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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 11, NOVEMBER 2008 3739

Particle Filtering Based Approach for LandmineDetection Using Ground Penetrating Radar

William Ng, Thomas C. T. Chan, H. C. So, Senior Member, IEEE, and K. C. Ho, Senior Member, IEEE

Abstract— In this paper, we present an online stochastic ap-proach for landmine detection based on ground penetrating radar(GPR) signals using sequential Monte Carlo (SMC) methods. Theprocessing applies to the two-dimensional B-scans or radargramsof 3-D GPR data measurements. The proposed state-space modelis essentially derived from that of Zoubir et al., which relies onthe Kalman filtering approach and a test statistic for landminedetection. In this paper, we propose the use of reversible jumpMarkov chain Monte Carlo in association with the SMC methodsto enhance the efficiency and robustness of landmine detection.The proposed method, while exploring all possible model spaces,only expends expensive computations on those spaces that aremore relevant. Computer simulations on real GPR measurementsdemonstrate the superior performance of the SMC method withour modified model. The proposed algorithm also considerablyoutperforms the Kalman filtering approach, and it is less sensitiveto the common parameters used in both methods, as well as thosespecific to it.

Index Terms—Ground penetrating radar (GPR), Kalman filter(KF), landmine detection, particle filter (PF), reversible jumpMarkov chain Monte Carlo (RJMCMC), sequential Monte Carlo(SMC).

I. INTRODUCTION

OWING to good penetration, depth resolution, and excel-lent detection of metallic and nonmetallic objects, ground

penetrating radar (GPR) has become an emerging techniquefor landmine detection [1]–[23]. A GPR system consists ofa transmitter emitting electromagnetic waves to the groundsurface and a receiver collecting returned signal from which thepresence of landmines can be indicated. However, the difficultyof using this technique for landmine detection remains, as thesignals originating from various types of ground surfaces, likesoil or clay, are nearly indistinguishable from those of genuinelandmines. Whereas the locations of landmines are usuallyunknown and positioned arbitrarily, the problem is further

Manuscript received January 24, 2008; revised May 23, 2008. Currentversion published October 30, 2008. This work was supported by a Grant fromthe Research Grants Council of the Hong Kong Special Administrative Region,China, under Project CityU 119605.

W. Ng and H. C. So are with the Department of Electronic Engineering,City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]).

T. C. T. Chan was with the Department of Electronic Engineering, CityUniversity of Hong Kong, Kowloon, Hong Kong. He is now with the Quan-titative Research Department, Nomura International (HK) Ltd., Hong Kong(e-mail: [email protected]).

K. C. Ho is with the Department of Electrical and Computer Engineer-ing, University of Missouri, Columbia, MO 65211 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2008.2002028

compounded as the signatures of the landmines and varioustypes of backgrounds are highly inconsistent, leading to the de-velopment of robust and intelligent approaches for the problem.

Typical landmine detection approaches are based on back-ground removal [7]–[14], and the corresponding techniquesinclude adaptive background subtraction [8] and backgroundmodeling using a time-varying linear prediction [9] and itsimprovement [10]. On the other hand, Zoubir et al. [1] haveproposed an approach that combines a Kalman filter (KF) forstate estimation and a detection method based on a comparisonbetween some threshold and test statistics. Whereas the modelin [1] is linear and Gaussian, this approach may suffer frominconsistent localization performance because its detection per-formance is subject to sensitivity to the selection of parameters,such as the size of the sliding window and threshold. Tang et al.have proposed in [14] using sequential Monte Carlo (SMC)methods [24]–[26] for landmine detection application. It is sug-gested that, prior to properly localizing landmine objects, theground bounce signals must be estimated and removed usingSMC methods, and it is shown that localization performancecan be improved by about 5% when compared with othercompeting methods.

In this paper, we essentially modify the data model in [1],which is focused on processing the two-dimensional (2-D)B-scans or radargrams extracted from the 3-D GPR data of[27] and [28]. In a given scan of surface, not only the numberof objects is unknown but it is also varying. To model thisrandomness of the existence of objects, a stochastic variableis introduced to indicate whether a landmine may be present.To facilitate a robust and efficient method for estimating thepresence of objects and hence estimating their locations, wepropose to devise a numerical approach using SMC methods,also known as particle filters (PFs) [24]–[26], in associationwith the reversible jump Markov chain Monte Carlo (RJM-CMC) [29] methods. Instead of adopting some data testingmethods that provide hard decisions as in [1], the RJMCMCgives soft decisions on where possible landmines are locatedby exploring all possible model spaces without specifying anythreshold in all the particles. Moreover, expensive computationswill only be expended on the most likely model spaces ratherthan blindly to all available spaces. In particular, three movesare proposed for every particle: birth, death, and update. Inthe birth (death) move, a landmine is proposed to be present(absent), whereas in the update move, the current state will beupdated in light of the current observations. Once the targetobjects have been detected, PF is then employed to track thetarget signals, by a set of random samples or particles, gen-erated by sequential importance sampling and their associated

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3740 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 11, NOVEMBER 2008

Fig. 1. Example layout of underground objects.

importance weights, which are then propagated through time togive predictions of the target posterior distribution function atfuture time steps. Note that some preliminary study and resultsof this paper have been presented in [30].

The rest of this paper is organized as follows. In Section II,we review the state-space model proposed in [1] and its stateupdate equations, followed by the formulation of our modifiedand compact models and their prior distribution functions inSection III. In Section IV, we present the development ofthe SMC method with the RJMCMC for landmine detectionapplication. Computer simulations and evaluations on real GPRdata are included in Section V, and conclusions are given inSection VI.

Notations: Bold upper case symbols denote matrices, andbold lower case symbols denote vectors. The superscript T

denotes the transpose operation, and the symbol “∼” means“distributed as.” The quantity π(·|·) denotes a posterior distri-bution, whereas qa(·|·) denotes a proposal distribution functionof parameter a. The notation (·)1:t indicates all the elementsfrom time 1 to time t. The quantity N (μ,Σ) indicates a realnormal distribution with mean μ and covariance matrix Σ.The quantity U(a, b) indicates a uniform distribution over theinterval [a, b], and UV indicates a uniform distribution withinthe volume V .

II. PROBLEM FORMULATION BASED ON KF

The GPR data, which are downloaded at [27] and [28], areinvestigated in our study. In particular, the data from [27] areobtained by measuring the response from an impulse GPR witha center frequency of around 1 GHz in the time domain. Eachsetup contains landmines and other anomalies including largestone, empty cartridge, and/or copper wire strip. A typical setupis shown in Fig. 1. For each value of y, denoted as channel,the operator sweeps the GPR device along the x-axis or scandirection, recording a response every 1 cm in space. In thisparticular example, a total of 51 channels, separated by 1 cm inspace, along with K = 197 measurements per channel, denotedas scans, complete a GPR data set for a setup. For example, theradargram or B-scan of the twenty-fifth channel, denoted byy(n, k), for the setup in Fig. 1 is shown in Fig. 2, where thehorizontal and vertical axes correspond to the distance indexk = {1, . . . , K} and time index n = {0, . . . , N − 1} samples,respectively, with N = 512. The same descriptions of theseterminologies are applied to the data from [28].

