## 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.