Biomedical Engineering Reference
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where
T
is the atlas we are estimating,
φ
i
is the deformation that maps the atlas to
the surfaces
T
i
,and
ε
i
represents the residuals (shape features not captured by the
atlas such as changes in topology etc.). The mean shape information is described in
the atlas
T
while the shape variability is encoded in the transformation
φ
i
.
The atlas is first initialized by taking the mean of the patient meshes. This initial
atlas is then registered to each of the patients individually. A new atlas
T
that
(
T
minimizes the error
Λ
)
is computed, with
M
(
T
φ
i
(
T
)
2
W
∗
.
Λ
)=
T
i
−
(5.3)
i
=1
We then register the updated atlas to the individuals, recompute the atlas and loop
until convergence (Algorithm
8
).
Algorithm 8
Atlas Estimation
Input:
M
surface meshes
T
i
extracted from patient images.
1: Rigidly align meshes to a reference patient using rotations and translations.
2: Create initial atlas
T
0
as the mean of the patient meshes(this is a current and no longer
amesh)
3:
loop
{
over
k}
4:
Estimate the
M
transformations
φ
i
and their initial momenta
μ
i
that register the atlas
T
k−
1
to the individual
T
i
.
5:
Update the atlas by minimizing the error in Eq.
5.3
using the estimated transformations
φ
i
and the atlas
T
k−
1
i
=1
μ
i
V
∗
7:
end loop
[
Stop whenever
μ
V
∗
<ησ
φ
where
η
=5
% typically
]
8:
return
Final atlas
T
k
and the related transformations
φ
i
parameterized by their momenta
μ
i
i
=1
μ
i
and variance
σ
φ
=
1
M
1
M
6:
Compute bias
μ
=
This algorithm produces an atlas which is a current and not any more a surface
mesh. To constrain the atlas to remain a mesh, one usually restrains the search of
the atlas to the subset of currents generated by all deformations of the data meshes.
This means that the atlas can be written
T
φT
l
for some diffeomorphism
φ
and
some index
l
. In practice, the atlas is initialized as the data mesh that has the smallest
variance (sum of square distance to all other data meshes without deformation), and
the update of the atlas becomes
T
k
=¯
=
μ T
k−
1
.
5.3
Shape Analysis of ToF Data
To demonstrate the usefulness of our statistical shape analysis in a clinical context,
we consider a population of patients with repaired tetralogy of Fallot. Recall that
shape here means an atlas, which is a surface represented by a current, along with a
deformation of that atlas. One of the key ideas of the “atlas framework” is to transfer
the analysis from the space of currents (shapes) to the space of deformations.
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