Biomedical Engineering Reference
In-Depth Information
field, but rather a field with the following properties: large deformations and
local perturbations that modify the topology of the structures, in order to vali-
date the basic hypothesis of our work. The “local” field is generated from 2,000
voxels which are randomly picked in the volume. For each voxel, each of the
three components of the deformation is the realization of a Gaussian random
variable of standard deviation 120 mm. We then perform a Gaussian smoothing
with a small average deviation in order to propagate this perturbation to a local
neighborhood while preserving discontinuities. The volumes and the results are
shown on Fig. 8.8. We compare the multigrid method with a global affine regis-
tration method, in which a 12-parameter deformation is estimated for the entire
volume.
To asses the quality of the registration, we compute the mean square er-
ror (MSE)
4
which is an indicator of the quality of the registration. However,
it would be unfair to evaluate the registration only with a measure that is the
underlying driving force of the estimation. Therefore, as we have the binary
classification of the phantom, we can also assess the quality of the registration
based on the overlap of two volumes: the first volume is the initial classification,
i.e., a gold standard (gray matter/white matter), the second volume is the de-
formed classification, registered with the estimated deformation field. We then
measure out overlapping ratios like the sensitivity, the specificity, and the to-
tal performance [142]. Results are presented in Table 8.1. Despite the use of
binary classes, the resulting measures that we obtain are very satisfactory. Par-
ticularly, the robustness of the method is demonstrated in critical conditions
(9% noise and 40% inhomogeneity), which are far tougher than in any realistic
acquisition.
The numerical evaluation also allows to study the sensitivity of the algorithm
with respect to the parameters of the algorithm, i.e., parameters of the robust
estimators. We have two parameters to fix,
σ
1
and
σ
2
.
σ
1
corresponds to the
hyperparameter of robust function
ρ
1
, associated with the similarity term, while
σ
2
corresponds to the hyperparameter of robust function
ρ
2
associated with the
regularization term. We made the parameters
σ
1
and
σ
2
vary in a cube of size
[1
.
0
e
4
,
1
.
0
e
5
]
×
[1
,
20] with step, respectively, of 1
.
0
e
4
and 1 (which means that
we performed the registration with 200 different sets of parameters), and we
observe that the final result (the mean square error between the source volume
N
i
=
N
1
4
MSE
i
=
1
(
I
1
(
i
)
I
2
(
i
))
2
, where
I
1
and
I
2
are the volumes to compare, and
N
is
=
−
the number of voxels.
