There also exist methods which do not impose requirements on r(x):
C) u is determined in such a way that the correlation coefficient
o(p) between f(x) and g(x*u) obtains a maximum:
Lf(x)g(x+u)
(3/9u) o(u) = (9/94) 3 3 = 0
ZJ[E£^(x)Eg^ (x*tu)]
or
M M 2
L f(üx)g(xtu).s.C | EG (ts) | (4)
x=1 x=1
The different requirements on r(x) given in a), b) and c) all result
in a relation of the form
M
(9/90) (1/M) E f(x)g(xtu) » c (8/8u) H [cum
x=1
il mix
een) | (5)
x=1
where H is a simple operator (H = 1*, Uds / ). No arguments appear
to exist in favour of one or the other Choice of a), b) or c).
Accordingly, the problem of matching two images is indefinite to the
extent of choosing the operator H. However, it should be noted that
under certain circumstances the right hand side of (5) is zero
independently of H. An obvious case, which will be discussed in this
paper, is when a variation of u produces a cyclical translation of
g(Xtu). It is also noted that for two-dimensional images and for their
projections onto one-dimensional arrays, the right hand side of
equation (5) tends to zero for large M. Due to these arguments, it
appears as 1f no great errors are introduced by simply neglecting the
right hand side of equation (5). The matching problem is then
simplified to solving for y in the equation
(8/8u) (1/M)
X
f(x)g(xtu) = © (6)
1
H =
It should be noted, that the procedures b) and c) but not a) involve
the u-derivative and therefore imply that y is real in spite of x
being integer. When working in image space, a digital image g must
therefore be resampled for each y involved in finding a solution to
(6). In the following section, the translation of the image g is
performed by means of a linear displacement operator. This method is
feasible, as the theory of functions of linear operators is general
enough to accept arbitrary real arguments.
MATCHING USING LINEAR OPERATOR THEORY
When matching the one-dimensional projections of two digital images f
and g, these can be considered as vectors in a linear space. Define
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