INTRODUCTION
Nowadays there are two strategies for target tracking. One is constructing
the target models, such as kernel histogram (Comaniciu et
al., 2000; Zuo et al., 2008) and invariant
moment (Taekyu and Sukbum, 2008), for matching and tracking;
the other is taking the differences between the target and background as the
features. Here, the centre image on which the tracking process is performed,
is referred to a confidence image (Avidan, 2005), likelihood
image (Chen et al., 2004) or weighted image (Collins
et al., 2005). The values of image pixels represent the probability
of the coordinate pixels belonging to some backgrounds or target (Kaawaase
et al., 2011; Benoudnine et al., 2009;
Kim, 2005).
The mean shift algorithm based on kernel histogram gradient matching, a model
matching and tacking algorithm, is widely applied in the practice. It takes
the kernel histogram of the features in the regions as the template for tracking
and the displacement for the centre of the targets is determined by the gradient
matching algorithm. The result has reliable performances under the case of rigid
moving object in visible light (Wu and Yang, 2007).
For the small target tracking in IR image on FLR, Yilmaz
et al. (2003) proposed a useful method. It describes the kernel histogram
using pixel grayscale and local standard deviation as 2D features and takes
the onedimensional kernel as the weighted factor of the histogram. Applying
the feature to the experiment has performed well in target tracking under the
meanshift frame. However, the meanshift algorithm has its implicit deficiency
in tracking the small IR target. Compared to the visible light image, the quality
of the IR is lower and this can be seen in the following two aspects: (1) For
IR image is of lower contrast, small SNR and fewer grayscales (Asaduzzaman
et al., 2010), kernel histogram cannot depict the features of the
small target with weak texture accurately; (2) The noise, inhomogeneity in space
and stray light can have great influence on the gray level distribution of the
small target to be tracked as well as the grayscale stability in different frames
which resulting the degeneration of the target model. The points above set some
limitations when we employ the meanshift algorithm based on model gradient
matching in the small target tracking. Further more, if the object to be tracked
is fast moving small target we must set the kernel window big enough as to cover
its moving range. Thus the influence of stray light and complex background is
increased and the respective features of kernel histogram and background can
cause aliasing which definitely diverges the algorithm and bring about bad tracking
results.
The second strategy has relatively strong stability, that is, if the selected
feature can distinguish the target from the background and other targets effectively,
it cannot be confused when we carry out the classification for the targets and
background. Collins takes the kernel density estimation on targets and background
grayscale histogram and forms grayscale probability density curve. Take the
log likelihood value in the curve as likelihood function which is coordinate
to the weighted image and the pixel value represents the probability of the
points to be discriminated as target pixel. Whereas in the process of tracking
the target, the weighted image in every frame is taken as data source and the
meanshift algorithm is adopted to seek the optimal model as the gradient matching
points (Collins et al., 2005). The method has
good tracking performance when the objects in the visible light image are of
rich color and of definite shapes but the performance is not good when the IR
objects are only of grayscale information. Besides, the implement is based on
the stability of grayscale value of the target image and thus it is not suitable
for small target tracking with the big SNR in spacetime domain.
In this study, a new approach is proposed to enhance the robustness of tracking
fast moving IR small target with unstable grayscale. Here we adopt the idea
of feature classification (Jun and Wang, 2008; Abed
and Zaoui, 2011; Ranjan and Khalil, 2007; Zhou
et al., 2007), propose a fast small target tracking algorithm based
on local background feature and mixture gussian models after extracting the
features of the small target and background. The Weighted Information Entropy
(WIE) is applied as the discrimination criterion of the local background complexity
to guide the updating of the model adaptively. The validity of this algorithm
is verified in the tracking experiment for the small targets in IR image.
Tracking algorithm based on gussian mixture clustering: The algorithm is summarized in the flowchart in Fig. 1, in which selecting the features for classification, implementing clustering method and classification method effectively are the key procedures.
Selecting the features for classification: The characters of IR image
with complex background f_{B} (x, y) can be summarized as follows (Anderson
and Iltis, 1997).
• 
Be of longer relevance length and usually of constant distribution
in large area 
• 
For the inhomogeneity in the thermal distribution in the scene and sensors,
the mean value of local grayscale may vary in a large range which is a twodimension
unstable random process with a slow change 
Compared to the background, the targets can be considered as a transient signal
and the emergence of the target destroys the constant consistency in the background.
If we neglect the effect of the optical aberration, the target image acquired
from Infrared focal plane arrays (IRFPA), for the diffraction that are caused
by the image size and atmospheric transmission, can be transformed into scattered
light points and represented by twodimension Gaussian distribution function.
In order to describe the spatial distribution of small target fully, the study
adopts the method of feature selection in study of Yilmaz
et al. (2003) and the extracted feature in pixel level can be represented
as:
where, f (x) is the grayscale value for the pixel point x in the search region;
g (x) is the grayscale standard deviation for the pixel point x in neighbor
regions.

