SUBPIXEL VISUAL TRACKING BASED ON ADAPTIVE
STRATEGIES
H
´
ector Barr
´
on, Janeth Cruz, Leopoldo Altamirano
National Institute of Astrophysics Optics and Electronics
Luis Enrique Erro No. 1
Keywords:
Target tracking, subpixel accuracy, probabilistic methods, adaptive tracking.
Abstract:
Several applications based on visual tracking need a better accuracy to perform a more reliable analysis of the
objects in scene. However, it is necessary to deal with environments with different atmospheric conditions.
Object dynamics can affect tracking throughout time. In this work, a tracking method with subpixel mea-
surements is described, where quality of the state estimate of the object is enhanced. The proposed scheme
is robust in scenes with occlusions and changes in appearance of the target. The target model is adapted to
size changes of the object, avoiding aperture problem and integration with false information. The state of the
object and its aspect along time are estimated. Each pixel is modeled by a random variable because the set of
pixels represents the non-observable surface of target where real value of pixels can be affected by noise. This
assumption allows the design of a gradual scheme for model updating. Subpixel precision in tracking is based
on an iterative method that uses the similitude surface between the target model and the current image of the
object on tracking.
1 INTRODUCTION
Object tracking allows to focus computer resources
for analysis on objects into dynamic environments.
When accuracy in tracking is increased, it is possi-
ble to perform an analysis more reliable on study ob-
jects in applications where availability of information
has a great importance (Awcock, 1996). High accu-
racy in tracking increases reliability at surveillance
systems whose functionality include intelligent mo-
tion detection, event analysis and domestic security.
However, it is required to deal with dynamic envi-
ronments where situations with different atmospheric
conditions are presented such as rain, haze and dust.
Furthermore, it is possible to deal withnoise produced
by sensor or data transmition, where tracking efficacy
is decreased. Another kind of problems are caused by
dynamic behavior of the object on tracking.
Image tracking analyzes a projection of the object
with respect to the observer. If object come near or far
way, object appearance may change. This situation
produces that the object model could be not enough
to maintain tracking or it could produce incorrect in-
formation on the state of tracking object. This paper
presents an integral solution for tracking with sub-
pixel accuracy, dealing with size changes on object
apparence under dynamic environment.
In section 2, problems and related approaches are
described. The proposed method of tracking is de-
tailed in section 3. Finally, results and conclusion are
found in sections 4 and 5, respectively.
2 BACKGROUND
A tracking system collects a set of sensor data that
contains attributes of objects with potencial interest.
This set of data is known as measurements and the
object on tracking is known as target. Internally, a tar-
get is represented by a state vector whose elements are
parameters that characterize its behavior such as posi-
tion, velocity, size and color. The state is updated by
each new measurement. In image-based tracking, it
is important for monitoring each spatial and temporal
change that an object suffers in a video sequence. So,
this process depends on temporal matching, where
two images represent the target state at two different
instant of time.
2.1 Target detection
Target detection is a process to search an interest ob-
ject into an image and get its position. This is done
286
Barrón H., Cruz J. and Altamirano L. (2006).
SUBPIXEL VISUAL TRACKING BASED ON ADAPTIVE STRATEGIES.
In Proceedings of the First International Conference on Computer Vision Theory and Applications, pages 286-293
DOI: 10.5220/0001377302860293
Copyright
c
 SciTePress

