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normalized cross correlation (Hsu et al., 2005). A
real-time implementation that adopts the two
images’ matching through the Fourier-Mellin
transformation has been reported in (Martinez et al.,
2004). The use of fuzzy logic for the global motion
vector computation can produce optimal results
(Güllü and Ertürk, 2004). In order to enhance the
compensated frame position Kalman filtering was
utilized (Hsu et al., 2005, Ertürk, 2002). The
estimation of the motion in a sequence is also
realized by optical flow techniques. The
approximation of the image flow field provides both
the translational and rotational information. The
undesired motion effects are calculated in (Suk et
al., 2005) by estimating the rotational center and the
angular frequency from the local translational
motion definition by fine-to-coarse multi-resolution
motion estimation. In (Pauwels et al., 2007) the
stabilization is accomplished by fixating at the
central image region, whilst optical flow estimation
optimizes this approximation. In most of the cases
the global motion vector is computed via a series of
local motion vectors. These describe the movement
in a particle of the image, which results to a better
estimation of the indented camera movement and the
undesired motion.
In this paper, a novel fuzzy Kalman digital
image stabilization technique in the log-polar plane
is proposed. First a transformation from the
Cartesian plane to the log-polar one takes place. The
acquired log-polar image sequence provides lesser
information in the background of the scenery than in
the foreground. This is due to the proper attribute of
the log-polar transformation to preserve high-
resolution at the center of the image, which
diminishes logarithmiticaly towards the periphery.
The motion estimation in the log-polar plane
provides a space-variant distribution of the local
motion vectors due to the aforementioned nature of
the log-polar plane. Consequently, the extracted
local motion vectors are imported into a recursive
fuzzy system based to the one presented in (Güllü
and Ertürk, 2004). However there are some distinct
differences. One lies to the fact that in this paper, the
fuzzy system utilizes the Kalman filter’s
mathematical model to filter the inputs
straightforwardly. Moreover, no mean operation
filtering takes place to the measured fluctuations.
Finally, the filtered vectors, define the global motion
vector from which the compensation vector is
calculated. The innovation of using log-polar images
for the motion field extraction provided optimal
results not only to the stabilization of each frame,
but also to the visual quality of the video output. The
advantages of the log-polar plane are well exploited,
as (i) the processing time is lesser, (ii) a single
motion estimation extraction provides information
for both the rotational and translational irregularities
and (iii) the center of attention has a higher impact
to the whole process without further preprocessing.
2 LOG-POLAR
TRANSFORMATION
The motion estimation process preserves high
computational burden, so it is normally improper for
real-time applications. One way to overcome the
computational burden is to sub-sample the images.
Yet, to estimate the motion field, all available
information is needed. Thus, a resolution decrease is
inappropriate as it causes loss of major information
and the provided results are sparse and inaccurate.
However, the volume of the image data can be
reduced by a topological arrangement, without loss
of information. Notably, a space-variant
arrangement such as log-polar provides lesser image
data without constraining the field of view, or the
image resolution at the fixation point. The log-polar
transformation is based on the human’s eyes
projections of the retina plane to the visual cortex. It
finds its origins into studies on the vision
mechanisms of the primates. The adoption of this
topology into artificial vision systems exhibits
several advantages as in visual attention, throughput
rate and real-time processing. Many applications of
the log-polar transformation have been reported,
such as the time-to-impact estimation (Tistarelli and
Sandini, 1993), wavelet extraction based on log-
polar mapping (Pun and Lee, 2003), tracking (Metta
et al., 2004) and disparity estimation and vergence
control in (Manzotti et al., 2001).
Figure 1: The log-polar transformation maps radial lines
and concentric circles into lines parallel to the coordinate
axes.
The mathematical model of the log-polar
mapping can be expressed as a transformation
between the polar plane (ρ, θ) (retinal plane), the
log-polar plane (η, ξ) (cortical plane) and the
Cartesian plane (x, y) (image plane).
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