Radio Science

Classification of closely spaced subsurface objects using electromagnetic induction data and blind source separation algorithms

Authors


Abstract

[1] Most research in the subsurface object identification area assumes that objects are well isolated from each other and thus that a single signature is measured by the sensing system. In the scenario where multiple closely spaced subsurface objects are present within the field of view of the sensor, the signals measured using electromagnetic induction sensors are mixed, and the mixed measurements cannot be used to determine the identity of each of the individual objects using conventional techniques. Since only the mixed observations are available, and these are usually available at multiple target/sensor orientations, separating individual signals from the set of mixtures can be posed as a blind source separation (BSS) problem. In this paper we consider two approaches to source separation, one based on the second-order statistics and the other based on the fourth-order statistics. Following the source separation, object classification performance is obtained using the separated sources and a Bayesian classifier. We analyze the strengths and weaknesses of each BSS approach and compare their performance.

1. Introduction

[2] In real-world discrimination problems involving subsurface unexploded ordnance (UXO) detection, the task is to determine whether a measured signal results from an object of interest (such as a UXO) or results from a clutter object. Simple anomaly detection approaches based on magnetometer or electromagnetic induction (EMI) sensors do not discriminate between UXO and clutter, and the resulting high false alarm rates drive the exceedingly high cost of UXO site remediation. Recent research [Won and Keiswetter, 1998; Norton and Won, 2001; Tantum and Collins, 2001; Gao et al., 2000; Shubitidze et al., 2002a; Keiswetter et al., 2000; Baum, 1999; Geng et al., 1997; Carin et al., 1998; Sower and Cave, 1995; Sun et al., 2002a; Bell et al., 2001; Collins et al., 2002; Won et al., 1997; Carin et al., 2001; Sun et al., 2002b; Won et al., 2001; Zhang et al., 2003; Collins et al., 2001; Nelson et al., 2001] suggests that UXO can be discriminated from clutter using physics-based statistical signal processing techniques applied to time domain [Tantum and Collins, 2001; Nelson et al., 2001] or frequency domain EMI data [Won and Keiswetter, 1998; Norton and Won, 2001; Gao et al., 2000; Shubitidze et al., 2002a; Keiswetter et al., 2000; Baum, 1999; Geng et al., 1997; Carin et al., 1998; Sower and Cave, 1995; Sun et al., 2002a; Bell et al., 2001; Collins et al., 2002; Won et al., 1997; Carin et al., 2001; Sun et al., 2002b; Won et al., 2001; Zhang et al., 2003; Collins et al., 2001]. In all of these studies, however, it was assumed the object to be identified occurred in isolation, i.e., that there were no other objects within the field of view of the sensor.

[3] In highly cluttered environments typical of impact zones the assumption that objects only occur in isolation is rarely valid. In the case where multiple subsurface objects are within the field of view of an EMI sensor, the sensor's measurements can be accurately modeled as a mixture of the responses from each of the objects [Sun et al., 2002b]. Therefore the raw (unprocessed) measurements cannot be used directly to determine the identity of each of the individual objects using traditional statistical signal processing techniques that are based on libraries of signatures from individual objects in isolation or on features derived from those signatures.

[4] Here we consider wideband electromagnetic induction data taken with the GEM-3 sensor developed by Geophex, Ltd. [Won et al., 2001]. The frequency domain wideband EMI response measured by the GEM-3 is a complex quantity with both in-phase and quadrature components recorded by the sensor. The units of the measured EMI response are “ppm,” which is 106 multiplied by the ratio between the secondary magnetic field at the receiver coil and the primary magnetic field at the receiver coil [Won et al., 2001].

[5] Since it is impossible to know in advance what subsurface objects are present, or what their orientations are, and how their individual signals are mixed, separating individual signals from the mixture can be posed as a blind source separation (BSS) problem. In typical BSS problems, mixtures of multiple signals are measured with different sensors. In the subsurface UXO discrimination problem a single sensor is deployed at different spatial locations to generate the multiple views of the mixture.

[6] In general, blind source separation can be approached in several ways. For example, the problem can be formulated by computing the fourth-order statistics (e.g., independent component analysis (ICA) [Comon, 1994]) or by computing the second-order statistics (e.g., an eigenvalue decomposition approach (EDA) [Chang et al., 2000; Belouchrani et al., 1997]). The goal of BSS in the subsurface discrimination problem is to recover the unknown signatures from each individual object from the spatially diverse sensor measurements of the mixtures. The objects, their positions, and their locations relative to each other are unknown.

[7] Once the individual signatures from the subsurface objects are recovered from the measured mixtures, each signature can be processed using a discrimination algorithm to determine whether it is associated with a UXO or a clutter item. A Bayesian classifier is an effective method for object identification, in which the optimal algorithm is to choose the object that maximizes the posterior probability of the hypothesis given the frequency domain data [Tantum and Collins, 2001]. In this paper, performance comparisons are made using linear combinations of simulated data and linear combinations of data measured from UXO.

