Geology Reference
In-Depth Information
attributed to OW area within the CAA as the Arctic ice
cover continues to retreat.
The main characteristics of the CIS‐ASTIS algorithm
are outlined in the following paragraphs, but details can
be found in
Komarov and Barber
[2013]. One of the main
motivations behind the development of this algorithm
was the ability to capture the rotational component of
sea ice motion, which cannot be captured by the MCC
mentioned above. To achieve this purpose the algorithm
employs the concept of phase correlation matching as
well as the familiar cross correlation. The combination of
these two concepts benefits from the advantage of each
one yet avoids the disadvantages. Phase correlation
matching was developed by
Bracewell et al
. [1993] to reg-
ister images based on the fast Fourier transform (FFT)
shift theory. This theory entails that a shift in the spatial
domain transforms into a phase shift in the frequency
domain. If the spatial function representing a 2D window
in an image
f
(
r
) has Fourier transform
F
1
(
u
), then the
shifted function of the same window in the next
image
f
(
r
−
r
0
) will have Fourier transform:
subimage by
θ
0
and
θ
0
+
π
and determining the transla-
tional component using equation (10.110).
While the phase correlation technique can capture the
rotational component between two matching subareas
when it reaches its peak value, the peak does not provide
a clear measure of similarity between the two subareas.
Therefore, the cross‐correlation MCC technique is used
to decide on this similarity. If
b
denotes a potential
rotated matching window in the second image that cor-
responds to the window
a
from the first image, then the
cross‐correlation coefficient
c
between
a
and
b
is given by
aabb
ij
.
ij
.
ij
.
c
(10.113)
2
2
aa bb
ij
.
ij
.
ij
.
ij
,
where
a
and
b
are average values of
a
and
b
, respectively.
The maximum value of
c
corresponds to the best match
(similarity) between
a
and
b
. The angle of rotation
θ
0
of
`
is known from the phase correlation. That is how the simi-
larity is established using both displacement (through the
cross correlation) and rotation (through phase correla-
tion). In addition, the cross‐correlation coefficient is used
for setting a level of confidence for the obtained drift vec-
tor. Vectors with scores below a certain level are rejected.
Similar to the RGPS, the CIS‐ASTIS uses hierarchical
data representation of SAR images in order to improve
the computational efficiency. This entails establishing a
first set of motion vectors using resampled images with
a 3 × 3 pixel window and then another set using images
with a 2 × 2 pixel window before ending with the original
image. Unlike the RGPS, CIS‐ASTIS does not calculate
the displacement or motion vectors at regular grid
points. Instead it performs the calculations at selected
control points over each hierarchical level. This means
that the product is Eulerian displacement maps that do
not include information on ice deformation. All control
points should be located in some distinctive areas that
have relatively high variance (i.e., visually recognizable
features such as floe edges or lead boundaries) and the
vertical and horizontal components of the distance
between any two points should not be lower than a pre-
scribed value. CIS‐ASTIS has proven to be successful in
mapping ice motion outside the melt season as long as
the separation time between the two images is within a
few days in areas of substantial ice drift and a few weeks
in areas of limited drift. The primary limitation of the
system, as in the case of the RGPS, is about its operation
during the melt season when the signature of the surface
at any given point changes rapidly due to changes in the
physical conditions of the ice rather than its dynamic
conditions. An example of rotated ice motion from
the
Komarov and Barber
[2013] study is shown in
iur
Fu eFu
(10.110)
0
2
1
where
r
0
is the shift vector between the matching windows
in the two images. This translational component can be
retrieved by finding the maximum value of the following
phase correlation:
*
*
CF
FF
FF
1
1
2
(10.111)
1
2
where
F
− 1
is the inverse Fourier transform operator and
the asterisk denotes complex conjugate. This basic for-
mulation of the phase correlation can be expanded for
detection of the rotational component based on the
Fourier shift and rotational theorem. In polar coordi-
nates the rotation transforms into a simple shift along the
rotation axis
θ
. If
P
1
and
P
2
are magnitudes of the spectra
of a feature window in the first image and its replica in
the second image, respectively, then the phase correlation
C
θ
can be calculated using the following equation (the
derivation is given in
Komarov and Barber
[2013] follow-
ing
Reddy and Chatterji
[1996]):
CF
Fp Fp
Fp Fp
*
*
(10.112)
1
1
2
1
2
The coordinates of the peak value of
C
θ
indicate the
angular shift
θ
0
. However, since the Fourier spectrum is
conjugate symmetric for real images, angular ambiguity
arises when determining the angular shift; namely
between
θ
0
and
θ
0
+
π
. This is resolved by rotating the