Fig. 1: 
The tracking algorithm based on the classification for the
targets and background 
GMM clustering based on the background feature: To complete the task of fast small target tracking, the search window should be enlarged and the pixel number in the background will be larger than that in the target pixel. Further more, the relevance of local background is strong, thus the background pixel in the feature space has good clustering feature as shown in Fig. 2.
Under the premise, the best classification criterion between targets and local background should be based on the clustering based on local background feature, that is we can use the distance between feature vector and background to discriminate the category the feature vector belongs to. The feature that is larger than some threshold is considered as the target feature, otherwise it is the background feature. For the background in the complex IR image is unstable in timespace domain we construct the new model using the Gaussian mixture model as to represent the local background accurately.
The EM algorithm is a supervised clustering method. It builds models of given
sample on the idea of probability and statistics, estimates the parameters by
maximizing the expectation of the model and acquires the description of the
original data set finally. It has applied to many fields, such as image recovery,
image classification and target tracking (He et al.,
2007; Wang et al., 2007). Considering that
constant iteration updating can affect the realtime featurv, the approach in
this study first initialize the parameters for the data set by using the standard
Kmean clustering algorithm and then model the clustering for every category
by using EM algorithm, ie. determine the weighted value, mean value and covariance
matrix for every category, finally the accurate description for the feature
of the local background is attained.
For the selected local background B_{0} in the reference frame, the feature vector of the current pixel x ε B_{0} is ξ1 (x) = [f (x), g (x)}, the Gaussian mixture probability model of ξ (x) attributed the background pixel is as follows:
where, the ω_{j} is the prior probability that the feature vector ξ of pixel x belong to the jth Gaussian component of the background and satisfy:
If we assume every mixture component distribution with mean vector μ and
covariance matrix Σ, then:

Fig. 2: 
Classification for small targets and local background in the
feature space 
where, d is the dimension of feature space.
Now, the Gaussian mixture model for the pixel feature in the local background B_{0} can be represented as:
where as, taking into account the idea of the second strategy for IR target tracking we should make sure that the established background model has the optimal capacity for distinguishing the targets and background, so the target feature should be added into the model. By Reallocating the ω_{j} on the basis of the target feature, the similarity between the target feature and background is reduced. Let the feature vector of the pixel y ε T_{0} of the small target in the reference frame is η, we can have the formula as follows:
The υ_{j} represents the mean confidence level that the small target feature η belong to the jth Gaussian component of the background. For the component with low confidence level, the corresponding weighted factor should be increased; for that with high confidence level, the weighted factor should be reduced. In this way, the description for the differences between the feature of background and targets is formed and the adjustment for the prior probability of the model is as follows:
According to this formula we normalize and
get the final weighted factors as follows:
Thus, the Eq. 1 is transformed into:
Now, the Gaussian mixture model based on background feature can be represented
as:
Classification and discrimination model for the feature of the targets and
background: For the respective pixel feature of local background and small
target, ie., ξ (x) and η (x), we can calculate the mean value and
mean square deviation of the probability density in every category, respectively
as:
and
by Eq. 1.
Set the test threshold of the probability density as:
where,
And the k is a set constant used to control the false alarm in the discrimination of small target feature.
For the feature vector ζ (x) in the search window of the following frames we construct two kinds of discrimination model for the classification, the model A^{0} being as:
Updating of the gaussian mixture model: The Eq. 7
represents the two classification and discrimination model constructed from
two training samples according to the feature vectors of the targets and background.
Use this discrimination criterion to classify the targets and background and
locate the target points. For the inhomogeneity in the detector and the timespace
complexity in actual scene (for example, the influence of the background and
stray light), destroy the stability of the grayscale in the target itself and
cause a gradual change in the statistics distribution of the feature of local
background, so it will result in tracking deviation when we apply Eq.
7 in the frames. Therefore, to improve the robustness of the algorithm we
employ the Weighted Information Entropy (WIE), the index of local background
complexity, as the discrimination function of updating Gaussian mixture model,
here WIE is defined as:
where, the S is the image of local back ground and p_{s} is the proportion of the pixel with the grayscale s to all the pixels in the image.
Figure 3 represents a updating process for Gaussian mixture model when applying WIE. In the process we adopt the discrimination model A^{0} in the classification of the feature vectors in all search windows before the kth frame and WIE^{0} is the grayscale Weighted Information Entropy (WIE) of local background B_{0} in the training samples; for the image feature of the kth frame, ie., ζ_{k} (x), we use the model A^{0} to classify and discriminate its feature subset in the search window Q_{k1}.
Figure 4 represents the shifting process of the small targets in two continuous frames and the definition for the local background and search window in the tracking process. The updating process for the model is specified as follows.
If the Weighted Information Entropy (WIE) of the local ground B'_{k1}, ie. WIE'_{k1} satisfies:
(
is a set empirical value), we consider the discrimination of high confidence
level and take centre of target pixel point, ie. x_{K} as the target
location point in the image of the kth frame while keep the discrimination
model A_{0} invariant. Finally, store the pixel feature vectors sample
of local background, ie., B'_{k1}.
If the WIE of the local background B'_{k1} satisfies

