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

58 Chapter 3 Hippocampal segmentation using multiple atlases
3.3.2 Least angle regression
LAR is a variable selection method in regression, which is closely related to other
model selection methods such as Lasso (Tibshirani, 1996) and stagewise forward se-
lection. The conceptual connections between them have been exploited, and made
it possible to implement all these methods within the same algorithm framework.
We re-formulate the atlas selection in the atlas-based segmentation approach as a
variable selection problem in which we estimate the image I from the transformed
atlases {Ik ◦ Tk} as covariates. Assuming the linear independence of the atlas set,
for the coefficients { ˆ βk} of each atlas, the estimated image Î is calculated from the
linear model
where Îb is the intercept.
Î = Îb +
n∑
ˆβk · Ik ◦ Tk, (3.26)
k=1
Let A ⊂ {1, 2, · · · , n} denote the active subset of atlases that is selected by the
LAR procedure, and
ÎA = Îb + ∑
k∈A
ˆβk · Ik ◦ Tk, (3.27)
is the estimate from the current selection A . We can compute the correlation ĉk
between the current residual I − ÎA and the transformed atlas Ik ◦ Tk
and the sign sk of ĉk
ĉk = C(Ik ◦ Tk, I − ÎA ) (3.28)
⎧
⎪⎨
sk =
1 if ĉk > 0,
0 if ĉk = 0,
⎪⎩ −1 if ĉk < 0.
(3.29)
LAR based atlas selection starts with A = {k1} such that the atlas Ik1 ◦ Tk1 is
most correlated with the image I
k1 = arg max C(Ik ◦ Tk, I) (3.30)
k