Statistical classification of buried objects from spatially sampled time or frequency domain electromagnetic induction data



[1] Methods for classifying objects based on spatially sampled electromagnetic induction data taken in the time or frequency domain are developed and analyzed. To deal with nuisance parameters associated with the position of the object relative to the sensor as well as the object orientation, a computationally tractable physical model explicit in these unknowns is developed. The model is also parameterized by a collection of decay constants (or equivalently Laplace-plane poles) whose values in theory are independent of object position and orientation. These poles are used as features for classification. The overall algorithm consists of two stages. First, we estimate the values of the unknown parameters and then we do classification. Classification is done by comparing either the raw data or some low-dimensional collection of features extracted from the data to entries in a library. The library can be constructed using either simulated or calibration data. A maximum likelihood method is developed and analyzed for the problem of joint pole, location, and orientation parameter determination. Here we examine and compare two classification schemes. The first classification method is based on data residuals generated from estimated signal parameters. This scheme performs well in low SNR cases. The second is based on estimated pole values themselves, which performs well in high SNR cases. We validate our methods on both simulated and field data taken from frequency and time domain sensors.

1. Introduction

[2] The problem of detecting and classifying buried objects using electromagnetic induction (EMI)–based sensing technologies has received considerable attention in recent years in a range of application areas including unexploded ordinance (UXO) and land mine remediation. In the last decade, considerable advances have been made in the area of EMI instrumentation, yielding sensors capable of providing data both in the time and frequency domains which convey far more information concerning the structure of buried objects than is the case with older metal detectors. Extracting information such as size, shape, orientation, and type of target requires the development of advanced signal processing methods which are tied directly to the physical model of the sensor. In this paper we consider a number of options for the classification of buried objects given EMI data obtained at multiple points in space in the vicinity of an already detected object with particular attention paid to UXO and demining applications. The problem of object detection from EMI data has received considerable attention in recent years [Collins et al., 2000] and is more or less solved. Thus we concentrate on the related classification problem, i.e., declaring the type of object in the sensor field of view.

[3] Generally, classification is done by comparing either the raw data or some low-dimensional collection of features extracted from the data to entries in a library [Tantum and Collins, 2001]. The library itself is built from data signature vectors [Riggs et al., 2001; Tantum and Collins, 2001] or feature vectors from all targets of interest in a given application [Tantum and Collins, 2001; Bell et al., 2001; Barrow and Nelson, 2001]. The classifier takes as the selected object that element of the library which in a sense “best fits” the data or the features.

[4] Most all of the recent work in the area of EMI classification has been based on a simplified physical model for the interaction of the fields with the unknown target. As we describe in greater mathematical depth in section 2, assuming that the target scatters the incident energy like a dipole [Barringer Research Ltd., 1976, 1979; Chesney et al., 1984; Das et al., 1990], information concerning the class and orientation of the object in space is encoded in the magnetic polarizability tensor (MPT) which is independent of the location of the object relative to the sensors. This location information is contained in two 3 × 1 field vectors. Mathematically, the MPT is a 3 × 3 matrix which has a functional dependence on time or frequency depending on the sensor being used. In theory, this matrix can be diagonalized by a time- or frequency-independent rotation matrix indicating the orientation of the object in space. Each element of the resulting 3 × 3 diagonal matrix (which carries all of the time or frequency dependence) provides the scattering characteristics of the object along each of its three principal axes and are used for classification purposes. We refer to these as the principal axis polarizability functions (PAPFs).

