Aerospace and Electronic Systems Magazine July 2017 - 46
Automatic Target Recognition in Missing Data Case
TARGET CLASSIFICATION
To demonstrate ATR performance with missing data, the multipleclass classification problem with three types of civilian vehicles
(namely sedans, SUVs, and vans) is considered. The local learning
based feature selection method [11], [12], which performs feature selection and classification simultaneously is applied to this problem.
I
Let D = ( xi , yi )
⊂ R z × {1, 2, 3} be a training data set with
i =1
three types of targets (sedan, SUV, and van), where I and Z denote
the number of training points and the vector size of xi, respectively.
Given a sample xi, we first identify two types of nearest neighbors
for xi; one from the same class which is termed the nearest hit or
NH, and the other from a different class which is termed as the
nearest miss or NM. The margin of xi is then given by
{
}
ρi = xi − NM ( xi ) 1 − xi − NH ( xi ) 1 ,
(24)
where ⋅ 1 is the 1 norm or the Manhattan distance. An interpretation of the margin in (24) is a measure of how much xi can be
corrupted by noise before being misclassified [15]. By adopting
large-margin theory [16], [17], a learning algorithm which reduces
a margin-based error function usually generalizes well on unseen
test data. The natural idea of this classification method is to scale
each feature in an original space and then obtain a weighted feature space parameterized by a nonnegative vector w. Therefore, the
margin-based error function in the feature space is minimized. The
margin of xi in the induced feature space is given by
ρi ( w ) = xi − NM ( xi ) w − xi − NH ( xi ) w = w T ui
1
1
(25)
where u i = xi − NM ( xi ) − xi − NH ( xi ) and ⋅ is an element-wise
absolute value operator. Since ρi(w) is a linear function of w, the
margin in (25) requires only information about the neighborhood
of xi, and no assumption is made about the underlying data distribution. This implies that an arbitrary nonlinear problem can be
transformed into a set of locally linear problems [11], [12]. The
local linearization of the nonlinear problem is available to estimate
the feature weights by using a linear model. However, the main
problem of the margin definition in (25) is that the nearest neighbors of a given sample are unknown before learning. With the existence of many irrelevant features, the nearest neighbors identified
in the original space may be different from those in the weighted
feature space. In order to overcome the uncertainty in defining local information, we make use of the probabilistic model where the
nearest neighbors of the given samples are handled as hidden variables. By adopting the expectation-maximization algorithm [18],
we estimate the margin by calculating the expectation of ρi(w) by
averaging out the hidden variables as follows:
Figure 11.
Feature alignment (in 30% missing data).
(
)
x − xi
ρ i ( w ) = wT E q ~ x q − xi − E q ~
q
i
i
= w T P x q = NM ( xi ) w x q − xi
q∈
i
(
Figure 12.
Final feature alignment of a sedan, an SUV, and a van before target classification (with 30% missing data).
46
IEEE A&E SYSTEMS MAGAZINE
−
P(x
q∈i
)
q
(26)
= NH ( xi ) w x q − xi = w T u i ,
)
JULY 2017
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