[8] This paper is organized as follows. The signal model and signal preprocessing performed prior to the application of BSS are introduced in section 2. The BSS implementation for separation of multiple objects is introduced in section 3. The effect of signal correlation is considered for a simple set of signatures in order to provide insight into the effect of correlation on discrimination performance. Object identification using the Bayesian approach is discussed in section 4. Section 5 describes results from both the simulated and the measured UXO data. Finally, in section 6, conclusions are presented.

2. Problem Overview, Signal Model, and Signal Preprocessing

[9] To simulate the multiple proximate subsurface object discrimination problem, we separately consider a library of four simulated and four real objects. The four real objects used in this work are a 155 mm projectile, an M42 submunition, an aluminum disk (simulating a clutter object), and an MK 118 Rockeye. Data from these objects were collected at an Army test site with the GEM-3 sensor. Simulated data for cylindrical objects was generated using the model described below, and object parameters were chosen to simulate UXO [Shubitidze et al., 2002a].

[10] The blind source separation algorithms will be applied to linear combinations or mixtures of both the simulated and experimental data from pairs of these objects. The two objects present in the field of view of the sensor are randomly selected from the set of four objects from trial to trial and can be placed at any depth (distance below the sensor) between 5 and 50 inches. A set of 100 trials is run, and average performance is calculated. The two objects are not constrained to be at the same depth. The inclination angle θi (i = 1 or 2) and the azimuth angle ϕi of each object are also randomly selected, in this case from a uniform distribution covering 0 to π.

[11] An EMI sensor is essentially a metal detector. It records the induced electromagnetic field due to an incident electromagnetic field that impinges on underground objects, clutter, etc. It has been shown that the measured frequency domain fields can differ significantly depending on the object shape and the object's constitutive parameters [Geng et al., 1997]. Since the objects of interest in the subsurface UXO classification problem can be approximated as cylinders [Tantum and Collins, 2001], we adopt a cylinder model in the simulations. Real UXO, not simple cylinders, are considered as well. For the cylindrically shaped objects in this work the object can be modeled as body of revolution (BOR) [Tantum and Collins, 2001]. A BOR has two principal coordinates, vertical (along the cylinder axis) and transverse (orthogonal to the cylinder axis). The corresponding resonant frequencies for these two modes are fv (strictly speaking, fv is the break frequency of a high-pass filter) and ft, and αv and αt are the amplitudes of the modes [Tantum and Collins, 2001; Kajfez and Guillon, 1986]. On the basis of the cylinder model, the resonant frequencies of objects in the transverse and vertical directions can be calculated. These frequencies do not change with target/sensor geometry although the EMI response does change with target/sensor geometry [Carin et al., 2001]. The magnetic tensors sv (f) and st (f) can be written as [Carin et al., 2001]:

equation image

where av, at are constants corresponding to ferrous objects and are zero for a nonferrous object [Baum, 1988]. We assume that fv, ft are the fundamental resonant frequencies that dominate the signals, i.e., that higher-order modes have amplitudes too small to be recorded by the sensor [Gao et al., 2000].

[12] We refer to these two components in equation (1) as sources and the entire signal, which consists of both a vertical and transverse component as x(f). Thus, for any given object, the signal measured by the sensor consists of two sources, with the amplitudes associated with those sources defined by the target/sensor geometry. Figure 1 shows the sources associated with each of the four objects in the simulated database. As the target/sensor geometry changes, different mixtures of the fixed sources are generated and measured. This is clarified in detail below.

Figure 1.

Frequency domain sources associated with the four simulated objects. In-phase component is shown with a solid line; quadrature component is shown by pluses.

[13] In the frequency domain the complex EMI sensor data x (f) is usually modeled as [Gao et al., 2000]

equation image

where so (f) is the response due to the object alone, and sb (f) is the response from the earth in the absence of any objects [Gao et al., 2000]. Direct coupling between transmitter and receiver is included in sb (f). Therefore, prior to object classification, an estimate of sb (f) is subtracted from the measured response since changes in sb (f) can impede accurate identification of so (f) [Gao et al., 2000]. To estimate the background response, a linear prediction model has been used [Collins et al., 2002]. After subtracting the estimated background response, the signal can be modeled as the response due to the objects plus a zero mean complex Gaussian noise whose variance is the variance of the estimated background response, i.e.,

equation image

where the g s are due to the variance of the estimated background response and noise associated with the sensor [Gao et al., 2000]. Here g(f) is a frequency-dependent complex Gaussian noise process with zero mean and variance σ2. Both Re(σ2) and Im (σ2) are functions of frequency [Gao et al., 2000]. We assume that g(f) is independent across frequency and across measurements. In practice, if the noise is not independent, the classification performance may not reach its optimum unless this fact is considered in the formulation of the detector [Gao et al., 2000].