by image registration, where an image represents the
target and another image is the search space. This
problem has been studied at different ways, but it is
difficult to find a robust and accuracy scheme and it
depends of the target representation and the condi-
tions to localize it. Image registration is one of the
fundamental tasks in vision systems. However, this
is not an easy topic because there are several factors
that affect performance of the vision systems as sen-
sor noise, different views of the observer, motion per-
turbations, changes on objects state by motion and at-
mospheric and illumination conditions.
Image registration can be defined as mapping
between images, considering spatial and intensity
differences. Let be I
1
and I
2
images related by
I
2
(x, y)=g(I
1
(f(x, y))) (1)
where f is a spatial transformation between two coor-
dinates and g is a radiometric transformation. Image
registration is defined as estimation of geometric and
radiometric transformations, such that two images
could be compared by detections of coincidences. It
is important to realize that if the number of parame-
ters that define the relation between two images of
the same object is increased, then the complexity of
searching is increased too.
Image registration depends of three elements
(Brown, 1992; Zitova, 2003):
• Characteristic space. In image registration, it is
important to determinate which set of characteris-
tics defines the best representation of image. Selec-
tion is affected by different factors and conditions,
such as quantity of obtained information, sensiti-
vity to properties of sensor and scene and compu-
tational cost. Sometimes, a characteristic space is
created with intensity levels at pixels or a transfor-
mation on them, as FFT. Another schemes are de-
fined by structural features (borders, contorns, in-
terest points, centroids) or texture properties (con-
trast, homogeneity, correlation).
• Similitude measure. This measure identifies the
compatibility degree between two images. Simi-
litude metric is used for finding the required para-
meters in a mapping between related images. Some
of the most used similitude measures are cross-
correlation, sum of absolute difference, sum of
square difference and phase correlation. Further-
more, it is posible to use methods more complex as
bayesian detectors and neural networks. According
with application, if characteristics space and simi-
litude metric are correctly selected, then it is pos-
sible to ignore some non-relevant distortions for a
correct matching.
• Search strategy. In case where only displacement is
required, it is sufficient with a sequential search to
determine mapping. However, if mapping requires
more parameters, searching must be more com-
plex. Some techniques used are hierarchy search,
relax labeling, dynamic programming and heuris-
tic search. The number of parameters that defines
the mapping and computational cost are the most
important factors to determine the search strategy.
2.2 Visual Tracking of Objects
Visualtracking is used in a wide range of applications,
but there is not an algorithm to be used in whatever
conditions. In general, tracking methods can be sepa-
red in two groups: tracking based on motion detection
and tracking based on models.
Tracking based on motion detection uses detection
algorithms as optical flow, gaussian mixture or image
difference. This approach has a good performance
and it is posible to work on no-rigid objects. However,
these schemes do not use a target model, they are sen-
sible to false detections and tend to loss tracking on
target, if displacements are very small.
Tracking based on models uses image registration
for detection because it is defined a target model.
These techniques are more robust and it is possible
to use image analysis more complex to obtain mea-
surements. This approach has major computational
cost, so deformation of objects must be considered.
However, information about behavior could help to
enhance efficacy (Hong, 2002).
In (Comaniciu, 2003), an object model is crea-
ted with a probability distribution function pdf from
histogram. The target position is defined by Bhat-
tacharyya coefficient as similitude metric. The search
strategy begins on previous position of target and it is
guided by a derivative kernel. This approach is fast,
an exhaustive search is not required and subpixel mea-
surement is obtained. But, size changes are not sup-
ported.
In (Son, 2002), a correlation window is adapted to
size changes of targets. Using temporal and spatial
gradient between two consecutive images, the occu-
pation ratio in window is obtained. If ratio is less then
window is decreased one pixel, otherwise the window
is increased. In (Chien, 2000), correlation window is
adapted by modeling with motion vectors. Direction
and magnitude of contraction or expansion depend on
simple image processing. This approach is useful if
displacement are small.
Probabilistic approaches have been suitable solu-
tions to above presented problems, such as adapta-
bility in tracking based on target behavior. In (Ras-
mussen, 2001), it is proposed to take advantage of
a set of ramdom samples around prediction of target
geometric parameters and to use a pdf as similitude
measure. The evaluation of the samples set defines the
measurement process, where samples with low prob-
SUBPIXEL VISUAL TRACKING BASED ON ADAPTIVE STRATEGIES
287

ability are eliminated. Remainder samples are inte-
grated to compute real position of the target.
In (Ross, 2004) a method similar to (Rasmussen,
2001) is proposed, where a sampling is performed
on geometric parameters of target. This approach in-
cludes a gradual updating of target model based on
PCA, allowing reliability due false detection.
2.3 Subpixel Measurements
Methods to obtain subpixel measurements can be di-
vided in two groups: techniques based on numeric
calculus and techniques based on image interpolation.
The computation of centroid is the simplest method
used, and extention is determinated by parabolic or
gaussian fitting. This approach produces a set of
equations whose result is the point of maximum simi-
litude with subpixel accuracy (Shortis, 1994). These
methods have good performance but they have con-
strain accuracy.
The second approach (Frischholz, 1995) is based
on image matching. In general, a sampling is per-
formed on a reference image and each sample image
is a possible real position. Later, each image is eva-
luated to obtain the nearest position with different si-
militude measures.
Work in (Thevenaz, 1998) presents a method with
subpixel accuracy based on interpolation. The search
space is formed by a set of parameters of an affine
transformation, and the similitude measure is based
on squared difference between the reference image
and an input image. The measure of similitude is de-
fined by
χ
2
(p)=
1
N
N

