Particle Filtering Based Approach for Landmine ... - Semantic Scholar

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

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Particle Filtering Based Approach for Landmine Detection 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 approach for landmine detection based on ground penetrating radar (GPR) signals using sequential Monte Carlo (SMC) methods. The processing applies to the two-dimensional B-scans or radargrams of 3-D GPR data measurements. The proposed state-space model is essentially derived from that of Zoubir et al., which relies on the Kalman filtering approach and a test statistic for landmine detection. In this paper, we propose the use of reversible jump Markov chain Monte Carlo in association with the SMC methods to 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 are more relevant. Computer simulations on real GPR measurements demonstrate the superior performance of the SMC method with our modified model. The proposed algorithm also considerably outperforms the Kalman filtering approach, and it is less sensitive to the common parameters used in both methods, as well as those specific to it. Index Terms—Ground penetrating radar (GPR), Kalman filter (KF), landmine detection, particle filter (PF), reversible jump Markov chain Monte Carlo (RJMCMC), sequential Monte Carlo (SMC).

I. I NTRODUCTION

O

WING to good penetration, depth resolution, and excellent detection of metallic and nonmetallic objects, ground penetrating radar (GPR) has become an emerging technique for landmine detection [1]–[23]. A GPR system consists of a transmitter emitting electromagnetic waves to the ground surface and a receiver collecting returned signal from which the presence of landmines can be indicated. However, the difficulty of using this technique for landmine detection remains, as the signals originating from various types of ground surfaces, like soil or clay, are nearly indistinguishable from those of genuine landmines. Whereas the locations of landmines are usually unknown and positioned arbitrarily, the problem is further Manuscript received January 24, 2008; revised May 23, 2008. Current version published October 30, 2008. This work was supported by a Grant from the 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: willing@ cityu.edu.hk; [email protected]). T. C. T. Chan was with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. He is now with the Quantitative Research Department, Nomura International (HK) Ltd., Hong Kong (e-mail: [email protected]). K. C. Ho is with the Department of Electrical and Computer Engineering, University of Missouri, Columbia, MO 65211 USA (e-mail: hod@ missouri.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2008.2002028

compounded as the signatures of the landmines and various types of backgrounds are highly inconsistent, leading to the development of robust and intelligent approaches for the problem. Typical landmine detection approaches are based on background removal [7]–[14], and the corresponding techniques include adaptive background subtraction [8] and background modeling using a time-varying linear prediction [9] and its improvement [10]. On the other hand, Zoubir et al. [1] have proposed an approach that combines a Kalman filter (KF) for state estimation and a detection method based on a comparison between some threshold and test statistics. Whereas the model in [1] is linear and Gaussian, this approach may suffer from inconsistent localization performance because its detection performance 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 suggested that, prior to properly localizing landmine objects, the ground bounce signals must be estimated and removed using SMC methods, and it is shown that localization performance can be improved by about 5% when compared with other competing 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 number of objects is unknown but it is also varying. To model this randomness of the existence of objects, a stochastic variable is introduced to indicate whether a landmine may be present. To facilitate a robust and efficient method for estimating the presence of objects and hence estimating their locations, we propose to devise a numerical approach using SMC methods, also known as particle filters (PFs) [24]–[26], in association with the reversible jump Markov chain Monte Carlo (RJMCMC) [29] methods. Instead of adopting some data testing methods that provide hard decisions as in [1], the RJMCMC gives soft decisions on where possible landmines are located by exploring all possible model spaces without specifying any threshold in all the particles. Moreover, expensive computations will only be expended on the most likely model spaces rather than blindly to all available spaces. In particular, three moves are proposed for every particle: birth, death, and update. In the birth (death) move, a landmine is proposed to be present (absent), whereas in the update move, the current state will be updated in light of the current observations. Once the target objects have been detected, PF is then employed to track the target signals, by a set of random samples or particles, generated by sequential importance sampling and their associated

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In this section, we reproduce the state-space model and the algorithm for landmine detection proposed in [1]. The statespace model serves as the basis on which our proposed method is developed in the next section. A. State-Space Model

Fig. 1. Example layout of underground objects.

importance weights, which are then propagated through time to give predictions of the target posterior distribution function at future time steps. Note that some preliminary study and results of 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 state update equations, followed by the formulation of our modified and compact models and their prior distribution functions in Section III. In Section IV, we present the development of the SMC method with the RJMCMC for landmine detection application. Computer simulations and evaluations on real GPR data are included in Section V, and conclusions are given in Section VI. Notations: Bold upper case symbols denote matrices, and bold lower case symbols denote vectors. The superscript T denotes the transpose operation, and the symbol “∼” means “distributed as.” The quantity π(·|·) denotes a posterior distribution, whereas qa (·|·) denotes a proposal distribution function of parameter a. The notation (·)1:t indicates all the elements from time 1 to time t. The quantity N (μ, Σ) indicates a real normal distribution with mean μ and covariance matrix Σ. The quantity U(a, b) indicates a uniform distribution over the interval [a, b], and UV indicates a uniform distribution within the volume V . II. P ROBLEM F ORMULATION B ASED ON KF The GPR data, which are downloaded at [27] and [28], are investigated in our study. In particular, the data from [27] are obtained by measuring the response from an impulse GPR with a center frequency of around 1 GHz in the time domain. Each setup contains landmines and other anomalies including large stone, empty cartridge, and/or copper wire strip. A typical setup is shown in Fig. 1. For each value of y, denoted as channel, the operator sweeps the GPR device along the x-axis or scan direction, recording a response every 1 cm in space. In this particular example, a total of 51 channels, separated by 1 cm in space, along with K = 197 measurements per channel, denoted as scans, complete a GPR data set for a setup. For example, the radargram or B-scan of the twenty-fifth channel, denoted by y(n, k), for the setup in Fig. 1 is shown in Fig. 2, where the horizontal and vertical axes correspond to the distance index k = {1, . . . , K} and time index n = {0, . . . , N − 1} samples, respectively, with N = 512. The same descriptions of these terminologies are applied to the data from [28].