Fig. 3: 
The updating process for GMM 

Fig. 4: 
The shifting process of the small targets in (k1)th and
kth frames and the related 
We consider there is a great change in the statistics features from the current
local background B'_{k1} to the local background B'_{k1} when
the discrimination model for clustering is established and further updating
process is needed; otherwise, the confidence level of the discrimination for
clustering is reduced. Considering that there would be few differences between
the statistics feature of local background in the two continuous frames including
the targets we adopt k2 frames to discriminate the features when updating the
model and get the feature vector sample of local background B'_{k1},
ie., V_{k–2}, to conduct the clustering and classification process
on Gaussian mixture model and then the Weighted Information Entropy (WIE) of
A^{1} and B'_{k1}, ie., WIE_{k1}, is attained. Finally,
combine the target feature vectors in k1 frames into a discrimination model
A^{1}, which means a model updating process is completed and save the
three parameters for clustering and tracking in the flowing frames.
EXPERIMENTAL RESULTS
To test the performance of the proposed algorithm for the small target tracking on the IR image sequence we use a uncooled infrared microbolometer with the size of 60x120 and conduct video collecting under the cloud background with the frequency in every frame of the FLR 25 frame per second. Figure 5 shows the reference image and local standard deviation image used in the training process for the classification algorithm and the selected target T_{0 }and local background B_{0 }are shown respectively in the two boxes.
Figure 6 represents the 4dimension Gaussian mixture clustering results of the target feature scattered points and background feature points. We can see that the feature vectors of the pixels in background B_{0 }are of higher clustering effect in every category while the dispersion in the target is bigger when compared to that of the local background.

Fig. 5: 
The data for reference frame: (a) the image (b) the local
standard deviation image 

Fig. 6: 
The scatter diagram for the feature of local background B_{0
}and target T_{0 }with 4dimension Gaussian mixture model 

Fig. 7: 
The tracking results of the fast moving small target 
Table 1: 
The prior probability before and after the adjustment 

The data for the prior probability before and after adjustment based on the target features, i.e., Pb (probability before adjustment) and Pa (probability after adjustment), is shown as in Table 1. It is noticeable that the algorithm reforms the prior probability effectively by the rule of the dispersion of the targets and local background.
The tracking results of the fast moving small target for the proposed algorithm are shown in Fig. 7. Obviously the target is a norigid body and it changes gradually in its shape and local background. The target is of 3x3 pixels size and have moved over 70 pixels in the time of 17 frames, i.e., the shifted pixels is more than its size and can be seen as a fast moving small target. Here the meanshift algorithm cannot track the target effectively while the method proposed in this study has good performance in tracking this kind of fast moving small target with unknown motion model.
CONCLUSIONS
In this study, we first analyze the limitations of the meanshift algorithm in tracking fast moving IR small target, then propose a tracking algorithm based on local background feature vectors Gaussian Mixture Model (GMM) clustering. The method extracts the grayscale of the small target and background as the feature vectors and treats the tracking process as two classification problems. In the process, the Weighted Information Entropy (WIE) is applied as the discrimination criterion of the local background complexity to guide the updating of the model adaptively. The experimental results show that this algorithm is verified effectively in the tracking for the small targets in IR image.