Biomedical Engineering Reference
In-Depth Information
position and orientation of the PF. The same transformation was also applied to the
landmarks and the surface data. This alignment minimised undesirable distortions
due to differences in scaling and shearing in the subsequent affine transformation.
The interpolation order of the fitted PF models was reduced to trilinear and a
generic model was generated using the mean nodal positions of the models.
2.1.2 Affine Transformation
To eliminate the difference in overall size and shape between the generic trilinear
PF muscles mesh and the surface data from individual subjects, an affine transfor-
mation was performed on the trilinear generic mesh to obtain a reasonable initial
estimate for the subsequent individual mesh fitting. The transformation can be
written as the following,
2
4
3
5
¼
2
4
3
5
2
4
3
5
2
4
3
5
2
4
3
5
þ
2
4
3
5
x
t
y
t
z
t
1
S
x
000
0
S
y
00
00
S
z
0
0001
R
xx
R
xy
R
xz
0
R
yx
R
yy
R
yz
0
R
zx
R
zy
R
zz
0
0001
1
Sh
xy
Sh
xz
0
x
y
z
1
T
x
T
y
T
z
1
Sh
yx
1
Sh
yz
0
Sh
zx
Sh
zx
10
0
0
0
1
(1)
Where
S
,
R
,
Sh
and
T
are the orthotropic scaling, rigid body rotation, shearing and
rigid body translation matrices, respectively.
The transformation was applied to the nodal positions (
x
,
y
,
z
) of the generic mesh
and produced transformed coordinates (
x
t
,
y
t
,
z
t
), prepared for the subsequent
mesh surface fitting. Since the affine transformation applies deformations to the
meshes globally, nodal points in the transformed generic meshes retained their
anatomical positions consistently across the population. For each subject, transforma-
tion components for the rigid body translation, rotation, orthotropic scaling and
shearing were calculated by matching positions of the generic landmarks and the
aligned landmarks identified fromMR images (target points), using a constrained non-
linear optimisation algorithm implemented in MATLAB 7.10.0. The sum of distance
(error) between the transformed generic landmarks
x
i
and their corresponding target
points
x
targe
i
was minimised by varying the independent variables (transformation
components) and a solution with the best global match was obtained.
Error =
X
N
x
target
i
x
i
(2)
i¼
1
where
N
and
i
refer to the total number and the index of landmarks, respectively.
The 16 landmarks utilised in the affine transformation were all defined on the
soft tissues to obtain a close estimation of the overall PF muscle geometry for each