Denote our state values at time index n = {1, . . . , N } and distance index k = {1, . . . , K} in a given radargram with size N × K by {α(n, k), β(n, k), γ(n, k)}, where α(n, k) is the background signal, β(n, k) is the target signal, and γ(n, k) is a random bias accounting for the changes in the target signal, if the landmine is present. The physical location of a measurement at distance index k is x(k) on the x-axis. Depending on whether a landmine is present, these state parameters 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) are assumed to be zero-mean white Gaussian random variables 2 2 and σv,1 . with their respective variances σv,0 At any particular time, the observed signal y(n, k) is characterized by α(n, k)+(n, k), target-free y(n, k) = (3) β(n, k)+γ(n, k)+(n, k), target present where (n, k) is assumed to be a zero-mean white Gaussian random variable with variance σ2 . Our objective, in light of observations y(n, k), is to detect where in the radargram a landmine is located by estimating the state vector x(n, k). Typical approaches collect a group of samples and use them jointly in order to enhance the reliability of detection and localization. Denote by M the strip size or successive time samples at a given distance index k and by L = N/M (the largest integer contained in N/M ) the number of nonoverlapping strips. Accordingly, we express the state evolution model in vector form for l ∈ {0, . . . , L − 1} as follows: αl,k = αl,k−1 + v 0,l,k β l,k = β l,k−1 γ l,k = γ l,k−1

(4)

if a landmine is absent, and αl,k = αl,k−1 β l,k = β l,k−1 + γ l,k γ l,k = γ l,k−1 + v 1,l,k


(5)

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

Fig. 2.

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Example radargram of the layout in Fig. 1.

if a landmine is present, where al,k = [a(M l + 1, k), a(M l + and v j,l,k = [vj (M l + 1, k), 2, k), . . . , a(M l + M, k)]T vj (M l + 2, k), . . . , vj (M l + M, k)]T with covariance matrix 2 IM for j = {0, 1} and I M being an M × M identity σv,j matrix. Likewise, the observation model now becomes αl,k + l,k , target-free (6) y l,k = β l,k + γ l,k + l,k , target present where l,k = [(M l + 1, k), . . . , (M l + M, k)]T is again a zero-mean white Gaussian random variable vector, distributed as follows: l,k ∼ N (0, Σl )

(7)

where 0M is an M × M matrix of zeros, and Σl = σ2l I M is known and constant covariance matrix of strip l. In short, to facilitate the localization of possible objects in a radargram, the observations are subdivided into L strips, and each strip contains M × K measurements. B. Zoubir’s Approach [1] The landmine detection based on the Zoubir’s approach is composed of two modules: detection and estimation. Whereas interdependent, these two modules are conducted separately. We first reproduce the schema of the KF update equations for both models and then the target detection module based on hypothesis testing methods [1]. A set of L filters is used to estimate the state values in light of observations. In each filter l = {1, . . . , L}, the state values in either models are assumed to be a Gaussian random variable with mean and covariance matrix which are then propagated from scan k to k + 1. It follows that a test statistic can be constructed from the estimation error or innovation between the updated states and

latest observations and decision can be made on comparing this statistic with a predetermined threshold. Details of these two modules are given as follows. 1) KF: Table I summarizes the KF update equations for both models: target-free and target present. In the case where a target is assumed absent in strip l, we only estimate the state φ0l,k = αl,k in light of observations y l,k with distribution function φ0l,k ∼ N (φ0l,k|k , Φ0l,k|k ), where αl,k|k and Φ0l,k|k are the mean state value and its estimation covariance matrix. On the other hand, if a target is detected in strip l, we need to estimate the target as well as the random bias signals, i.e., T T φ1l,k = [β T l,k , γ l,k ] , while keeping the background signal intact. The estimated state φ1l,k is again assumed to be a Gaussian random variable with mean and covariance matrix, i.e., φ1l,k ∼ N (φ1l,k|k , Φ1l,k|k ). 2) Target Detection Module: Because it is assumed that the observations in all strips are corrupted with white Gaussian noise, the residual between the observation y l,k and its esti l,k in (6), depending on whether a target is detected, mate y can be used as an indication of how well is the explanatory l,k , which power of an assumed model. Let ν l,k = y l,k − y is a zero-mean Gaussian random variable whose distribution j −1 is ν l,k ∼ N (0, Σl ), as in (7), and let el,k = ν T l,k Φl,k|k ν l,k for j = {0, 1}. It is clear that the random variable el,k is chisquared distributed with M degrees of freedom, and hence, one may easily set up a hypothesis testing on the variable el,k as a measure on the appropriateness of an assumed model. That is, for each scan k, we first compute the values of {el,k } for l = {1, . . . , L} and then conduct the χ2 test with an appropriate significance level κ. The hypothesis is that the current model is assumed appropriate with sufficient explanatory power with probability Pr el,k ≥ χ2M,κ = κ


(8)

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TABLE I SCHEMA OF KALMAN FILTERING UPDATE EQUATIONS FOR TARGET-FREE AND TARGET PRESENT SCENARIOS

sections, we will present a numerical based method that jointly detects and localizes the landmines, given a radargram, and that performs more consistently and less sensitively to the variations of parameters than the Zoubir’s approach. III. F ORMULATION OF P ROPOSED S TATE -S PACE M ODEL We now present the state-space model for the proposed method. Denote the state vector by φl,k , containing M successive time samples, as follows: T T φl,k = αT l,k , β l,k

(9)

where αl,k and β l,k are as before, and a random variable by sl,k ∈ {0, 1}, which is an existence variable, which indicates whether 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 ) + B sl,k v l,k

(10)

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

IM 0M

0M . j × IM

(11)