[14] In the general case the background corrected response measured from K sources (or M objects, K = 2M) in one of the N spatial positions may be modeled as:

equation image

where x = (x1 (f), x2 (f), …, xN (f))equation image; L is the number of frequencies measured by the EMI sensor; M = [mn,k]; s = (s1 (f), s2 (f), …, sK (f))equation image; sk = 1,3.K−1 (f) = aequation image + αequation image × equation image; sk=2,4.K (f) = aequation image + αequation image × equation image; M = [mn,k]; and mn,2k−1, mn,2k are the position coefficients corresponding to the kth object, which can be calculated as described in the work of Carin et al. [2001] and is summarized below:

equation image
equation image

where fz (equation image, θ, ϕ) corresponds to the orientation coefficient in the cylinder axis direction, z, which is a function of the vector equation image pointing from the object to the observation, the inclination angle, θ, and the azimuth angle, ϕ [Carin et al., 2001]. Here f(x,y) (equation image, θ, ϕ) corresponds to the orientation coefficient in the plane (x, y) that is perpendicular to the cylinder axis, which is also a function of equation image, θ, and ϕ [Carin et al., 2001].

[15] In this study, M = 2, K = 4, and N = 10. In the case where more objects are present in the field of view of the sensor, the BSS implementation would be identical except that the number of the sources would increase [Comon, 1994; Chang et al., 2000; Belouchrani et al., 1997]. In the real world the number of subsurface objects present is not known. To address this issue and estimate M, we can take the singular value decomposition of xxH and determine the number of significant eigenvalues (i.e., those significantly different from zero), which corresponds to the number of sources present [Comon, 1994]. In this application, based on the EMI model we have adopted, the number of sources K is two times the number of objects M in our simulations.

[16] To construct the library, we choose every object from the set and calculate the transverse and vertical response based on the model discussed in equation (1) for the numerical simulations and measured response directly from the objects for the real data case. Then we normalize each response by subtracting the mean and dividing it by its standard deviation to obtain the corresponding normalized source.

3. Blind Source Separation of the Mixed EMI Responses

[17] Many algorithms that address the BSS problem have been developed in the last decade. Possibly the most well known of these algorithms is independent component analysis (ICA), but many other approaches have also proven effective. The most common assumptions associated with BSS algorithms are:

[18] 1. Sources are statistically independent random signals, and at most one source is Gaussian, for algorithms utilizing fourth-order statistics, such as independent component analysis (ICA) [Comon, 1994].

[19] 2. Sources are uncorrelated and stationary for algorithms utilizing second-order statistics [Belouchrani et al., 1997].

[20] 3. Sources are colored deterministic sequences that are uncorrelated for algorithm utilizing second-order statistics, such as the eigenvalue decomposition approach [Chang et al., 2000].

[21] On the basis of the analysis presented below, we hypothesize that the sources considered in this work can be modeled most accurately as colored deterministic sequences that are nearly uncorrelated, which comes closest to matching the assumption listed above in point 3. Thus, for this particular problem, we hypothesize that second-order statistical methods may be more suitable and effective [Chang et al., 2000; Belouchrani et al., 1997] and may have better source separation performance than the separation performance based on higher-order statistics [Chang et al., 2000]. To test this hypothesis, we will consider both ICA and eigenvalue decomposition approaches, analyze their strengths and weaknesses, and evaluate their performance. In the sections below, these two approaches are briefly described, and their application to the subsurface sensing problem using EMI data is outlined.

3.1. ICA Algorithm

[22] Suppose that N linear mixtures x1, x2, …, xN are observed of K-independent components s1, s2, …, sK (in this study, N = 10 and K = 4). It is convenient to use vector-matrix notation to describe the ICA model as follows [Comon, 1994]:

equation image

where M denotes the N × K mixing matrix, s is a K × L source matrix (L is 25 for numerical simulations and 10 for real data in this study), and x is an N × L matrix of measured data or observations. Observations x are known, while M and s are unknown. The data x are the input to the ICA algorithm described below.

[23] The assumptions governing the application of the ICA approach are [Comon, 1994]:

[24] 1. The components or sources are statistically independent.

[25] 2. At most one component follows a Gaussian distribution.

[26] The goal of the ICA method is to invert the mixing matrix or function and recover the sources given only the mixed signals. A key assumption underlying ICA is that the sources are statistically independent [Comon, 1994], which is satisfied in this application assuming an independent noise process. Another assumption is that all but one independent component must follow a non-Gaussian distribution. Since our sources are deterministic and have nonzero kurtosis (discussed in the following paragraphs), the noise source is the only Gaussian component, so this assumption is generally met.

[27] Generally, the key to estimating the ICA model is non-Gaussianity [Comon, 1994]. According to the central limit theorem, the distribution of a sum of independent random variables tends toward a Gaussian distribution. Thus a sum of two independent random variables usually has a distribution that is closer to Gaussian than any of the two original random variables. Below, we summarize the development of the ICA algorithm provided in the work of Comon [1994].