i=1
(f
R
(x
i
) − f
T
(Q
p
(xi))
2
(2)
where N is the number of pixels, x
i
is the coordinate
of each pixel, f
R
(x) is the reference image, f
T
(x)
is the input image, Q
p
is the geometric transforma-
tion defined by vector P . So, the search strategy is
an optimization of a no-linear problem that is solved
by Marquardt-Levemberg method. In this approach,
gradient of the image is used to find the set of para-
meters that minimize eq. 2. This method converges
quickly to an accuracy solution, but it depends on in-
terpolation scheme and a correct initialization of the
transformation parameters is required.
3 PROPOSED METHOD
Kalman filter allows to obtain a suitable estimation of
the target position, due to its dynamics satisfies with
the linear-gaussian assumption. However, when ima-
ges are used in some application, the discrete nature
of pixels causes measurements to get some additive
noise no-gaussian, then the estimation is more diffi-
cult. When measurements are supplied with subpixel
accuracy, filter input has not constrains, hoping to re-
duce estimation error.
The proposed tracking system allows to get sub-
pixel accuracy and it is reliable due changes of tar-
get apparence, in dynamic environments. Taking into
account that object localization is non-observable be-
cause measurements are affected by noise, then it is
required to estimate the target position by means of a
random variable. This estimation is provided as pre-
diction to use in real applications.
A general description of the proposed system is
given in Fig. 1. An object model is defined to
enhance measurements of tracking. At each input
image, the target model is searched to get its new posi-
tion. When position measurement is already obtained,
size is measured and subpixel accuracy is calculated.
Prediction by Kalman filter allows to know the target
position in next image and to define the search space
for next measure process.
Detection
Size measure
Subpixel
measure
Model update
Predictor
Search
region
Figure 1: General diagram of the proposed method.
3.1 Representation and Localization
Position measurement of a target into an input image
is obtained from a detector based on model. So, it
is required to define the target representation, search
space and similitude measure.
3.1.1 Target Representation
The target representation is given by two elements: a
state vector that describes the target behavior and a
model image that represents aspect of target through-
out time. First, dynamics of target is resumed by the
state vector x
t
at time t. Measurement vector is re-
lated to state by means of function z
t
= h(x
t
). Mea-
surements vector is determinated by
z
t
=[x, y] (3)
VISAPP 2006 - MOTION, TRACKING AND STEREO VISION
288

where x, y is the target position on time t. Second,
target surface is presented as a set of pixels in image
of the target. However, images could be noised and
gray levels at pixels can vary on time. So, the real
surface of target may be non-observable then
z
I
(x, y)=I
R
(x, y)+w
i
(4)
where z
I
is the pixel value (x, y), I
R
(x, y) is the pixel
value that represents the real value of target surface
and w
i
is gaussian noise. At the target model, value of
each pixel is represented by a random variable defined
as a gaussian pdf I
R
, with mean
¯
I
R
and variance σ
I
R
.
This model must be adaptive to changes on aspect of
target.
3.1.2 Search Space
In (Rasmussen, 2001), it is suggested that many pro-
blems in vision may be solved with a MAP estimator.
This measurement process is based on maximization
of p(z
t
|I,x
t−1
) with measurements z
t
, due to an in-
put image and the previous state of target, the most
probable measurement is found according to
p(z
t
|I,x
t−1
) ∝ p(z
t
|x
t−1
)p(I|z
t
) (5)
where p(z
t
|x
t−1
) is a pdf that describes prediction
of the current measurement and p(I|z
t
) is a likeli-
hood function that defines probability where a specific
image is observable, due position z
t
. The likelihood
function determines similitude between target model
and an input image. The pdf p(z
t
|x
t−1
) is defined by
its mean ¯z
t
and variance σ
z
t
, such as
p(z
t
|x
t−1
)=N(z
t
, ¯z
t
,σ
z
t
) (6)
To determinate if a position (x, y) is located into
the set of posible positions, (Stauffer, 1999) defines
an approximation for gaussian functions. A gaussian
distribution considers a position if this one is located
to 2.5 standard deviation around the mean. Approxi-
mation is given by M
t
, where
M
t
(x, y)=