The quantity v l,k = [v(M l + 1, k), . . . , v(M l + M, k)]T is a zero-mean white Gaussian random variable vector with covariance matrix Σlv for strip l, defined as where Pr(·) is the probability of a given event. Otherwise, the hypothesis is rejected, implying that the current model is inappropriate, and other model should be adopted. However, any decision solely based on the hypothesis testing on a strip at one scan is not only premature but also unreliable, particularly when M is not sufficiently large. As suggested in [1], not one but a few hypothesis testings on the same strip l over a window of consecutive scans Kτ should be based. As a result, if a preset number of hypothesis out of Kτ scans is accepted, then the current model is considered appropriate. Nevertheless, unless carefully selected, the size of Kτ certainly affects the performance of the detection. Alternatively, one can plot the contour lines of these residual energies el,k for l = {0, . . . , L − 1} and k = {1, . . . , K} to identify the possible locations of landmines. To summarize, one may realize that this approach has a few potential drawbacks. First, whereas the detection and localization are considered as a joint problem, they are solved in completely separate modules. Second, the overall performance of the algorithm heavily relies on the outcomes of the hypothesis testings, which, in turn, critically depend on a careful selection of the strip size M , the window size Kτ , and the significance level κ. Finally, because the detection results obtained from the hypothesis testings on every strip are mutually exclusive, i.e., the probability of detection being either zero or one, there is a discontinuity of detection or “gaps” among adjacent strips of a genuine target, particularly when the observation noise is large or the strip size is small (see Section V for a vigorous computer evaluation of the Zoubir’s approach). In the next

Σlv =

2 σv,0,l IM 0M

0M . 2 σv,1,l IM

(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 stochastic relationship sl,k = sl,k−1 + s [31], where s is a discrete independently and identically distributed random variable such that the prior distribution function of sl,k is as follows: ⎧ ⎨ Pr(s = −1) = pd p(sl,k |sl,k−1 ) = Pr(s = 0) = 1 − pb − pd ⎩ Pr(s = 1) = pb

(14)

where pb and pd ∈ {0, 1} are the probabilities of incrementing and decrementing the number of targets, respectively, such that pb = 0 if sl,k−1 = 1 and pd = 0 if sl,k−1 = 0. From this point onward, our parameters of interest are denoted by θ l,k = {φl,k , sl,k } for strip l, whose prior distribution function can then be expressed as follows: p(θ l,k |θ l,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 prior distribution functions of φl,k and sl,k in (13) and (14), respectively. There are two major differences between the proposed model and the one in (6). First, the bias term γ l,k is removed, because it is believed that the quantity is redundant and its contribution to the entire state vector can be absorbed into the target signal β l,k . Second, in reality, the background signal αl,k changes in every distance index k, regardless whether a target exists or not. 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 according to the selection of the matrix B sk,l in (11). Collectively, the proposed model, in theory, describes the situation more realistically and reduces the computational load, as well as enhances the state estimation performance in terms of error variance. It is further assumed that the states at different strips are statistically independent, as considered in [1], such that we may express the joint prior distribution function of θ k as p(θ k |θ k−1 ) =

L−1

p(θ l,k |θ l,k−1 )

(16)

l=0

where θ k = [θ 0,k , . . . , θ L−1,k ]T and p(θ l,k |θ l,k−1 ) is the prior distribution function in (15). Likewise, we extend the observation model of y l,k in (6) as follows: y l,k = g(θ l,k ) + l,k

(17)

where the function g(·) may be linear or nonlinear, and l,k is identical to that in (6). Accordingly, the likelihood of the observation y l,k due to θ l,k can be written as (18) p(y l,k |θ l,k ) = N y l,k |g(θ l,k ), Σl where the observations from a given strip l have its own covariance matrix Σl ∀l. Denoting y k = [y 0,k , . . . , y L−1,k ]T and θ k = [θ 0,k , . . . , θ L−1,k ]T , we would like to estimate θ l,k = {φl,k , sl,k } ∀l sequentially upon the receipt of y k . IV. SMC M ETHODS In the context of online parameter estimation, we are interested in the posterior distribution π(θ k |y 1:k ) with θ k , which can be recursively obtained from two steps according to the Bayesian sequential estimation framework described by the following two equations: π(θ k |y 1:k−1 ) = p(θ k |θ k−1 )π(θ k−1 |y 1:k−1 ) dθ k−1 (19) π(θ k |y 1:k ) ∝ p(y k |θ k )π(θ k |y 1:k−1 ).

In very limited scenarios, the models of interest are “weakly” nonlinear and Gaussian in which one may utilize the KF and its derivatives, including the extended KF, to obtain an approximately optimal solution. It is well known that the update expression in (20) is analytically intractable for most models of interest. We therefore turn to SMC methods [24]–[26], [32]–[34], also known as PFs, to provide an efficient numerical approximation strategy for recursive estimation of complex models. These methods have gained popularity in recent years, due to their simplicity, flexibility, ease of implementation, and modeling success over a wide range of challenging applications. A. Sequential Importance Sampling Because it is assumed that the states from different strips are statistically independent, we separately rather than jointly compute the particles and their associated weights for every individual strip. In other words, we will have L independent PFs, each of which estimates the posterior distribution function of θ l,k of a particular strip l. The basic idea behind PFs is very simple: The target distributions are represented by a weighted set of Monte Carlo samples. These samples are propagated and updated using a sequential version of importance sampling, as new measurements become available. Hence, statistical inferences, such as expectation, maximum a posteriori estimates, minimum mean square error (MMSE), etc., can be computed from these samples. (i) s From a large set of Ns particles {θ l,k−1 }N i=1 with their as(i)

s sociated importance weights {wl,k−1 }N i=1 , we approximate the posterior distribution function π(θ l,k−1 |y l,1:k−1 ) as follows:

