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Comparison of Sparse Signal Separation Algorithmsfor Maritime Radar Target Detection

Brian Ng∗, Luke Rosenberg∗† and Paul Berry†

∗University of Adelaide, Australia, †Defence Science and Technology Group, AustraliaEmail: brian.ng@adelaide.edu.au

Abstract—Due to the non-stationary nature of sea clutter,traditional maritime radar detection schemes utilise non-coherentprocessing. To further enhance the detection performance, onealternative is to use sparse signal separation. This is an alternativeparadigm, whereby the different spatio-temporal characteristicsof the radar signal are exploited to separate targets from thebackground interference. In previous work, the sparse signalseparation problem has been posed in a compressive sensingframework so as to improve detection of small maritime targets.This paper investigates the performance of three different algo-rithms for solving the signal separation problem. These includethe Split Augmented Lagrangian Shrinkage Algorithm (SALSA),adaptive Complex Approximate Message Passing (CAMP) andthe Fast Sparse Functional Iteration Algorithm (FSFIA). Thefirst contribution is to reformulate the CAMP algorithm to theframework of sparse signal separation. The suitability of eachalgorithm is then assessed using real data from the Ingararadar, and is based on the quality of the solutions obtained,the computational speed and the robustness to the user’s choiceof ‘tuning’ parameters.

Index Terms—Sea clutter, detection, sparse signal processing.

I. INTRODUCTION

Detection of small or slow moving targets in the maritimedomain is difficult due to the motion of the sea, which resultsin the Doppler spectrum varying over both range and time,and the presence of sea-spikes which can resemble targetsand result in false detections [1]. For a high altitude airborneplatform, the problem is even more difficult as the sea clutterbackscatter is stronger and the performance of traditional non-coherent detection techniques is greatly reduced. This hasgenerated interest in alternative detection schemes based onsingle and multi-channel coherent processing [2]–[4]. Otherapproaches focus on exploiting differences in the spatio-temporal structure of returns from the sea surface (clutter) andpossibly the targets. In particular, joint time-frequency tech-niques such as wavelets have been proposed and investigatedby researchers [5]. One of the more promising techniquesis to exploit the tuned Q wavelet transform (TQWT), whichis a linear transformation that uses analysing functions withtuneable, constant fractional bandwidths for decomposition ofthe radar signals ([6], [7]).

Previous work has shown that returns from slow movingpoint targets are sparsely represented using low oscillatorysignals or ‘low Q’ wavelets, whilst fast targets are sparse with‘high Q’ wavelets [8]. This led to the formulation of a signal

separation problem, in which the collected radar signals y aredecomposed into a sum of two components,

y = x + r, (1)

where x is a target component and r is an interference (clutterplus thermal noise) component. The goal is to make thetarget component ‘cleaner’, effectively raising the signal-to-interference ratio (SIR) and improving the detection perfor-mance. Another approach to this problem is to suppress thesea-clutter using a whitening filter. However, this techniquerelies on estimation of the covariance matrix which implicitlyassumes the underlying processes are stationary. This is notthe case with sea clutter, which can contain strong spikes overthe coherent processing interval (CPI). The approach adoptedhere is to formulate the problem as a basis pursuit denoising(BPD) problem, which seeks to find a sparse representation ofthe target signal in a transformed domain that best describesthe observation.

The BPD formulation has strong roots in the compressivesensing literature, and for this reason, there are many algo-rithms that solve the BPD problem [9]–[11]. These have greatvariability in the quality, speed of the solution and most impor-tantly in how it can applied in a real system. Another practicalquestion associated with the signal separation approach is theselection of the penalty parameter in the BPD. It has beenshown that the value of this parameter has a strong influenceon the effectiveness of the separation, and is thus of greatimportance in practice [7].

The focus of this paper is to compare the suitability ofthree algorithms for solving the signal separation problem.The first algorithm is the iterative Split Augmented LagrangianShrinkage Algorithm (SALSA) which has been established asan effective and robust algorithm in previous work [9]. Thesecond algorithm is Fast Sparse Functional Iteration Algorithm(FSFIA) which has recently been developed. It offers similarsolutions to SALSA but with lower computational cost in adirection-of-arrival estimation problem ([11], [12]). The lastalgorithm is adaptive Complex Approximate Message Passing(CAMP). As presented in the literature, the CAMP radardetector works directly on the radar backscatter, essentiallyincorporating range processing into the formulation. Thisformulation is advantageous in that it is able to link theBPD penalty parameter to the radar’s false alarm rate [10]if the returns are corrupted by Gaussian noise. However, thisformulation does not apply to the current problem where slow-

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time returns at a given range bin are separated by solving (3).One of the contributions in this paper is to adapt CAMP inorder to solve the signal separation problem.