In this section, we reproduce the state-space model and thealgorithm for landmine detection proposed in [1]. The state-space model serves as the basis on which our proposed methodis developed in the next section.

A. State-Space Model

Denote our state values at time index n = {1, . . . , N} anddistance index k = {1, . . . , K} in a given radargram with sizeN ×K by {α(n, k), β(n, k), γ(n, k)}, where α(n, k) is thebackground signal, β(n, k) is the target signal, and γ(n, k) isa random bias accounting for the changes in the target signal, ifthe landmine is present. The physical location of a measurementat distance index k is x(k) on the x-axis.

Depending on whether a landmine is present, these stateparameters follow two different models, given as follows:

α(n, k) = α(n, k − 1) + v0(n, k)

β(n, k) = β(n, k − 1)

γ(n, k) = γ(n, k − 1) (1)

if a landmine is absent, and

α(n, k) = α(n, k − 1)

β(n, k) = β(n, k − 1) + γ(n, k)

γ(n, k) = γ(n, k − 1) + v1(n, k) (2)

if a landmine is present. The noises v0(n, k) and v1(n, k) areassumed to be zero-mean white Gaussian random variableswith their respective variances σ2

v,0 and σ2v,1.

At any particular time, the observed signal y(n, k) is charac-terized by

y(n, k) ={

α(n, k)+ε(n, k), target-freeβ(n, k)+γ(n, k)+ε(n, k), target present

(3)

where ε(n, k) is assumed to be a zero-mean white Gaussianrandom variable with variance σ2

ε . Our objective, in light ofobservations y(n, k), is to detect where in the radargram alandmine is located by estimating the state vector x(n, k).

Typical approaches collect a group of samples and use themjointly in order to enhance the reliability of detection andlocalization. Denote by M the strip size or successive timesamples at a given distance index k and by L = �N/M� (thelargest integer contained in N/M ) the number of nonoverlap-ping strips. Accordingly, we express the state evolution modelin vector form for l ∈ {0, . . . , L− 1} as follows:

αl,k = αl,k−1 + v0,l,k

βl,k = βl,k−1

γl,k = γl,k−1 (4)

if a landmine is absent, and

αl,k = αl,k−1

βl,k = βl,k−1 + γl,k

γl,k = γl,k−1 + v1,l,k (5)


NG et al.: PARTICLE FILTERING BASED APPROACH FOR LANDMINE DETECTION USING GROUND PENETRATING RADAR 3741

Fig. 2. Example radargram of the layout in Fig. 1.

if a landmine is present, where al,k = [a(Ml + 1, k), a(Ml +2, k), . . . , a(Ml + M,k)]T and vj,l,k = [vj(Ml + 1, k),vj(Ml + 2, k), . . . , vj(Ml + M,k)]T with covariance matrixσ2

v,jIM for j = {0, 1} and IM being an M ×M identitymatrix. Likewise, the observation model now becomes

yl,k ={

αl,k + εl,k, target-free

βl,k + γl,k + εl,k, target present(6)

where εl,k = [ε(Ml + 1, k), . . . , ε(Ml + M,k)]T is again azero-mean white Gaussian random variable vector, distributedas follows:

εl,k ∼ N (0,Σεl) (7)

where 0M is an M ×M matrix of zeros, and Σεl= σ2

εlIM is

known and constant covariance matrix of strip l. In short, tofacilitate the localization of possible objects in a radargram,the observations are subdivided into L strips, and each stripcontains M ×K measurements.

B. Zoubir’s Approach [1]

The landmine detection based on the Zoubir’s approach iscomposed of two modules: detection and estimation. Whereasinterdependent, these two modules are conducted separately.We first reproduce the schema of the KF update equationsfor both models and then the target detection module basedon hypothesis testing methods [1]. A set of L filters is usedto estimate the state values in light of observations. In eachfilter l = {1, . . . , L}, the state values in either models areassumed to be a Gaussian random variable with mean andcovariance matrix which are then propagated from scan k tok + 1. It follows that a test statistic can be constructed from theestimation error or innovation between the updated states and

latest observations and decision can be made on comparing thisstatistic with a predetermined threshold. Details of these twomodules are given as follows.

1) KF: Table I summarizes the KF update equations forboth models: target-free and target present. In the case wherea target is assumed absent in strip l, we only estimate thestate φ0

l,k = αl,k in light of observations yl,k with distributionfunction φ0

l,k ∼ N (φ0l,k|k,Φ0

l,k|k), where αl,k|k and Φ0l,k|k are

the mean state value and its estimation covariance matrix. Onthe other hand, if a target is detected in strip l, we need toestimate the target as well as the random bias signals, i.e.,φ1

l,k = [βTl,k,γT

l,k]T, while keeping the background signal in-tact. The estimated state φ1

l,k is again assumed to be a Gaussianrandom variable with mean and covariance matrix, i.e., φ1

l,k ∼N (φ1

l,k|k,Φ1l,k|k).

2) Target Detection Module: Because it is assumed that theobservations in all strips are corrupted with white Gaussiannoise, the residual between the observation yl,k and its esti-mate yl,k in (6), depending on whether a target is detected,can be used as an indication of how well is the explanatorypower of an assumed model. Let νl,k = yl,k − yl,k, whichis a zero-mean Gaussian random variable whose distributionis νl,k ∼ N (0,Σεl

), as in (7), and let el,k = νTl,kΦ

j−1

l,k|kνl,k

for j = {0, 1}. It is clear that the random variable el,k is chi-squared distributed with M degrees of freedom, and hence, onemay easily set up a hypothesis testing on the variable el,k asa measure on the appropriateness of an assumed model. Thatis, for each scan k, we first compute the values of {el,k} forl = {1, . . . , L} and then conduct the χ2 test with an appropriatesignificance level κ. The hypothesis is that the current modelis assumed appropriate with sufficient explanatory power withprobability

Pr(el,k ≥ χ2

M,κ

)= κ (8)



TABLE ISCHEMA OF KALMAN FILTERING UPDATE EQUATIONS FOR

TARGET-FREE AND TARGET PRESENT SCENARIOS

where Pr(·) is the probability of a given event. Otherwise,the hypothesis is rejected, implying that the current model isinappropriate, and other model should be adopted. However,any decision solely based on the hypothesis testing on a strip atone scan is not only premature but also unreliable, particularlywhen M is not sufficiently large. As suggested in [1], not onebut a few hypothesis testings on the same strip l over a windowof consecutive scans Kτ should be based. As a result, if a presetnumber of hypothesis out of Kτ scans is accepted, then thecurrent model is considered appropriate. Nevertheless, unlesscarefully selected, the size of Kτ certainly affects the perfor-mance of the detection. Alternatively, one can plot the contourlines of these residual energies el,k for l = {0, . . . , L− 1} andk = {1, . . . , K} to identify the possible locations of landmines.