[28] Given a mixture of independent components, to estimate one of the independent components, we denote y = hTx and define z = MTh. Then

equation image

We find h which maximizes the non-Gaussianity of hTx by using the fixed-point method (FastICA algorithm) [Hyvarinen, 1999]. Such h makes hTx = zTs, and only one element of z1, z2, …, zM is nonzero; therefore hTx equals one of the independent components. The measure of non-Gaussianity used in this version of ICA is kurtosis or the fourth-order cumulant. The kurtosis of y is defined by

equation image

Kurtosis of a Gaussian random variable is zero, and the non-Gaussianity is usually measured by the absolute value of the kurtosis [Hyvarinen, 1999]. Let us assume that there are two independent components s1 and s2. We can compute their kurtosis values kurt(s1) and kurt(s2). We also have y = hTx = hTMs = z1s1 + z2s2. On the basis of the additivity and scaling properties of kurtosis [Hyvarinen, 1999], then it can be shown that

equation image

where we assume y, s1, and s2 have zero means and unit variances, which can be satisfied by using the whitening process of ICA [Comon, 1994].

[29] Then we obtain a constraint on z1, z2:

equation image

Our goal is to maximize equation (11) given the constraint in equation (12). Thus this turns out to be a constrained optimization problem. It is not hard to show [Delfosse and Loubaton, 1995] that the maxima are at the points when one of the z1 and z2 is zero and the other is 1 or −1. In any case, y equals one of the independent components ±si, and except for the fact that the sign may be opposite of the original unmixed source, the problem has been solved.

3.2. Eigenvalue Decomposition Analysis (EDA) Approach

[30] Suppose again that N linear mixtures x1, x2, …, xN are observed of K signals s1, s2, …, sK. Vector-matrix notation can be used to describe the BSS problem as follows [Chang et al., 2000; Belouchrani et al., 1997]:

equation image

where s is a vector of source signals of dimension K, x is the mixed signal vector of dimension N, and g is additive colored Gaussian noise. M denotes the N × K mixing matrix which has full column rank (NK). Observations x are known, while M and s are unknown.

[31] The assumptions regarding this formulation are:

[32] 1. M must be of full column rank. When M is not full rank, the signals can only be separated as classes [Chang et al., 2000].

[33] 2. Sources are uncorrelated.

[34] 3. Sources have different autocorrelation sequences since if only second-order statistics are used, any unitary transform of two uncorrelated sources with identical autocorrelation sequences will maintain the autocorrelation sequences and the uncorrelated property.

[0] BSS techniques that utilize second-order statistics (e.g., the eigenvalue decomposition approach) do not require that the sources to be statistically independent. These techniques only consider the structure of sources via computation of the correlation matrix.

[35] For deterministic signals s = s(n) = (s1 (n), s2 (n), …, sN (n))equation image, the correlation matrix of the sources is defined as:

equation image

Then

equation image

where Rx (0) denotes the autocorrelation of x(x = x(n) = (x1 (n), x2 (n), …, xN (n))T), Rs (0) denotes the autocorrelation of s, and Rg is the correlation matrix of the noise g(g = g(n) = (g1 (n), g2 (n), …, gN (n))T). Without loss of generality, we can assume that Rs (0) = I, then

equation image

which can be whitened by a matrix W such that

equation image

Then U = WM is unitary.

[36] We obtain a whitened data vector (y = y(n) = (y1 (n), y2 (n), …, yN (n))T) via:

equation image

and

equation image

where the columns of U are the eigenvectors of Ry (k) (lag k is usually 1 or 2).

[37] If Ry (k) has identical eigenvalues, then sources can be extracted by generating subclasses of the signals [Chang et al., 2000]. If Ry (k) has distinct eigenvalues, then

equation image

On the basis of equations (17)(20), the matrices W, U, and M are unique.

[38] In addition, we may obtain the estimates of all of the sources: equation image = (equation image1, equation image2, …, equation imageK)T.

equation image
equation image

where tk = 1 or −1 and k = 1, 2, …, K.

3.3. Application of BSS to EMI Data From Multiple Subsurface Objects: The Effect of Source Correlation

[39] In section 2 we noted that a UXO object may be modeled as a BOR. On the basis of the BOR model the existence of Gaussian noise (equation (4)), and using equation (5), the N spatial measurements can be modeled as:

equation image

where g is Gaussian noise vector with dimension N and has correlation matrix Rg = diag2 (f1), σ2 (f2), …, σ2 (fL)}. The signal to noise ratio (SNR) definition used is:

equation image

where Eng() denotes the average energy of the signals, xn is the nth measurement, N is the number of measurements, and g is the colored noise with statistics defined by the experimental data [Gao et al., 2000].

[40] Since the measured response is a complex signal, from equation (23) we obtain

equation image

where gR = (gequation image, gequation image, …, gequation image)T (or gI = (gequation image, gequation image, …, gequation image)T) is a N × L matrix, and gequation image (or gequation image) (i = 1, 2, …, N) is a zero mean Gaussian noise corresponding the ith spatial measurement, with variance Re(σ2) (or Im(σ2)), which is a function of frequency [Collins et al., 2002]. We assume that gequation image, gequation image, …, gequation image (or gequation image, gequation image, …, gequation image) are independent. L is the number of frequencies measured by the EMI sensor.

