Biomedical Engineering Reference
In-Depth Information
at the beginning of the registration and fine matching toward the end of the
minimization.
The next step is to search for the transformation parameters using genetic
algorithms (GAs). GAs, pioneered by Holland [23], are adaptive, domain inde-
pendent search procedures derived from the principles of natural population
genetics. GAs borrow its name from the natural genetic system. In natural ge-
netic system whether a living cell will perform a specific and useful task in a
predictable and controlled way is determined by its genetic makeup, i.e., by the
instructions contained in a collection of chemical messages called genes [24].
Genetic algorithms are briefly characterized by three main concepts: a Dar-
winian notion of fitness or strength which determines an individual's likelihood
of affecting future generations through reproduction; a reproduction operation
which selects individuals for recombination to their fitness or strength; and
a recombination operation which creates new offspring based on the genetic
structure of their parents.
Genetic algorithms work with a coding of a parameter set, not the parameters
themselves and search from a population of points, not a single point. Also ge-
netic algorithms use payoff (objective function) information, not derivatives or
other auxiliary knowledge and use probabilistic transition rules, not determinis-
tic ones. These four differences contribute to genetic algorithms' robustness and
resulting advantage over other more commonly used search and optimization
techniques (see Fig. 1.3).
Since the genetic algorithm works by maximizing an objective function, the
fitness function,
F
r
(
R
,
t
), can be defined as in Eq. (1.3).
Combining (1.3) with the GCP transform to find matching points between
the two data sets, Eq. (1.3) can be rewritten as
n
d
2
(
y
,
GCP
(
y
i
)
+
y
i
)
F
r
(
R
,
t
)
=−
(1.14)
i
=
1
where
y
i
=
Ry
i
+
t
. By maximizing (1.14) we effectively minimize (1.3).
The objective of the registration process is to obtain the rotation matrix
R
and the translation vector
t
. Thus in 3D space, there are six parameters that
need to be evaluated; the three angles of rotation around the three principal
axes
α
,
β
, and
γ
, and the displacement
x
,
y
, and
z
in the
x
,
y
,
z
directions,
respectively.