1(x, y) is included
0 otherwise
(7)
≈ p(z
t
|x
t−1
) (8)
The search space is defined by a circular area
around the position prediction, as it is showed in Fig.
2. This region is set by a gaussian function on the
positions space. As the measurements are affected by
gaussian noise and real movement of target, detection
can be described by a linear dynamic model.
Figure 2: Search space.
3.1.3 Similitude Measure
Target on tracking could have different patterns of in-
tensity, where horizontal and vertical gradients pro-
vide information on target aspect. So, the similitude
measure must take advantage of all information ava-
liable on pixels, such as squared difference or norma-
lized correlation. However, it is possible that changes
on illumination, low contrast and present noise affect
those criterions. Correlation with full sustracted mean
(Ronda, 2000) is more robust and allows a larger
range of values, and similitude measure is distributed
correctly to avoid false detection. A likelihood func-
tion based on correlation is defined as
p(I|z
t
)=sig (FSMC(x, y))) (9)
with
FSMC(x, y)=


x
I
R
(x)I(x) − M
¯
I
¯
I
R

⎛
⎜
⎝
1


x
I
2
R
(x) − M
¯
I
R
2

1
2
⎞
⎟
⎠
⎛
⎝
1


x
I
2
(x) − M
¯
I
2

1
2
⎞
⎠
(10)
where sig is the sigmoid function, x is a position
into a image, I
R
(x) represents each pixel at image of
the target model, I(x) represents each pixel at input
image defined as projection of z
t
, M is the number of
pixels in each the image,
¯
I
R
and
¯
I are the mean of the
target model and the input image, respectively.
3.2 Size Adaptive Tracking
Size and change ratio are parameters that describe dy-
namics of target. Let be x
s,t
a state vector that des-
cribes the target behavior on scale space. The mea-
surement vector z
c,t
contains target size related with
SUBPIXEL VISUAL TRACKING BASED ON ADAPTIVE STRATEGIES
289

state vector by
z
s,t
= h(x
s,t
) (11)
The measure process is based on obtaining mea-
surement z
s,t
that maximize p(z
s,t
|I,x
s,t−1
). Ac-
cording to Bayes theorem
p(z
s,t
|I,x
s,t−1
) ∝ p(z
s,t
|x
s,t−1
)p(I|z
s,t
) (12)
where p(z
s,t
|x
s,t−1
) is a pdf that describes prediction
on current scale of target and p(I|z
s,t
) is a likelihood
function that defines probability of a possible projec-
tion of z
s,t
. Further details may be found in (Barron,
2004).
3.3 Precision Subpixel
A pixel-based measure takes advantage of all infor-
mation on pixels, as interpolation approaches. How-
ever, this scheme has a wide search space because
it is possible to perform comparations using the full
range of real displacements. The method used in this
paper is based on search of an optimum geometric
transformation using gradients, whose functionality
is enhanced by the information intrinsic of a tracking
problem.
3.3.1 Similitude measure
The similitude measure is based on difference
between image of the target model and an input
image, defined by the discrete position of target. The
similitude measure is defined as
ε
2
=

z∈R
q
(I
R
(z) − I
E
(Q
p
(z)))
2
dz (13)
= ||I
R
(z) − I
E
(Q
p
(z))||
2
(14)
where Q
p
is a linear transformation whose parameters
are into p,q is the dimension of the coordinate vector
z; I
E
is a test image and I
R
is the target model. Q
p
is
defined as
Q
p
(z)=

a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33

x
y
1

(15)
where z =(x, y). The transformation that considers
displacements of the target is given by
Q
p
(z)=

