Herramientas de Desarrollo

As mentioned in 4.1, features are a means of making samples separable into classes to facilitate classification. A good feature should extract relevant information from the data

4.2. Features 29

and/or the underlying phenomenon that the data describe. Features can, first, describe the data in a compact way, make raw data unnecessary, and enable simple comparison between the feature values, and, second, characterize the target by representing its intrinsic properties. Most parameters of the physical models described in Chapter 2 act as features because they describe the measured EMI signal in a compact manner and characterize the target. There are many examples of this in the literature. Tarokh et al. [71] have used the Laplace-plane pole representation of MPT eigenvalues to classify buried objects with a CW EMI device. In a pulsed EMI scenario, Fernandez et al. [95], used some decay parameters as features while applying the NSMS model, whereas Shubitidze et al. [96] have shown with real UXO test scenario data that use of such features can help accurately distinguish between UXO and clutter.

In the case of WTMD, raw time-domain measurement data consisting of the induced voltage of each coil pair is not very useful for classifying metallic objects, because it does not characterize the target adequately because it contains background noise and information related, e.g., to the speed of the walk-through and the gait of the candidate. Therefore, the MPT is a step towards a better characterization of the target, containing relevant information for classification, such as its intrinsic properties. In theory, the MPT is independent of target location but orientation-dependent, making it unsuitable for use as a feature. Therefore, eigenvalues λ, as defined in Section 2.2, have been introduced, and as shown in Publication II, they constitute a rotation- (and location-) invariant [54] representation of the MPT, and hence a good feature. Moreover, according to Bell et al. [62], the eigenvalues contain all the information within the MPT that can be used for classification. 3

In this thesis, eigenvalues λ as such refer to their Cartesian presentation, i.e., to complex two-dimensional values whose real part is on the X-axis and imaginary part on the Y-axis. However, for practical purposes, a polar presentation of the eigenvalues is defined. The

magnitudeof an eigenvalue λi is given by

τ(λi) = τ = ||λi||=

q

λi· λi, (4.4)

where λi is the complex conjugate of λi. In polar presentation, the magnitude is on the

Y-axis. Similarly, the angle ϕ(λi) of an eigenvalue is given by

ϕ(λi) = ϕ = atan(<(λi), =(λi)), (4.5)

where atan is the four-quadrant arctangent function. In polar presentation, the angle is on the X-axis, and its range is usually [−π . . . π]. Correspondingly, the magnitude is on the Y-axis. These concepts are visualized in Figure 4.2.

It is known a priori that some materials are typical of certain types of objects, and similarly, e.g., knives are usually long and thin objects. Therefore, heuristic features for metallic target characterization might include descriptors of material, size, and shape. There are many examples in the literature that indicate the usefulness of MPT eigenvalues for this purpose. Norton et al. [61] state that the MPT represents the object as a uniform and ellipsoidal shape, and that its eigenvalues can be related to the lengths of

3_{Because the eigenvectors ofM are complex, they do not contain real orientation information for}↔ the object. It has been suggested by Prof. Lionheart that the real and imaginary parts of

↔

M may be considered separately, hence yielding two sets of real eigenvectors. The relationship between these orientations may then be of value. However, SNR-related problems are likely to arise in WTMD portals.

Figure 4.2: Eigenvalues and their angles and magnitudes, shown on a Cartesian plot.

the semi-major axes of the ellipsoid. 4 _{Furthermore, according to Bell et al. [40, 62],}

the eigenvalues of the MPT are related to the strength of the induced field along the principal axis of the target, and there is a strong correlation between the size of the target and the magnitudes of the eigenvalues. Moreover, they state that measurements of the longitudinal vs. transverse field ratio of the object show a strong correlation between the physical aspect ratio of the object and the corresponding measurements [40].

Many studies have used this aspect ratio as a feature for classification. In these studies, the elements of the time domain diagonalized MPT were often given by ΛX = m1,1,ΛY = m2,2,ΛZ = m3,3; these values correspond to the eigenvalues λ. In case of a cylindrical

object, there are two distinct values: Λlongitudinal= m1,1 and Λtransverse= m2,2 = m3,3

(see, e.g., Khadr et al. [97] for details).

Khadr et al. [97] and Bell et al. [40] have used this representation for UXO/clutter discrimination using a pulsed EMI system. They propose a feature for modeling the length-to-diameter aspect ratio of the detected object, assumed to be a prolate spheroid, given as the ratio |Λlongitudinal/Λtransverse|of two time-domain eigenvalues of the MPT.

However, this approach may not be reliable. Indeed, in the magnetostatic case, the above ratio is known to be nonlinear [55]. Furthermore, for the eddy current case, there are examples of nonlinearity in the literature. Bell et al. [40] consider a pulsed EMI system and argue that since the time window is of fixed length, the ratio of the transverse and longitudinal responses depends not only on the shape of the target but also on its size. More importantly, they have shown that the ratio is also frequency-dependent. In a more recent study, Bell et al. [62] state that in the time-domain, the transverse and longitudinal responses of the targets have different decay rates. This results in different ratios of the above eigenvalues at different times, and hence frequencies. A detailed analysis of the relationship between the eigenvalues and target size appears in another study by Bell et al. [41]. Barrow and Nelson [98] have used a similar method and ratios of the ΛX,Y,Z-values

to determine the shape of targets. However, they found that these values are related to object dimensions in a nonlinear fashion, and that there is a typical variance of 20-30%

4_{There exists mathematical proof of this in the magnetostatic case [55]. However, in the eddy current} case, i.e., for conductive objects, this remains hypothetical.