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Chapter 3 Hippocampal segmentation using multiple atlases 59
. Î0
.
. Î{k1}
.Ik2 ◦ Tk2
.U{k1,k2}
.Ik1 ◦ Tk1
.Ik2 ◦ Tk2
. Î{k1,k2}
Figure 3.3: Least angle regression (LAR) with the first 2 cavariates/altases.
Î {k1,k2} is the projection of I into span(Ik1 , Ik2 ). The initial residual Î {k1,k2} − Î0
has greater correlation with Ik1 ◦ Tk1 than Ik2 ◦ Tk2 ; the next LAR estimate is
Î {k1} = Î0 + ˆγ1Ik1 ◦ Tk1 , where ˆγ1 is chosen such that Î {k1,k2} − Î {k1} bisects the
angle between Ik1 ◦ Tk1 and Ik2 ◦ Tk2 . Then Î {k1,k2} = Î {k1} + ˆγ2U2, where U2 is
the unit bisector. Adapted from Efron et al. (2004).
The coefficient ˆ βk1 is increased until a second atlas image Ik2 has the same cor-
relation ĉk2 with the current residual as ĉk1. We use ˆγ1 to denote the value of
coefficient ˆ βk1 at this point. Thus
Î{k1} = Ib + ˆγ1Ik1 ◦ Tk1
(3.31)
and k2 is selected and added to A . The LAR then proceeds in along the direction
U2 equiangular to both Ik1 ◦ Tk1 and Ik2 ◦ Tk2
ÎA = Î{k1} + γ2U2. (3.32)
A third covariate/atlas is added when it has the strongest correlation with the
residual equaling the correlation with the two selected variables. The algorithm
continues in the direction of the least angle to all the selected variables, and so on.
When more than two atlases are selected, the data matrix of the selected atlases
XA can be defined as
XA = (· · · , sj · Ij ◦ Tj(Ω), · · · ), j ∈ A (3.33)