Mutuallnformation:A Similarity Measure for Intensity Based Image Registration
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measure (Cover and Thomas 1991). MI is related to entropies by the relationship
I(A, B) = H(A) + H(B) - H(A, B) ,
(3.2)
where H(A) and H(B) are the entropies of A and Band H(A, B) is their joint
entropy. Considering A and B as two images, floating image (F) and reference
image (R) respectively, the MI based image registration criterion states that
the images shall be registered when I(F, R) is maximum. The entropies and the
joint entropy can be computed from,
H(F) = - L Pp (t) log Pp (t) ,
f
H(R) = - LPR(r) 10gPR(r) ,
H(F,R) = - LPP,R (t,r) 10gPp,R (t,r),
f,r
(3.3)
(3.4)
(3.5)
where Pp (t) and PR(r) are the marginal probability mass functions, and
Pp,R (t, r) is the joint probability mass function of the two images F and R. The
probability mass functions can be obtained from,
h (t, r)
Pp,R (t, r) = L h (t, r) ,
f,r
Pp (t) = L Pp,R (t, r) ,
PR(r) = L Pp,R (t, r) ,
f
(3.6)
(3.7)
(3.8)
where h is the joint histogram of the image pair with each image being of size
(M x N). It is a 2D matrix given by
h(O,1)
h(l,1)
h(M -1, 1)
h(O,N -1) ]
h(l,N -1)
h(M -1,N-1)
(3.9)
The value h(a, b), a E [O,M - 1], bE [O,N - 1], is the number of pixel pairs
having intensity value a in the first image (i. e. F) and intensity value b in the
second image (i. e. R). It can thus be seen from (3.2) to (3.8) that the joint
histogram estimate is sufficient to determine the MI between two images.
To interpret (3.2) in the context of image registration with random variables
F and R, let us first assume that both H(F) and H(R) are constant. Under this
assumption, maximization of MI in (3.2) is equivalent to the minimization of