To summarize, one may realize that this approach has a fewpotential drawbacks. First, whereas the detection and localiza-tion are considered as a joint problem, they are solved in com-pletely separate modules. Second, the overall performance ofthe algorithm heavily relies on the outcomes of the hypothesistestings, which, in turn, critically depend on a careful selectionof the strip size M , the window size Kτ , and the significancelevel κ. Finally, because the detection results obtained from thehypothesis testings on every strip are mutually exclusive, i.e.,the probability of detection being either zero or one, there isa discontinuity of detection or “gaps” among adjacent stripsof a genuine target, particularly when the observation noise islarge or the strip size is small (see Section V for a vigorouscomputer evaluation of the Zoubir’s approach). In the next

sections, we will present a numerical based method that jointlydetects and localizes the landmines, given a radargram, and thatperforms more consistently and less sensitively to the variationsof parameters than the Zoubir’s approach.

III. FORMULATION OF PROPOSED STATE-SPACE MODEL

We now present the state-space model for the proposedmethod. Denote the state vector by φl,k, containing M succes-sive time samples, as follows:

φl,k =[αT

l,k,βTl,k

]T(9)

where αl,k and βl,k are as before, and a random variable bysl,k ∈ {0, 1}, which is an existence variable, which indicateswhether within the current M samples a landmine is present.Then, we express a general state evolution model as follows:

φl,k = f(φl,k−1) + Bsl,kvl,k (10)

where f(·) may be a linear or nonlinear dynamical function,and Bj for j = {0, 1} is given as follows:

Bj =[

IM 0M

0M j × IM

]. (11)

The quantity vl,k = [v(Ml + 1, k), . . . ,v(Ml + M,k)]T is azero-mean white Gaussian random variable vector with covari-ance matrix Σl

v for strip l, defined as

Σlv =

[σ2

v,0,lIM 0M

0M σ2v,1,lIM

]. (12)

Note that an individual strip l has its own covariance matrix Σlv .

Accordingly, the prior distribution function of φl,k for strip l,which is conditional on sl,k, is given by

p(φl,k|φl,k−1, sl,k). (13)

For the random variable sl,k, we model it by the stochasticrelationship sl,k = sl,k−1 + εs [31], where εs is a discreteindependently and identically distributed random variable suchthat the prior distribution function of sl,k is as follows:

p(sl,k|sl,k−1) =

⎧⎨⎩Pr(εs = −1) = pd

Pr(εs = 0) = 1− pb − pd

Pr(εs = 1) = pb

(14)

where pb and pd ∈ {0, 1} are the probabilities of incrementingand decrementing the number of targets, respectively, such thatpb = 0 if sl,k−1 = 1 and pd = 0 if sl,k−1 = 0.

From this point onward, our parameters of interest are de-noted by θl,k = {φl,k, sl,k} for strip l, whose prior distributionfunction can then be expressed as follows:

p(θl,k|θl,k−1) = p(φl,k, sl,k|φl,k−1, sl,k−1)

= p(φl,k|φl,k−1, sl,k)× p(sl,k|sl,k−1) (15)



where p(φl,k|φl,k−1, sl,k−1) and p(sl,k|sl,k−1) are the priordistribution functions of φl,k and sl,k in (13) and (14),respectively.

There are two major differences between the proposed modeland the one in (6). First, the bias term γl,k is removed, becauseit is believed that the quantity is redundant and its contributionto the entire state vector can be absorbed into the target signalβl,k. Second, in reality, the background signal αl,k changesin every distance index k, regardless whether a target exists ornot. To reflect this situation, the state dynamical function f(·)in (10) is made independent of the existence variable sk,l ∀k,which determines the magnitude of the driving noise accordingto the selection of the matrix Bsk,l

in (11). Collectively, theproposed model, in theory, describes the situation more realis-tically and reduces the computational load, as well as enhancesthe state estimation performance in terms of error variance.

It is further assumed that the states at different strips arestatistically independent, as considered in [1], such that we mayexpress the joint prior distribution function of θk as

p(θk|θk−1) =L−1∏l=0

p(θl,k|θl,k−1) (16)

where θk = [θ0,k, . . . ,θL−1,k]T and p(θl,k|θl,k−1) is the priordistribution function in (15).

Likewise, we extend the observation model of yl,k in (6) asfollows:

yl,k = g(θl,k) + εl,k (17)

where the function g(·) may be linear or nonlinear, and εl,k

is identical to that in (6). Accordingly, the likelihood of theobservation yl,k due to θl,k can be written as

p(yl,k|θl,k) = N (yl,k|g(θl,k),Σεl

)(18)

where the observations from a given strip l have its owncovariance matrix Σεl

∀l.Denoting yk = [y0,k, . . . ,yL−1,k]T and θk = [θ0,k, . . . ,

θL−1,k]T, we would like to estimate θl,k = {φl,k, sl,k} ∀lsequentially upon the receipt of yk.

IV. SMC METHODS

In the context of online parameter estimation, we are inter-ested in the posterior distribution π(θk|y1:k) with θk, whichcan be recursively obtained from two steps according to theBayesian sequential estimation framework described by thefollowing two equations:

π(θk|y1:k−1) =∫

p(θk|θk−1)π(θk−1|y1:k−1) dθk−1 (19)

π(θk|y1:k) ∝ p(yk|θk)π(θk|y1:k−1). (20)

The term π(θk−1|y1:k−1) in (19) is the posterior distributionfunction at k − 1, and the term p(yk|θk) in (20) refers tothe likelihood function of the observations yk from all Lstrips. The recursion is initialized with some distribution, forexample, p(θ0).

In very limited scenarios, the models of interest are “weakly”nonlinear and Gaussian in which one may utilize the KFand its derivatives, including the extended KF, to obtain anapproximately optimal solution. It is well known that the updateexpression in (20) is analytically intractable for most modelsof interest. We therefore turn to SMC methods [24]–[26],[32]–[34], also known as PFs, to provide an efficient numer-ical approximation strategy for recursive estimation of com-plex models. These methods have gained popularity in recentyears, due to their simplicity, flexibility, ease of implementa-tion, and modeling success over a wide range of challengingapplications.

A. Sequential Importance Sampling

Because it is assumed that the states from different stripsare statistically independent, we separately rather than jointlycompute the particles and their associated weights for everyindividual strip. In other words, we will have L independentPFs, each of which estimates the posterior distribution functionof θl,k of a particular strip l.

