Digital Signal Processing Reference
In-Depth Information
4.3 PSO-Based Object Tracking
The visual object tracking can be perceived as a dynamic optimization problem. In
the PSO-based tracking, in each frame, the object's state is determined using a fit-
ness function expressing the object's appearance. In order to cover possible state
changes between consecutive images, the particles are propagated according to a
weak transition model. In this section, we show how single object tracking can be ac-
complished by PSO. We present the fitness function as well as the re-diversification
of the swarm to cover the object state changes between the consecutive images.
4.3.1 Multi-patch Based Object Tracking Using Region
Covariance
The fitness function is based on the region covariance matrix (RC). The object is rep-
resented by an image template consisting in several non-overlapping image patches.
For every pixel
i
in such a patch of size
M
×
N
, we calculate a feature vector
b
i
b
i
=
xyRGB
x
I
y
)
T
(4.3)
where
x,y
represent the Cartesian coordinates of pixel
i
, whereas
R,G,B
stand for
color components, and
I
x
,I
y
are image derivatives. The RC descriptor is given by:
MN
1
MN
−
(b
i
−
b)(b
i
−
b)
T
C
=
(4.4)
1
i
=
1
where
b
denotes the vector of means of the corresponding features for the pixels in
the template. The region covariance descriptor has many advantages. In particular,
RC indicates both spatial and statistical properties of the objects, it allows combin-
ing multiple modalities and features, and last but not least, it is capable of relating
regions of different sizes. This descriptor is also robust to the variations in illumi-
nation conditions, pose, and view. Although the covariance matrices are positive
semi-definite in general, in practice they should be regularized by adding a small
constant multiple of the identity matrix, making them strictly positive.
In [
5
], a Log-Euclidean Riemannian metric has been introduced to obtain statis-
tics on symmetric positive definite matrices. The Singular Value Decomposition
(SVD) of a symmetric matrix
A
of size
n
n
is
UΣU
T
, where
U
is an or-
×
thonormal matrix, and
Σ
diag
(λ
1
,...,λ
n
)
is diagonal matrix with nonnega-
tive eigenvalues. The matrix exponential exp
(A)
of a symmetric matrix is given
by: exp
(A)
=
U
T
; conversely, the matrix loga-
rithm of a symmetric positive definite matrix is calculated according to: log
(A)
=
U
·
diag
(
exp
(λ
1
),...,
exp
(λ
n
))
·
=
U
T
. Each symmetric matrix is associated to a tensor
by the exponential, conversely, a tensor has a unique symmetric matrix logarithm.
U
·
diag
(
log
(λ
1
),...,
log
(λ
n
))
·
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