Image Registration Methods in MRI - Advance Image Processing in Magnetic Resonance Imaging - page 195

low-resolution images are registered. The result of this step is used as initial estimate
for the next registration that involves images at higher resolution. The process is
iterated until the best available resolution is reached. The main problem encountered
in the registration process is the presence of local maxima of the similarity metric.
Many similarity metrics, such as MI, are irregular and rough and are often trapped
in local optima. In particular, this problem affects the multiresolution approach,
because the global optimum may not be present in lower resolutions.
In conclusion, the choice of the appropriate optimization technique is a com-
promise between the effectiveness of the method (i.e., the ability to find the global
optimum) and the processing time required for the optimization process. Local
methods, such as the Powell method [21], conjugate gradient [22], and the Leven-
berg-Marquard or simplex algorithms [23] provide good performance but do not
guarantee that the global optimum of the similarity function will be reached. On
the other hand, global optimization methods such as simulated annealing [24],
genetic algorithms [25], tabu search [26], and particle swarm optimization [27] are
generally more expensive in terms of processing time although they ensure the
convergence to a global optimum under some conditions. An extensive description
of the optimization is beyond the scope of this chapter. In the following text, we
describe two representative methods of the two classes, the Nelder-Mead simplex
algorithm and the genetic algorithm approach.

7.5.1 N ELDER -M EAD S IMPLEX A LGORITHM

The Nelder-Mead algorithm is one of the most well-known optimization algo-
rithms, also known as AMOEBA from the name of its implementation in the
book Numerical Recipes [28]. It is often able to find reasonably good solutions
quickly with only a few function evaluations per iteration. The convergence
properties are less than satisfactory, so a number of variants have been proposed
in the literature that attempt to address such issues [29].
The amoeba algorithm maintains at each iteration a nondegenerate simplex,
a geometric figure in n dimensions (the number of dimensions is equal to the
number of optimization parameters), that is, the convex hull of n

+ 1 vertices,

x 0 , x 1 , , x n in the search space, and their respective function values. In the

solution of the registration problem, the function is the similarity metric. In
each iteration, new points are computed, along with their function values, to
form a new simplex. The algorithm terminates when the function values at the
vertices of the simplex satisfy a predetermined condition. In the case of 2-D
rigid registration (N = 3), one iteration of the amoeba algorithm consists of the
following steps:

…

Order: Order and relabel the four vertices as x 0 , x 1 , x 2 , and x 3 , such that
f( x 0 ) f( x 1 ) f( x 2 ) f( x 3 ); x 0 is defined the best vertex , while x 3 is the
worst point , and x 2 is the next-worst point . The algorithm computes x