Image Processing Reference
In-Depth Information
Lie group exponential
ˆ
(
x,
1) = exp(
v
). Due to the Baker-Campbell-Hausdorff
(BCH) formula [
27
],
v
can be computed directly in the log-domain by minizing
the energy functional
E
(
v
,
v
u
)=
T
−
S
ⓦ
exp(
v
u
)
2
L
2
2
V
+
v
−
v
u
V
˃
x
+
∇
v
(2)
˃
i
˃
s
where
˃
i
,
˃
x
and
˃
s
are parameters related to image noise, matching uncer-
tainty and spatial smoothness respecitely, whereas exp(
v
u
) is an unregularized
correspondence field between
T
and
S
1
.
2.2 Enforcing Temporal Consistency with Limited Image
Information
Lack of image information between time-points in homogeneous regions and
tangentially to image gradients can lead to spurious deformations that adversly
affect the overall optimality of the resulting deformation
ʦ
1
N
=
ˆ
1
ⓦˆ
2
ⓦ···ⓦˆ
N−
1
.
We propose a solution to this problem by means of a temporal smoothing prior
defined locally in time. This enables us to define a spatial weighting of the prior
based on the local model residual, thereby retaining important deformation cues
from the underlying images.
Temporal Smoothing Prior.
A velocity field
v
can be used as a prior to
control regularization in the LogDemons registration [
24
] by replacing the update
field
v
u
in the regularization of Eq. (
2
)by
˃
x
v
u
+
˃
t
v
˃
x
+
˃
t
,
(3)
where
˃
t
defines the weight of the prior. In [
24
], the prior field
v
was obtained
from a series of registrations between observations
S
t
n
(
x
)
,n
=2
,...,N
to the
baseline
S
1
(
x
) by fitting a linear model over
t
to the sequence velocity fields
v
t
(
x
)atevery
x
.
When considering quickly changing morphologies, registration of all images to
a common reference can be dicult and a linear model might be too restrictive.
Instead, we enforce temporal smoothness by transporting the SVF
v
t−
1
betwen
S
t−
1
and
S
t
to the space of
S
t
as
v
t
(
x
)=
v
t−
1
(
ˆ
t−
1
(
x
))
,x
=
ˆ
t−
1
(
x
). This
corresponds to imposing constant velocity/no acceleration prior at every point
of the deformation field.
Spatially Adaptive Prior.
The update step (
3
) assumes the same amount of
temporal consistency over the whole image domain
ʩ
. However, this assumption
is often violated due to the complex nature of biological processes. We therefore
propose to weigh the influence of the temporal smoothness prior depending on
the registration error at the previous time-step.
1
Without loss of generality, the intensities of all images are assumed to be scaled to
[0
,
1].
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