Digital Signal Processing Reference
In-Depth Information
Since we have to consider only translations (the scale as well as
the rotation of the image patches can be assumed to be the same due
to the experimental setup), image-stitching of two patches
I
1
and
I
2
is performed by minimizing
d
p
(
I
1
,τ
δ
(
I
2
)), where
τ
δ
(
I
)(
x, y
):=
I
(
x
−
δ
1
,y
−
δ
2
) denotes the translation of the image patch
I
by the vector
2
(possible additional zero-padding of the images assumed):
δ
∈ R
δ
0
:= argmin
d
p
(
I
1
,τ
δ
(
I
2
))
.
(13.3)
δ
Various minimization algorithms can be employed to find or approx-
imate
δ
0
. A simple solution is, for example, given by (discrete)
gradient
descent
to determine local minima: the update rule is defined by
d
p
(
I
1
,τ
δ
(
I
2
))
p
,
δ
new
=
δ
old
−
η
∇
(13.4)
where
η
denotes a fixed or adaptive learning rate and
is the discretized
gradient of the cost function (taken to the power
p
to avoid roots) with
respect to
∇
δ
. The latter can easily be calculated as
p
(x,y),I
1
(x,y)=−1
τ
δ
I
2
(x,y)=−1
p−1
sgn(
τ
δ
I
2
(
x, y
)
−
I
1
(
x, y
))
|
τ
δ
I
2
(
x, y
)
−
I
1
(
x, y
)
|
·
τ
(δ
1
+1,δ
2
)
(
I
2
)(
x, y
)
.
−
τ
δ
(
I
2
)(
x, y
)
(13.5)
τ
(δ
1
,δ
2
+1)
(
I
2
)(
x, y
)
−
τ
δ
(
I
2
)(
x, y
)
FFT-based speed-up
In practice, we use a previously selected feature from each image to
restrict the search space spanned by
δ
in equation (13.3), and then search
for translations
within this restricted region. This is necessary because
evaluation of the image distance equation (13.2) is computationally
expensive; an exhaustive search for all possible 4
hw
translations is not
feasible, and local update algorithms such as equation (13.4) need good
starting values in order to avoid local minima.
In the following, we will describe an easy-to-calculate approximation
of image similarity which allows us to estimate a fusion parameter
δ
ˆ
δ
0
,
from which we can start the above algorithm. The idea is to determine a
δ
with maximal crosscorrelation between
I
1
and
τ
δ
(
I
2
); in other words,
we want to maximize the autocorrelation between the images. This can
be interpreted as a second-order approximation of the distance equation
(13.2), ignoring scaling and especially ROI parameters. In order to
Search WWH ::
Custom Search