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Chapter 3 Hippocampal segmentation using multiple atlases 55
where
pi = 1
{
x ∈ Ω : Pl(x) =
|Ω|
1
}
(3.18)
n
is the probability mass function (pmf) of voxels which have i votes on the given
label l. The difference between the definition of HP and H in (3.7) is that in HP
we only use the positive part of the histogram, thus excluding the background
with zero vote (i = 0). In the ideal case when all the atlases agree unanimously,
the entropy HP = 0.
3.2.2.2 Partial moment of histogram
In addition to the entropy, the second order partial moments of histogram with
respect to the reference point 1 is used. It measures the overall deviation of the
distribution of the votes from unanimous agreement. It can be defined as follows,
n∑
µ2 = pi ·
i=1
In the ideal case of unanimous agreement, µ2 = 0.
3.3 Atlas selection by re-ranking
(
1 − i
)2
. (3.19)
n
When using the locally based methods, the segmentation accuracy on the ranked
atlases does not converge as quickly as the simple majority voting on the ranked
atlases, and keeps increasing as long as new atlases are added (see, e.g. Sdika,
2010). Since locally based methods requires both the image and the label map of
the atlases in the fusion step, it is computationally expensive in terms of both the
computation time and the memory footprint, if we increase the number of atlases
until the segmentation accuracy converges. In the context of locally based label
fusion, we propose the atlas selection strategies other than the similarity ranking.
Using the LWV for the label fusion, we re-rank the atlases by maximal marginal
relevance (MMR), and use least angle regression (LAR, Efron et al., 2004) to