π(θ l,k−1 |y l,1:k−1 ) ≈

Ns

(i) (i) wl,k−1 δ θ l,k−1 − θ l,k−1

(21)

i=1

where δ(·) is the Dirac delta function. We would like to generate (i) s a set of new particles {θ l,k }N i=1 from an appropriately selected proposal function, i.e., (i) (i) i = {1, . . . , Ns }. θ l,k ∼ q θ l,k |θ l,k−1 , y l,1:k , (22) Among many practical PFs, we choose to use the boot(i) (i) strap PF [24] and assign q(θ l,k |θ l,k−1 , y l,1:k ) = p(θ l,k |θ l,k−1 ), as in (15). (i) With the set of state particles {θ l,k } obtained from (22), the (i)

importance weights wl,k are recursively updated as follows: (i) (i) (i) p y l,k |θ l,k p θ l,k |θ l,k−1 (i) (i) wl,k ∝ wl,k−1 × (23) (i) (i) q θ l,k |θ l,k−1 , y l,1:k

(20)

The term π(θ k−1 |y 1:k−1 ) in (19) is the posterior distribution function at k − 1, and the term p(y k |θ k ) in (20) refers to the likelihood function of the observations y k from all L strips. The recursion is initialized with some distribution, for example, p(θ 0 ).

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Ns

(i) i=1 wl,k = 1. It follows that the (i) Ns {θ l,k }i=1 with the associated importance

with

new set of particles (i)

s weights {wl,k }N i=1 is then approximately distributed according to π(θ l,k |y l,1:k ). (i) Accordingly, once the set of particles θ l,k and their associ(i) ated weights wl,k for i ∈ {1, . . . , Ns } have been computed, we


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obtain target state estimation according to the MMSE estimation, given by ˆ l,k = E[θ k |y l,1:k ] ≈ θ

Ns

(i) (i)

wl,k θ l,k

(24)

i=1

In our application for a given strip l at every k and particle, we randomly generate a new candidate θ l,k = {φl,k , sl,k } from a distribution function d(·|·), which may be conditional on (i) (i) (i) θ l,k = {φl,k , sl,k }. The candidate will be accepted with probability ξ = min{1, r}, where r is the acceptance ratio which is computed as

where E denotes the expectation operator for l ∈ {0, . . . , L − 1}. Likewise, we also compute the residual energies for all strips that indicate whether possible landmines are present by comparing the observation y l,k with its estimate solely based on the background signal as follows: η l,k =

Ns

(i) (i) wl,k × VAR y l,k − g φl,k , 0

∀l

(25)

(i)

where g(φl,k , 0) is an estimate of an observation y l,k based solely on the background signals, and VAR[·] is the variance operator. That is, if the observation y l,k contains a background signal only, the residual energy η l,k is expected to contain small values. On the other hand, if the observation y l,k contains a target signal, using the background signal alone to estimate y l,k is going to yield much larger value in η l,k . We also denote the aggregate residual energy signal by η k as follows: L

η l,k

(27)

(i)

i=1

ηk =

(i) π (θ l.k |y) d θ l,k |θ l,k ×J r= (i) (i) π θ l,k |y d θ l,k |θ l,k

(26)

l=1

where the residual energy signals are summed over all strips l along the x-axis. This quantity will be used later when receiver operating characteristic (ROC) curves are prepared for performance evaluation in Section V. As the PF operates through time, only a few particles contribute significant importance weights in (23), leading to the well-known problem of degeneracy [26], [33]. To avoid this, one needs to resample the particles according to their importance weights. That is, those particles with more significant weights will be selected more frequently than those with less significant weights. More detailed discussion of degeneracy and resampling can be found in [26]. B. RJMCMC Prior to properly estimating the locations of landmines, it is necessary to tell whether they are present in a given region in the radargram. Existing methods such as [1] deterministically decide whether a landmine is present in a given area, solely based on a comparison between predefined thresholds with some forms of test statistics computed from the data. On the contrary, the proposed RJMCMC strategy explores all possible model spaces and softly determines the presence of a landmine with a probability in the same area. The RJMCMC process is a variation of Metropolis–Hastings (MH) algorithm [35], [36], which inherently sets up a Markov chain whose invariant distribution corresponds to the posterior of interest. The states sampled according to the MH algorithm at successive iterations represent samples from the distribution of interest.

with J is the Jacobian of the transformation from θ l,k to θ l,k . In effect, the proposed RJMCMC method jumps between parameter subspaces, thus visiting all relevant models. In every iteration, a candidate with a particular model is proposed from a set of proposal distribution functions, which will be randomly accepted according to an acceptance ratio that ensures reversibility and, therefore, invariance of the Markov chain with respect to the desired posterior distribution. In particular, three different moves are randomly selected to enable the exploration of the parameter subspace: 1) birth move, chosen with probability pb , for which the presence of a landmine is proposed, i.e., sl,k = 1; 2) death move, chosen with probability pd , for which the absence of a landmine is proposed, i.e., sl,k = 0; 3) update move, chosen with probability 1 − pb − pd , for which the state parameters are updated given sl,k = sl,k−1 . It can be shown in [37] that the proposed MCMC sampling procedure does not require a burn-in period in this application because the particles before the MCMC step are already distributed according to the limiting distribution of the chain. In other words, only one MCMC iteration is needed for each particle at each time. In short, in the proposed method, all Ns particles assume different models according to the RJMCMC at a given scan k; a histogram of the available models can be constructed (i) from {sl,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, it is 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 candidate particle {φl,k , sl,k = 1} is generated according to (15) and is accepted with probability ξbirth = min{1, rbirth }, where rbirth is computed as rbirth

1 T −1 T −1 T e Σ e − e Σ e = exp − 2


(28)


with e = y l,k − g(φl,k , sl,k ) and e = y l,k − g(φl,k , sl,k ), and φl,k ∼ p(φl,k |φl,k−1 , sl,k = 0, sl,k−1 = 0). When the death move is selected, similar procedures in the birth move are taken in which a candidate particle φl,k with sl,k = 0 and another particle φl,k with sl,k = 1 are generated. The candidate particle will be accepted with probability ξdeath as follows: 1 ξdeath = min 1, . (29) rbirth (i)