The problem formulation is presented in Section II withdetails of the three algorithms given in Section III. To testthe algorithms, we are using the Ingara sea clutter data setwhich is described in Section IV. Section V then comparesthe algorithms in terms of their accuracy, speed and theirsuitability for use in a detection scheme.

II. SPARSE SIGNAL SEPARATION FOR RADAR RETURNS

Joint time-frequency decompositions are often used tosparsely represent non-stationary signals. Linear transforma-tions such as short-time Fourier and wavelets are popular, asthey do not give rise to cross-terms and efficient algorithmsare readily available. In particular, wavelets offer sparse multi-resolution decompositions for many natural signals. Classi-cally, discrete wavelet transforms utilise dyadic re-samplersin the multi-resolution decomposition, which corresponds tothe analysing filters being approximately half-band. This is alimitation that is lifted in the more general tuned Q wavelettransform. TQWTs also use filters with a constant fractionalbandwidth (Q factor) but this is fully adjustable to achieve dif-ferent tilings of the time-frequency plane. As a result, TQWTscan be more readily adapted to the intrinsic time-frequencycharacteristics of different signals. In previous work [8], it wasfound that TQWT atoms are well-matched to a targets’ radarreturns across pulses (i.e. slow-time). In particular, returnsfrom stationary targets were more sparsely represented usinglow Q wavelets, while moving target returns were more sparsefor increasingly high Q, depending on the target velocity.

The sparsity of the target returns in the wavelet domaincan be exploited using a denoising framework where thecollected radar return y is modelled as a sum of a target andinterference components in (1). Unlike the interference, thetarget component x is typically sparse in the TQWT domain,which leads to the optimisation

min ‖w‖1, such that y = Aw + r, (2)

where A represents the inverse TQWT, and ‖·‖1 denotes the `1norm. The optimisation in (2) can be solved using the methodof Lagrange multipliers, leading to Basis Pursuit Denoising(BPD) formulation:

w = arg min1

2‖Aw − y‖22 + λ‖w‖1, (3)

where λ is a penalty parameter. The separated target com-ponent is obtained from x = Aw. The `1 term on theright is used to promote sparsity in the TQWT domain. Thepenalty parameter λ value presents a balance between thedegree of sparsity in the wavelet coefficients and the fidelityof the representation of the observed signal. It usually needsa judicious choice for each problem. In the data analysed inthis paper, the residue component may not be very small asit contains the majority of the clutter. As a result, the valueof λ needs to be larger to separate the weaker targets. As theforward and inverse TQWT transform matrices can be quitelarge, it is important that the solution of (3) be as efficient as

possible. Note that in the equations above, the normalisationfactors in [13] are absorbed into the matrices for brevity.

III. ALGORITHMS FOR SIGNAL SEPARATION

It is not possible to solve the BPD problem analytically.Instead, numerical algorithms must be used to obtain nearoptimal solutions. Being convex, the BPD is amenable tosolving with general techniques such as interior point methods,but these are usually slow. Methods for speeding up thesolution of BPD have been reported by numerous researchers,with many of these using variations of iterative techniques.Prominent examples include the Iterative Shrinking Threshold-ing Algorithm (ISTA), Fast ISTA (FISTA) and SALSA, withthe latter being shown to be fast and reliable for most problems[9]. For this reason, SALSA was adopted by Selesnick et al.[13] for solving (3) in several applications and has been usedin our previous work [8].

A. SALSA

The SALSA algorithm uses variable splitting to solve theBPD problem. A summary of this algorithm is given in [14],with notation adapted to our problem:

Initialise τ > 0, d, count = 0; choose maxIts ≥ 1while count < maxIts do

u← soft(w + d, τ)− dd← AH(y −Au)w← d + uincrement count

end whileAt the core of SALSA is a quadratic minimisation and a

soft thresholding function due to the use of the `1 norm in theBPD formulation. Note that the algorithm can be computedefficiently when the TQWT dictionary is replaced by an FFT-based transform.

