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Chapter 3 Hippocampal segmentation using multiple atlases 49
two images acquired from different modalities, or in MRI, images of different char-
acteristics depending on the acquisition protocols. It is based on the information
theoretic definition of the entropy
H(I) = −
∫
pI(s) log pI(s)ds
R
(3.7)
where pI(·) is the probability density of the image intensity estimated from the
histogram of image I. The joint entropy of two images I and J
∫
H(I, J) = −
R2 pI,J(s, t) log pI,J(s, t)dsdt (3.8)
is defined by the joint density pI,J(·, ·) of the two images. The mutual information
is defined as
MI(I, J) = H(I) + H(J) − H(I, J) (3.9)
measuring the dependency between two images, and the NMI (Studholme et al.,
1996)
NMI(I, J) = 1 +
MI(I, J)
H(I, J)
H(I) + H(J)
= . (3.10)
H(I, J)
When image I and J are independent, the mutual information MI(I, J) = 0 and
NMI(I, J) = 1. When the two images are identical, the mutual information is
maximized and NMI(I, J) = 2.
3.1.2.2 Correlation coefficient
Correlation coefficient is another image similarity metric, which is defined as
C(I, J) =
1
|Ω| · σIσJ
∑
x∈Ω
(
(
I(x) − Ī) J(x) − ¯ J )
, (3.11)
where Ī and σI, ¯ J and σJ are the mean and the standard deviation of the gray
level in image I and J respectively.