[5] The various EMI classification methods developed to date differ according to factors related to the sensors being studied and the manner in which the dipole model is employed in the processing. Aside from Bell et al. [2001], classification algorithms have been concerned strictly with either time domain EMI sensors [Barrow and Nelson, 2001; Tantum and Collins, 2001] or frequency domain [Norton and Won, 2001; Riggs et al., 2001] but not both. Riggs et al. [2001] and Tantum and Collins [2001] employ a parametric model for the PAPFs which takes the form of a sum of decaying exponentials [Sower et al., 1999; Collins et al., 2000; Carin et al., 2001] in the time domain or a sum of one-pole rational system functions each with a zero at DC in frequency. The poles and/or decay constants are then used as the features for classification. Alternatively, the methods of Bell et al. [2001], Barrow and Nelson [2001], and Norton and Won [2001] use no such models and treat the time or frequency samples of the individual PAPFs as independent quantities using the entire time or frequency domain signals for purposes of object determination.

[6] The current collection of processing methods also differ in how they treat the fact that, in addition to unknown object class, the orientation and location of the target may not be known or may be known imprecisely. In the work of Bell et al. [2001] and Barrow and Nelson [2001], for example, the location of the object is estimated as part of the processing, while in the work of Riggs et al. [2001] and Norton and Won [2001], the unknown location effects are included in an overall scaling of the data and not treated explicitly. A similar approach is taken to the orientation issue in the work of Tantum and Collins [2001], while in the work of Riggs et al. [2001], a different target signature is used for each orientation of each target in the library. Finally, in the work of Bell et al. [2001], Barrow and Nelson [2001], and Norton and Won [2001], the rotation matrix is in fact determined as part of the eigen-analysis of the polarizability tensor; however, the authors do not map this rotation matrix back to an explicit orientation of the object in space.

[7] In this work we construct two classification methods based on a physical model that is the fusion of the dipole scattering model in the work of Das et al. [1990] and a parametric PAPF model elucidated in the work of Carin et al. [2001]. This model is analytical in the parameters of the PAPF, the (x, y, z) location of the object, and the three Euler angles [Hassani, 1991] describing the rotation matrix. The relatively simple closed form nature of the model with respect to these parameters leads us to classification methods in which the orientation and location of the object are explicitly estimated along with the parameters needed for classification. Thus our approach provides information regarding these geometric characteristics of the object. Also, the closed form nature of the PAPF model allows our approach to be applied with equal ease to both time or frequency domain sensor data.

[8] The model of Geng et al. [1999] indicates that in theory the PAPFs are composed of an infinite number of decaying exponentials in time domain, which in the frequency domain, translate to an infinite number of one-pole transfer functions. Unfortunately, it is both impossible and unnecessary to estimate an infinite number (or even a large number) of poles or decay rates. First, pole estimation in the presence of noise is known to be a very delicate signal processing problem [Apollo, 1991]. Second, because objects do not scatter exactly as dipoles, it makes sense to consider reduced order models for the purpose of processing. Indeed, experimentally it has been shown that one or two poles can typically be used to match the model to measured data [Geng et al., 1999]. Thus in this paper, we posit a model in which each PAPF is represented as a single decaying exponential or one-pole transfer function. Since there are three principal axes, we estimate three poles/decay constants as part of the classification routine. This is similar to the one- or two-pole methods of Geng et al. [1999]; however, where these approaches disregard the information in the coefficients multiplying the decaying exponentials in time (or poles in the frequency domain), we are able to relate these coefficients directly and analytically to the location and orientation of the object in space.

[9] Moreover, our method explicitly accounts for modeling error introduced by the fact that we are not using an exact scattering model. Because a three-pole model cannot (in general) exactly represent the data, the pole values will not be independent of object position and orientation. Rather, for a given type of target, there will be a “spread” of pole values as a function of these geometric nuisance parameters. Thus we introduce a simple quadratic-form classifier which compensates for this effect of model mismatch. For the purpose of comparison, classifiers based on data residuals are also constructed.

[10] The physical model employed in this work was introduced in the work of Miller [2001], where a preliminary set of results demonstrated the effectiveness of the model for a limited set of simulated data. In this work we study the model and the processing schemes more extensively and develop an improved understanding of the physical model. In particular, we investigate the effect of SNR on the performance of our residual-based and pole-based classifiers. We find that the residual-based classifiers perform well under low SNR, whereas under high SNR the pole-based classifiers perform well. We validate all claims using simulated as well as field data.