The basic idea behind PFs is very simple: The target distribu-tions are represented by a weighted set of Monte Carlo samples.These samples are propagated and updated using a sequentialversion of importance sampling, as new measurements becomeavailable. Hence, statistical inferences, such as expectation,maximum a posteriori estimates, minimum mean square error(MMSE), etc., can be computed from these samples.

From a large set of Ns particles {θ(i)l,k−1}Ns

i=1 with their as-

sociated importance weights {w(i)l,k−1}Ns

i=1, we approximate theposterior distribution function π(θl,k−1|yl,1:k−1) as follows:

π(θl,k−1|yl,1:k−1) ≈Ns∑i=1

w(i)l,k−1δ

(θl,k−1 − θ

(i)l,k−1

)(21)

where δ(·) is the Dirac delta function. We would like to generatea set of new particles {θ(i)

l,k}Nsi=1 from an appropriately selected

proposal function, i.e.,

θ(i)l,k ∼ q

(θl,k|θ(i)

l,k−1,yl,1:k

), i = {1, . . . , Ns}. (22)

Among many practical PFs, we choose to use the boot-strap PF [24] and assign q(θl,k|θ(i)

l,k−1,yl,1:k) = p(θl,k|θ(i)l,k−1),

as in (15).With the set of state particles {θ(i)

l,k} obtained from (22), the

importance weights w(i)l,k are recursively updated as follows:

w(i)l,k ∝ w

(i)l,k−1 ×

p(yl,k|θ(i)

l,k

)p

(θ

(i)l,k|θ(i)

l,k−1

)q(θ

(i)l,k|θ(i)

l,k−1,yl,1:k

) (23)

with∑Ns

i=1 w(i)l,k = 1. It follows that the new set of particles

{θ(i)l,k}Ns

i=1 with the associated importance weights {w(i)l,k}Ns

i=1 isthen approximately distributed according to π(θl,k|yl,1:k).

Accordingly, once the set of particles θ(i)l,k and their associ-

ated weights w(i)l,k for i ∈ {1, . . . , Ns} have been computed, we



obtain target state estimation according to the MMSE estima-tion, given by

θl,k = E[θk|yl,1:k] ≈Ns∑i=1

w(i)l,kθ

(i)l,k (24)

where E denotes the expectation operator for l ∈ {0, . . . ,L− 1}. Likewise, we also compute the residual energies for allstrips that indicate whether possible landmines are present bycomparing the observation yl,k with its estimate solely basedon the background signal as follows:

ηl,k =Ns∑i=1

w(i)l,k ×VAR

[yl,k − g

(φ

(i)l,k, 0

)]∀l (25)

where g(φ(i)l,k, 0) is an estimate of an observation yl,k based

solely on the background signals, and VAR[·] is the varianceoperator. That is, if the observation yl,k contains a backgroundsignal only, the residual energy ηl,k is expected to contain smallvalues. On the other hand, if the observation yl,k contains atarget signal, using the background signal alone to estimate yl,k

is going to yield much larger value in ηl,k.We also denote the aggregate residual energy signal by ηk as

follows:

ηk =L∑

l=1

ηl,k (26)

where the residual energy signals are summed over all stripsl along the x-axis. This quantity will be used later whenreceiver operating characteristic (ROC) curves are prepared forperformance evaluation in Section V.

As the PF operates through time, only a few particles con-tribute significant importance weights in (23), leading to thewell-known problem of degeneracy [26], [33]. To avoid this,one needs to resample the particles according to their impor-tance weights. That is, those particles with more significantweights will be selected more frequently than those with lesssignificant weights. More detailed discussion of degeneracy andresampling can be found in [26].

B. RJMCMC

Prior to properly estimating the locations of landmines, it isnecessary to tell whether they are present in a given region inthe radargram. Existing methods such as [1] deterministicallydecide whether a landmine is present in a given area, solelybased on a comparison between predefined thresholds withsome forms of test statistics computed from the data. On thecontrary, the proposed RJMCMC strategy explores all possiblemodel spaces and softly determines the presence of a landminewith a probability in the same area. The RJMCMC processis a variation of Metropolis–Hastings (MH) algorithm [35],[36], which inherently sets up a Markov chain whose invariantdistribution corresponds to the posterior of interest. The statessampled according to the MH algorithm at successive iterationsrepresent samples from the distribution of interest.

In our application for a given strip l at every k and particle,we randomly generate a new candidate θ�

l,k = {φ�l,k, s�

l,k} froma distribution function d(·|·), which may be conditional onθ

(i)l,k = {φ(i)

l,k, s(i)l,k}. The candidate will be accepted with prob-

ability ξ = min{1, r}, where r is the acceptance ratio which iscomputed as

r =π (θ�

l.k|y) d(θ

(i)l,k|θ�

l,k

)π

(θ

(i)l,k|y

)d

(θ�

l,k|θ(i)l,k

) × J (27)

with J is the Jacobian of the transformation from θ(i)l,k to

θ�l,k. In effect, the proposed RJMCMC method jumps between

parameter subspaces, thus visiting all relevant models. In everyiteration, a candidate with a particular model is proposed froma set of proposal distribution functions, which will be ran-domly accepted according to an acceptance ratio that ensuresreversibility and, therefore, invariance of the Markov chain withrespect to the desired posterior distribution. In particular, threedifferent moves are randomly selected to enable the explorationof the parameter subspace:

1) birth move, chosen with probability pb, for which thepresence of a landmine is proposed, i.e., sl,k = 1;

2) death move, chosen with probability pd, for which theabsence of a landmine is proposed, i.e., sl,k = 0;

3) update move, chosen with probability 1− pb − pd, forwhich the state parameters are updated given sl,k =sl,k−1.

It can be shown in [37] that the proposed MCMC samplingprocedure does not require a burn-in period in this applicationbecause the particles before the MCMC step are already distrib-uted according to the limiting distribution of the chain. In otherwords, only one MCMC iteration is needed for each particleat each time. In short, in the proposed method, all Ns particlesassume different models according to the RJMCMC at a givenscan k; a histogram of the available models can be constructedfrom {s(i)

l,k}, and a detection can be made if necessary.The selection of the move can be described by the following

schema:

1) for a given strip l at location k;2) we select each move for each particle i = {1, . . . , Ns};

a) sample u ∼ U(0, 1);b) if (u < pb), then “birth move;”c) else, if (u < pd + pb), then “death move;”d) else, update all parameters.

3) k ← k + 1, go to step 2).

1) Birth/Death Move: When the birth move is selected, itis assumed that a landmine is present in the current region,i.e., sl,k = 1 given sl,k−1 = 0. In particular, we choose d(·|·)in (27) to be the state prior function in (15). A candidateparticle {φ�

l,k, s�l,k = 1} is generated according to (15) and is

accepted with probability ξbirth = min{1, rbirth}, where rbirth

is computed as

rbirth = exp{−1

2

(e�T

Σ−1ε e� − e′TΣ−1

ε e′T)}

(28)



with e� = yl,k − g(φ�l,k, s�

l,k) and e′ = yl,k − g(φ′l,k, s′l,k),and φ′l,k ∼ p(φl,k|φl,k−1, s

′l,k = 0, sl,k−1 = 0).