[41] We consider the real part and imaginary part separately and perform source separation on equation (25) to obtain the estimates of all of the sources: equation imageequation image, equation imageequation image, …, equation imageequation image and equation imageequation image, equation imageequation image, …, equation imageequation image

equation image

where H = (h1, h2, …, hN)T and each hk is a vector of dimension K. When we obtain equation imageequation image, we use both equation imageequation image and − equation imageequation image for object identification, one of which should correspond to the true source in the library. Without loss of generality we only consider the situation tk = 1. Then we have

equation image
equation image

where gequation image and gequation image are also zero mean Gaussian noise vectors. From equations (27) and (28), adding the real part and the imaginary part together, we obtain the overall expression for the estimate of the kth source:

equation image

where gequation image is a zero mean complex Gaussian noise with variance σequation image and

equation image

where {hequation image (i)} are elements of hequation image and {hequation image (i)} are elements of hequation image.

[42] On the basis of the estimates equation image1, equation image2, …, equation imageK, the goal is to identify the corresponding objects based on the signatures that make up the library. If we assume that each of the objects in the library corresponds to a hypothesis and suppose that all the hypotheses are equally likely a priori, the posterior probability of the hypothesis given the estimated signatures can be obtained from Bayes theorem. The hypothesis that maximizes the posterior probability corresponds to the object of interest [Tantum and Collins, 2001]. As a result of BSS, the index order of the estimated sources, 1, 2, …, K, may not correspond to the index of the objects. This means that equation image1, equation image2 may not correspond to the first object, equation image3, equation image4 may not correspond to the second object, …, equation imageK−1, equation imageK may not correspond to the Mth (M = K/2) object. Thus we need to consider all possible orders of the sources and calculate the posterior probability associated with each pair prior to determining the maximizing case.

[43] When the sources are correlated with each other it has been suggested that the EDA source separation procedure based can be modified as follows [Chang et al., 2000; Belouchrani et al., 1997]. From equation (15) we know that

equation image

Rs (0) can be whitened by matrix B (which is also unique) such that

equation image

then

equation image

and

equation image

where z is composed of uncorrelated sources and Rz (0) = I then

equation image
equation image

where U = WMB−1 is a unitary matrix and W is the whitening matrix corresponding to Rx (0) − Rg.

[44] From equations (18) and (19),

equation image
equation image

where the columns of U are eigenvectors of Ry (k). From U = WMB−1 we can obtain MB−1 = W−1U and then estimate the sources via z = inv(MB−1)x.

[45] In the classification problem we assume that we have a library consisting of the sources associated with the UXO objects, and the task is to separate the sources from the mixed measurements of unknown objects and to compare them with the sources in the library to determine whether the subsurface object is a UXO or a clutter object. On the basis of the above, we can compute B for each object (2 × 2 matrix) and each pair of two objects (4 × 4 matrix) from the library set in advance. Suppose two objects exist, then we obtain the separated sources z which are uncorrelated as in equation (33), use each precomputed Bpq(corresponding to objects p and q) from the library of UXO, and obtain Bpqs (s are the original sources associated with objects p and q) to compare with the separated sources z. If the two subsurface objects are objects p and q, then Bpqs should match z well. If not, the decision is that the subsurface objects are not objects p and q. Since the total number of objects in the set is four, then the four matrices B1, B2, B3, and B4 (2 × 2 matrices) correspond to objects 1, 2, 3, and 4 and B12, B13, B14, B23, B24, and B34 (4 × 4 matrices) correspond to objects 1 and 2, 1 and 3, 1 and 4, 2 and 3, 2 and 4, and 3 and 4. We call B the correction matrix. (We can also apply ICA to s (the sources of objects p and q) and obtain uncorrelated sources z and corresponding correction matrices Bpq′ (Bpq′ ≈ Bpq)).

[46] Since one assumption of BSS is that the sources are uncorrelated, it is important to determine whether this assumption is valid for the EMI signatures considered in this work, and the impact of source correlation if present. The average correlation coefficient was computed for the eight simulated and real sources separately (two sources per object) and is 0.220 for the simulated sources (Figure 1) and 0.255 for the sources associated with the real objects. Separation of signals with zero correlation has been studied extensively in the literature [Chang et al., 2000; Belouchrani et al., 1997]. Here we consider the relationship between source separation performance and source correlation.

[47] To analyze the effect of correlation on source separation, we consider a series of mean square error (MSE) versus SNR curves as a function of average correlation coefficient for four simulated source signals for which we can control the average correlation coefficient. The average correlation coefficients between these four sources are 0.061, 0.221, and 0.516. A mixing matrix

equation image

is assumed. Additive colored noise with statistics defined by the experimental data (equation (4)) is added to the mixture prior to analysis by ICA or EDA. In order to obtain the MSE estimates, we generate the noisy mixed signals 100 times for each value of correlation and SNR and calculate the MSE of the reconstructed sources. The MSEs of the four individual sources are averaged to obtain the average MSE. The results of this analysis are shown in Figure 2, where ACC denotes the average correlation coefficient.

Figure 2.

Average MSE versus SNR at different average correlation coefficients for ICA and EDA (with additive colored noise).