[11] The remainder of this paper is organized as follows. In section 2 the scattering model is discussed. A description of the processing method is given in section 3. Experiments using both simulated as well as real data from time and frequency domain sensors are presented in section 4. Conclusions and directions for future work are the subjects of section 5.

2. Physical Model

[12] We consider a combination of the physical EMI model of Das et al. [1990] describing the scattering of low-frequency electromagnetic radiation by spherical or spheroidal objects with the model of Geng et al. [1999], which rigorously justifies the use of decaying exponentials in time or one-pole models in frequency for problems of this type. As seen in Figure 1, the transmitters and receivers are taken to be coils (not necessarily colocated) with sides of length 2A and 2B, respectively. (We assume square coils for convenience; with a little work, the model could be generalized to circular coils.) The target center is located at r0 = (x0, y0, z0) in the xyz coordinate system. We are concerned with processing methods based on time-sampled or frequency domain–sampled data obtained from multiple transmitter/receiver locations. Assuming we collect M time or frequency samples from each of N combinations of transmitters and receivers positions, then under the model the kth sample at the jth position is

equation image

Here g is a 3 × 1 vector holding the (x, y, z) components of the magnetic field produced at r0 by a current I flowing through the receiver coil. Also, f represents the excitation fields vector evaluated at the dipole position. The variable nj,k is a zero mean unit variance random variable, and σ is the standard deviation of the assumed additive white Gaussian measurement noise. Functional forms for f and g are provided by Das et al. [1990, Appendix A]. The quantity Λk is the complex-valued polarizability tensor for the kth frequency and has the following form:

equation image

where λ1, λ2, λ3 are associated with one of each of the principle axes of the object and ω is the operating frequency. Replacing ω by t will give the time domain version of this equation. Here we consider a form of the model provided by Geng et al. [1999]:

equation image

where j = equation image, pi,l is the lth pole for the ith axis, and ai,l is the expansion coefficient. An inverse Fourier transform yields the time domain version of λ:

equation image

with u(t) the unit step function. For cylinders and disks, Carin et al. [2001] provide a fast numerical method for computing the pi,l. The model in equations (3) and (4) strictly holds for nonferrous objects. In the case of ferrous objects, one must add a DC offset in frequency or a Dirac delta function in time. For notational ease in what follows, we concentrate on the nonferrous case with the understanding that these small changes need to be made for ferrous objects.

Figure 1.

One sensor comprising sensor coils and target object.

[13] In equation (1), R is a rotation matrix which orients the object in the space and is used to transform field quantities between a global frame of reference and the local frame of the object. Here R is parameterized by three Euler angles [Hassani, 1991] and explicitly takes the form:

equation image

[14] The magnetic polarizability tensor Λ can be diagonalized by R. Each element of the resulting matrix holds the scattering characteristics of the object along each of the three-principle axes.

[15] Going back to equation (1), by gathering the data together from all sensors, the overall model in compact notation can be written as

equation image

where y is the vector composed of the data from all sensor locations and time/frequency samples, s is the signal vector, n the noise vector, p is the vector of all poles in the model, a is the vector of expansion coefficients, and θ is the six-dimensional vector composed of the object coordinates and Euler angles.

[16] As stated, this model assumes that the object behaves electromagnetically like a dipole. The three λis fully summarize the scattering behavior of the object and only depend on the size, shape, and material of the object and not on the orientation and position of the object relative to the sensor. Thus the pole and expansion coefficients make good candidates for use in a classification routine. The orientation information is explicitly contained in the matrix R, while the field vectors f and g convey position information. Owing to the simple analytical nature of this model, it is quite well suited for use in a signal processing routine where operations like pole fitting and parameter estimation are accomplished using optimization routines. The complexity of these routines is substantially reduced owing to our ability to use the model to compute closed form sensitivity information; essentially, the derivative of the data with respect to any of the unknowns (poles, expansion coefficients, Euler angles, or location coordinates). Such calculations are at the heart of any parameter fitting scheme employing, e.g., a gradient descent, conjugate gradient, or Newton type of optimization scheme.

