Image Processing Reference
In-Depth Information
2M thod
Let
S
t
n
(
x
)beasetof
N
observations of a continuous process affecting the shape
of the observation space
ʩ
at time points
t
n
with
n
=1
,...,N
. As the group
of diffeomorphisms is closed under composition, we can obtain a diffeomorphic
model of the overall deformation between time-points
t
1
and
t
N
by concatenat-
ing pair-wise diffeomorphic registrations
ˆ
n
,sothat
S
t
n
(
ˆ
n
(
x
))
≈ S
t
n
+1
(
x
). A
mapping from any image
S
i
to another
S
j
with
j>i
canbeexpressedasa
concatenation of transforms
ʦ
ij
=
ˆ
i
ⓦ···ⓦˆ
j
(Figure
1
). Each
ˆ
j
is calculated
based on an image pair and a prior on the deformation field that acts as regular-
ization. Our aim is to find a spatio-temporal regularization prior that accurately
models the deformation over time.
Fig. 1.
Sketch of a continuous deformation and an intermediate shape between
t
2
and
t
3
computed from three possible registrations
φ
2
The properties of the SVF parametrization of diffeomorphisms can be
exploited to formulate a prior on temporal smoothness in a chain of pairwise
image registrations. Also, it enables to use the model residual as a weighting
function that reduces the influence of the prior in regions where the assumption
of constant deformation does not hold.
2.1 Pair-Wise LogDemons Registration
We calculate diffeomorphic mappings
ˆ
n
between images using the LogDemons
algorithm [
26
]. The algorithm computes a diffeomorphism
ˆ
(
x
) defined on
x ∈
ʩ ↂ
R
d
,d ∈{
2
,
3
}
and parametrized by an SVF
v
via the ODE
dˆ
(
x, ˄
)
d˄
=
v
(
ˆ
(
x, ˄
))
,
ˆ
(
x,
0) = id
(1)
Equation (
1
) represents a geodesic curve between a source image
S
(
x
) and a
target image
T ≈ S
(
ˆ
(
x,
1)) in the one-parameter subgroup generated by SVF
of the Lie group of diffeomorphisms
D
. The velocity field
v
is an element of
the tangent space at the identity
T
id
D
and the flow
ˆ
(
x,
1) is defined as the
Search WWH ::
Custom Search