Image Registration Methods in MRI - Advance Image Processing in Magnetic Resonance Imaging - page 200

some constraints have to be introduced about the nature of nonrigid transforma-
tion in order to reduce the number of unknown parameters involved in the
registration operation.
To simplify the optimization problem, we can suppose that W ij = 0 for some
( i , j ) couples. In particular, we can state that W ij = 0 if | i-j |

R. This means that the
similarity function is computed only for near slices that present greater similarity.
The optimization algorithm described is based on random selection of a slice I i in
the sequence followed by local registration in respect to I i of all other slices in the
neighborhood of i . The process is iterated until all slices have been processed.
The local registration is performed by extracting contours for involved images
and minimizing the distance between extracted contours by an iterated conditional
models (ICM) optimization algorithm. Note that image contours have to be
extracted just one time at the beginning of the registration process. The described
methodology clearly shows how the solution of the global registration problem
will require the introduction of some hypothesis to simplify the problem, and the
use of similarity metrics that can be computed in a very short time.
A 3-D data set can be also acquired several times following the evolution
of a phenomenon under investigation, as happens in monitoring the perfusion
of a contrast medium in tissues (e.g., the brain on cardiac perfusion) with an
endogen signal change (e.g., fMRI). These data sets are usually defined as
dynamic 3-D or 4-D acquisitions . In these cases, misalignment in time acquisi-
tion will result in artifacts in signal monitoring. Registration of a 4-D data set
is often reduced to a number of registrations of image pairs in order to reduce
the required computation time and to exploit the available registration algo-
rithms. However, global registration of multiple data sets can lead in general to
better results.
In theory, it is possible to extend the concept of similarity metrics to more than
two images. As an example, in square root metric, the similarity S along N images
can be defined as the squared root of the sum of quadratic distances of each
corresponding point in the definition field

>

, where the distance along the N
corresponding points is a suitable distance metric defined in an N -dimensional
space. In this approach, the similarity metric depends on all the registration param-
eters (e.g., 6N parameters in 3-D rigid registration) and can be optimized as pre-
viously described. Following information theory, a higher dimensional MI metric
can also be defined.
In the case of three images MI becomes:

Ω

MI (A:B:C) = H (A) + H (B) + H (C) H (A,B) H (A,C) H (B,C) + H (A, B,C)

(7.14)

The MI defined in this manner may not necessarily be nonnegative, so it is
not a true metric. An alternate definition is often used in medical image registra-
tion [31]:

MI (A:B:C) = H (A) + H (B) + H (C) H (A,B,C) (7.15)