[17] While the utility of the model described here has been validated using real sensor data [Das et al., 1990; Sower et al., 1999; Riggs et al., 2001] as described in section 1, generally objects do not behave exactly as dipoles. Moreover, one cannot practically use an infinite number of poles for each λi. Rather, a single pole per axis is the most that is typically supported by the data [Riggs et al., 2001; Carin et al., 2001; Sower et al., 1999]. In such a case the “effective” pole for each axis will be dependent on the object position and orientation. The end result is that for all practical purposes, model mismatch or required model reduction for the physical model described above will force us to consider pole-based classifiers which explicitly account for variations in the feature values. If such variations are small, then one expects success in using poles (really effective poles) for classification.

3. Processing Method

[18] Our approach to classification starts with the construction of a target signature library which will be used in the actual processing. For each target of interest, this library will be composed of the three effective pole and expansion coefficients which define the PAPFs. Given that library, classification is a two-step process. First, for each target in the library the data are used to estimate the unknown parameters associated with that model: poles, expansion coefficients, object location, and object orientation. Second, using these estimates, we examine two classification schemes. The first classifier is based on using the pole estimates alone and is expected to work well in high signal to noise cases when we can get accurate estimates of these quantities. The second classifier is based on the fit of the kth model in the library to the available data. As explained more fully in section 3.3, this approach is likely to be of use when the noise level is relatively high. We begin by discussing the construction of this library.

3.1. Library Construction

[19] As discussed in section 2, the pole estimates which we use for classification will have some orientation and position dependence which should be accounted for in the construction of the library and in the processing. Let us suppose that we have data from a known target in a known position and orientation which either has been computed using an exact computational model or measured using an actual sensor. For the kth target in the library we are going to use one effective pole per λi defined in a best fit manner as the solution to the following optimization problem

equation image

where θ0 holds the true position and orientation information, yk is the true data vector, and p and peff are vectors of three-pole parameters, one per λi. The symbol “equation image” above quantities indicates that these are fitted to data. We note that to be consistent with the estimation scheme developed in section 3.2, here we do fit a and θ; however, we care only about the effective pole values in constructing the library. Additionally, the effective pole parameters are implicitly dependent on the specifics of the sensing system we use, including frequencies of operation, time gates measured, and spatial sampling strategy. Hence, in theory, each sensing configuration will require a separate library.

[20] While we could construct a library holding pkeff(θ) for a dense sampling of points in θ space, here we choose a simpler approach. For the classifiers considered in section 3.3, we look only at the first two moments of the effective pole vector averaged over θ. Mathematically, we define the mean pole vector and the associated covariance matrix respectively via

equation image
equation image

where the index i ranges over a grid of points in θ space. Thus the feature library we employ for classification based on pole estimates is composed of one three-dimensional vector and one 3 × 3 matrix for each target of interest and each sensing system under investigation.

3.2. Parameter Estimation

[21] The first stage of processing is to estimate the parameters of our model for each target in the library. We actually do this twice: the first time to obtain parameter estimates to be used in a classifier based on data fit and the second time to obtain estimates for a pole-based classifier.

