W., Schlick, T.: ‘HF radar wave measurements in the presenceof ship echoes-problems and solutions’, OCEANS 2005 – Eur., 2005, 2,pp. 937–941

2 Ponsford, A.M., Sevgi, L., Chan, H.C.: ‘An integrated maritimesurveillance system based on high-frequency surface-wave radars, part


www.ietdl.org

2: operational status and system performance’, IEEE Antennas Propag.Mag., 2001, 43, pp. 52–63

3 Rohling, H.: ‘Radar cfar thresholding in clutter and multiple targetsituations’, IEEE Trans. Aerosp. Electron. Syst., 1983, 19, (4),pp. 608–621

4 Turley, M.: ‘Hybrid CFAR techniques for HF radar’. Radar 97 (Conf.Publ. No. 449), 1997, pp. 36–40

5 Dzvonkovskaya, A.L., Rohling, H.: ‘Adaptive thresholding for HF radarship detection’. Sixth Int. Radiowave Oceanography Workshop (ROW-2006), 2006

6 Hongbo, L., Yiying, S., Yongtan, L.: ‘Estimation of detection thresholdin multiple ship target situations with HF ground wave radar’, J. Syst.Eng. Electron., 2007, 18, (4), pp. 739–744

7 Jangal, F., Saillant, S., Helier, M.: ‘Wavelet contribution toremote sensing of the sea and target detection for a high-frequencysurface wave radar’, IEEE Geosci. Remote Sens. Lett., 2008, 5, (3),pp. 552–556

8 Baussard, A., Grosdidier, S.: ‘Detection de cibles par radarHFSW: utilisation des curvelets et des ondelettes continues’, GRETSI,2009

9 Starck, J.-L., Elad, M., Donoho, D.: ‘Redundant multiscale transformsand their application for morphological component separation’, Adv.Imag. Electron Phys., 2004, 132, pp. 288–348


10 Starck, J.-L., Moudden, Y., Bobin, J., Elad, M., Donoho, D.:‘Morphological component analysis’. Proc. SPIE Conf. Wavelets,2005, vol. 5914

11 Gurgel, K.W., Antonischki, G., Essen, H.-H., Schlick, T.: ‘Wellen radar(WERA), a new ground-wave based HF radar for ocean remote sensing’,Coast. Eng., 1999, 37, (3), pp. 219–234

12 Grosdidier, S., Baussard, A., Khenchaf, A.: ‘HFSW radar model:Simulation and measurement’, IEEE Trans. Geosci. Remote Sens.,2010, 48, (9), pp. 3539–3549

13 Donoho, D.L., Elad, M.: ‘Maximal sparsity representation via l1minimization’, Natl. Acad. Sci., 2003, 100, p. 21972202

14 Fadili, J., Starck, J.-L.: ‘Encyclopedia of complexity and system science’(Springer, 2009), ch. Curvelets and Ridgelets

15 Starck, J.-L., Murtagh, F., Fadili, M.-J.: ‘Sparse image and signalprocessing – wavelets, curvelets, morphological diversity’ (CambridgeUniversity Press, 2010)

16 Sardy, S., Bruce, A.G., Tseng, P.: ‘Block coordinate relaxation methodsfor nonparametric signal denoising with wavelet dictionaries’,J. Comput. Graph. Stat., 2000, 9, pp. 361–379

17 Candes, E., Demanet, L., Donoho, D., Ying, L.: ‘Fast discrete curvelettransforms’, Multiscale Model. Simul., 2007, 5, (3), pp. 861–899

18 Peyre, G., Fadili, J., Starck, J.-L.: ‘Learning the morphologicaldiversity’, SIAM J. Imaging Sci., 2010, 3, (3), pp. 646–669

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