Biomedical Engineering Reference
In-Depth Information
Constraints on the allowed deformation must be applied to make the prob-
lem computationally tractable and physically plausible. These constraints
will depend on the application. Examples range from the addition of a rela-
tively small number of extra degrees of freedom (typically five to ten) where
the variation across a population can be described by parameters derived from
principal component analysis as used in the active shape model
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to the many
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tens of thousands implicit in the algorithm of Rueckert et al.
In studies of soft
tissue deformation, understanding the physics of the deformation process
might reduce the number of degrees of freedom of the transformation (see
Chapter 15). Models based on the laws of physics have also been used to
reduce the degrees of freedom of the matching process in intersubject regis-
tration. The example in Figure 2.4 was generated using a “thin-plate spline”
deformation, a well known algorithm described in Chapter 3, that calculates
the deformation expected in a thin plate that is anchored at a number of tie
points or points of correspondence. As stated above, these quasi-physical mod-
els, unlike the biomechanical models discussed in Chapter 15, do not represent
any real physical process within the patient.
2.4
Image Registration Algorithms
Registration algorithms compute image transformations that establish corre-
spondence between points or regions within images, or between physical
space and images. This section briefly introduces some of these methods.
Broadly, these divide into algorithms that use corresponding points or corre-
sponding surfaces, or operate directly on the image intensities.
2.4.1
Corresponding Landmark-Based Registration
One of the most intuitively obvious registration procedures is based on iden-
tification of corresponding point landmarks or “fiducial markers” in the two
images. For a rigid structure, identification and location of three landmarks
will be sufficient to establish the transformation between two 3D image vol-
umes, provided the fiducial points are not all in a straight line. In practice it
is usual to use more than three. The larger the number of points used, the
more any errors in marking the points are averaged out. The algorithm for
calculating the transformation is well known and straightforward.
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It
involves first computing the average or “centroid” of each set of points. The
difference between the centroids in 3D tells us the translation that must be
applied to one set of points. This point set is then rotated about its new cen-
troid until the sum of the squared distances between each corresponding
point pair is minimized. The square root of the mean of this squared distance