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TABLE II TRUE 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

(i)

(i)

(i)

TABLE III COMMON PARAMETERS FOR COMPUTER SIMULATIONS IN EXPERIMENT 1

(i)

The particle θ l,k = {φl,k , sl,k } becomes φl,k , sl,k , if accepted (i) (i) φl,k , sl,k = φl,k , sl,k , otherwise

(30) (i)

and will then be used to update the importance weights wl,k . 2) Update Move: If this move is selected, we simply gener(i) (i) (i) (i) ate the particle φl,k ∼ q(φl,k |φl,k−1 , y l,1:k ) with sl,k = sl,k−1 ,

TABLE IV PARAMETERS USED IN THE PROPOSED METHOD FOR COMPUTER SIMULATIONS IN EXPERIMENT 1

(i)

which will then used to compute the importance weights wl,k . C. Adaptivity of Noise Covariance Matrices As we do not have any exact knowledge of the state and 2 observation noise variances, i.e., σv,j,l for j = {0, 1} and σ2l , we also need to update their values in light of new observations. Here, we propose to update these variances for each strip l as follows: s 2 (1 − λ) (i) Δφl,k M Ns i=1

N

2 2 (k) = λσv,0,l (k − 1) + σv,0,l

2 2 σv,1,l (k) =λσv,1,l (k − 1) +

Ns 2 (1 − λ) (i) Δφl,k M Ns i=1

s (1 − λ) (i) 2 Δy l,k M Ns i=1

(31)

(32)

N

σ2l (k) = λσ2l (k − 1) + (i)

(i)

(i)

(i)

(33) (i)

where Δφl,k = φl,k − f (φl,k−1 ), Δy l,k = y l,k − g(θ l,k ), and 0 < λ < 1 is a forgetting factor. Note that it is assumed that (i) the state particles {θ l,k } have been resampled with uniform weights. V. S IMULATION R ESULTS In this section, a vigorous evaluation on the performance of the proposed algorithm on two different sets of real GPR measurements is given. They are obtained from [27] and [28]. Furthermore, the results from the proposed method will be compared with those from the Zoubir’s method in [1]. Here, we specify the form of the dynamical function f (·) in (10) and observation function g(·) in (18). In particular, we choose to use the same linear model as in [1] in order to carry out a fair performance comparison. Accordingly, the dynamical function is f (φl,k−1 ) = I 2M φl,k−1 , and the observation function becomes g j (θ l,k ) = H sl,k φl,k , where H j is given by H j = [I M

j × I M ],

j = {0, 1}.

TABLE V PARAMETERS USED IN THE ZOUBIR ’ S APPROACH FOR COMPUTER SIMULATIONS IN EXPERIMENT 1

Note that, for other forms of functions, for example, nonlinearity, chosen for f (·) and g(·) will not drastically and adversely affect the performance of the proposed SMC algorithms, as they are developed to tackle problems that are nonlinear and nonGaussian [24]–[26], [32]–[34]. For the RJMCMC, in the absence of additional information about the locations of genuine target objects, the probabilities of death and birth are set identical, i.e., pd = pb , in all the experiments. It is worthy to note that our selection of pb and pd is essentially arbitrary, unless more prior information is available to guide their assignment. Therefore, in the absence of additional information for the real GPR measurements, it is reasonable to assume that these two probabilities are equally probable. Furthermore, as will be seen shortly, we have conducted a sensitivity evaluation on the two parameters for different values, and the results show that they do not appear to have significant impact to the performance of the proposed method. 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 objects located at x = 25, 75, 125, and 175 cm, respectively, and were buried 5 cm deep in clay mixed with small rocks. The GPR data were collected after 23 days while the clay was still moist. The data set had K = 192 distance samples along the x-axis and N = 512 time samples along the y-axis (see Fig. 2). The true landmines were located at x = 25 cm and x = 125 cm, and the


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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 independent trials 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 over 100 independent trials 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.

other two objects were a large stone and a copper strip (see Fig. 1). Note that, due to possible time delays of the returned signals, the indicated locations of these four objects observed in the data set were indeed shifted to the right, as shown in Fig. 2. For evaluation purposes, we assume that the two genuine landmines were located at x = 34 cm and x = 138 cm. Table II summarizes the information regarding the positions of these true objects. Under this setup, we examine the performance of joint landmine detection and localization of the proposed method and compare it with that of the Zoubir’s approach as described in

Section II-B. Table III summarizes the parameters shared by the two methods in this evaluation, whereas Tables IV and V list the individual parameters used in the proposed and Zoubir’s methods. We will first evaluate and compare the sensitivity of the performance of these two methods on a range of strip sizes M = {5, 10, 15, 20, 25} over 100 independent trials. Figs. 3–7 show a comparison between the detection results plotted in contour lines obtained from the Zoubir’s approach and the proposed method for different strip sizes. It is fairly obvious that, when the strip size is M ≤ 10, the detection results from the Zoubir’s approach are very nonconclusive and



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Fig. 5. Comparison of the detection results between (upper) the Zoubir’s and (lower) the proposed methods with strip size M = 15 averaged over 100 independent trials in Experiment 1.