[22] As explained more fully in section 3.3, the classifier based on the data vector directly makes use of the kth residual vector, yequation imagek, where equation imagek is an estimate of the signal vector for the kth object in the library. A first approach to generating equation imagek is to solve a problem similar to that of equation (7). We could then make use of the fact that if the data did in fact come from the kth object (which is what we will ultimately be testing), then the poles should be equation imagek. Thus under this scheme we would not need to estimate the poles (and the expansion coefficients if we were to keep track of these as well), and we would only need to determine the elements of θ. While such an approach is feasible, it ignores the fact that we have information concerning the behavior of the pole estimates in the form of a mean vector and a covariance matrix. Hence, rather than fixing the poles in the estimation scheme, we let them float but impose some bounds on their values in recognition of the fact that since we are going to be testing the fitness of the data to the kth model, the poles should be constrained to be close to the average pole value for this model. Specifically, we solve the constrained optimization problem:

equation image

where [equation imagek]i is the ith element of the vector equation imagek and [σk]i is the square root of the ith element along the diagonal of Rk. Hence [σk]i is the estimated standard deviation of [equation imagek]i. The above optimization problem essentially restricts the estimates of the poles to stay within plus or minus two standard deviations of their expected value. Again, the philosophy underlying this choice is that since we will be using these estimates to test the goodness of fit of the data vector to the kth model, we should encourage the parameter estimates to stay “close” to the model.

[23] To solve the problem in equation (10), we use a nonlinear least squares solver that makes use of a constrained Gauss-Newton algorithm for finding the a local minimizer of the objective function in the neighborhood of an initial guess. We initialize the algorithm as follows. For the poles we use equation imagek, and we initially take the expansion coefficients to be equal to 1.0. The initial (x, y) location of the object is taken to be that point in space with the largest magnitude response in the data (a heuristic but one which seems to work well), while the initial depth is 1.0 m from the sensor. The Euler angles are initialized all to 0.0.

[24] The second classifier discussed in section 3.3 is based on estimates of the poles. To allow the maximum flexibility in determining these quantities, we use the following second estimation scheme in which the bound constraints are lifted

equation image

Again, a nonlinear least squares solver is used. This time the algorithm is initialized with equation image1,k, equation image1,k, and equation image1,k. We have found that by constraining the poles in the first estimation stage, we obtain high-quality estimates of the position and orientation parameters. These estimates are then used to obtain strong overall estimates of all relevant parameters in the second estimation step. Thus this appears to provide an effective means to avoid the problem of reaching a local minimum, which is often associated with nonlinear parameter estimation problems.

3.3. Classification

[25] Given the model-based approach we have developed in this paper, there are two natural classification schemes that can be employed. The first is based on the idea that if the kth model is the true model, then according to equation (6) ρk = ys (equation image1,k, equation image1,k, equation image1,k) should be a zero mean uncorrelated Gaussian random vector. The second compares the pole estimates, equation image2, to the elements of the library in a way which accounts for the known “spread” in the pole estimates for a given target.

[26] Consider the low SNR case where ρk is dominated by the additive white noise. For a fixed σ, this scenario would arise when the object under investigation is deeply buried in which case the signal strength is significantly lowered owing to the 1/r3-type one-way amplitude loss seen in the field strength for problem of this type. In such situations, when the kth object is in fact the true object, the statistical distribution of the residuals is dominated by the zero mean, white σn term and goodness of fit classification tests based on the zero mean nature of the residuals are expected to do well. Alternatively, when the kth target is not the correct object, δsk = s(p0, a0, θ0) − s(equation image1,k, equation image1,k, equation image1,k) will be significant thereby adding to the mean of the ρk. Hence a classifier constructed to test that the mean of ρk is in fact zero would correctly reject this hypothesis.

[27] Next, consider the case of shallowly buried objects where the SNR is high so σn is fundamentally small. In these cases, even when the kth object is in fact correct, the errors caused by even slight inaccuracies in the estimates of p, a, and θ will dominate the noise, so that a classifier wishing to exploit the expected zero mean nature of ρk under the true hypothesis will fail. These high SNR situations, however, are exactly those where we anticipate the ability to obtain good estimates of the pole structure. Hence classification schemes based on the pole estimates themselves are expected to perform well here.