10p
x
01p
y
00 0

x
y
1

(16)
where p
x
and p
y
are parameters of transformation that
represent displacements applied to image and they
must be computed in subpixel matching.
3.3.2 Search Strategy
Optimization of the similitude measure defined by eq.
13 is obtained by solving
∂ε
2
(p)
∂p
=0 (17)
using a method no-linear based on gradients, known
as Levemberg-Marquard. At each iteration of that
method, a comparation between I
R
and I
E
is per-
formed, applying the transformation Q
p
. The para-
meter vector p is updated by
p
t+1
= p
t
+ δp
t
(18)
with
M

i=1
α
kl
δp
l
= β
k
(19)
where α
kl
is the kl element of a hessian matrix and β
k
is proportional to gradient of ε
2
. The finite approxi-
mation of eq. (13) is given by
ε
2
∼
=
χ
2
(p)=
1
N
N

i=1
(I
R
(x) − I
E
(Q
p
(x)))
2
(20)
where N is the pixel number in reference image. To
solve the linear-optimization, β
k
is computed as
β
k
=
−1
2
∂χ
2
(p)
∂p
k
(21)
=
1
N
N

i=1
(I
R
(x) − I
E
(Q
p
(x)))
∂I
E
(Q
p
(x))
∂p
k
(22)
and each element of the hessian matrix is defined as
α
kl
=
1
2
∂
2
χ
2
(p)
∂p
k
∂p
l
(23)
=
N

i=1
(
∂I
E
(Q
p
(x))
∂p
k
∂I
E
(Q
p
(x))
∂p
l
(I
R
(x) − I
E
(Q
p
(x)))
N
∂
2
I
E
(Q
p
(x))
∂p
k
∂p
l
) (24)
The second derivative terms are usually ignored, so
b
kl
is defined as
b
kl
=
1
N
N

i=1

∂I
E
(Q
p
(x))
∂p
k
∂I
E
(Q
p
(x))
∂p
l

(25)
VISAPP 2006 - MOTION, TRACKING AND STEREO VISION
290

and the set of equation is given by
{
α
kl
= b
kl
(1 + λ) k = l
α
kl
= b
kl
k = l
(26)
where λ ≥ 0 defines the updating step for p, given the
gradient direction.
3.3.3 Algorithm of Subpixel Matching
In this section, the iterative method is presented to ob-
tain the subpixel measure of the target position.
1. The parameter vector p = p
0
is initialized and
∆p =0is defined.
2. Image I
E
(Q
p
) is obtained applying transformation
Q
p
to the input image.
3. The set of equation is solved to get ∆p and p is
updated by p
0
+∆p.
4. RMSE is computed according with the similitude
measure.
5. ε
2
is evaluated.
• If ε
2
is decreased, λ is decreased and p
0
= p.
• Otherwise, λ is increased
6. If current value of ε is equal to above, the process
is stopped. Otherwise, it continues with step 2.
It is required to use a reliable scheme of interpola-
tion, such as B-Spline. Indeed, the parameter vector p
must be initialized correctly to avoid local minimum,
but the above problem is solved with the discrete de-
tection of the proposed system.
3.4 Updating of the Target Model
Such as it is realized in section 3.1.1, each pixel is
considered as a gaussian distribution that includes the
possible values of the pixel. These values reveal the
real surface of the target, considering gradual changes
of ilumination or gaussian noise.
So, it is necessary to consider {z
I
(t)} as the set
of values that the pixel has had throughout sequence.
The mean and variance that define the gaussian dis-
tribution produce a large computational cost. Due to
that, an approximation is defined as
¯
I
R
=(1− α)
¯
I
R
+ αz
I
(27)
σ
I
R
=(1− α)σ
I
R
+ α(z
I
−
¯
I
R
)
T
(z
I
−
¯
I
R
) (28)
with α as learning ratio. The updating algorithm is
performed as follows:
1. After subpixel and size measure, an image used for
updating is obtained as a projection of prediction
x
t|t−1
.
2. The image is scaled to the model size.
3. A comparation pixel-to-pixel is performed.
(a) If pixel is located to 2.5 standard deviation, dis-
tribution is updated according with eq. (27-28).
(b) Otherwise, distribution is not affected.
4 EXPERIMENTAL RESULTS
A set of synthetic and real image sequences are used
to evaluate the performance of the proposed algorithm
and estimate accuracy of the method due to changes in
appearance of target, . Three different basis synthetic
sequences with 120-180 frames were created, where
a target maintains linear or nearly linear movements
with displacements between 2-8 pixels. The target
size changes from 15 × 12 to 110 × 90 pixels and
the size change rate is from 1 to 4 pixels per frame.
Gaussian noise is applied to each sequence with five
different signal-to-noise ratios (SNR) defined as fol-
lows:
SNR = 20 log
| µ
T
− µ
B
|
σ
N
(29)
where | µ
T
− µ
B
| is the absolute difference of the
intensity average between the target and the back-
ground and σ
N
is the standard deviation of the added
Gaussian noise.
It is defined E
p
as the error between the correct
position of the target and estimated position with the
proposed method (eq. 30), X is the value of the coor-
dinate obtained by predictor and X
r
is the value of
the coordinate in the real position.
E
p
= ||X − X
r
|| (30)
Furthermore, error of size in tracking window with
respect to real size of target is defined with eq. 31,
where A is the area of estimated window and
¯
A is the
real area of a window to be set around the target.
E
s
=
| A −
¯
A |
¯
A
(31)
In Fig.3, results of the proposed tracking algorithm
on a synthetic sequence is shown, where the SNR
value is 6dB. In this sequence, the target is affected
by size changes and occlusion along of the sequence.
The adaptive algorithm maintain the tracking even un-
der occlusion, because it has an estimation of the ob-
ject appearance when the object appears again.
In Table 1, statistics of displacement error on syn-
thetic sequences are shown with varying SNR from
0.0 to 10 dB. The mean error describes accuracy and
standard deviation shows precision of our adaptive al-
gorithm.
SUBPIXEL VISUAL TRACKING BASED ON ADAPTIVE STRATEGIES
291

