around the mean of each value – -even for a steel sphere. They state that this variation in the values may arise from a measurement error due to sensor noise and inaccuracies in location estimation. To complicate things further, the coil geometry used affects the ratio; therefore, Grimm and Sprott [99] have studied different transmitter-receive coil geometries to be used with the above method.
However, a variety of advanced features exploiting the aspect ratio have been proposed. Pasion and Oldenburg [73] use the decay parameters in (2.8) to determine the shape of the object. They use ratios κ1/κ2and ψ1/ψ2to decide whether the geometry is plate-like
or rod-like. Similarly, Norton et al. [54] have proposed ordnance-likeness as a feature to distinguish between UXO and clutter. First, they ordered the eigenvalue vector by their real parts in such a way that <(λ1) < <(λ2) < <(λ3). Then, a ratio was calculated by
OrdnanceLikenessRatio=<(λ3) − <(λ2)
<(λ2) − <(λ1)
. (4.6)
This ratio was then calculated at multiple excitation frequencies, and the feature for each object was the mean of log(OrdnanceLikenessRatio) across all frequencies. The imaginary parts of the eigenvalues were found less suitable for this purpose, as shown in the results [54]. Ambrus et al. [100], while using a pulsed EMI landmine detection system, took this idea further by constructing a 3-by-3 matrix of the ratios of all three eigenvalues, each of which was measured at three time gates of the detector, i.e., at three points in time, to capture the signal decay characteristics.
In addition to methods using the aspect ratio, some approaches in the literature classify metallic targets based on their volumetric size. However, these methods use more advanced classifiers. Zhang et al. [64] have used support vector machines (SVM) and neural networks
(NN)for this (SVM and NN are described in Section 4.4). In addition, Fernandez et al.
[101] have shown that spheroids of different sizes can be discriminated by their volume and radius by using an SVM. These studies suggest that features can be generated that enable use of size and shape information as features for classification.
Extracting material information from EMI data has also been reported. Huang and Won [102] have used CW EMI and modeled metallic targets as permeable and conductive spheres and used the model parameters to determine the conductivity and permeability of the targets. However, the results show that solving permeability and conductivity is feasible only in a noise-free scenario. Furthermore, using pulsed EMI, Pasion and Oldenburg [73] have used the parameter ψ in (2.8) to determine whether the target is magnetic or non-magnetic. Furthermore, Trang et al. [103] have studied the relationship of the phase angle and magnitude of a CW EMI response and the material of metallic targets. According to them, the permeability of the target can be seen at very low frequencies of the in-phase component of the response. Moreover, the phase angle of various metals, namely magnetic steel, non-magnetic steel, copper, brass, lead, and aluminium, was found to be independent of object orientation [103].
4.3
Statistical classification and generative methods
Bayesian decision theory lays the basis for optimal classification rules. The Bayes formula relates the posterior probability to the priors as
P(Ωi|x) =
P(x|Ωi)P (Ωi)
where P (Ωi|x) is the posterior probability for the state of nature being Ωi given that the
feature vector is x, P (x|Ωi) is the prior conditional probability for the feature vector x
given that the state of nature is Ωi. P (x) and P (Ωi) are the independent probabilities for xand Ωi, respectively. The prior (a priori) probabilities P (Ωi) for the classes in Ω are
unknown, i.e., it is not possible to know how many, e.g., threats are encountered compared to innocuous items. Therefore, it is common to assume that each class is equally probable, but this also makes it impossible, in most cases, to use (4.7) directly for classification. If the prior probability density function (PDF) for each class is known, the likelihood ratio test (LRT) is the optimal classifier [33]. The LRT states the ratio between how likely the given data x is under the class Ωi compared to the null hypothesis. A decision threshold
for the ratio Υ is used to classify the samples. However, in most cases, the true PDFs
P(x|Ωi) for each class Ωi are unknown, though they may be estimated by using training
data: the true PDF is replaced with a maximum likelihood estimate (MLE) of the PDF. As a visual example, Figure 4.3 shows a scenario of modeling the PDF for some class using its eigenvalues λ. For each of the three eigenvalues, there is a cloud of data points within the training values. This cloud can be used to find an approximation of the PDF, or three PDFs, should each cloud be considered a PDF of its own.
Figure 4.3: A PDF has been estimated for each eigenvalue of some category. The eigenvalues of an unknown sample x are shown in red. It is a statistical problem to estimate the probability of the red points having been created by the PDFs shown.
Collins et al. [72] were among the first to incorporate Bayesian decision theory in MSI, namely in landmine detection, using both CW EMI and pulsed EMI. For classification, they used the so-called generalized likelihood ratio test (GLRT). The GLRT is a generalized version of the LRT, which can be used even if there are unknown parameters in the PDFs. The GRLT is given by
PM LE(x|Ω1)