Biomedical Engineering Reference
In-Depth Information
4.5.1.3 Transformation-Based Features
A third group of not very common approaches are transformation-based features.
They exploit transformation parameters that can be used for the statistical deforma-
tion model. They have been extended to also support non-rigid transformations.
4.5.2 Construction
The creation process of statistical or probabilistic models of feature variation can be
divided into two parts. Initially, the training models have to be aligned before the
construction can be performed.
The alignment process aims at eliminating pose changes that are not feature re-
lated. To do this, commonly the iterative generalized Procrustes approach [
73
]is
applied.
T
∗
=
2
argmin
T
||
y
i
−
T
(
x
)
||
(4.8)
i
The transformation
T
∗
is obtained by finding a transformation that minimizes the
distance between a model
x
and all other models
y
i
. The resulting transformation
T
∗
is combination of a translation, a rotation and scaling. Once this transformation is
applied, all models are in a common coordinate frame: the model space of the point
distribution model. Although aligned, these models still have a substantial amount
of shape variability.
The actual construction process uses the corresponding and aligned features and
performs a principal component analysis (PCA). PCA has been introduced by Pear-
son [
74
] in 1901. This technique has been used for the point distribution model
(PDM) by Cootes and Taylor [
70
]. Given
n
shapes
Y
1
,
Y
2
,…,
Y
n
, the mean shape can
be defined as
n
1
n
Y
=
Y
i
.
(4.9)
i
=
1
The covariance matrix is defined as
n
1
T
S
=
1
(
Y
i
−
Y
)(
Y
i
−
Y
)
(4.10)
n
−
1
i
=
and its eigenvectors correspond to the largest eigenvalues of
S
. The eigenvectors
describe the most significant variation modes from which a subset (much smaller
than the number of points) is used.
This approach needs a sufficiently high number of training models. Otherwise a
poor generality is given, leading to poor adaptation to new data and enforcement of
