Docteur de l'université Automatic Segmentation and Shape Analysis ...

84 Chapter 4 Statistical shape model of Hippocampus
position in the landmark Si(x) has been derived by Ericsson and Åström (2003)
based on the SVD decomposition of the data matrix, and by Hladůvka and Bühler
(2005) on the eigen-decomposition of the covariance matrix Σ.
The variation based on eigen-decomposition is used in the implementation. The
variation of the cost function with respect to the displacement ui at pixel (ı, j) in
i-th shape image can be calculated by applying the chain rule
δLMDL
δui(xıj)
∂LMDL ∂λm
= · ·
∂λm ∂Σjl
δΣjl
δSi(xıj)
δSi(xıj)
· , (4.31)
δui(xıj)
where Einstein summation convention applies to the repeated subscripts m, j, l.
The partial derivative of the cost function with respect to the eigenvalues is
∂LMDL
=
∂λm ⎧
⎪⎨ 1
λm
⎪⎩
1
λc
if λm ≥ λc,
if λm < λc.
(4.32)
The partial derivative of the eigenvalue with respect to the element of covariance
matrix Σjl is calculated from the eigen-decomposition
Σ = WΛW T , (4.33)
where Λ is the diagonal matrix of eigenvalues, and W is the matrix of eigenvectors,
which gives the derivative
∂λ m
∂Σjl
= WjmWlm, (4.34)
where Wlm, Wjm are elements of W. The variation of the covariance matrix with
respect to the position of the point Si(xıj) is derived from (4.29)
δΣjl
δSi(xıj) =
1
(2N + 1) 2
[(
δij − 1
) (
Sl(xıj) + δil −
n
1
) ]
Sj(xıj) ∈ R
n
1×3 . (4.35)
The variation of the point Si(xıj) with respect to the displacement u
[ ]
δSi(xıj)
δui(xıj)
∈ R 3×2
(4.36)