Mutuallnformation:A Similarity Measure for Intensity Based Image Registration
101
then (3.14) can be written as,
h (F (xi,xj),R (Yi + P,Yj + q)) + = f (p -L1i)f (q -L1j) Vp,q E Z, (3.16)
where Z is the set of all integers. Notice that in (3.16), the increments are all
zero except when p, q E {O, I}. In fact, when p = 0 and q = 0, the increment
is WI> when p = 0 and q = 1, the increment is W2, when p = 0 and q = 1,
the increment is W3 and when p = 1 and q = 1, the increment is W4. Next the
GPVE algorithm for the 2D case is described in terms of a more general kernel
function.
Let f be a real valued function that satisfies the following two conditions:
1) f(x) 2: 0, where x is a real number,
(3.17)
00
2) L f(n + L1) = 1, where n is an integer, 0 ~ L1 < 1 ,
(3.18)
1/=-00
then for each grid point x = (Xl, X2) E X in the image F, the joint histogram h
is updated in the following manner:
h(F(Xi,Xj),R(Yi+p,Yj+q))+=f(p-L1i)f(q-L1j) Vp,qEZ, (3.19)
where f is referred to as the kernel function of GPVE and Z is the set of
all integers. The first condition on f ensures that the joint histogram entries
are non-negative while the second condition makes the sum of the updated
amounts equal to one for each corresponding pair of points (Xi,Xj) in F and
(yi +L1i,Yj +L1j) inR.
From this generalization, it can be seen that the PVI algorithm proposed
by Maes et.al (1997) is a special case when f is a triangular function defined
by (3.15). In GPVE, a family of functions called B-splines may be used as the
kernel function f as it satisfies both the conditions in (3.17) and (3.18) and
furthermore, it has finite support (the domain within which the function value
is not zero). It is also interesting to note that the PVI algorithm corresponds to
using a first order B-spline as the kernel function, which is a triangular function. Figure 3.12 shows the shapes of first, second, and third order B-splines.
Notice that they have different supports. The size of the support determines
0.5
0.5
0.5
0
0
0
-3 -2
-1
3
-3
2
3
-3 -2
-1
0
2
3
a
b
c
Fig.3.12a-c. B-sp1ine functions of different orders. a First order. b Second order. c Third
order
101
then (3.14) can be written as,
h (F (xi,xj),R (Yi + P,Yj + q)) + = f (p -L1i)f (q -L1j) Vp,q E Z, (3.16)
where Z is the set of all integers. Notice that in (3.16), the increments are all
zero except when p, q E {O, I}. In fact, when p = 0 and q = 0, the increment
is WI> when p = 0 and q = 1, the increment is W2, when p = 0 and q = 1,
the increment is W3 and when p = 1 and q = 1, the increment is W4. Next the
GPVE algorithm for the 2D case is described in terms of a more general kernel
function.
Let f be a real valued function that satisfies the following two conditions:
1) f(x) 2: 0, where x is a real number,
(3.17)
00
2) L f(n + L1) = 1, where n is an integer, 0 ~ L1 < 1 ,
(3.18)
1/=-00
then for each grid point x = (Xl, X2) E X in the image F, the joint histogram h
is updated in the following manner:
h(F(Xi,Xj),R(Yi+p,Yj+q))+=f(p-L1i)f(q-L1j) Vp,qEZ, (3.19)
where f is referred to as the kernel function of GPVE and Z is the set of
all integers. The first condition on f ensures that the joint histogram entries
are non-negative while the second condition makes the sum of the updated
amounts equal to one for each corresponding pair of points (Xi,Xj) in F and
(yi +L1i,Yj +L1j) inR.
From this generalization, it can be seen that the PVI algorithm proposed
by Maes et.al (1997) is a special case when f is a triangular function defined
by (3.15). In GPVE, a family of functions called B-splines may be used as the
kernel function f as it satisfies both the conditions in (3.17) and (3.18) and
furthermore, it has finite support (the domain within which the function value
is not zero). It is also interesting to note that the PVI algorithm corresponds to
using a first order B-spline as the kernel function, which is a triangular function. Figure 3.12 shows the shapes of first, second, and third order B-splines.
Notice that they have different supports. The size of the support determines
0.5
0.5
0.5
0
0
0
-3 -2
-1
3
-3
2
3
-3 -2
-1
0
2
3
a
b
c
Fig.3.12a-c. B-sp1ine functions of different orders. a First order. b Second order. c Third
order