Biomedical Engineering Reference
In-Depth Information
2.3 Statistical Shape Analysis
A principal component analysis (PCA) approach was adopted to extract the variation
in PF muscle shapes of the sample population, where each subject-specific model
could be regenerated as a linear combination of shape components. A pre-requisite
of PCA to preserve its validity is that all parameters in the analysis have to be in the
same spatial scales [
19
]. To satisfy this requirement, all 12 models (interpolated by a
bicubic Hermite-linear Lagrange scheme where the nodal coordinates and
derivatives had different scales) were converted into equivalent models interpolated
by a bicubic Bezier-linear Lagrange scheme.
2.3.1 Conversion from the Hermite to Bezier Interpolation
The Bezier scheme interpolates the model surfaces using control points which
collectively define the nodal position and derivatives. All control points in a Bezier
scheme are established by spatial coordinates which essentially have the same
spatial scales and are appropriate to be used in PCA [
20
].
The model conversion from the Hermite to Bezier representation was achieved
by multiplying the array of Hermite global nodal parameters to the conversion
matrix which encapsulated their linear transformations, as illustrated below.
2
4
3
5
2
4
3
5
B
1
B
2
B
3
.
B
N
2
B
N
1
B
N
H
1
H
2
H
3
.
H
N
2
H
N
1
H
N
2
3
M
HB
M
HB
4
5
.
.
.
¼
.
.
(3)
M
HB
M
HB
where
B
,
M
HB
and
H
are the Bezier parameters, constant local conversion matrix of
each element and Hermite parameters, respectively.
2.3.2 Principal Component Analysis
Principal component analysis was performed on the Bezier PFmuscle models whose
parameters were arranged into an M
¼
number of DOF per model) after their means have been removed. The singular
values and mode shapes of the matrix were obtained using singular value decompo-
sition. The mode shapes which described the principal modes of variations were
ranked according to the magnitude of their corresponding singular values. The
degree of variations explained by each shape mode was calculated as the ratio
N matrix (M
¼
number of models, N