[28] Motivated by the considerations outlined in the above discussion, here we define two statistics to be used for classification. The first is based on the data residuals and is taken as:

equation image

where N is the dimensionality of ρk. The normalization of the residuals in this way ensures that ε1,k is asymptotically distributed as a zero mean unit variance Gaussian random variable when the k corresponds to the true object. Thus classifiers based on tests of the closeness of ε1 to zero are appropriate here. To generate a classifier based on the estimates of the poles, we use

equation image

This Mahalanobis-type distance metric is expected to be close to zero when k is true and larger than zero when the true object is not the kth.

[29] Using ε1,k and ε2,k, the classification rule is defined as follows: Choose the k* object in the library for which the magnitude of εk* is minimum. Here k* is selected in one of two ways. If we just want a classifier based on the residuals, we let k* be the index of the smallest ε1,k. For a classifier based only on the pole estimates, k* is that index minimizing ε2,k over all k.

4. Numerical Examples

[30] Here we consider numerical tests of the classification methods described in section 3 using two different object libraries. The first sensing system was composed of colocated square transmit and receive coils 1/2 m on a side. These coils sampled a 1 m square area on an equally spaced 5 × 5 grid of measurement points. Frequency domain versions of the sensor collected complex valued data (in-phase and quadrature) at 30 logarithmically spaced frequencies between 10 and 30 kHz. For time domain we employed a sensor that collected 60 equally spaced samples between 10−6 and 10−3 s.

[31] The simulated object library is comprised of four objects: a 3 inch long by 1 inch diameter stainless steel cylinder (S1), a 6 inch long by 1 inch diameter stainless steel cylinder (S2), a 3 inch long by 1 inch diameter aluminum cylinder (A1), and a 6 inch long by 1 inch diameter aluminum cylinder (A2). The “ground truth” model for the scattering characteristics of these objects was obtained using the method of Carin et al. [2001], in which the dipole model was taken to be exact, four terms were kept in each of the λi summations in equation (4), and all expansion coefficients were taken to be 1. As cylinders are symmetric about their primary axis, these objects have two unique λ′is. The range of values for the minimum and maximum poles for each of the four objects in the library are given in Table 1. The effective pole parameters for the frequency domain version of the sensing system as a function of the object position and orientation are plotted in pole space in Figure 2. Each point on the plot corresponds to the (p1eff, p2eff, p3eff) value computed from equation (7) for the ith term in the summation of equations (8) or (9). The top plots in this figure show the effective pole distributions for the steel targets, while the bottom plots do the same but for the aluminum targets. Note that the axes for the top and bottom plots are distinctly different and that the points for the different objects cluster reasonably well in pole space. Thus, as is evident from the table as well as the figure, the pole characteristics of the steel objects are quite distinct from those of the aluminum; however, the differences between the 6 and 3 inch versions of the same material are a bit more subtle. Hence it is anticipated that we will be able to distinguish material type better than precise object.

Figure 2.

Effective pole distribution for steel and aluminum objects in frequency domain.

Table 1. Pole Characteristics for Objects in First Library
TargetMinimum p1, kHzMaximum p1, kHzMinimum p2, kHzMaximum p2, kHz
3 inch steel3.45.63.610.3
6 inch steel3.
3 inch aluminum0.
6 inch aluminum0.

[32] A Monte Carlo approach is used to analyze the performance of our algorithms for this library. Two sets of simulations were carried out to compare the performance of our classifiers for high and low SNR cases. In each case, 100 separate data sets were generated where we randomized uniformly over object type, object location, orientation, and additive sensor noise (by selecting a new set of parameters according to a uniform distribution at each Monte Carlo round). The bounds on the various quantities are provided in Table 2. The corresponding values of the Euler angles were chosen uniformly over their full range of definition (either 0 to 2π or 0 to π, depending on the angle [Hassani, 1991]). For our purposes, SNR is calculated according to:

equation image

where y denotes the signal vector, ly is its length, and σ2 denotes the noise variance. At each Monte Carlo run, we select the burial depths of the targets randomly from a specific interval, while keeping the noise variance σ2 fixed. For the low SNR case this interval consists of burial depths in the range of 1.00 to 2.00 m, while for the high SNR, case this interval consists of burial depths in the range of 0.1 to 0.3 m.

