Image Processing Reference
In-Depth Information
The computation of the analytic gradient of the bending energy and the
Jacobian-based penalty is identical to the classical FFD. The derivatives of these
terms are given in [
13
,
14
].
2.2
Spatio-Temporal Atlas Construction
To avoid bias of the atlas towards a given anatomical configuration, we create the
spatio-temporal atlas using pairwise inter-subject transformations. All pairwise
transformations are hereby computed using the symmetric and inverse consistent
diffeomorphic registration presented in the previous section. The anatomy of each
subject is mapped by the inverse of the age-dependent Log-Euclidean mean [
2
]
of the transformations relating it to the other subject anatomies of similar ages.
This average transformation minimizes the weighted sum of squared distances to
the observed inter-subject transformations, given the squared distance between
two diffeomorphisms generated by stationary velocity fields as defined in [
2
], i.e.,
d
2
(
T
1
,
T
2
)=
2
log(
T
1
)
−
log(
T
2
)
(8)
In particular, given
n
transformations
T
ij
, which map an anatomical refer-
ence point
x
i
of subject
i
to their corresponding points
x
j
=
T
ij
(
x
i
) of subjects
j
[1
,n
] (incl. the identity for
j
=
i
), the average transformation which maps
x
i
to its atlas coordinate at age
t
is given by
∈
T
i
(
t
)=exp(
v
i
(
t
))
(9)
where
v
i
(
t
)=
j
=1
w
(
t
j
,t
)
v
ij
j
=1
w
(
t
j
,t
)
(10)
with Gaussian kernel weights for the temporal regression, i.e.,
˃
√
2
ˀ
e
−
(
t
j
−t
)
2
1
w
(
t
j
,t
)=
(11)
2
σ
2
The atlas template of mean shape and mean intensities at age
t
is then
estimated as the image which minimizes the weighted sum of squared differences
to the observations close to
t
, after mapping these into the respective atlas
coordinate system, i.e.,
I
(
t
)=
i
=1
w
(
t
i
,t
)
I
i
◦
T
−
1
i
(
t
)
i
=1
w
(
t
i
,t
)
(12)
Possibly available hard segmentation labels or probability maps can be trans-
ferred into the atlas space as well, using the computed inverse average trans-
formations. Propagated hard segmentation masks are averaged using the same
weights as in (
12
) to obtain separate mean probability maps for each class.
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