B. FSFIA

Recently, Berry et al. ([11], [12]) proposed a fast sparsefunctional iteration algorithm for solving the BPD algorithm.It is based on the observation that the objective function in (3)is continuously differentiable at all points except w = 0 andtherefore use of the well-known gradient descent techniquesis not possible. FSFIA replaces the `1 regularisation term inthe BPD formulation by a function f(·) that is differentiableat all points. For a scalar argument, this is defined by

f(x) =

{|x|, if |x| > r

2r −√

2r2 − |x|2, otherwise, (4)

where r is a user-defined parameter. For the vector argumentsuch as in (3), f(·) is evaluated component-wise. This functionis identical to the `1 norm term everywhere except in theproximity of zero, where f(·) is a smooth function. Geomet-rically, it replaces the sharp tip of the `1 norm with a smallspherical surface. If r is small, then this approximation allowsthe algorithm to avoid numerical ill-conditioning problemsarising from singularities, while preserving the essence offinding a sparse representation of the collected signal. Solving

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this problem then requires iterating a simple least squares errorsolver:

wn+1 ← (Λ(wn) + µAHA)−1(µAHy),where the matrix Λ(w) is diagonal with each non-zero entrybeing the derivative of that element in w. It has been shownthat this algorithm requires few iterations to converge for anumber of applications. However, unlike SALSA the presenceof the AHA term precludes the use of fast FFT-based routinesto compute each iteration. The matrix inversion lemma can beused to reduce the computational requirement of the iterations.

C. Adaptive CAMP

Complex approximate message passing has been developedas a technique for solving the BPD problem. Like SALSA,it is iterative, but crucially modifies the estimation of the‘state’ during its iterations. Specifically, the variance of theerror in the estimated sparse representation is computed inevery iteration. This extra step allows the algorithm to accountfor the noise distribution of the residual term, which leadsto better quality solutions. Previous work has shown thatthis approach can improve the sparsity-undersampling tradeoff[15]. In practice, the truth is not known, and so the errorvariance must be estimated. Maleki et al. [16] named thisfurther refinement adaptive CAMP. Anitori et al. [10] thenapplied adaptive CAMP to the radar detection problem. Animportant contribution of this work was the establishmentof a direct link between the BPD penalty parameter to thefalse alarm rate of the detector, which meant that a suitablepenalty parameter can be chosen without an exhaustive search.This attractive property arises from the problem formulationin [10], where the sensing matrix Φ is a partial Fourier matrix,chosen to model stepped-frequency waveforms for processingfast-time samples. However, for our problem, we want toperform signal processing in the slow-time, so the sensingmatrix or sparsifying dictionary is replaced by a TQWTmatrix. A corollary of this necessary change is the loss of thelink between the false alarm rate and the penalty parameter.Whether this relationship can be restored with a re-formulationremains a question for further research. Instead, we investigatethe use of the adaptive CAMP algorithm to solve our signalseparation problem. Another attractive property of adaptiveCAMP is that it selects the optimal threshold multiplier, basedon the estimated standard deviation of the achieved solution.In practice, this saves the user from choosing a parameter, asrequired by SALSA and FSFIA.

IV. INGARA SEA CLUTTER DATA SET

The signal separation is demonstrated with real sea clutterdata from the DST Group Ingara X-band airborne radar. Thisdata was collected during the trials in August 2004 and July2006, where fine resolution (0.75 m) fully polarimetric datawas collected in open seas off the coasts of Port Lincolnand Darwin, Australia respectively. The aircraft flew in acircular track with a side-looking antenna pointing at a patchof sea, providing continuous data over 360◦ of azimuth lookdirections and a range of grazing angles between 15◦ and 45◦.

A full description of the data set and the collection campaignsare given in [17].

The data analysed in this paper is collected in the dualpolarimetric mode with a pulse repetition frequency of 578 Hz.It is from the upwind direction at approximately 30◦ grazingusing a horizontal polarisation. A block of 200 range bins and128 pulses is selected for analysis with synthetic Swerling1 targets of varying strength injected into the time domainsignal post range processing. The signal separation algorithmsare then applied to the slow-time signal for each range binseparately. An example of the data set with an injected targetis shown in Fig. 1.

Fig. 1. Radar backscatter intensity data used for the experiments with anexample synthetic target.