(a) Image 79
(b) Image 123
Figure 3: Results in a synthetic sequence.
To evaluate the proposed algorithm in real envi-
ronments, six different sequences are used. Different
objects were selected as target where appearance and
size change over time. Targets have linear and nearly
linear movements with some atmospheric factors such
as snow and haze. In Fig. 4 is shown that adaptive
method maintains tracking over sequence Kw,even
if target changes of appearance by changing on orien-
tation and size.
The general performance of the adaptive algorithm
was of 19-27 frames per second, depending on tar-
get size. The subpixel optimization converges with
4-7 iterations throughout time. We demostrated re-
Table 1: Error in estimation of position.
Error in synthetic sequence
Position
Pixel Subpixel
SNR Mean Std. Dev. Mean Std. Dev.
10 0.1407 0.1003 0.0451 0.0191
8 0.1601 0.1082 0.0887 0.0674
6 0.2017 0.1607 0.1366 0.0603
4 0.2923 0.1949 0.1545 0.1232
2 0.3057 0.2507 0.1908 0.1345
liability of the proposed method to be compared with
two schemes more. A dynamic strategy takes into ac-
count changing on the reference image at each frame
and a static scheme consists on maintain the reference
image without changes along the sequence. To eva-
luate target position error, we use the absolute diffe-
rence between real position and the estimated posi-
tion in this algorithm. Tracking with dynamic strategy
over this sequence is affected by the drift problem in-
duced by the discrete nature of the visual tracking.
Meanwhile, when the target changes its appearance,
the static strategy is obsolete. The adaptive strategy
maintains tracking to the end of the sequence.
(a) Image 10
(b) Image 151
Figure 4: Results in real sequence Kw.
Fig. 5 shows visual results over sequence Kw.
Fig 5(a) shows as the dynamic strategy lost the ori-
ginal object because it actually follows a corner of the
target. However, the adaptive strategy maintains the
tracking over the original object.
5 CONCLUSIONS
We have presented a visual tracking method based on
probabilistic methods where tracking is seen as an
image registration problem. Matching between the
VISAPP 2006 - MOTION, TRACKING AND STEREO VISION
292

(a) Dynamic strategy
(b) Adaptive strategy
Figure 5: Results in real sequence Nevel.
reference image and the input image is defined by es-
timation over the target behavior. Results of experi-
ments over synthetic and real video sequences ensure
reliability of the adaptive proposal when the object
size changes and occlusions are presented. This work
presents comparisons with two techniques to update
the reference image where the adaptive probabilistic
method gives satisfactory results. The proposed al-
gorithm was tested over sequences in dynamic envi-
ronments and Gaussian noise. As it was shown, the
accuracy was increased in the measurements process,
allowing to enhance the estimation of the target in
tracking. Another advantage was a scheme to update
gradually the target model for improve tracking due
to appearance changes. As future work, particle fil-
ter could be used to enhance estimation for maneu-
vering targets and geometric transformation could be
extended in optimization method.
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