Table 2. Bounds for Monte Carlo Analysis
ParameterValue, m
Minimum x coordinate0.25
Maximum x coordinate0.75
Minimum y coordinate0.25
Maximum y coordinate0.75
Minimum depth−0.10
Maximum depth−2.00

[33] The classification results of this example, for each of the above cases using the two classification methods, are summarized in the confusion matrices of Tables 310. The i, jth element of each matrix demonstrates the number of times that object i was the true target, and object j was declared by our processing scheme. The results presented in these tables verify our claim with respect to the two different classifiers. As shown here, when operating under high SNR (on the order of 20 dB or higher), pole-based classifiers operate well, whereas residual-based classifiers operate well under low SNR (on the order of 0 dB).

Table 3. Frequency Domain Classification Results for Monte Carlo Analysis Based on Pole Estimates Under High SNR
 Estimated A1Estimated A2Estimated S1Estimated S2
True A1811900
True A2168400
True S100928
True S2002278
Table 4. Frequency Domain Classification Results for Monte Carlo Analysis Based on Data Residual Under High SNR
 Estimated A1Estimated A2Estimated S1Estimated S2
True A1563275
True A22957311
True S120899
True S2203563
Table 5. Frequency Domain Classification Results for Monte Carlo Analysis Based on Pole Estimates Under Low SNR
 Estimated A1Estimated A2Estimated S1Estimated S2
True A13419452
True A234114312
True S115192145
True S22116549
Table 6. Frequency Domain Classification Results for Monte Carlo Analysis Based on Data Residual Under Low SNR
 Estimated A1Estimated A2Estimated S1Estimated S2
True A1592786
True A2335296
True S113116412
True S28142058
Table 7. Time Domain Classification Results for Monte Carlo Analysis Based on Pole Estimates Under High SNR
 Estimated A1Estimated A2Estimated S1Estimated S2
True A1792100
True A2188200
True S1007723
True S2002872
Table 8. Time Domain Classification Results for Monte Carlo Analysis Based on Data Residual Under High SNR
 Estimated A1Estimated A2Estimated S1Estimated S2
True A15425912
True A23741715
True S1416139
True S2303367
Table 9. Time Domain Classification Results for Monte Carlo Analysis Based on Pole Estimates Under Low SNR
 Estimated A1Estimated A2Estimated S1Estimated S2
True A12016604
True A232173813
True S123213224
True S22818477
Table 10. Time Domain Classification Results for Monte Carlo Analysis Based on Data Residual Under Low SNR
 Estimated A1Estimated A2Estimated S1Estimated S2
True A15422195
True A217571511
True S1136234
True S2144055

[34] The misclassification presented in our confusion matrices can be attributed to the fact that the scattering characteristics of the two aluminum (or steel) targets are very similar. Note that the target material is rarely misclassified under high SNR, while under low SNR the residual-based algorithm, though misclassifying target type more often, performs significantly better than the pole-based method. It is also clear that similar results are obtained from our algorithm regardless of the characteristics of the sensor employed (time or frequency domain).