V. RESULTS

Each TQWT is specified by three parameters:

1) Q is the quality factor which determines the degree of‘oscillations’ and relates the signal bandwidth to thecentre frequency. A high value of Q implies a highlyoscillatory signal;

2) r controls the redundancy of the wavelet. If the signal xhas N samples, then the number of coefficients is rN .For a given Q, increasingly larger values of r generatewavelets with increasingly compact time support;

3) J specifies the number of bandpass sub-bands to use inthe TQWT.

As discussed in Section II, TQWTs with low Q are favourablefor representing stationary or slow moving targets. This paperonly considers stationary targets, and so a low Q factor TQWTis used throughout. However, the findings in this work alsoapply to fast moving targets with high Q TQWTs, based onevidence and simulations from earlier studies [8].

The three algorithms are compared in three principal ways:(1) the sensitivity of the solution to the penalty parameter, (2)the accuracy and (3) the computation time.

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A. Sensitivity to penalty parameter

The quality of separation can be measured using the energyratio of the separated components,

Er =‖X‖22‖R‖22

. (5)

Ideally, the residue component R captures the interference andthe target is confined to the X component. When a target ispresent and is successfully separated from the interference, theenergy ratio is large, otherwise the energy ratio is small. Notethat the separation is not energy preserving, so a rise in the Xcomponent energy does not necessarily imply a fall in the Rcomponent energy. For the same data, the obtained energy ratiodepends on the quality of the signal separation. This is affectedby the choice of the penalty parameters in the each algorithm.These are related to the BPD regularisation parameter (λ in(3)) in different ways. In adaptive CAMP’s case, the penaltyparameter is estimated as part of the algorithm’s adaption,which uses a grid search. For SALSA and FSFIA, the penaltyparameters are freely set by the user.

To compare the different algorithms, a number of synthetictargets with varying strengths are injected into range bins 35,40, 45, 75, 80 and 85. The target SIRs decrease from 10 to0 dB in steps of 2 dB, respectively. The penalty parametersfor SALSA and FSFIA are then varied in order to examinetheir effect on the separation performance. Since these twoalgorithms use different conventions with normalisation, thepenalty parameter values extend over different parameter val-ues.

Signal separation is now performed over all range bins. Theresultant energy ratios, as defined by (5), are shown in Fig. 2.For the stronger targets (SIR ≥ 6 dB), it can be observedthat the energy ratios in the target range bins show greateruniformity across the range of penalty parameter values forFSFIA, compared to SALSA. For the weaker targets, there arenot many differences between the two algorithms. There arealso some fairly strong energy ratios in range bins where thereare no targets. Referring to Fig. 1, it can be seen that theseappear in regions with relatively strong clutter returns, such asthe regions around range bins 5, 40, 80, 120 and 160. AdaptiveCAMP does not admit a penalty parameter, so it is not possibleto perform a similar comparison. Instead, the achieved energyratios are plotted against range bins and shown in Fig. 3. It isclear from this result that some strong targets do not show largecontrasts in the energy ratios compared with the interference.This may be attributed to CAMP’s assumption that the residuenoise (interference in this case) is Gaussian distributed, whichis definitely not the case for sea clutter data.

B. Accuracy

A successful signal separation is achieved if the targetcomponent is well-separated from the underlying data. In thatcase, the X component should resemble the injected target.To evaluate the relative merits of the different algorithms,stationary synthetic targets Xt are injected into all range bins,and signal separation is performed. For the FSFIA and SALSAalgorithms, the most favourable penalty parameters for each

Fig. 2. Energy ratios, in dB, for FSFIA and SALSA. Artificial targets ofvarying SIRs are injected into the data at range bins 35, 40, 45, 75, 80 and85.

Fig. 3. Energy ratios, in dB, for CAMP. The same artificial targets as inFig. 2 are injected into the data.

range bin are selected by grid search in order to achieve thebest possible performance. The separated X components arethen compared to the injected target signals and the meansquared differences (i.e. `2 error), normalised by the injectedenergy, is computed at each range bin:

L =‖X−Xt‖2‖Xt‖2

. (6)

We call L the relative target leakage. This is used to comparethe three algorithms’ outputs with 0 indicating perfect targetseparation, and 1 is total failure to separate (e.g. X = 0).The results are shown in Fig. 4 where the average leakagefor FSFIA, SALSA and CAMP are 0.0521, 0.0680 and 0.134,respectively. There is little difference between the FSFIA andSALSA algorithms with only a small performance advantagefor FSFIA. More importantly, leakage variations largely followthe clutter intensity along range bins. This is not surprising,as the stronger clutter results in a lower target SIR and weused a constant power level for all the injected targets. On theother hand, CAMP has a higher mean leakage, but does notseem to be affected by the underlying clutter. Its failure canbe attributed to the noise estimates within CAMP’s iterations.