ambiguous, as there are a number of seemingly independent objects around the genuine landmine locations. In order to reveal where the possible targets are located, convex hulls [38], [39] for these individual sets of objects are formed (see Figs. 3 and 4). On the other hand, the genuine landmines are unambiguously located by the proposed method even though when M = 5. The inferior performance by the Zoubir’s approach may be explained as follows. First, where a small strip size is chosen, the depth of a genuine landmine may be oversubdivided into many tiny, nonoverlapping strips, and hence, insufficient observations are included in each strip, thereby limiting the performance of the state estimation on which the detection

algorithm relies. Second, the target detection algorithm, as described in Section II-B, may fail to construct appropriate and reliable test statistics as a result of insufficient observations in every strip. Finally, and more importantly, because this ad hoc detection algorithm only gives mutually exclusive detection in every strip on the basis of a hypothesis testing scheme, unless the test statistics representing the presence of a target is strong and uncorrupted, numerous tiny detected objects will be sparsely located around the genuine landmines as if they had originated from different landmines. As a result, the overall poor localization performance from the Zoubir’s approach is expected, as inconsistent and inaccurate detection results lead


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to poor target tracking performance by the KF even though the model is linear and Gaussian. Note the “gaps” among the adjacent strips around the location of a genuine landmine (see Fig. 3). In the case where M = 10 is employed, this problem alleviates considerably, but the detection performance of the Zoubir’s approach is still poor, as seen in Fig. 4. On the other hand, the proposed method, however, allows every strip to have detection probability between zero and one. Accordingly, more continuous and much smoother contours of the detected objects can be obtained even though a small strip size is utilized (see Figs. 3 and 4). Furthermore, as a result of this continuity, the spread or width of every detected object by the proposed method is wider than that by the other approach and increases as strip size increases. As the strip size increases, for example, M ≥ 15, the detection performance for genuine landmines of both methods is generally improved, as shown in Figs. 5–7, but one can easily see that spurious objects and noises are introduced in the detection results obtained from the Zoubir’s approach (see the upper figures of Figs. 5–7). On the contrary, with larger strip sizes, not only does the proposed method obtain much smoother contours of the detected landmines without introducing significant noise but it also has a very consistent performance for all sizes. Nevertheless, the proposed method does suffer from a performance degradation when M > 20 due to oversmoothing. In short, according to the contour plots of the detected objects, including genuine landmines and spurious objects, the performance of the proposed method is clearly superior to Zoubir’s approach and less sensitive to different values of M . We now conduct a series of evaluation on both methods in terms of the ROC curves (see Appendix A for details). Prior to constructing ROC curves for the evaluation on both methods, we need to first define the assumed width of a genuine landmine. In the absence of this piece of information, we consider that the strongest detection value occurs within ±W

Fig. 8. Comparison of ROC curves obtained from the Zoubir’s method with M = 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 parameters Kτ and κ in Table V used in Zoubir’s method, because they critically determine its performance. As discussed in [1], the right parameter values are obtained empirically but vary from one scan or medium type to another. In particular, the performance is very sensitive to the value of κ, which is the significance level used for hypothesis testing [1]. Fig. 8 shows a comparison of ROC curves obtained from Zoubir’s method for different values of κ at M = 15. When other things are kept unchanged, the number of spurious objects increases with the value of κ. It can be seen that, when κ becomes relatively large, for example, κ > 0.10, the performance deteriorates in the sense that, given a detection



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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 in Experiment 1.

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

probability PD , the false alarm probability PF increases as κ increases. This is another area where the proposed method has an edge over Zoubir’s algorithm and other methods whose performance heavily relies on right parameter values. In the following, we will examine the sensitivity and consistency of both approaches to parameter selection, namely, the strip size M for both methods, the birth and death probabilities pb and pd , and the number of particles Ns for the proposed method. Figs. 9–11 show the ROC curves of the evaluation results. 1) Performance Evaluation With Different Strip Sizes: To evaluate the sensitivity and consistency of the methods to different values of strip size M , we conduct 100 independent trials on both methods on a range of strip sizes M = {5, 10, 15, 20, 25}. In particular, Ns = 300 and pb = pd = 0.3 are employed in the proposed method. According to Fig. 9, it can be seen that the ROC performance of the proposed method [see Fig. 9(b)]

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

is far less sensitive and fairly consistent to the entire range of strip sizes M in the evaluation when compared with that of the Zoubir’s approach [see Fig. 9(a)]. These findings are exactly consistent with the explanation given earlier for the contour plots in Figs. 3–7. Other things being equal, the proposed method clearly outperforms the Zoubir’s approach in terms of ROC curves. This indicates a disadvantage of Zoubir’s approach, namely, that it requires one to seek the right parameter values in order to perform properly and consistently. Furthermore, it seems that, for both approaches, the strip sizes of 10 and 15 give superior performance to other sizes, and from this point onward, the evaluation on the proposed method is conducted with M = 15. Another point worth for discussion is that, when the strip size M ≤ 10, the performance of Zoubir’s algorithm is exceptionally poor, as evident in Figs. 3, 4, and 9a. In spite


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of being out of the scope of this paper, we present a brief description of the steps that may lead to an improvement of the Zoubir’s performance for small M . These steps are intended to artificially fill the gaps around the tiny objects that are closely spaced, as shown in Figs. 3 and 4. The first step involves a grouping of objects that are closely located with each other, and this objective can be readily achieved by a number of established algorithms, including convex hull [38], [39] and K-means clustering [40]. Examples are shown in Figs. 3 and 4, where the objects are inscribed by polygons constructed by convex hulls. Nonetheless, these grouping algorithms may require their own sets of parameters for proper operations. The second step is to fill the gaps of all the tiny objects grouped within a polygon via the smoothing of their residual energies with respect to a reference point. We assume that the center of a polygon is the reference point, where the peak of the residual energy is located, and that the energy value of an object is gradually decreasing as the object moves away from the center. It follows that a smooth 2-D window can then be applied to the area inscribed by the polygon such that every object within the area can have an interpolated energy above the 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 the grouped objects, as well as the shape of the 2-D window. It is expected that, the narrower is the spread or the faster the window rolls off, the smaller are the effects caused by spurious objects. 2) Performance Evaluation With Different Values of pb and pd : In this evaluation, we examine the sensitivity of the proposed method on a set of prior birth and death probabilities used in the proposed method, i.e., pb and pd , over 100 independent trials with M = 15 and Ns = 500. It is assumed that both probabilities share common values in each scenario. Three different sets of values pb = pd = {0.1, 0.2, 0.3} are considered in this evaluation. According to the ROC curves in Fig. 10, the proposed method performs nearly the same with different values of pb and pd . 3) Performance Evaluation With Different Values of Ns : Finally, we examine the performance of the proposed method when different number of particles Ns is employed. Three different values are considered: 100, 200, and 500. Other parameters are chosen as M = 15 and pb = pd = 0.2. The comparison results over 100 independent trials are shown in Fig. 11. Once again, the proposed method performs fairly consistent with different values of Ns , and as expected, the performance in terms of ROC curves is improved, as Ns increases at the expense of increased computations, but only a small marginal gain 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 proposed method using another set of real GPR data containing a number of different landmine targets and some clutter objects [28]. In the data set, there are a total of 91 channels, and in this evaluation, we consider three subsets of channels, each of which