[35] The second object library is composed of nine targets. Their scattering characteristics were obtained from the work of Huang and Won [2003]. The data for this sensing system come from the GEM-3 sensor developed by Geophex. This sensor has been used successfully in many environmental sites and can detect small targets, such as UXO and land mines, providing high spatial resolution [Huang and Won, 2003]. The current GEM-3 operates in a bandwidth from 30 to 24 kHz. The Geophex test site in Raleigh, NC, specially designed by Geophex Inc., is a 10 m ×10 m test site. This test site, detailed by Huang and Won [2003], contains a total of 21 metal pipes of various types, lengths, and diameters. Huang and Won [2003] obtained the ground truth data by placing the targets at two different orientations at a single known position. Because of the lack of multiple measurements, we were not able to generate the standard deviations of the estimates of the poles. Hence, for our purposes, Rk was assumed to be the identity matrix. The effective pole parameters of these objects are plotted in pole space in Figure 3. Each point on the plot reflects the (equation image1, equation image2, equation image3) value computed from the corresponding target in the library.

Figure 3.

Pole locations for GEM-3 data library.

[36] We have separated the aforementioned nine targets into three sets of targets, designated by letters L, M, and S. The letter L corresponds to targets that are deeply buried at a depth ranging from 90 to 110 cm. The letter M corresponds to targets buried at a depth ranging from 50 to 80 cm. Finally, the letter S corresponds to shallowly buried targets at a depth ranging from 10 to 30 cm.

[37] The scattering characteristics of these targets were also acquired by Huang and Won [2003] at a line spacing of 25 cm using the dead reckoning method at a height of about 20 cm above the ground. The GEM-3 in this case collected about 8 to 10 data points per second, which resulted in a data interval of about 15 cm [Huang and Won, 2003]. The position error for such data could be as big as 20 cm owing to uneven walking speed and incorrect walking path. Also, the errors associated with the sensor height could be more than 5 cm [Huang and Won, 2003].

[38] Table 11 summarizes the classification results for this target library. For the measurement data corresponding to each true target, the column labeled “depth” corresponds to the burial depth of each target in centimeters. Clearly, the results presented in this table suggest a correlation between object depth and the dominating classification method. Specifically, we find that our L series targets are classified perfectly using the residual-based method (and poorly using the pole-based method); the M series targets show mixed results with correct classification under the residual-based method alone for particularly deeply buried targets, both methods classifying for an object of moderate depth, and correct classification under the pole-based method alone for a particularly shallowly buried target. For the majority of the S series targets, correct classification is achieved under the pole-based method alone. These results closely correspond to our claim that the pole-based classifier is better suited for shallowly buried objects, while the residual-based classifier is better suited for deeply buried targets.

Table 11. GEM 3 Pipe Data Results
True TargetDepth, cmPole-Based ClassifierResidual-Based Classifier
Estimated TypeResultEstimated TypeResult

5. Conclusions and Future Work

[39] In this paper we have explored a number of statistically motivated model-based options for object classification using spatially sampled time and frequency domain EMI data. Preliminary results using synthetic data are promising and indicate there is much work to be done in the future. Of specific interest are the following items.

[40] 1. Extending the classifier algorithm to be able to distinguish objects that are not in the library (clutter items). This step can be very important for the UXO and demining problems where there is a strong desire to correctly reject clutter items.

[41] 2. Testing the approach on field data from other EMI sensors.

[42] 3. Analytical performance evaluation of the proposed methods. By obtaining closed form or computable values or bounds on the relevant probabilities, we can begin to explore issues such as optimizing sensor configuration (time gates, frequencies collected, spatial sampling rate) to maximize performance.

[43] 4. More rigorous error analysis. It would be useful to take a more careful quantitative look at the role of parameter estimation errors as well as sensor noise in the ultimate performance of the classification scheme.

[44] 5. Alternate classification techniques. The pole-based classification statistic, ε2,k in equation (13), implicitly assumes that the effective pole parameters as a function of object location and orientation cluster into an ellipsoid. The plots in Figure 2 show that while the effective poles for different objects do cluster, they are far from ellipsoidal. Thus it may be useful to consider alternate classification schemes such as a linear discriminant function.


[45] This work was supported by funding from the Strategic Environmental Research and Development Program under project CU-1217. The authors also wish to thank Lawrence Carin of Duke University for providing us with the data used in this work.