Image Registration Methods in MRI - Advance Image Processing in Magnetic Resonance Imaging - page 189

where ( i, j ) denotes the couple of corresponding voxels in the two images, and S

is the domain where both images are defined. The introduction of the domain

means that the metric must be computed only in the geometrical region where both
images are defined. This approach holds in all voxel-based metrics. This simple
metrics can be effectively used when the images to be registered are sufficiently
similar.

S

7.3.1 M I

UTUAL NFORMATION

Much of the current work on biomedical image registration utilizes information
theoretic voxel similarity measures, in particular, mutual information (MI) based
on the Shannon definition of entropy[13,14]. The MI concept comes from infor-
mation theory, measuring the dependence between two variables or, in other
words, the amount of information that one variable contains about the other. The
MI concept measures the relationship between two random variables, i.e., inten-
sity values in two images: if the two variables are independent,

is equal to
zero. If one variable provides some information about the second one, the

MI

MI

becomes greater than zero. MI is related to the image entropy by:

I ( X; Y ) = HHH ( X ) + ( Y ) − ( X, Y ) (7.4)

( . ) is the entropy of a random variable, where X and Y are the two images and H

and is defined as:

∑ Ω

H ( X ) =− log[ P ( X = x )] ⋅ P ( X = x ) (7.5)

i i

x i ∈

X

The joint entropy of two images X and Y is:

∑ ∑

H ( X, Y ) =− log[ P ( X = x , Y = y )] ⋅ P ( X = x , Y = y ) (7.6)

i j i j

x ∈ Ω y ∈ Ω

i X j Y

All entropies must be evaluated on the domain where both images are defined,
usually as overlapping areas. Normalized mutual information (NMI), given as

HH
H

() X + ( )

I

N (

X, Y ) = (7.7)

2 ( X, Y)

is less sensitive to the size of the overlap [15] and can be used instead of MI.
The probability distribution for the evaluation of can be estimated

with Parzen windows, histograms, or other probability density estimators. The
most common method uses images histograms. Let Q and K be images with

MI and NMI

M