Biomedical Engineering Reference
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Fig. 5.7 Shape variation analysis pipeline with both clinically relevant model reduction and
growth model generation. Using PCA modes and standard correlation analysis, the pathological
shape patterns can be identified. Using a combination of PLS regression and CCA a statistical
generative growth model can be derived
clinical features to determine the severity of the disease, which requires a consistent
representation of the patient shapes. For this we use firstly principal component
analysis (PCA) to extract the main modes of shape variation followed by standard
statistical design to exhibit those that are correlated to the pathology (Fig. 5.7
top row).
5.3.2.1
Model Reduction Using Principal Component Analysis
Since statistical shape analysis is a high dimensional problem with a large number
of parameters and variables to solve for (despite the matching pursuit reduction,
a shape can still be represented by hundreds of moments), we first reduce the
dimension of the problem by applying principal component analysis (PCA). This
gives the modes of deformation that describe the amount of variation of shape
observed in the population.
PCA is applied on the initial velocity fields v ( i )
0
to extract the main deformation
modes observed in the population. PCA finds basis vectors, the modes, of the space
of variables (here the initial velocities) that best explain their variance. The modes
p are calculated by solving the eigenvalue problem Σp
μp , where the elements
σ ij of the covariance matrix Σ are calculated in the kernel space W . Assuming that
the deformation from the atlas to patient i is parameterized by the initial vector field
v ( i 0 ( x )= k K W ( x, x k ) β ( i )
=
,wherethe x k
are the point positions of the delta
k
Dirac currents of the atlas, and β ( i )
k
the moment vector at x k , then the mean initial
)= k
x, x k ) β k and the covariance is
vector field is
¯
v 0 (
x
K W (
β ( i k − β k )
− β l )
v 0 ,v 0 ¯
β ( j )
l
<v 0 ¯
σ ij =
v 0 > V =
(
K W (
x k ,x l )(
.
(5.4)
k,l
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