Digital Signal Processing Reference
In-Depth Information
Fig. 16.14
Tracking of
interest points in translated
images via visuo-inertial
method
images where interest points have been tracked (Fig.
16.14
) for integration in an
assistive device for the visually impaired [
71
].
Position estimation errors in the presence of occlusions are the main problem
for all the above algorithms. Statistical methods can help to solve these tracking
problems.
16.3.2 Bayesian Approaches
The position of an object in an image at times
t
=
1
,
2
,...
is defined by a series of
states
X
t
,
t
=
1
,
2
,...
. The state evolution is modeled by dynamic Eq. (
16.6
):
X
t
=
f
t
X
t
−
1
+
W
t
(16.6)
where
W
t
is white noise. The relationship between a measurement
Z
t
and the state
X
t
is given by Eq. (
16.7
):
h
t
X
t
,N
t
Z
t
=
(16.7)
where
N
t
is white noise independent of
W
t
.
The state
X
t
is estimated using all the measurements
Z
s
,
s
0
,
1
,...,t
, obtained
up to an including time
t
. The information in the measurements is summarized by
the probability density function (pdf) for the state conditional on the measurements,
p(X
t
=
Z
1
,...,Z
t
)
.
A theoretical optimal solution can be obtained using a recursive Bayesian fil-
ter which consists of two steps: prediction and correction (update). The predic-
tion step uses the dynamic Eq. (
16.6
) to infer the pdf
p(X
t
|
Z
1
,...,Z
t
−
1
)
.The
|
update step uses this pdf and the likelihood function
p(Z
t
X
t
)
to estimate the pdf
|
p(X
t
Z
1
,...,Z
t
)
. If the scene contains only a single moving object, if the functions
f
t
and
h
t
are linear, and if the initial state
X
1
and the noise both have Gaussian dis-
tributions, then the pdf
p(X
t
|
|
Z
1
,...,Z
t
)
is given by the Kalman filter [
13
].
In the general case, the pdf for the state is not assumed to be Gaussian. The pdf
can be approximated using particle filters [
4
,
88
]. There are four steps in particle
filtering: particle sampling, prediction, pdf updating, and particle re-sampling, in
order to eliminate particles in regions with very low probabilities. The particle filter
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