Biomedical Engineering Reference
In-Depth Information
similar intensities and the transformation required to establish correspon-
dence is small. In this case, the transformation can be approximated by the
first term in an expansion of the function relating one image to the other as a
series of terms, i.e., the Taylor series. An approximate transformation can be
calculated directly.
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These algorithms are described in more detail in Chapter 3.
In all other algorithms, a process of optimization is required. This means
that the algorithm takes a series of guesses from an initial starting position.
The starting position has to be sufficiently close for the algorithm to converge
to the correct answer, i.e., it has to be within what is known as its “capture
range.” This first guess can be set automatically or with a simple user inter-
action. The algorithm computes a number, known as the cost function or sim-
ilarity function, relating to how well the two images are registered. Mutual
information, correlation coefficient, and sum-of-squared-intensity differences
are all examples of cost functions. Some cost functions (e.g., the correlation
coefficient) increase as the images come into alignment; others (e.g., sum-of-
squared-intensity-differences) decrease. The registration algorithm pro-
ceeds by taking another guess and recalculating the cost function. Progres-
sion towards an optimal registration is then achieved by seeking
transformations that increase (or decrease) the cost function until a maxi-
mum (or minimum) of the cost function is found. The best registration that
can be achieved is defined by this maximum (or minimum). The strategy for
“optimization”, i.e., guessing subsequent alignment transformations, is an
important subdiscipline in the area of computing known as numerical meth-
ods. The next chapter contains a more detailed treatment of optimization of
cost functions.
2.7
Transformation of Images
Registration algorithms are designed to establish correspondence. In many
applications this is sufficient. All that is required is an indication of what
point in one image corresponds to a particular point in the other. In some
applications, however, we need to transform an image into the space of the other.
This process requires resampling one image on the grid corresponding to the
voxels or pixels in the other. To do this, interpolation is required. The accu-
racy with which this interpolation is done depends on the motivation for reg-
istering the images in the first place. In most applications simple nearest
neighbor or trilinear interpolation will suffice. In nearest neighbor interpola-
tion, as the name suggests, the location of each voxel in the transformed
image is transformed back to the appropriate location in the original image
and the nearest voxel value is copied into the transformed voxel. In trilinear
interpolation the linearly weighted average of the eight nearest voxels is
taken. For the highest accuracy, sampling theory tells us that a sinc ((sin x)
x)
weighting function applied to all voxels should be used.
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This is particularly