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Fig. 4. Target energy leakage in X component for all range bins. The leakageis expressed as a fraction of the injected energy.

As currently formulated, the algorithm and underlying theoryrequires the residue distribution to be zero-mean and Gaussian,which is not the case with the sea clutter considered here. Itremains an interesting future direction of research to amendCAMP to handle more general distributions of the residue.

C. Computation time

All three algorithms used in this work use the same sparsedictionary, and share the same computational complexity withrespect to the forward/inverse transform computations. Com-pared with SALSA, FSFIA has is an additional computationalstep due to the inversion of (Λ(wn) + µAHA). This matrix,while not diagonal, is quite sparse, and is solved using a linearsystem of equations. CAMP is itself similar to SALSA interms of computation steps, but the adaptive noise estimationand grid search means a greater number of computationsare needed. In practice, the real computation time for allthree algorithms strongly depends on the number of iterationsrequired to achieve the desired accuracy.

To determine the relative computational time, a numberof experiments were conducted using Matlab R2017b scriptson a late-2013 MacBook Pro with a 2.3 GHz Intel Core i7CPU and 16 GB of 1600 MHz DDR3 RAM. The total runtimes for the FSFIA and SALSA are plotted against thedifferent penalty parameter values (represented by an index)in Fig. 5. It is clear that SALSA runs significantly fasterfor all values of penalty parameters. The average run timesover 20 iterations of SALSA and FSFIA are 1.84 and 3.70seconds, respectively. In comparison, adaptive CAMP’s runtime for the same block of data is almost 900 seconds withover 500 iterations required for the grid search, or 1.8 secondsper iteration. The sub-routines for the common components(e.g. TQWT calculations) can be optimised, or replaced withcompiled code, and that will affect the run times, but not therelative speed of the algorithms.

Fig. 5. Run times for SALSA and FSFIA, averaged for each selection ofthe penalty parameter. Adaptive CAMP’s run time is 1.8 seconds over 500iterations for the penalty parameter grid search.

The three algorithms generally reduce the objective functionin (3) as the number of iterations increase. However, thismay not directly lead to continued improvements in the targetleakage. To examine this aspect of performance, we measurethe number of iterations needed for each algorithm to achievethe minimum target leakage, as indicated in Fig. 4. It wasfound that SALSA and FSFIA require an average of 3.2 and4.2 iterations, respectively, to reach minimum leakage, whileCAMP requires 405 iterations. This massive discrepancy isthe reason why adaptive CAMP is much slower than the othertwo algorithms in practice.

D. Discussion

Empirical results show SALSA and FSFIA to be moreeffective in separating the target signal component from theinterference, with CAMP further behind. The target energyleakage is under 15% on average for all three algorithms,which suggests that the target components separate quitecleanly from the background clutter. The results here show thatthere is little difference in the best-case separation performancebetween SALSA and FSFIA. However, it may be observedin Fig. 2, that for the FSFIA algorithm there is a greatercontrast between the target and background interference whenthe penalty parameter is low. This implies that the FSFIAalgorithm may be more robust in a real world detectionscheme. The computational time per iteration and the timerequired to achieve the same accuracy, puts SALSA in front ofFSFIA, while CAMP lags far behind. However, for a practicaldetection scheme, both SALSA and FSFIA require an estimateof the penalty parameter. A technique for achieving this hasbeen previously presented in [7] where an additional 9 blocksof data was required to determine the penalty parameter fora given probability of false alarm. The total time to achievethis would be 18.4 seconds for SALSA and 37 seconds forFSFIA. This is still far less than the 900 seconds required forthe grid search in CAMP.

VI. CONCLUSIONS

This paper presents a comparison of three different sparsesignal separation algorithms for extracting small targets fromsea clutter. The comparison is based on results using the

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Ingara sea clutter data set with synthetically injected pointtargets. The metrics used to judge the performance includethe algorithms’ ability to effectively separate the targets, thecomputational speed and the sensitivity to the choice ofpenalty parameter. It was found that CAMP is less able toseparate the target from interference, while SALSA and FSFIAhave very similar performance when using optimal choices ofpenalty parameters. On balance of separation accuracy andcomputational speed, SALSA offers the best solution to thesparse signal separation problem.

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