When the death move is selected, similar procedures in thebirth move are taken in which a candidate particle φ�

l,k with

s(i)l,k = 0 and another particle φ′l,k with s

(i)l,k = 1 are generated.

The candidate particle will be accepted with probability ξdeath

as follows:

ξdeath = min{

1,1

rbirth

}. (29)

The particle θ(i)l,k = {φ(i)

l,k, s(i)l,k} becomes{

φ(i)l,k, s

(i)l,k

}=

{{φ�

l,k, s�l,k

}, if accepted{

φ′l,k, s′l,k}

, otherwise(30)

and will then be used to update the importance weights w(i)l,k.

2) Update Move: If this move is selected, we simply gener-ate the particle φ

(i)l,k ∼ q(φl,k|φ(i)

l,k−1,yl,1:k) with s(i)l,k = s

(i)l,k−1,

which will then used to compute the importance weights w(i)l,k.

C. Adaptivity of Noise Covariance Matrices

As we do not have any exact knowledge of the state andobservation noise variances, i.e., σ2

v,j,l for j = {0, 1} and σ2εl

,we also need to update their values in light of new observations.Here, we propose to update these variances for each strip l asfollows:

σ2v,0,l(k) =λσ2

v,0,l(k − 1) +(1− λ)MNs

Ns∑i=1

∥∥∥Δφ(i)l,k

∥∥∥2

(31)

σ2v,1,l(k) =λσ2

v,1,l(k − 1) +(1− λ)MNs

Ns∑i=1

∥∥∥Δφ(i)l,k

∥∥∥2

(32)

σ2εl

(k) =λσ2εl

(k − 1) +(1− λ)MNs

Ns∑i=1

∥∥∥Δy(i)l,k

∥∥∥2

(33)

where Δφ(i)l,k = φ

(i)l,k − f(φ(i)

l,k−1), Δy(i)l,k = yl,k − g(θ(i)

l,k),and 0 < λ < 1 is a forgetting factor. Note that it is assumed thatthe state particles {θ(i)

l,k} have been resampled with uniformweights.

V. SIMULATION RESULTS

In this section, a vigorous evaluation on the performanceof the proposed algorithm on two different sets of real GPRmeasurements is given. They are obtained from [27] and [28].Furthermore, the results from the proposed method will becompared with those from the Zoubir’s method in [1]. Here,we specify the form of the dynamical function f(·) in (10) andobservation function g(·) in (18). In particular, we choose touse the same linear model as in [1] in order to carry out a fairperformance comparison. Accordingly, the dynamical functionis f(φl,k−1) = I2Mφl,k−1, and the observation function be-comes gj(θl,k) = Hsl,k

φl,k, where Hj is given by

Hj = [IM j × IM ], j = {0, 1}.

TABLE IITRUE POSITIONS OF THE OBJECTS STATED IN [27], WHERE THE

QUANTITIES IN THE BRACKETS ARE ACTUAL POSITIONS

OBSERVED IN THE DATA SET AND USED IN EXPERIMENT 1

TABLE IIICOMMON PARAMETERS FOR COMPUTER SIMULATIONS IN EXPERIMENT 1

TABLE IVPARAMETERS USED IN THE PROPOSED METHOD FOR

COMPUTER SIMULATIONS IN EXPERIMENT 1

TABLE VPARAMETERS USED IN THE ZOUBIR’S APPROACH FOR

COMPUTER SIMULATIONS IN EXPERIMENT 1

Note that, for other forms of functions, for example, nonlinear-ity, chosen for f(·) and g(·) will not drastically and adverselyaffect the performance of the proposed SMC algorithms, as theyare developed to tackle problems that are nonlinear and non-Gaussian [24]–[26], [32]–[34].

For the RJMCMC, in the absence of additional informationabout the locations of genuine target objects, the probabilitiesof death and birth are set identical, i.e., pd = pb, in all theexperiments. It is worthy to note that our selection of pb andpd is essentially arbitrary, unless more prior information isavailable to guide their assignment. Therefore, in the absenceof additional information for the real GPR measurements,it is reasonable to assume that these two probabilities areequally probable. Furthermore, as will be seen shortly, we haveconducted a sensitivity evaluation on the two parameters fordifferent values, and the results show that they do not appearto have significant impact to the performance of the proposedmethod.

A. Experiment 1: Evaluation Using GPR Data in [27]

The proposed method had been tested on the setup in Fig. 1.According to the documentation in [27], there were four objectslocated at x = 25, 75, 125, and 175 cm, respectively, and wereburied 5 cm deep in clay mixed with small rocks. The GPR datawere collected after 23 days while the clay was still moist. Thedata set had K = 192 distance samples along the x-axis andN = 512 time samples along the y-axis (see Fig. 2). The truelandmines were located at x = 25 cm and x = 125 cm, and the



Fig. 3. Comparison of the detection results between (upper) the Zoubir’s and (lower) the proposed methods with strip size M = 5 averaged over 100 independenttrials in Experiment 1. Note the polygons in the upper figure groups, those points that are closely spaced with each other based on the convex hull.

Fig. 4. Comparison of the detection results between (upper) the Zoubir’s and (lower) the proposed methods with strip size M = 10 averaged over100 independent trials in Experiment 1. Note the polygons in the upper figure groups, those points that are closely spaced with each other based on theconvex hull.

other two objects were a large stone and a copper strip (seeFig. 1). Note that, due to possible time delays of the returnedsignals, the indicated locations of these four objects observedin the data set were indeed shifted to the right, as shown inFig. 2. For evaluation purposes, we assume that the two genuinelandmines were located at x = 34 cm and x = 138 cm. Table IIsummarizes the information regarding the positions of thesetrue objects.

Under this setup, we examine the performance of joint land-mine detection and localization of the proposed method andcompare it with that of the Zoubir’s approach as described in

Section II-B. Table III summarizes the parameters shared bythe two methods in this evaluation, whereas Tables IV and Vlist the individual parameters used in the proposed and Zoubir’smethods. We will first evaluate and compare the sensitivity ofthe performance of these two methods on a range of strip sizesM = {5, 10, 15, 20, 25} over 100 independent trials.