TABLE VI TRUE 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 from the proposed method averaged over 100 independent trials in Experiment 2. (a) Radargram of a target object collected from channel 15. (b) Results from the proposed method.

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




purposes. As a result, every data set in this experiment has a size of 150 × 91 measurements. The proposed method is once again investigated in this experiment for localizing possible objects in these data subsets. A total of 100 independent trials are conducted on every channel in these subsets. The parameters used in the method are nearly identical to those used in the previous computer simulations in Experiment 1, except when the strip size is M = 10 and the number of particles is Ns = 500. According to the plots in Figs. 12–14, it is clearly seen that the proposed method clearly localizes the objects when compared with the true radargrams. To quantitatively evaluate the localization performance on the real measurements, the ROC curves are constructed for the results from the proposed method. For comparison purposes, the Zoubir’s approach that reuses the parameters in Experiment 1 except M = 10 is also studied with these sets of

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measurements. From Fig. 15, the proposed method once again outperforms the Zoubir’s approach when real measurements are considered, and the same reasons (inconsistent and inaccurate detection results) concluded in Experiment 1 are applied here for the inferior results. VI. C ONCLUSION A stochastic online landmine detection method for GPR data using the SMC approach with RJMCMC has been presented. The proposed method takes the advantage of the RJMCMC to explore different model spaces and to expend the extensive computation only on the most possible model space. One benefit is that no hard or predetermined thresholds are needed to decide which model should be used, given the data. Computer simulations demonstrate that the proposed approach is able to successfully localize landmine objects from two different sets of real GPR measurements. Furthermore, owing to its relative


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Fig. 16. True aggregate signals of two objects in a scan: The objects are ˜1 and k ˜2 . located at k

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

robustness and insensitivity to parameter values, the proposed method outperforms the Zoubir’s method in terms of ROC curves, according to the evaluation on the two sets of real GPR measurements. A PPENDIX A C ONSTRUCTION OF ROC C URVES In this section, the steps on the construction of ROC curves are presented. To construct an ROC curve, the entire set of residual energy obtained from the state vectors for all strips l is needed, and the true positions of No objects in a radargram must be known, for example, k˜n for n = 1, . . . , No . As an example, Fig. 16 shows the aggregate signals of a radargram containing two objects at distance indices k˜1 and k˜2 or at locations x(k˜1 ) and x(k˜2 ). The signals are summed over the depth along the xaxis. Note that the peaks of these two objects are not the same, implying that, in this particular scan, the signals originating

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

from the object at distance index k˜1 or location x(k˜1 ) are stronger than those from the other one. Given any localization method, it is expected that the estimated locations of the available objects will deviate from their true positions. Thus, in order to accommodate possible localization errors in the performance evaluation, a small window U (k) with size W centered at the true position of every object is usually applied when constructing an ROC curve. This type of window can be described as follows: ˜ U (k) = 1, |k − k1 | ≤ W ∀n 0, otherwise. Fig. 17 shows the signals of the true objects with the application of U (k) with W = 3. The entire signals have been divided into two sets of regions: genuine and spurious objects. Here, we



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TABLE VII STEPS TO CONSTRUCT AN ROC

Fig. 18. Aggregate (normalized) residual energy of the estimated objects with the 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 to one is also plotted in dashed lines.

o define the genuine object regions by Xg = N n=1 Xn , where Xn = {k k − k˜n | ≤ W }, and, likewise, the spurious regions / Xg }. After the residual energy of a scan has by Xs = {k|k ∈ been estimated by a selected approach, the former regions enable the calculation of the probability of detection, or PD , when an ROC is constructed, whereas the latter enable the calculation of the probability of false alarm, or PF . Fig. 18 shows an example 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, as shown in Fig. 19, where the upper portion corresponds to the residual energy residing in Xg and the lower portion would be considered spurious signals.

To obtain the detection probability PD of an object, the residual energy in a region in Xg will be compared with a range of threshold levels, for example, ϕj , where j = 1, . . . , J and J is the number of levels. The larger the portion of the residual energy above a given threshold in a particular region in Xg , the higher is the value of PD at that level. In similar spirit, the probability of false alarm PF can be obtained. Any residual energy η k in (26) for k ∈ {1, . . . , K} that falls in Xs contributes to PF for a given threshold level ϕj . In other words, the larger the portion of residual energy that is above a 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 be available, and the corresponding ROC curve can be obtained as a graph of PD versus PF . However, in practice, the residual energy η k is a set of discretized signal points; thus, to compute PD and PF , we will follow the steps in Table VII. As an example, in Fig. 19, there are J = 10 levels represented by the dashed lines, spanning from zero to one. It can be 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, its signal values are always larger than the threshold levels until j = 7. However, the object on the right in the same scan seems to have weaker signals that fall between the threshold levels from one to six. In other words, it is expected that PD (j) = 1 for j = 1, . . . , 6, and the values of PD (j) are dropping from one as j ≥ 7 and, subsequently, PD (J = 10) = 0. Likewise, one can determine the values of PF in a similar fashion from the lower graph in Fig. 19. One can see that none of the (K − 2W ) signal points is above the threshold level j = 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, giving rise to PD (j = 1) = 1. In other words, it is intuitive to expect that, when PF is low (for example, less than 0.1), PD should be high (for example, larger than 0.7), and vice versa. With the set of {PF , PD }, the ROC can then be obtained, as shown in Fig. 20. Fig. 21 shows a family of ROC curves in the same example in J = 100 levels with a range of W = {3, 5, 7, 9, 11}. It is clearly seen that wider windows yield inferior values of PD for