[48] As shown in Figure 2, when the SNR is 1e + 3 or higher, all of the MSEs reach a plateau and do not decrease even with further increases in SNR. In general, the MSE of ICA is a slightly larger than the MSE of EDA at all SNRs, suggesting that when the sources are deterministic signals EDA is more appropriate than ICA for blind source separation since the estimates of second-order statistics are more accurate than the estimates of fourth-order statistics [Chang et al., 2000; Belouchrani et al., 1997]. As the average correlation coefficient increases, the MSE increases, which suggests that source separation performance becomes worse. This effect is more pronounced for ICA than for EDA.

[49] To investigate the effect of this poorer source separation with increased source correlation more carefully, we consider the effect of MSE on probability of correct classification for these same four simulated sources. Since the goal is simply to understand the effect of correlation on performance, object classification is performed on four simulated sources in this simulation. In Figure 2 the average MSE ranged from 0.12 to 0.44 when the average correlation coefficient was at a level similar to the simulated and real UXO source data (0.221). Using the 100 groups of reconstructed sources obtained in the procedure used to generate Figure 2, we utilize a Bayesian classifier [Tantum and Collins, 2001; Gao et al., 2000] to determine the percent correct classification performance for these reconstructed sources (see section 4 for a description of Bayesian classifiers). The average percent correct classification of these four sources as a function of average MSE is shown in Figure 3. As expected, the probability of correct classification decreases as the average MSE increases.

Figure 3.

Average probability of correct classification versus MSE.

[50] These results suggest that as signal correlation increases, signal separation performance, as measured by the MSE of the reconstructed sources, becomes poorer. This poorer signal separation is reflected in a reduction of the ability of a Bayesian classifier to accurately discriminate one source from another. These results suggest that EDA is more robust to source correlation effects, particularly at higher average source correlations. The simulated and real EMI UXO data have a relatively low level of correlation at which these differences between the two approaches are small but still present.

4. Bayesian Object Identification

[51] In a Bayesian framework, object identification may be viewed as a multiple hypothesis problem, in which each of the objects of interest corresponds to a hypothesis. The Bayes decision rule is to choose the hypothesis that maximizes a posterior probability [Whalen, 1984] of the hypothesis given the estimated signal produced by the BSS algorithm:

equation image

where p(equation imageHi) is the likelihood of the data given the hypothesis; p(Hi) is the prior distribution of the hypothesis; and p(equation image) is the probability of the estimate.

[52] Suppose that in the library, there are P objects and each object corresponds to a hypothesis. We assume that all of the hypotheses are equally likely (i.e., p(Hi) = 1/P, i = 1, 2, …, P). Given the estimated sources equation image = (equation image1, equation image2, …, equation imageK) of K/2 objects, we assume that sources s1, s2, …, sK correspond to K/2 of the P hypotheses in the library: Hequation image, Hequation image, …, Hequation image.

[53] Then p(equation imageHequation image, Hequation image, …, Hequation image can be expressed as

equation image

where

equation image

[54] As discussed in the previous section, for more accurate classification in the case where the sources are correlated, the correction matrix Bpq can be used. Then the likelihood function (in the case where two objects are present in the field of view of the sensor) is rewritten as:

equation image

where s(pq) = Bpqs(pq), and s(pq) = (sequation image, sequation image, sequation image, sequation image)T are the sources corresponding to objects p and q in the library.

[55] The order of the sequence of the estimated sources generated by ICA is arbitrary; thus we consider all possible orders of the sequence. We evaluate equation (39) for all possible order and then find the maximum and declare the objects which maximize equation (39) to be the objects present. When the conclusions from the analysis for the real and imaginary parts disagree, the source separation algorithm is reinitialized and rerun.

[56] From equation (29) the estimate of the kth source is given by

equation image

where gk′ has a zero mean complex Gaussian noise with variance σk2.

[57] Our estimate of the source is unbiased since gk′ has zero mean,

equation image

Using the Cramer-Rao lower bound, we obtain

equation image

[58] When the positions and orientations of the objects are known, we can use equation (43) to obtain the lower bound on the variance of the estimated sources and further obtain the classification performance bound (by using the estimates of the sources obtained under the lower bound).

5. Numerical Simulations and Experimental Results

[59] This section presents the simulation results for the multiple subsurface object classification problem and then evaluates performance for mixtures of measured data collected at a government test site. Simulations consider cylindrical BOR objects; measured data are from real UXO. We consider both independent components analysis and the eigenvalue decomposition approaches in the simulations. The object identification algorithm used is the Bayesian method described in the previous section.

