Vol 2 Issue 6

1. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
237
Road-Map Estimation using Kalman Filter
Suman Chaudhary1, Rajiv Dhaiya2
Electronics and communication1, 2, PDM college of Engg1, 2
Email: suman.tarar@gmail.com1, rajiv_engg@pdm.ac.in2
Abstract—An extended target tracking framework with polynomials in order to model extended objects in the
scene of interest from imagery sensor data is presented in this paper. State-space models enables the use of
Kalman filters in tracking for proposed extended target objects. A general target tracking algorithm that utilizes
a specific data association method for the extended targets is presented. In order to detect and initialize extended
tracks from the point tracks some form of prior information always use by the overall algorithm.
Index Terms—Extended target tracking, parabola, polynomial, Kalman Filter.
1. INTRODUCTION
The Kalman filter is essentially a set of mathematical
equations that implement a predictor-corrector type
estimator that is optimal in the sense that it minimizes
the estimated error covariance when some presumed
conditions are met. Since the time of its introduction,
the Kalman filter has been the subject of extensive
research and application, particularly in the area of
autonomous or assisted navigation. This is likely due
in large part to advances in digital computing that
made the use of the filter practical, but also to the
relative simplicity and robust nature of the filter itself.
e Kalman filter has been used extensively for tracking
in interactive computer graphics. We use a single-constraint-
at-a-time Kalman filter [1-2].
Fig.1. Block Diagram of the Kalman Filter
The Gauss-Markov signal model discussed had the
form
y[m] = Py[m-1] + W[m] m≥0
…(1)
We describe a sequential MMSE estimator which
allow us to estimate y[m] based on the data {x[0],
x[1]…..x[m]} as m increases. Such an operation is
referred to as filtering. It compute the estimator y(m)
based on the estimator for the previous time sample
y(m-1) and so is recursive in nature. This is called as a
kalman filter.
2. SIMULATION SETUP
The block diagram of the kalman filter is shown in fig
1 [3]. The dynamic model for the signal is an integer
part of the estimator. The versatility of the Kalman
filter accounts for its widespread use. It can be applied
to estimation of a scalar Gauss-Markov signal as well
as to its vector extension. Furthermore, the data
consisted of a scalar sequence such as
{x[0],x[1]…..x[n]}, can be extended to vector
observations or {x[0],x[1]…..x[n]} [4].
Consider the scalar state equation and the scalar
observation equation
s[n] = as[n-1] + u[n]
...(2)
x[n] = s[n] + w[n]
…(3)
Where u[n] is zero mean Gaussian noise with
independent samples and
E (u2 [n]) = σu
2, w[n] is zero mean Gaussian noise with
independent samples & the value E (w2 [n]) = σn
2. We
finally assume that s [-1], u[n], and w[n] are all
independent. Finally it assume s [-1] ~ N (μs, σs
2). The
noise process w[n] differs from WGN only in that its
variance is allowed to change with time. To estimate
s[n] based on the observations {x[0],x[1]…..x[n]} or
to filter x[n] to produce ŝ[n] [5-7].
More generally, the estimator of s[n] based on the
observations {x[0],x[1]…..x[m]} will be denoted by
ŝ[n|m]. Our criterion of optimality will be the
minimum Bayesian MSE or
E[(s[n]-ŝ[n|n])2
…(4)
where the expectation is with respect to
p(x[0],x[1]…..x[n], s[n]). But the MMSE estimator is
just the mean of posterior PDF or
ŝ[n|n] = E(s[n]|x[0],x[1,……,x[n]])
…(5)
with zero means this becomes
ŝ[n|n] = CθxC-1
xx x
…(6)
where θ = s[n] and x = [x[0],x[1]…..x[n]]T are jointly
Gaussian. Assuming Gaussian statistics for the signal
and noise, the MMSE estimator is linear and is

2. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
238
identical in algebraic form to the LMMSE estimator
[9]. The implicit linear constraint does not detract
from the generality since already know that the
optimal estimator is linear. Further from above
equations and the orthogonality principle we will have
ŝ[n|n] = E(s[n]|x[0],x[1,……,x[n-1]) + E(s[n]|x[n])
…(7)
ŝ[n|n] = ŝ[n|n-1] + E(s[n]|x[n])
…(8)
Which has the desired sequential form.
Let X[n] = [x[0]x[1]…..x[n]]T and x~[n] denote the
innovation. The innovation is the part of x[n] that is
uncorrelated with the {x[0],x[1]…..x[n-1]} or
x~[n] = x[n] - x~[n|n-1] …(9)
Further to determine E(s[n]| x~[n]) note that it is the
MMSE estimator of s[n] based on x~[n]. As such it is
linear, and because of the zero mean assumption of
s[n], it takes the form
E(s[n]| x~[n]) = K[n] x~[n]
…(10)
E(s[n]| x~[n]) = K[n](x[n] - x~[n|n-1])
…(11)
Where,
Hence,
K[n] = E[s[n](x[n] – ŝ[n|n-1])]
…(12)
E[(x[n] - ŝ[n|n-1])2]
or
K[n] = E[(s[n] – ŝ[n|n-1])(x[n] - ŝ[n|n-1])]
E[(s[n] - ŝ[n|n-1] + w[n])2]
K[n] = E[(s[n] – ŝ[n|n-1])2]
σn
2 + E[(s[n] - ŝ[n|n-1])2]
…(13)
The numerator is just the minimum MSE incurred
when s[n] is estimated based on the minimum one step
prediction error. We denote this by M[n|n-1], so that
…(14)
Hence, we can summarize it below for n≥0 as
Prediction:
ŝ[n|n-1] = aŝ[n-1|n-1]
…(15)
Minimum Prediction MSE:
M[n|n-1] = a2M[n-1|n-1] + σu
2
…(16)
Kalman Gain
Correction:
ŝ[n|n] = ŝ[n|n-1] + K[n](x[n] - ŝ[n|n-1])
…(17)
Minimum MSE:
M[n|n] = (1 – K[n])M[n|n-1]
…(18)
Although derived for μs = 0, the same equations result
for μs ≠ 0. Hence, to initialize the equations we use ŝ[-
1|-1] = E(s[-1]) = μs, and M[-1|-1] = σ
2, since this
amounts to the estimation of s[-1] without any data.
A block diagram of Kalman filter is shown. It is
interesting to note that the dynamic model for the
signal is an integral part of the estimator. Furthermore,
the output of the gain block as an estimator of u[n] [8].
3. SIMULATION RESULTS WITH
KALMAN FILTER
Object tracking using conventional Method
In this experiment, we are performing the object
tracking based on the conventional method that is
using the approach of central difference. Fig. 2 depicts
the Object tracking using conventional method. Fig. 3
shows the true position of the object with
conventional method. Fig. 4 demonstrates the
estimated position of the object with conventional
method. Fig. 5 shows the comparison of true and
estimated position of the object with conventional
method.

3. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
239
Fig. 2 Object tracking using conventional method
Fig. 3 True position of the object with conventional
method
Fig. 4 Estimated position of the object with
conventional method
Fig. 5 Comparison of true and estimated position of
the object with conventional method
4. CONCLUSION
Kalman filter estimates a process by using a form of
feedback control i.e. the filter first estimates the state
using the previous state and then obtain feedback in
the form of measurements. Thus the filter equations
are of two groups. The time update equations that
projects current state estimate ahead in time and
measurement. The main application of the Kalman
filter in robot vision is the following object, also
called tracking. To carry out this, it is necessary to
calculate the object position and speed in each instant.
As input is considered a sequence of images captured
by a camera containing the object. Then using a
image processing method the object is segmented and
later calculated their position in the image.
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[5]. Nickels, K. ; Hutchinson, S., “Model-based
tracking of complex articulated
objects”, IEEE Transactions on Robotics and
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4. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
240
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