Figs. 3–7 show a comparison between the detection resultsplotted in contour lines obtained from the Zoubir’s approachand the proposed method for different strip sizes. It is fairlyobvious that, when the strip size is M ≤ 10, the detectionresults from the Zoubir’s approach are very nonconclusive and



Fig. 5. Comparison of the detection results between (upper) the Zoubir’s and (lower) the proposed methods with strip size M = 15 averaged over100 independent trials in Experiment 1.


ambiguous, as there are a number of seemingly independentobjects around the genuine landmine locations. In order toreveal where the possible targets are located, convex hulls [38],[39] for these individual sets of objects are formed (see Figs. 3and 4). On the other hand, the genuine landmines are unam-biguously located by the proposed method even though whenM = 5. The inferior performance by the Zoubir’s approachmay be explained as follows. First, where a small strip size ischosen, the depth of a genuine landmine may be oversubdividedinto many tiny, nonoverlapping strips, and hence, insufficientobservations are included in each strip, thereby limiting theperformance of the state estimation on which the detection

algorithm relies. Second, the target detection algorithm, asdescribed in Section II-B, may fail to construct appropriate andreliable test statistics as a result of insufficient observationsin every strip. Finally, and more importantly, because thisad hoc detection algorithm only gives mutually exclusive detec-tion in every strip on the basis of a hypothesis testing scheme,unless the test statistics representing the presence of a target isstrong and uncorrupted, numerous tiny detected objects will besparsely located around the genuine landmines as if they hadoriginated from different landmines. As a result, the overallpoor localization performance from the Zoubir’s approach isexpected, as inconsistent and inaccurate detection results lead




to poor target tracking performance by the KF even thoughthe model is linear and Gaussian. Note the “gaps” among theadjacent strips around the location of a genuine landmine (seeFig. 3). In the case where M = 10 is employed, this problemalleviates considerably, but the detection performance of theZoubir’s approach is still poor, as seen in Fig. 4.

On the other hand, the proposed method, however, allowsevery strip to have detection probability between zero and one.Accordingly, more continuous and much smoother contours ofthe detected objects can be obtained even though a small stripsize is utilized (see Figs. 3 and 4). Furthermore, as a result ofthis continuity, the spread or width of every detected object bythe proposed method is wider than that by the other approachand increases as strip size increases.

As the strip size increases, for example, M ≥ 15, the de-tection performance for genuine landmines of both methodsis generally improved, as shown in Figs. 5–7, but one caneasily see that spurious objects and noises are introduced in thedetection results obtained from the Zoubir’s approach (see theupper figures of Figs. 5–7). On the contrary, with larger stripsizes, not only does the proposed method obtain much smoothercontours of the detected landmines without introducing signif-icant noise but it also has a very consistent performance forall sizes. Nevertheless, the proposed method does suffer from aperformance degradation when M > 20 due to oversmoothing.

In short, according to the contour plots of the detectedobjects, including genuine landmines and spurious objects, theperformance of the proposed method is clearly superior toZoubir’s approach and less sensitive to different values of M .We now conduct a series of evaluation on both methods in termsof the ROC curves (see Appendix A for details).

Prior to constructing ROC curves for the evaluation on bothmethods, we need to first define the assumed width of a genuinelandmine. In the absence of this piece of information, weconsider that the strongest detection value occurs within ±W

Fig. 8. Comparison of ROC curves obtained from the Zoubir’s method withM = 15 and a range of κ = {0.05, 0.10, 0.15, 0.20, 0.25} in Experiment 1.

scans around the center, where W is set to three in this study.(see Appendix A for details).

A point worth for discussion is the selection of the parame-ters Kτ and κ in Table V used in Zoubir’s method, becausethey critically determine its performance. As discussed in [1],the right parameter values are obtained empirically but varyfrom one scan or medium type to another. In particular, theperformance is very sensitive to the value of κ, which is thesignificance level used for hypothesis testing [1]. Fig. 8 showsa comparison of ROC curves obtained from Zoubir’s methodfor different values of κ at M = 15.

When other things are kept unchanged, the number of spu-rious objects increases with the value of κ. It can be seen that,when κ becomes relatively large, for example, κ > 0.10, theperformance deteriorates in the sense that, given a detection



Fig. 9. Performance comparison between the Zoubir’s and proposed methodswith different strip sizes (M = 5, 10, 15, 20, 25) using 100 independent trialsin Experiment 1. (a) Zoubir’s method. (b) Proposed method.

probability PD, the false alarm probability PF increases asκ increases. This is another area where the proposed methodhas an edge over Zoubir’s algorithm and other methods whoseperformance heavily relies on right parameter values.

In the following, we will examine the sensitivity and consis-tency of both approaches to parameter selection, namely, thestrip size M for both methods, the birth and death probabilitiespb and pd, and the number of particles Ns for the proposedmethod. Figs. 9–11 show the ROC curves of the evaluationresults.

1) Performance Evaluation With Different Strip Sizes: Toevaluate the sensitivity and consistency of the methods to differ-ent values of strip size M , we conduct 100 independent trials onboth methods on a range of strip sizes M = {5, 10, 15, 20, 25}.In particular, Ns = 300 and pb = pd = 0.3 are employed inthe proposed method. According to Fig. 9, it can be seen thatthe ROC performance of the proposed method [see Fig. 9(b)]

Fig. 10. Performance comparison between different values of pb = pd ={0.1, 0.2, 0.3} used in the proposed method with 100 independent trials inExperiment 1.

Fig. 11. Performance comparison between different values of Ns ={100, 200, 500} used in the proposed method with 100 independent trials inExperiment 1.

is far less sensitive and fairly consistent to the entire rangeof strip sizes M in the evaluation when compared with thatof the Zoubir’s approach [see Fig. 9(a)]. These findings areexactly consistent with the explanation given earlier for thecontour plots in Figs. 3–7. Other things being equal, theproposed method clearly outperforms the Zoubir’s approachin terms of ROC curves. This indicates a disadvantage ofZoubir’s approach, namely, that it requires one to seek the rightparameter values in order to perform properly and consistently.Furthermore, it seems that, for both approaches, the strip sizesof 10 and 15 give superior performance to other sizes, and fromthis point onward, the evaluation on the proposed method isconducted with M = 15.

Another point worth for discussion is that, when the stripsize M ≤ 10, the performance of Zoubir’s algorithm is ex-ceptionally poor, as evident in Figs. 3, 4, and 9a. In spite



of being out of the scope of this paper, we present a briefdescription of the steps that may lead to an improvement of theZoubir’s performance for small M . These steps are intended toartificially fill the gaps around the tiny objects that are closelyspaced, as shown in Figs. 3 and 4.

The first step involves a grouping of objects that are closelylocated with each other, and this objective can be readilyachieved by a number of established algorithms, includingconvex hull [38], [39] and K-means clustering [40]. Examplesare shown in Figs. 3 and 4, where the objects are inscribedby polygons constructed by convex hulls. Nonetheless, thesegrouping algorithms may require their own sets of parametersfor proper operations.