5.1. Numerical Simulations

[60] We consider four simulated cylindrical objects for a total of eight sources, the model for which has been discussed in section 2. Two objects are randomly selected from the set of four objects and placed in a three-dimensional coordinate system where x and y measure downtrack and crosstrack distance and z measures depth. Simulated measurements are obtained spatially from −10 to +10 inches in 5 inch increments along both the x and y axes at z = 0. Therefore the total number of sampled positions is 10. The measurements in the frequency domain are taken at 30, 130, 210, 330, 600, 930, 2010, 2790, 4170, 5010, 6990, 8190, 9990, 11,010, 12,990, 14,010, 15,990, 16,950, 17,970, 18,990, 19,950, 20,970, 21,950, 22,950, and 23,970 Hz respectively (25 sample points). Mixtures included additive colored noise with statistics defined by previously collected experimental data [Gao et al., 2000]. Note that it was necessary to incorporate additional noise to achieve similar SNRs to those considered for the simulated cylinder data as this particular experimental data set had extremely low noise levels. We evaluate the classification performance in several different ways.

[61] First, we consider performance bounds where all parameters associated with the problem with the exception of the objects that are present are known. The depths and locations of the two objects are fixed: one is located at a depth of 10 inches, while the other is located at a depth of 15 inches. Objects are located at y = 0 and centered about x = 0, and the distance between the centers of the objects in the x direction is 10 inches. θ and ϕ are also known. In this case the mixing matrix can be calculated a priori, and we can invert it to obtain the lower bound discussed in section 4. From this bound we simulate sources with additive noise assuming σk2 and compare them with the sources in the library to obtain the theoretical classification performance. Next we use correction matrix B and the Bayesian classifier for object identification. Performance curves plotting probability of correct classification for objects 1 and 2 as a function of SNR are provided for these two cases in Figure 4. Similar patterns of results were obtained for other pairs of objects. Correct classification occurs when objects p and q are present and the Bayesian classifier correctly identifies both sources as being present.

Figure 4.

Probability of correct classification versus SNR for two objects (1 and 2) (with additive colored noise).

[62] In the next set of simulations we evaluate performance as we remove prior knowledge regarding the positional data associated with the two objects. We consider the situation in which only the positions of the two objects were fixed (at 10 and 15 inches with 10 inches between them), but θ and ϕ were randomly selected from 0 to π. Next, the position of each object was randomly selected (depth ranges from 5 to 50 inches, relative distance between objects ranges from 2 to 20 inches), but θ and ϕ were fixed. Finally, all the location and orientation parameters were randomly selected. In these simulations the correction matrix was not utilized but will be considered again below. The classification performance versus SNR is given in Figure 4.

[63] From Figure 4 we can see that EDA is more robust to noise than ICA, especially when the SNR is less than 1e + 2, because EDA utilizes second-order statistics and considers the noise within the formulation of the model. Since the sources are not completely uncorrelated, the sources cannot be perfectly reconstructed and the probability of correct classification plateaus at about 0.85. However, as shown below, when we utilize the correction matrix, the classification performance improves dramatically and approaches the theoretical performance.

[64] In the final set of simulations we fix the noise variance to the level defined by the experimental data [Gao et al., 2000], which results in an average SNR of about 1e + 4. No prior information regarding object position or orientation is utilized. The two objects can be located at any depth between 5 and 50 inches. The distance between them is randomly selected from 2 to 20 inches. θ and ϕ are randomly selected from 0 to π. Mixtures had a colored noise added. On the basis of these above conditions, the confusion matrices given in Table 1 (EDA) and Table 2 (ICA) are obtained. The average probabilities of correct classification from Tables 1 and 2 are 0.870 and 0.819. Table 3 presents the confusion matrix obtained with the correction matrix is utilized, which increases the average probability of correct classification to 0.953. All three confusion matrices indicate that confusions tend to cluster along the diagonal, especially when the B matrix is utilized. This suggests that in most cases the classification algorithm is correctly identifying one of the two objects present when both are not identified correctly.

Table 1. EDA-Based Confusion Matrix for the Simulated Data, Average Probability of Correct Classification = 0.870
Objects
1 and 21 and 31 and 42 and 32 and 43 and 4
0.870.030.000.040.060.00
0.040.860.000.050.010.04
0.060.060.870.000.010.00
0.010.040.020.900.000.03
0.030.020.000.070.840.04
0.010.000.010.080.020.88
Table 2. ICA-Based Confusion Matrix for the Simulated Data, Average Probability of Correct Classification = 0.819
Objects
1 and 21 and 31 and 42 and 32 and 43 and 4
0.800.020.120.010.050.00
0.050.820.040.090.000.00
0.020.000.830.020.050.08
0.030.060.110.790.000.01
0.050.000.010.100.830.01
0.000.020.080.010.050.84
Table 3. EDA-Based Confusion Matrix for the Simulated Data, Using the B Matrix, Average Probability of Correct Classification = 0.953
Objects
1 and 21 and 31 and 42 and 32 and 43 and 4
1.000.000.000.000.000.00
0.040.900.000.060.000.00
0.000.040.950.000.000.01
0.000.040.000.960.000.00
0.000.000.030.030.940.00
0.000.000.020.010.000.97

5.2. Measured UXO Data Results

[65] Data were collected from ordnance items buried at a U.S. government test site using the GEM-3 sensor. Data were collected across both the x and y axis of each object from −10 to +10 inches in 5 inch increments. The sample size in the frequency domain was L = 10 (210, 330, 930, 2010, 2790, 4170, 8190, 12,990, 15,990, and 20,010 Hz) as opposed to L = 25 in the simulated data. The four real objects had sources that were correlated (ACC = 0.255). The 155 mm projectile was located at a depth of 15, 17.5, or 18 inches, the M42 submunition was located at a depth of 2.75, 3, or 4 inches, The Alu disk was located at a depth of 12.5, 23.7, or 48 inches, and the MK 118 Rockeye was located at a depth of 12, 12.5, or 13 inches. Measured data were used to calculate the sources, then mixtures were generated as in the previous section.