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

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

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[31] J. R. Larocque, J. P. Reilly, and W. Ng, “Particle filters for tracking an unknown number of sources,” IEEE Trans. Signal Process., vol. 50, no. 12, pp. 2926–2937, Dec. 2002. [32] O. Cappe, S. Godsill, and E. Moulines, “An overview of existing methods and recent advances in sequential Monte Carlo,” Proc. IEEE, vol. 95, no. 5, pp. 899–924, May 2007. [33] A. Doucet, N. de Freitas, and N. Gordon, Eds., Sequential Monte Carlo in Practice. New York: Springer-Verlag, 2001. [34] J. Liu and R. Chen, “Sequential Monte Carlo methods for dynamic systems,” J. Amer. Stat. Assoc., vol. 93, no. 443, pp. 1032–1044, 1993. [35] J. Bernardo and A. Smith, Bayesian Theory. New York: Wiley, 1994. [36] W. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika, vol. 57, no. 1, pp. 97–109, 1970. [37] A. Doucet, N. Gordon, and V. Krishnamurthy, “Particle filters for state estimation of jump Markov linear systems,” IEEE Trans. Signal Process., vol. 49, no. 3, pp. 613–624, Mar. 2001. [38] F. P. Preparata and S. J. Hong, “Convex hulls of finite sets of points in two and three dimensions,” Commun. ACM, vol. 20, no. 2, pp. 87–93, 1977. [39] H. Masum, “Review of computational geometry: Algorithms and applications (2nd ed.) by M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf,” SIGACT News, vol. 31, no. 4, pp. 10–12, 2000. [40] A. Rekik, M. Zribi, M. Benjelloun, and A. ben Hamida, “A k-means clustering algorithm initialization for unsupervised statistical satellite image segmentation,” in Proc. IEEE Int. Conf. E-Learning Ind. Electron., Hammamet, Tunisia, Dec. 18–20, 2006, pp. 11–16.

William Ng was born in Hong Kong. He received the B.Eng. degree in electrical engineering from the University of Western Ontario, London, ON, Canada, in 1994, the M.Eng. and Ph.D. degrees in electrical engineering from McMaster University, Hamilton, ON, in 1996 and 2004, respectively, and the M.M.Sc. degree in management sciences from the University of Waterloo, Waterloo, ON, in 2004. From 1996 to 1999, he was with Forschungszentrum Informatik, Karlsruhe, Germany, developing an expert system using neural networks for nondestructive pipeline evaluation, and from 1999 to 2002, he was with the Pressure Pipe Inspection Company Ltd., Missisauga, ON, where he was the Head of Software and IT Department. In 2004, he was with the Signal Processing Group, University of Cambridge, Cambridge, U.K., as a Research Associate. Since 2007, he has been with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, as a Research Fellow. His research interests include financial engineering, statistical signal processing for sensor arrays and multitarget tracking, and multisource information fusion. Dr. Ng is a Registered Professional Engineer in the province of Ontario and British Columbia, Canada.

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H. C. So (S’90–M’96–SM’07) was born in Hong Kong. He received the B.Eng. degree in electronic engineering from the City University of Hong Kong, Kowloon, Hong Kong, in 1990 and the Ph.D. degree in electronic engineering from The Chinese University of Hong Kong, Shatin, Hong Kong, in 1995. From 1990 to 1991, he was an Electronic Engineer with the Research and Development Division, Everex Systems Engineering Ltd., Hong Kong. During 1995–1996, he was a Postdoctoral Fellow with The Chinese University of Hong Kong. From 1996 to 1999, he was a Research Assistant Professor with the Department of Electronic Engineering, City University of Hong Kong, where he is currently an Associate Professor. His research interests include adaptive filter theory, detection and estimation, wavelet transform, and signal processing for communications and multimedia.

K. C. Ho (S’89–M’91–SM’00) was born in Hong Kong. He received the B.Sc. degree with First Class Honors in electronics and the Ph.D. degree in electronic engineering from The Chinese University of Hong Kong, Shatin, Hong Kong, in 1988 and 1991, respectively. From 1991 to 1994, he was a Research Associate with the Royal Military College of Canada, Kingston, ON, Canada. He was with the BellNorthern 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 of Electrical 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 Computer Engineering. His research interests are statistical signal processing, source localization, subsurface object detection, wavelet transform, wireless communications, and the development of efficient adaptive signal processing algorithms for various applications including landmine detection and echo cancellation. He has been active in the development of the ITU Standard Recommendation G.168 since 1995. He is the Editor of the ITU Standard Recommendations G.168: Digital Network Echo Cancellers and G.160: Voice Enhancement Devices. He is the inventor/coinventor of three United States patents, three Canadian patents, two patents in Europe, and four patents in Asia on mobile communications 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 Award from the College of Engineering, University of Missouri, in 2003.

Thomas C. T. Chan was born in Hong Kong. He received the B.A.Sc. degree in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 2005 and the M.Phil. degree from the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, in 2008. He is currently with the Quantitative Research Department, Nomura International (HK) Ltd., Hong Kong. His research interest includes statistical signal processing techniques for efficient land mine detection, and mathematical finance.


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