The second step is to fill the gaps of all the tiny objectsgrouped within a polygon via the smoothing of their residualenergies with respect to a reference point. We assume that thecenter of a polygon is the reference point, where the peak ofthe residual energy is located, and that the energy value of anobject is gradually decreasing as the object moves away fromthe center. It follows that a smooth 2-D window can then beapplied to the area inscribed by the polygon such that everyobject within the area can have an interpolated energy abovethe decision threshold, alleviating the issue caused by the gaps.Doing so yields an improvement of the ROC performance,where the degree of improvement relies on the spread of thegrouped objects, as well as the shape of the 2-D window. Itis expected that, the narrower is the spread or the faster thewindow rolls off, the smaller are the effects caused by spuriousobjects.

2) Performance Evaluation With Different Values of pb andpd: In this evaluation, we examine the sensitivity of the pro-posed method on a set of prior birth and death probabilities usedin the proposed method, i.e., pb and pd, over 100 independenttrials with M = 15 and Ns = 500. It is assumed that bothprobabilities share common values in each scenario. Three dif-ferent sets of values pb = pd = {0.1, 0.2, 0.3} are consideredin this evaluation. According to the ROC curves in Fig. 10,the proposed method performs nearly the same with differentvalues of pb and pd.

3) Performance Evaluation With Different Values of Ns:Finally, we examine the performance of the proposed methodwhen different number of particles Ns is employed. Threedifferent values are considered: 100, 200, and 500. Other para-meters are chosen as M = 15 and pb = pd = 0.2. The compar-ison results over 100 independent trials are shown in Fig. 11.Once again, the proposed method performs fairly consistentwith different values of Ns, and as expected, the performancein terms of ROC curves is improved, as Ns increases at theexpense of increased computations, but only a small marginalgain of performance is obtained from Ns = 200 to Ns = 500.

B. Experiment 2: Evaluation Using GPR Data in [28]

Here, we present a performance evaluation of the proposedmethod using another set of real GPR data containing a numberof different landmine targets and some clutter objects [28]. Inthe data set, there are a total of 91 channels, and in this eval-uation, we consider three subsets of channels, each of which

TABLE VITRUE POSITIONS OF THE OBJECTS IN SETS 1, 2, AND 3 IN EXPERIMENT 2,

WHERE x POSITIONS ARE THE OBSERVED POSITIONS OF THE TRUE

OBJECTS IN THE DATA SETS

Fig. 12. Comparison between the true object and the localization results fromthe proposed method averaged over 100 independent trials in Experiment 2.(a) Radargram of a target object collected from channel 15. (b) Results fromthe proposed method.

clearly shows the presence of target object(s). Table VI summa-rizes the information regarding these data subsets. Figs. 12–14show the radargrams collected by one of the channels in thesedata subsets. Each data file has K = 91 distance samples andN = 200 time samples. The first 50 rows of data are takenaway as the signals in this block correspond to the responsesto the ground surface, which are inappropriate for localization




purposes. As a result, every data set in this experiment has asize of 150 × 91 measurements.

The proposed method is once again investigated in this ex-periment for localizing possible objects in these data subsets. Atotal of 100 independent trials are conducted on every channelin these subsets. The parameters used in the method are nearlyidentical to those used in the previous computer simulationsin Experiment 1, except when the strip size is M = 10 andthe number of particles is Ns = 500. According to the plotsin Figs. 12–14, it is clearly seen that the proposed methodclearly localizes the objects when compared with the trueradargrams.

To quantitatively evaluate the localization performance onthe real measurements, the ROC curves are constructed forthe results from the proposed method. For comparison pur-poses, the Zoubir’s approach that reuses the parameters inExperiment 1 except M = 10 is also studied with these sets of


measurements. From Fig. 15, the proposed method once againoutperforms the Zoubir’s approach when real measurements areconsidered, and the same reasons (inconsistent and inaccuratedetection results) concluded in Experiment 1 are applied herefor the inferior results.

VI. CONCLUSION

A stochastic online landmine detection method for GPR datausing the SMC approach with RJMCMC has been presented.The proposed method takes the advantage of the RJMCMCto explore different model spaces and to expend the extensivecomputation only on the most possible model space. Onebenefit is that no hard or predetermined thresholds are needed todecide which model should be used, given the data. Computersimulations demonstrate that the proposed approach is able tosuccessfully localize landmine objects from two different setsof real GPR measurements. Furthermore, owing to its relative



Fig. 15. ROC curves obtained from the proposed and Zoubir’s methods onthe data sets over 100 independent trials in Experiment 2. (a) Proposed method.(b) Zoubir’s method.

robustness and insensitivity to parameter values, the proposedmethod outperforms the Zoubir’s method in terms of ROCcurves, according to the evaluation on the two sets of real GPRmeasurements.

APPENDIX ACONSTRUCTION OF ROC CURVES

In this section, the steps on the construction of ROC curvesare presented. To construct an ROC curve, the entire set ofresidual energy obtained from the state vectors for all strips l isneeded, and the true positions of No objects in a radargram mustbe known, for example, kn for n = 1, . . . , No. As an example,Fig. 16 shows the aggregate signals of a radargram containingtwo objects at distance indices k1 and k2 or at locations x(k1)and x(k2). The signals are summed over the depth along the x-axis. Note that the peaks of these two objects are not the same,implying that, in this particular scan, the signals originating

Fig. 16. True aggregate signals of two objects in a scan: The objects arelocated at k1 and k2.

Fig. 17. Illustration of a window Uk with width W = 7 applied to the signalsof the true objects and the division of these signals into the true object regionsXg and the spurious regions Xs.

from the object at distance index k1 or location x(k1) arestronger than those from the other one.

Given any localization method, it is expected that the esti-mated locations of the available objects will deviate from theirtrue positions. Thus, in order to accommodate possible local-ization errors in the performance evaluation, a small windowU(k) with size W centered at the true position of every objectis usually applied when constructing an ROC curve. This typeof window can be described as follows:

U(k) ={

1, |k − k1| ≤W ∀n0, otherwise.

Fig. 17 shows the signals of the true objects with the applicationof U(k) with W = 3. The entire signals have been divided intotwo sets of regions: genuine and spurious objects. Here, we



Fig. 18. Aggregate (normalized) residual energy of the estimated objects withthe window.

Fig. 19. Division of the residual energy signals into the true object regions Xg

and the spurious regions Xs. A range of threshold levels spanning from zero toone is also plotted in dashed lines.

define the genuine object regions by Xg =⋃No

n=1 Xn, whereXn = {k‖k − kn| ≤W}, and, likewise, the spurious regionsby Xs = {k|k /∈ Xg}. After the residual energy of a scan hasbeen estimated by a selected approach, the former regions en-able the calculation of the probability of detection, or PD, whenan ROC is constructed, whereas the latter enable the calculationof the probability of false alarm, or PF . Fig. 18 shows anexample of the localization results from the proposed method.When this window is applied to the estimated residual energy,two different sets of filtered residual energy can be obtained, asshown in Fig. 19, where the upper portion corresponds to theresidual energy residing in Xg and the lower portion would beconsidered spurious signals.