[66] Using this data, we obtain the confusion matrix shown in Table 4 (EDA) and Table 5 (ICA). The average probabilities of correct classification are 0.852 and 0.810. When we utilize the correction matrix form of the EDA algorithm, we obtain the results shown in Table 6, with an average percent correct classification rate of 0.925. One reason that performance on real data is not as good as that obtained in the numerical simulations may be that the number of the measurements taken in the frequency domain was relatively small, which makes the source separation less accurate [Comon, 1994; Chang et al., 2000]. When we separate deterministic signals and the data length is short, EDA is more effective than ICA since the estimates of second-order statistics are more accurate than the estimates of fourth-order statistics.

Table 4. EDA-Based Confusion Matrix for the Experimental Data, Average Probability of Correct Classification = 0.852
Objects
1 and 21 and 31 and 42 and 32 and 43 and 4
0.830.060.070.010.030.00
0.000.860.110.000.030.00
0.030.000.840.020.080.03
0.010.000.090.870.020.01
0.090.000.030.010.850.02
0.000.050.060.020.010.86
Table 5. ICA-Based Confusion Matrix for the Experimental Data, Average Probability of Correct Classification = 0.810
Objects
1 and 21 and 31 and 42 and 32 and 43 and 4
0.840.110.020.030.000.00
0.040.820.140.000.000.00
0.020.040.850.010.050.03
0.010.000.130.760.070.03
0.000.000.010.180.790.02
0.000.000.010.040.150.80
Table 6. EDA-Based Confusion Matrix for the Experimental Data, Using the B Matrix, Average Probability of Correct Classification = 0.925
Objects
1 and 21 and 31 and 42 and 32 and 43 and 4
0.940.000.030.030.000.00
0.010.960.020.000.010.00
0.010.030.940.000.020.00
0.000.010.000.920.050.02
0.040.010.020.010.890.03
0.000.040.010.000.050.90

6. Conclusions

[67] To address the blind source separation problem, computing the fourth-order statistics (e.g., ICA [Comon, 1994]) or computing the second-order statistics (e.g., EDA [Chang et al., 2000; Belouchrani et al., 1997]) are two popular approaches. In the subsurface UXO discrimination problem we can assume that the sources are deterministic signals, and when the data length is short, our results suggest that it is more appropriate to utilize a procedure based on second-order statistics (e.g., EDA) than fourth-order statistics. We considered both the ICA approach and the eigenvalue decomposition approach in the simulations and analyzed their strengths and weaknesses and evaluated their performance. For identification, we used a Bayesian method that provides an optimal classifier [Gao et al., 2000]. Specifically, in this study we have ignored the coupling known to exist between closely spaced highly conducting and permeable objects [Miller et al., 2001; Shubitidze et al., 2002b]. Such coupling would certainly reduce the “independence” of the measured signals as well as the linearity and could affect performance. Finally, we used both numerical simulations and real data to evaluate classification performance. From the simulation results and the real data results, we find that source separation is less than ideal since the sources are not uncorrelated. However, the reconstructed sources can still be processed to allow some level of classification. Comparing these results with traditional methods that do not utilize BSS (average percent correct performance on the simulated data is 0.345) [Tantum and Collins, 2001; Gao et al., 2000; Zhang et al., 2003], we observe that source separation based on second-order statistics results in improved performance in multiple closely spaced subsurface object classification problems.

[68] In this study we assumed that we know the signatures of all objects that could be presented in a general UXO classification scenario. It is possible to build a library of M expected target signatures or of physics-based features associated with UXO. However, since clutter is essentially an infinite set and is often site dependent, our signature libraries cannot reasonably be expected to incorporate clutter signatures. Thus, in the general case, the Bayesian classification must be posed as a 1 of M hypothesis testing scenario with an option of “none of the above.” As an alternative, features can be utilized, and statistics for each of the M UXOs, as well as statistics for clutter features, can be developed from site specific data. This will be investigated in future efforts.

[69] We have also assumed that the mixed signal obtained for closely spaced objects is a simple linear mixture of the signatures of the individual objects. Recent modeling of experimental data has suggested the interaction occur that yield a nonlinear relationship [Miller et al., 2001; Shubitidze et al., 2002b]. While the nonlinear interactions are small, they will result in extracted signatures which are not correct. Thus classification performance will be impaired negatively. The magnitude of these effects and approaches for accounting for the uncertainty introduced by these nonlinearities will be considered in a follow-up study.

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