TABLE VIISTEPS TO CONSTRUCT AN ROC

To obtain the detection probability PD of an object, theresidual energy in a region in Xg will be compared with arange of threshold levels, for example, ϕj , where j = 1, . . . , Jand J is the number of levels. The larger the portion of theresidual energy above a given threshold in a particular regionin Xg , the higher is the value of PD at that level. In similarspirit, the probability of false alarm PF can be obtained. Anyresidual energy ηk in (26) for k ∈ {1, . . . , K} that falls inXs contributes to PF for a given threshold level ϕj . In otherwords, the larger the portion of residual energy that is abovea given threshold level in Xs, the higher is the value of PF

at that level. Once all threshold levels have been considered,{PF (j), PD(j)} for all levels, where j = 1, . . . , J , will beavailable, and the corresponding ROC curve can be obtainedas a graph of PD versus PF . However, in practice, the residualenergy ηk is a set of discretized signal points; thus, to computePD and PF , we will follow the steps in Table VII.

As an example, in Fig. 19, there are J = 10 levels repre-sented by the dashed lines, spanning from zero to one. It canbe seen that there are W = 7 signal points in each region in Xg

in the upper portion in Fig. 19. For the object on the left, itssignal values are always larger than the threshold levels untilj = 7. However, the object on the right in the same scan seemsto have weaker signals that fall between the threshold levelsfrom one to six. In other words, it is expected that PD(j) = 1for j = 1, . . . , 6, and the values of PD(j) are dropping fromone as j ≥ 7 and, subsequently, PD(J = 10) = 0.

Likewise, one can determine the values of PF in a similarfashion from the lower graph in Fig. 19. One can see that noneof the (K − 2W ) signal points is above the threshold levelj = 5, indicating that the values of PF (j) = 0 from j ≥ 5.When j < 5, PF gradually increases from zero, and eventually,at level j = 1, all points are above that threshold level, givingrise to PD(j = 1) = 1. In other words, it is intuitive to expectthat, when PF is low (for example, less than 0.1), PD shouldbe high (for example, larger than 0.7), and vice versa. Withthe set of {PF , PD}, the ROC can then be obtained, as shownin Fig. 20.

Fig. 21 shows a family of ROC curves in the same examplein J = 100 levels with a range of W = {3, 5, 7, 9, 11}. It isclearly seen that wider windows yield inferior values of PD for



Fig. 20. ROC curve obtained in this example with W = 3 or 7 distancesamples.

Fig. 21. Family of ROC curves with a range of W in this example.

any given value of PF > 0.1. On the contrary, when PF < 0.1,wider windows yields superior values of PD. Thus, as long asthe genuine target has been included within the window beingconsidered, a smaller value of W is preferred.

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William Ng was born in Hong Kong. He received theB.Eng. degree in electrical engineering from the Uni-versity of Western Ontario, London, ON, Canada,in 1994, the M.Eng. and Ph.D. degrees in electricalengineering from McMaster University, Hamilton,ON, in 1996 and 2004, respectively, and the M.M.Sc.degree in management sciences from the Universityof Waterloo, Waterloo, ON, in 2004.

From 1996 to 1999, he was with Forschungszen-trum Informatik, Karlsruhe, Germany, developing anexpert system using neural networks for nondestruc-

tive pipeline evaluation, and from 1999 to 2002, he was with the PressurePipe Inspection Company Ltd., Missisauga, ON, where he was the Head ofSoftware and IT Department. In 2004, he was with the Signal ProcessingGroup, University of Cambridge, Cambridge, U.K., as a Research Associate.Since 2007, he has been with the Department of Electronic Engineering, CityUniversity of Hong Kong, Kowloon, Hong Kong, as a Research Fellow. Hisresearch interests include financial engineering, statistical signal processing forsensor arrays and multitarget tracking, and multisource information fusion.

Dr. Ng is a Registered Professional Engineer in the province of Ontario andBritish Columbia, Canada.

Thomas C. T. Chan was born in Hong Kong. Hereceived the B.A.Sc. degree in electrical engineeringfrom the University of British Columbia, Vancouver,BC, Canada, in 2005 and the M.Phil. degree from theDepartment of Electronic Engineering, City Univer-sity of Hong Kong, Kowloon, Hong Kong, in 2008.

He is currently with the Quantitative ResearchDepartment, Nomura International (HK) Ltd., HongKong. His research interest includes statistical signalprocessing techniques for efficient land mine detec-tion, and mathematical finance.

H. C. So (S’90–M’96–SM’07) was born in HongKong. He received the B.Eng. degree in electronicengineering from the City University of Hong Kong,Kowloon, Hong Kong, in 1990 and the Ph.D. degreein electronic engineering from The Chinese Univer-sity of Hong Kong, Shatin, Hong Kong, in 1995.

From 1990 to 1991, he was an Electronic En-gineer with the Research and Development Divi-sion, Everex Systems Engineering Ltd., Hong Kong.During 1995–1996, he was a Postdoctoral Fellowwith The Chinese University of Hong Kong. From

1996 to 1999, he was a Research Assistant Professor with the Department ofElectronic Engineering, City University of Hong Kong, where he is currentlyan Associate Professor. His research interests include adaptive filter theory,detection and estimation, wavelet transform, and signal processing for com-munications and multimedia.

K. C. Ho (S’89–M’91–SM’00) was born in HongKong. He received the B.Sc. degree with First ClassHonors in electronics and the Ph.D. degree in elec-tronic engineering from The Chinese University ofHong Kong, Shatin, Hong Kong, in 1988 and 1991,respectively.

From 1991 to 1994, he was a Research Asso-ciate with the Royal Military College of Canada,Kingston, ON, Canada. He was with the Bell-Northern Research, Montreal, QC, Canada, in 1995,as a Member of the Scientific Staff. From September

1996 to August 1997, he was a member of the faculty in the Department ofElectrical Engineering, University of Saskatchewan, Saskatoon, SK, Canada.Since September 1997, he has been with the University of Missouri, Columbia,where he is currently a Professor in the Department of Electrical and ComputerEngineering. His research interests are statistical signal processing, sourcelocalization, subsurface object detection, wavelet transform, wireless communi-cations, and the development of efficient adaptive signal processing algorithmsfor various applications including landmine detection and echo cancellation.He has been active in the development of the ITU Standard RecommendationG.168 since 1995. He is the Editor of the ITU Standard RecommendationsG.168: Digital Network Echo Cancellers and G.160: Voice EnhancementDevices. He is the inventor/coinventor of three United States patents, threeCanadian patents, two patents in Europe, and four patents in Asia on mobilecommunications and signal processing.

Dr. Ho has served as an Associate Editor of the IEEE TRANSACTIONS ON

SIGNAL PROCESSING from 2003 to 2006, and the IEEE SIGNAL PROCESSING

LETTERS from 2004 to 2008. He received the Junior Faculty Research Awardfrom the College of Engineering, University of Missouri, in 2003.


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